633867 research-article2016

JMEXXX10.1177/1052562916633867Journal of Management EducationWilliams-Pierce

Rejoinder

On Reading and Digital Media: Rejoinder to “Digital Technology and Student Cognitive Development: The Neuroscience of the University Classroom”

Journal of Management Education 2016, Vol. 40(4) 398–404 © The Author(s) 2016 Reprints and permissions: sagepub.com/journalsPermissions.nav DOI: 10.1177/1052562916633867 jme.sagepub.com

Caroline Williams-Pierce1 The article “Digital Technology and Student Cognitive Development: The Neuroscience of the University Classroom,” by Cavanaugh, Giapponi, and Golden (2015), succeeded tremendously in its goal to provoke conversation, and I enjoyed the opportunity to be one of the first respondents! My expertise lies particularly in how digital media can be used productively in formal and informal contexts, and how designing with different representations and interactions can increase learning and interest. I have also conducted research on the power of interest-driven play in commercial games—that is, games not designed with academic learning goals in mind—and the communities that emerge online to discuss and create around those game worlds. Consequently, the following commentary serves as an introduction to multiple scholarly fields about the value of digital media for providing contexts for—and provoking—learning.

On Reading Cavanaugh et al. (2015) set digital reading and book reading as opposed activities, with the former aligned with quick sips of information in a constant 1University

at Albany, State University of New York, Albany, NY, USA

Corresponding Author: Caroline Williams-Pierce, University at Albany, State University of New York, ED 127A, 1400 Washington Ave, Albany, NY 12222, USA. Email: [email protected]

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waterfall of information overload, and the latter aligned with the familiar and comforting image of sitting cuddled in an oversized chair, surrounded by the smells of a crackling fire and hard-bound books. While the latter image is deeply welcoming to me personally, I cannot help but think: What then are the cognitive implications of sitting in that oversized chair with a Kindle in my hands, reading Fahrenheit 451? I am using a technological object to read a digital book about physical books being destroyed such that their only trace is left within the heads of individuals who have memorized their content. This illustration pushes back against the distinction between digital media and reading by Cavanaugh et al., but perhaps this Kindle-as-digital-media fruit is too low-hanging to make my point. Another digital media project, Bedtime Math, takes the form of free mobile apps with word problems that are designed for parents to read—mathematize?—with their young children at night, and were shown to have a significant influence on their children’s mathematical achievement, with the most significant improvements with parents who suffered from math anxiety (Berkowitz et al., 2015). David Landy (2015), a cognitive scientist at Indiana University, cheerfully warns other parents to avoid this product because his “n = 2 field study” (para. 2) has shown that children enjoy the math activities too much, and his do not always fall asleep on time when Bedtime Math is in the picture. But perhaps this is still too close to what physical books can provide, so I will instead veer into a high school English classroom led by Paul Darvasi. Darvasi, who has designed and implemented augmented reality games in his classroom for books such as One Flew Over the Cuckoo’s Nest (Darvasi, 2016, in press-a), discovered a short Indie videogame, Gone Home (Darvasi, 2014, in press-b), that was universally loved by game critics. Curiosity piqued, he played the game, which involves exploring an abandoned house and seeking to find out what had happened to the family that lived there. Darvasi realized that the nonlinear form of player-driven exploration meant that “this type of dynamic would not work as well in a novel or a film. This video game had staked out narrative territory where its traditional forerunners could not follow” (Darvasi, 2014, para. 7), and he promptly decided to use reframe the game as a text for use in his senior English class. He had his students play the game and analyze it using the traditional strategies of annotation and close reading, while expanding the normal analytic evidence of cited quotes with screenshots and video clips as a way to more deeply express the experience of playing Gone Home, gently merging traditional analytic methods with ones more suited to this new form. In other words, Darvasi used this videogame as a substitute for a more traditional text, supporting widely accepted methods of deep reading and analysis, while expanding the

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methods to align more clearly with the new digital media we are surrounded by. He concludes that English class can become a sort of theater where everyone pretends the texts are being read. . . . Our duty as educators is to design our courses to prepare students to think critically and succeed in their current communication context, as that is the environment where they must survive and, hopefully, prosper. (Darvasi, in press-b)

Naturally, Darvasi is not saying that we should get rid of books, but rather that as new types of texts come into existence, we can—and should—support students in using powerful reading strategies for both books and digital media. Consequently, I propose that instead of considering a dichotomy between reading physical books and reading digital media, as encouraged by Cavanaugh et al. (2015), we consider a scale of sorts between reading as pertaining to linear written endeavors and reading as situated experiences within designed spaces. Then, reading Fahrenheit 451 on a Kindle is fully placed within the former, while reading Bedtime Math is closer to the middle of the scale, as it is a written product that does not require linear engagement, since parents and their children can flip through the word problems in any order. Gone Home, however, is designed to be a situated (digital) experience that readers can apply respected, new, and adapted analytic methods to understand and deconstruct, enriching their ability to apply deep reading methods to both the old and new experiences that we are surrounded by.

Beyond Reading Numerous online communities actively engage in reading, reviewing, creating, and analyzing texts from this proposed spectrum. For example, the Hunger Games and Harry Potter books have powerful online presences, where readers voluntarily engage in a wide variety of practices that we value in the classroom, such as reading, writing, reviewing, arguing, discussing, and so on (Curwood, 2012; Jenkins, 2004). On the other end of the spectrum, the same practices are engaged in by game players who were inspired by game worlds, characters, or experiences. For example, the epic fantasy game World of Warcraft involves considerable reading within the game by even the most casual player, both in traditional text-based form and in terms of the situated reading of complex symbols and indicators within the virtual world (what Gee [2003] would call the game’s semiotic domain). The incredible popularity of the game, combined with its complexity and difficulty, has led to considerable production of text- and video-based information resources by

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players, and Steinkuehler has termed the game and its attendant player-created resources a constellation of literacies (Steinkuehler, 2007, 2008). How players navigate this constellation relates to various factors, such their identity (Martin, 2012) or level of expertise (Martin et al., 2012),1 but engagement in literacy practices are an integral part of the game experience and surrounding community. One small part of this constellation is termed theorycrafting, which involves collecting data during gameplay in order to further understand the game mechanics, and it “is a rich, compelling intellectual activity involving hypothesis generation, testing, numerical analysis, logical argumentation, rhetoric and writing” (Choontanom & Nardi, 2012, p. 187). In fact, theorycrafting has a lot in common with scientific fields in our physical worlds: Unknown laws and patterns undergird our natural or virtual world experiences, and we can further understand those laws and patterns by interrogating collected data. Unsurprisingly, then, when Steinkuehler and Duncan (2008) examined randomly selected posts on World of Warcraft forums, they determined that the vast majority of posts were engaging in scientific reasoning about the game world. They concluded that “forms of inquiry within play contexts such as these are authentic although synthetic: even though the worlds themselves are fantasy, the knowledge building communities around them are quite real” (p. 541). One particularly mathematically rich post from that data corpus was written by an author who combined a theorycrafting argument with a narrative structure to emphasize his/her argument, transforming mathematical algorithms into characters within a story (Steinkuehler & Williams, 2009; Williams, 2011). Such forum posters and theorycrafters provide examples of voluntary reading, writing, creating, and analyzing that we most desire from our students—provoked specifically by digital media experiences and presented in online spaces that we may have no awareness of. In short, our students may be walking into their first class on their first day with some powerful and relevant, albeit unrecognized, skills, thanks to the very technology that we often denigrate! Online reading-related activities, some very few of which I highlighted above, often take place in what James Paul Gee calls affinity groups or spaces (Gee, 2003, 2005). These spaces are where people with common endeavors work together to discuss, create, argue, invent, share, review, and so on. Gee and Hayes (2012) identified some important features of affinity spaces, but particularly relevant here is one that can occur specifically thanks to the anonymity available online—participants are not segregated by age, race, gender, expertise, and so on. Regardless of these characteristics of their human bodies or past history, they can create and critique content, develop further general and specialized knowledge to contribute to the group, gain status

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through multiple routes, earn the role of a respected leader in various ways, and so on—all important experiences that can lend themselves to online or offline interactions. I highlight this point, and these spaces, particularly to emphasize to the reader one of the distinct differences between traditional education contexts and digital media contexts; in the latter spaces, students may already be deeply engaged in activities that we are planning to teach them. In fact, they may already be considered respected leaders, despite their “student” label in our classroom.

Conclusion There are numerous points relating to learning, in general, and management education, in particular, in this commentary. However, I want to push back specifically against Cavanaugh et al.’s (2015) claim that today’s students will struggle with effective experiential learning exercises. I have presented considerable evidence that digital media can provide situated experiences that provoke reading, creating, evaluating, and so on through a variety of forms, which means that we cannot blame the students alone if they struggle with our educational versions of experiential learning. Rather, we need to provide sufficient autonomy such that students can engage in deep interest-driven learning, bring their own knowledge to the table, and contribute unique content and vision to the class. This is what our students do outside of class on a regular basis, for no reason rather than to create and contribute to their affinity spaces, so we must learn to see—and respect—how their new approaches to learning are enacted and displayed. Give the younger generation some space, they may surprise us! Declaration of Conflicting Interests The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author received no financial support for the research, authorship, and/or publication of this article.

Note 1.

As an aside about multitasking, Martin et al. (2012) found in their case study of a World of Warcraft player that his ability to multitask “across multiple timescales and in multiple spaces, real and virtual” (p. 242) is what enabled him to engage in advanced play at an expert level. In other words, what would traditionally be seen as drawback to efficiency was instead a necessary component of his successful engagement in a complex activity.

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Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement in school. Science, 350, 196-198. Cavanaugh, J. M., Giapponi, C. C., & Golden, T. D. (2015). Digital technology and student cognitive development: The neuroscience of the university classroom. Journal of Management Education, 40(4), 374-397. doi:10.1177/1052562915614051 Choontanom, T., & Nardi, B. (2012). Theorycrafting: The art and science of using numbers to interpret the world. In C. Steinkuehler, K. Squire, & S. Barab (Eds.), Games, learning, and society: Learning and meaning in the digital age (pp. 185-209). Cambridge, England: Cambridge University Press. Curwood, J. S. (2012). Cultural shifts, multimodal representations, and assessment practices: A case study. E-Learning and Digital Media, 9, 232-244. Darvasi, P. (2014, March 5). Prologue: A video game’s epic-ish journey to a high school English class. Retrieved from http://www.ludiclearning.org/2014/03/05/ gone-home-in-education/ Darvasi, P. (2016, January 17). The Ward game: How McMurphy, McLuhan and MacGyver might help free us from McEducation. Retrieved from http://www. ludiclearning.org/2016/01/17/the-ward-game-how-mcmurphy-mcluhan-andmacgyver-might-help-free-us-from-mceducation/ Darvasi, P. (in press-a). The Ward game: How McMurphy, McLuhan and MacGyver might free us from McEducation. In C. Williams (Ed.), Teacher pioneers: Visions from the edge of the map. Pittsburgh, PA: ETC Press. Darvasi, P. (in press-b). Gone home and the apocalypse of high school English. In C. Williams (Ed.), Teacher pioneers: Visions from the edge of the map. Pittsburgh, PA: ETC Press. Gee, J. P. (2003). What video games have to teach us about learning and literacy. Computers in Entertainment (CIE), 1(1), 20-20. Gee, J. P. (2005). Learning by design: Good video games as learning machines. E-Learning and Digital Media, 2(1), 5-16. http://doi.org/10.2304/elea.2005.2.1.5 Gee, J. P., & Hayes, E. (2012). Nurturing affinity spaces and game-based learning. In C. Steinkuehler & K. Squire (Eds.), Games, learning, and society: Learning and meaning in the digital age (pp. 123-153). Cambridge, England: Cambridge University Press. Jenkins, H. (2004, February 6). Why heather can write. Retrieved from http://www. technologyreview.com/news/402471/why-heather-can-write/?p=1 Landy, D. (2015). What is bedtime math really good for. Retrieved from https://davidlandy. net/the-dangerous-lie-of-bedtime-math/ Martin, C. (2012). Video games, identity, and the constellation of information. Bulletin of Science, Technology & Society, 32, 384-392. doi:10.1177/0270467612463797 Martin, C., Williams, C., Ochsner, A., Harris, S., King, E., Anton, G., . . .Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh, & J. Davies (Eds.), Virtual

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literacies: Interactive spaces for children and young people (pp. 226-243). London, England: Routledge. Steinkuehler, C. (2007). Massively multiplayer online gaming as a constellation of literacy practices. eLearning, 4, 297-318. Steinkuehler, C. (2008). Cognition and literacy in massively multiplayer online games. In J. Coiro, M. Knobel, C. Lankshear, & D. Leu (Eds.), Handbook of research on new literacies (pp. 611-634). Hillsdale, NJ: Erlbaum. Steinkuehler, C., & Duncan, S. (2008). Scientific habits of mind in virtual worlds. Journal of Science Education and Technology, 17, 530-543. doi:10.1007/s10956008-9120-8 Steinkuehler, C., & Williams, C. (2009). Math as narrative in WoW forum discussions. International Journal of Learning and Media, 1(3). Retrieved from http:// www.mitpressjournals.org/doi/abs/10.1162/ijlm_a_00028 Stone, J. C. (2007). Popular websites in adolescents’ out-of-school lives: Critical lessons on literacy. In C. Lankshear & M. Knobel (Eds.), A new literacies sampler (pp. 49-65). New York, NY: Peter Lang. Williams, C. (2011, April). “Shadow has crap scaling—FACT”: The intertwining of mathematics and narrative on a game forum. Paper presented at the 2011 American Educational Research Association Annual Meeting and Exhibition, New Orleans, LA.

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Apples and Coconuts: Young Children ‘Kinect-ing’ with Mathematics and Sesame Street Meagan Rothschild and Caroline C. Williams

Abstract The ability to count objects is a crucial skill for young children. We report on an experimental study that utilized a Kinect Sesame Street TV intervention designed to support two types of counting activities. We conducted quantitative as well as open-coding based analyses, on video data with 3- and 4-year-olds. The complexity of interactive digital media contexts for mathematical learning is unpacked with the assistance of literature from the fields of mathematics education and cognitive science. We conclude by making recommendations for interactive educational design in general. Keywords Common Core Standards for mathematics · Kindergarten · Microsoft · Kinect · Sesame Street · Embodied cognition · Early education · Number knowledge · Interactive television · Informal learning · Xbox

Introduction A foundational skill that young children need to develop for mathematics learning is counting. The United States Common Core State Standards for Mathematics Kindergarten standards state that students should learn the number names and count sequence, and be able to count objects (National Governors Association Center for Best Practices 2010). The National Council of Teachers of Mathematics (2000) include in the pre-kindergarten to second grade-band the requirement that all students learn to count with understanding, be able to determine the size of sets of objects, and use numbers to count quantities. Being able to count and connect the counting specifically to specific objects is a crucial part of learning how to mathematize M. Rothschild ( ) WIDA, Wisconsin Center for Education and Research, University of Wisconsin-Madison, Madison, WI, USA e-mail: [email protected] C. C. Williams Department of Curriculum and Instruction, Mathematics Education, University of WisconsinMadison, Madison, WI, USA e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 T. Lowrie, R. Jorgensen (Zevenbergen) (eds.), Digital Games and Mathematics Learning, Mathematics Education in the Digital Era 4, DOI 10.1007/978-94-017-9517-3_8

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the world, as well as to continue in further mathematics learning trajectories. This project focuses specifically on supporting 3- and 4-year-old children in counting by using a Sesame Street episode made interactive through the medium of the Microsoft Kinect 2012, a motion capture device for the Xbox console. In the following section, we briefly review literature on video games and learning, and on embodied cognition and mathematics. We then further describe the relationship between the Kinect and Sesame Street, before transitioning to describe our study design, implementation, and analysis. Considerably varied research indicates that video games can be powerful vessels for learning (Barab et al. 2010; Fisch et al. 2011; Gee 2003; Squire 2011; Steinkuehler and Duncan 2008). By leveraging some elements of video game design and transforming the traditionally televised one-way information flow into an interactive learning experience, the Sesame Street Kinect series has the potential to increase the engagement and learning of its participants. In particular, this multimodal design aligns with embodied cognition research that suggests that cognition and action are intertwined (Shapiro 2011). Theories of embodied cognition contend that thinking and learning are not based on amodal symbol systems, but rather are inextricably woven into action and perception systems (Barsalou 1999, 2008). Researchers examining the relationship of action and gesture to mathematics learning have found promising results (Alibali and Goldin-Meadow 1993; Glenberg et al. 2007a; Nathan et al. 1992), including interventions in which actions and gestures are designed to be related to successful solving of specific conjectures (Dogan et al. 2013; Walkington et al. 2012, 2013). In summary, physical action can influence mathematical cognition and, consequently, using the Kinect in conjunction with episodes designed to support mathematical learning may leverage action as a way to support cognition. The questions about the nature of learning with Kinect Sesame Street TV led to a research project conducted at Microsoft Studios in which the first author began to investigate the nature of participant experiences in two-way episodes and traditional television episode viewing. Two interconnected questions formed the focus of the project: How are mathematical concepts learned in each context, and how may interactivity relate to concept learning? The episode follows Sesame Street’s emphasis on literacy and STEM; it includes a word of the day, a number of the day, and—to connect to the interactive elements—a move of the day. The internal white paper produced as a result of the initial analysis of the study (Rothschild, internal Microsoft white paper 2012) presents preliminary results showing all students that watched the episode (both experimental and comparison groups) showed statistically significant learning gains when all the tests were collapsed. This included assessment items related to letter recognition, relational concepts, and number knowledge. Initial analysis of the assessment total did not, however, demonstrate a statistically significant difference by condition or gender. A review of observational notes and engagement data indicates that there may be more nuanced issues to explore within the data set in order to more deeply understand the experiences of participants engaging with the episode and related assessment. This paper uses the data collected in the earlier study conducted at Microsoft, and goes deeper into a quantitative and

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qualitative analysis of the questions specifically related to number knowledge, and presents the investigation of the number knowledge component of the episode studied within the frames of current math education and cognition research.

Sesame Street and Kinect Television Research and position papers by leading early education organizations recognize that varied media use is becoming ubiquitous in early childhood, and when used within developmentally appropriate frameworks, can effectively promote learning and development for young children (National Association for the Education of Young Children [NAEYC] and Fred Rogers Center for Early Learning and Children’s Media 2012). Sesame Street is a proven television format with an extended media legacy of success. The format has been shown to produce learning gains in younger viewers across studies over the last 43 years, including a longitudinal study that supports the findings of learning gains (Ball and Bogatz 1970; Bogatz and Ball 1971; Fisch and Truglio 2001). Additional studies suggest that as children form parasocial relationships with the characters in the Sesame Street narrative world, they are more apt to learn targeted video content (Lauricella et al. 2011). The Sesame Street Workshop leverages multiple media to extend educational content and play-based connections to the Sesame Street narrative world, including web-based games and resources, character toys, and video game console products. In 2010, Microsoft Studios released the Kinect, an Xbox peripheral device for motion-sensing input. Since the release, Microsoft has worked on ways to engage audiences beyond their traditional core gamer, producing titles like Dance Central, Kinect Sports, Disneyland Adventures, and Nike+ Kinect Training in order to engage kids and families. Using the Kinect, participants are not bound in their play experience by holding a controller, as the Kinect peripheral device uses gesture, facial, and voice recognition that turns the player’s body and physical participation into the controlling agent. Among the products that Microsoft has released to push the boundaries of traditional gaming and television viewing experiences is Kinect TV (2012), featuring initial product lines that include a uniquely developed set of Sesame Street and National Geographic interactive television episodes. For the developers of Kinect Sesame Street TV, the goal was to extend an already successful media property and viewing format. The designers wanted to design their products based on firm research, in order to make sure that the added Kinect interactivity would not adversely disrupt the potential for learning gains found in the linear television format (Rothschild, internal Microsoft white paper 2012). This included understanding situated learning theory and the role of learning in the context of relevant activity (Gee 2003; Barsalou et al. 2003), as well as scrutinizing the potential learning through a lens of embodied cognition, connecting concepts to a learner’s own perceptions—which includes relationships between the content and themselves/their own bodies (Glenberg et al. 2007a, b).

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Informal Mathematical Learning Cross et al. (2009) note that while young children are capable of becoming competent in mathematics, a lack of appropriate formal instruction or informal opportunities in the home or community often prevents the learning of foundational concepts. The authors go on to recommend two areas of focus, Number as well as Geometry and Measurement, and further suggest that one way to remedy suboptimal learning situations is to provide various informal opportunities for learning mathematics outside of school. Baroody et al. (2005) further suggest that informal mathematical learning is a key part of successful trajectories in learning mathematics, specifically for developing number sense in young children. Consequently, the Kinect Sesame Street TV episode format is poised to fit this gap by providing content-driven informal opportunities to engage in mathematical learning.

The Number Core The mathematics included in the study design and assessment involve what Cross et al. (2009) call the ‘number core’, in this case by modeling and asking participants to coordinate cardinality, the number word list, and one-to-one counting correspondences. They define each as following: cardinality involves perceptual or conceptual subitizing; the number word list involves knowing the order of number words (i.e., 1, 2, 3, …); and one-to-one correspondences requires matching the two such that, for example, each object being counted requires one and only one number word in the appropriate list order. Cross et al. (2009) note that practicing all three of these activities, as well as coordinating between them, will improve the ability of young children to be successful—for example, 2- and 3-year-old children are considerably more likely to be able to count five objects successfully if they have had repeated practice at this task. The methodology, reported in the next section, was designed specifically to support repeated practice, and the open-coding analysis, reported later in this paper, found subitizing, the number word list, and one-to-one correspondences to be an integral part of understanding the results.

Methods This chapter analyzes data that was collected in an earlier study that took place at Microsoft, led by the first author. Forty-two 3- and 4-year-olds participated in the study. The group was composed of a mix of boys and girls from Seattle and its surrounding areas. The requirements for participant families were that they needed to have regular access to an Xbox 360 and Kinect in their home, that they had not previously viewed the episodes, and that the child was proficient in English. Data was

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collected at the Microsoft User Research Labs, and consisted of video footage, observation notes, pre-tests and post-tests, and parent surveys (including demographic data). Participants were divided into two groups of 21 by a process of stratified random sampling, accounting for gender and known family annual income. One group of participants was designated as the KINECT group, in which Kinect Sesame Street TV experiences took place as designed with all interactions on. The other group was the TRADITIONAL group, in which interactions were still elicited but the episode did not require interactions to progress. In other words, the participant experienced the same content but edited to a non-interactive, linear format. Participants came in to the research lab with a parent or caregiver, and participated in a pre-test, watched the episode, then completed a mid-test. The child and guardian left the lab with a copy of the episode in the format that they viewed (KINECT or TRADITIONAL) and then played the same episode at home over the next couple weeks. Parents logged their child’s play and made observations. The child and a parent or guardian returned to the lab one more time to view the episode and then participate in a post-test. For the purposes of this paper, analysis is specifically targeting the questions regarding the number five (the number of the day for the episode), and comparing pre- and post-test scores, with mid-test scores used to interpret the open-coding analysis. The nature of these tests will be discussed in detail in a later section.

Number Knowledge in the Episode In the episode, the scene opens with Cookie Monster dropping a banana peel on the ground, which a bustling Grover then slips on, dropping his delivery of five coconuts. Grover then asks the audience member to please help him collect his five coconuts by throwing them into his box. For each throw, an image of the box is displayed with a visual of how many coconuts are now in the box. The number of coconuts in the box is displayed in the lower right corner of the box (see Fig. 1). Grover states, “Now I have ( number) coconuts in the box.” At the end, the box with five coconuts and the number five in the bottom right corner is displayed as Grover cheers, “Hooray! Now I have FIVE coconuts!” In the KINECT group, when the participants threw, the Kinect motion sensor would respond to their movement in the system, and the coconut would fly into the screen and into Grover’s box, sometimes in silly and surprising ways (see Fig. 2). If the child did not throw the coconut, Cookie Monster would come into the scene having ‘found’ one, and drop it into Grover’s box. Grover would then ask the audience member to try throwing the next one. The TRADITIONAL group would get the verbal prompts from Grover to throw the coconut, however, their activity did not affect the way the show progressed, and for each coconut, the show would progress as if the child had made a successful throw.

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Fig. 1 Throwing coconuts into Grover’s box

The Performance Assessment Games, interactive media, and playful learning are taking a prominent role in educational dialogue. Consequently, the issue of assessing these media must be raised. For early learners, design foundations should meet Developmentally Appropriate Practice (DAP), articulated by the National Association for the Education of Young Children (NAEYC). The 2009 policy statement describes the ways that knowledge Fig. 2 Participant throwing coconut

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of child development and learning and methods that are adaptive and responsive to individual children should be linked to social and cultural contexts of a child’s life. DAP contends that the domains of development and learning are interrelated. Understanding child development, trajectories of learning, and the contexts of media and play for learning became paramount in the development of the assessment activities for this research. The assessment activities were designed to feel playful and both match the spirit of the episode and align with the sorts of performance elicited in the show. Because the study that took place at Microsoft was a pilot study for Kinect Two-Way TV, the net cast in the research was wide and would encompass a broad variety of participants with a multitude of media and learning contexts, from home languages other than English to specific behavioral needs, to experiences with media and the narrative world of Sesame Street. This meant that the assessment tasks needed to be designed to allow a variety of levels of conceptual knowledge to be demonstrated. The protocol needed to remain reflexive to the behaviors and abilities of an individual child, particularly given the long time period of each study session (a 40-min episode and time for assessment activities). In addition, the move from a visual TV format of participation to a paper and analog manipulative format of assessment represented a shift in modality. Thus the characters and playful nature of episode activity were important for connecting the episode’s learning stimulus to the assessment performance activities. The activities related to letter recognition, number knowledge, and relational concepts were designed to include participant feedback and decision-making in the hopes of increasing participant agency in the activities without detracting from the ability to elicit specific modes of content knowledge demonstration. Because this new analysis focuses on the nuances of the mathematical activities, this chapter will describe number knowledge assessment items in details. The researcher began by asking the participant to pretend with her, imagining that they had been walking through an apple farm together (situating the activity). The researcher said, “Oh look! We found some apples on the ground!” and displayed a page with five apples on it (see Fig. 3). The researcher then asked, “Can you count how many apples we found?” and, if needed, prompted with “Point to and count each apple that you see”. If the child counted to five, it was initially coded as correct; anything other than counting exactly to five was initially coded as incorrect. Immediately following the Enumeration activity, the researcher segued into the Number Application activity by telling the child that Cookie Monster loves apples, and that today they were going to help him cook. The participant helped decide what should be cooked (e.g., apple cookies, apple cake, applesauce), further situating the activity with recognizable characters from the episode and providing the participant with an opportunity to determine elements of the assessment narrative. The researcher brought out a bowl, seven foam core apples, and an image of Cookie Monster, placing them in front of the child (see Fig. 4). The researcher then told the participant that Cookie Monster needed exactly five apples to make his recipe, and asked, “Can you put five apples in Cookie Monster’s

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Fig. 3 Enumeration activity

bowl?” If the participant placed five apples in the bowl, it was initially coded as correct. Anything other than five apples in the bowl was initially coded as incorrect. The goal of these two activities was to provide the child with avenues to convey a range of number knowledge abilities. Enumeration was most directly modeled in the episode, and matched a developmentally appropriate benchmark for 3- and 4-year-olds. The second activity, counting apples for Cookie Monster’s recipe, deviated from the episode. In the episode, there was no way in the interactive system to miscount—the activity was physically tied to throwing five and only five coconuts. The designed interactive system limited the participant to throwing or not throwing, and did not support actively applying number knowledge to a situation and receiving corresponding feedback. However, understanding the depth of the participants’ understanding of the number five required an assessment inquiry of more than enumeration ability.

Fig. 4 Number Application activity

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Results In this section, we share the results we anticipated, and then share the actual empiric results. We then explore the data using qualitative open-coding in order to further understand the quantitative results. We conclude by making recommendations to the researchers and designers interested in interactive digital learning experiences, in order to give insight into designs that more deeply support the desired types of learning.

Expected Results As a consequence of playing the Sesame Street episode, and due to the general benefit of this intervention discovered in Rothschild’s internal white paper (internal Microsoft communication 2012), we expected the children to improve in their ability to count to five, which requires attending to cardinality, the number word list, and one-to-one correspondences. Based on the existing success of the Sesame Street platform and the earlier preliminary results of the overall assessment (Rothschild, internal Microsoft communication 2012), we theorized that both groups would show learning gains, with the possibility of the KINECT group showing greater gain due to increased activity and engagement.

Empiric Results The actual results did not unilaterally fulfill our expectations. Regarding our hypothesis that the KINECT group would perform better than the TRADITIONAL group, no significant difference was found between the two conditions according to Fisher’s Exact Test for the Enumeration ( p > 0.05) or the Number Application ( p > 0.05) tasks. Furthermore, no significant difference was found when the conditions were collapsed ( p > 0.05). However, the results of our qualitative analyses align quite well with literature on child development and mathematics learning, and suggest that the lack of significance is due to considerably different reasons for each test. (Because of the lack of statistical differences between the conditions for this intervention, we collapse the conditions for the remainder of this chapter). For the Enumeration test, 38 children contributed complete data to our analysis. Of those 38 children, 28 were successful in enumerating five apples during the pre-test, indicating that counting to five was a skill that these participants were already quite competent at. At post-test, 32 participants were successful (which included all 28 who replied accurately during the pre-test). Given that nearly 75 % of participants came into the study with the target skill, it is hardly unexpected that a ceiling effect occurred.

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The Number Application test, on the other hand, suffered from no ceiling effect but similarly demonstrated few gains. Sixteen of the 38 participants were successful during the pre-test, and only 20 were successful during the post-test (again, all the participants who performed correctly during the pre-test continued to be correct in their post-test). Intriguingly, as an exact but nonverbal task, this performance assessment appears to be quite achievable, even for participants of this age (e.g., Baroody et al. 2006), so the study did not accidentally include a task with achievable content but overly challenging performance demands (as Gelman and Meck 1983 so eloquently warn us about). Consequently, we more deeply examined the data qualitatively to determine exactly why there were no significant results.

Open-Coding Analysis We analyzed the video data of each participant by coding each action undertaken. Since each participant was assessed three times (before encountering the intervention, after encountering the intervention once, and after encountering the intervention multiple times), we analyzed each of the two tasks during each of the three performance assessments per participant. If three or more of the six data points per participant were absent, the participant was dropped from this analysis, leaving a total of 35 participants from the original 43. Our coding schemes were emergent, achieved through open-coding the participants’ actions and merging similar codes into several core codes, presented in Tables 1 and 2. The schemes are structurally quite different, as the Enumeration coding scheme is built of categories, while the Number Application coding scheme has multiple codes which are applied in a concatenated fashion. The codes for the Enumeration task (Table 1) are applied singly, except for the ( circuitous) code, which is appended to the primary code when appropriate. The codes for the application task (Table 2) are broken down further than the codes for the enumerating task (Table 1), due to the increased complexity of the concept under examination. The physical process of moving the apples to the bowl is labeled as conducted in a sequential or grouping fashion, and appended to this code is whether the participant counted aloud (verbal or nonverbal) and whether the participant is correct or incorrect. The NOTHING code, used in both the Enumeration and Number Application tasks, is used during analysis as though data were absent. Following the completion of coding, an additional round of open-coding was conducted that built upon the patterns observed by each participant. For example, the majority of participants were already adept at the Enumeration task at pre-test (25 of the 35 in this analysis), and those who demonstrated their proficiency at all three assessments were coding more broadly as “All Correct—No Change.” A similar code was also applied to the smaller number of participants who were always able to perform correctly during the Number Application task (13 out of the 35)— and in both cases, the method with which each participant showed their knowledge was irrelevant to our needs. They came to the intervention knowing how to count,

Enumeration Code Definition The participant sequentially touches the apples (or points at them) while verbally following the appropriate number word list, and concludes with the accurate count of ‘five’ The participant verbally follows the appropriate number word list, but does not physically touch nor gesture towards the apples The participant sequentially touches the apples while verbally following the appropriate number word list, but over- or under-counts to x The participant sequentially touches the apples while verbally following the appropriate number word list, but divides the set into two subsets of x and y while counting. The most frequent occurrence of this code included the subsets of 2 and 3, likely due to the visual stimuli layout (see Fig. 3) Participant does not verbally enumerate, but concludes by stating the number x The participant follows an incorrect number word list, and ends the count with the number x, after speaking y number words in some order The participant follows the correct number word list, but ends the count with the number word of x, putting y objects under a single number word This is a subcode, appended to the previous codes, merely to indicate that the participant does not follow a path that makes it easy to remember which numbers have already been counted. For example, most correct participants follow a ‘loop’ while they’re counting; but participants who received this subcode may have followed, for example, a ‘W’ shape on the stimuli (see Fig. 3) Participant says and does nothing, or refuses to cooperate

Enumeration Code

One-to-one counting correspondence

One-to-one counting correspondence—no movement

One-to-one counting correspondence over-counts to x

One-to-one counting correspondence Subsets ( x, y)

Nonverbal enumeration to x

Without accurate number word list ( x, y)

With accurate number word list ( x, y)

(Circuitous)

NOTHING

Table 1 Enumeration codes and definitions

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Table 2 Number Application codes, types, and definitions Application Code

Code Type

Application Code Definition

Sequential

Apple Movement

The participant placed the apples in the bowl in a sequential fashion, one by one

Grouping

Apple Movement

The participant grouped the apples in some fashion before placing them in the bowl. Primary types of grouping included moving apples into clumps on the ground, and placing more than one apple in their hand at once

Verbal( x)

Communication

The participant speaks the number word list out loud, correctly or incorrectly. The x refers to how many number words were spoken in total

Nonverbal

Communication

The participant does not speak while completing the assessment

_Correct

Solution

Participant accurately placed exactly five apples in the bowl

_x

Solution

Participant placed a number of apples other than five in the bowl, with a total of x placed

NOTHING

Participant says and does nothing, or refuses to cooperate

and they left in the same condition. (And, unsurprisingly, the same 13 who were consistently successful at the Number Application task were also always accurate at the Enumeration task.) Once the “All Correct—No Change” participants in the Enumeration task were removed, additional patterns emerged, but with the weakness one expects when over 70 % of the participants are absent. Of the remaining nine participants, three showed no consistent improvement, four improved in their ability to perform oneto-one correspondences, one became more accurate in her use of the number word list and one-to-one correspondences, and two appeared to become worse. Thankfully, once the “All Correct—No Change” participants in the Number Application task were removed, a more interesting pattern emerged among the 22 participants remaining. A full 14 of those participants (64 %; 40 % of total) did not deviate from their initial response, doing the exact same action during the pre-test, mid-test, and post-test: placing all seven of the foam core apples in the bowl. We termed this the “All the Objects” rule, and unlike the “All Correct—No Change” participants, the “All the Objects” participants showed a wide variety of abilities, strategies, and trajectories in their Enumeration task performances, perhaps signifying that the “All the Objects” solution method is a particularly sticky one that requires deeper understanding than Cross et al.’s (2009) number core. Lastly, of the remaining eight participants in this grouping, two showed no improvement (but did

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not consistently put all seven apples in the bowl), two appeared to worsen, and the remaining four improved.

Discussion As 40 % of the participants used the “All the Apples” strategy to complete the Number Application task, and varied widely in their ability to succeed in the Enumeration task, some skill or level of conceptual understanding appears to be missing. Upon first glance, the two tasks seem quite similar: in both, you are asked to count five apples. Upon further reflection, however, the Number Application task requires that the participant carve out a subset of five apples from a set of seven—which was only partially modeled by the intervention. For example, the intervention modeled that every time a coconut is added to the box, the number of coconuts in the box goes up by one (e.g., the intervention gently focuses on the impact of the addition to the group). However, the intervention does not model the division of a set into subsets (e.g., how the whole can be separated into new wholes). Consequently, whether the 40 % of participants are not grasping the mathematical concept or whether they are merely following the social training of the episode (i.e., putting all the coconuts in Grover’s box), is difficult to tell. Given that carving out five apples from seven appears to be so difficult, we became interested in the few participants who used a grouping strategy prior to putting the apples into the bowl. (However, as the number of participants using grouping is so small, we include this in the Discussion section as a thought-provoking mention instead of in Analysis as a more significant finding.) Since some participants would use a grouping strategy during one assessment and a sequential strategy during the next, we broke up the assessments into individual ‘clips’, so that they stood alone (for example, Participant T001 is now broken up into T001Pretest, T001Midtest, and T001Postest, and grain size is now the clip). This resulted in a total of 105 clips, and of those, 87 were coded as sequential, while merely 18 were coded as grouping. The sequential clips had approximately a 40 % chance of being correct—while the grouping clips, remarkably, had approximately an 82 % of being correct! We do not have the data to conclude whether children who grouped were more successful because of the strategy, or because they knew more (and consequently knew to use the grouping strategy), but a possible next direction for teaching young children mathematics in this multimodal context would be to have the grouping strategy specifically modeled for them on screen in some fashion. As a brief note prior to concluding the Discussion section, the children who became worse are particularly interesting, but unfortunately too few in number to glean much from. We tentatively hypothesize that they were attempting to adjust their strategies, not knowing why their strategies were wrong but knowing that they were, and adjusted them in the incorrect direction.

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Conclusion The interactive media industry is saturated with products and applications targeting basic math and literacy skills for early childhood. A strong conceptual foundation requires that children have the ability to move from basic knowledge to content application. This analysis shows that for an older preschool target audience, interactive media developers would be well advised to move beyond enumeration activities and look into supporting the transition from enumeration to number application. Additionally, this analysis shows that what may appear (particularly to adults) to be a simple cognitive progression may be riddled with complexities for a young child who is learning higher order number sense. Interactive media tools hold promise for providing meaningful learning experiences for children, but the complex nuances of learning, particularly in mathematics education, may require specific forms of scaffolding, like that suggested above. While it is quite simple to merely discard results that, like ours, show no significant difference between pre- and post-tests, it is through more qualitative analyses that we—as members of many fields interested in similar design and research—can unpack the complications of learning and design more powerful interactive educational opportunities. Our design recommendations are broad and go beyond the scope of this particular study. It is quite easy to examine the findings of the second performance assessment and make particular design recommendations. For example, based on the literature cited above, the finding that participants struggled to count five apples into the bowl is not surprising—and fixing it may be as simple as re-designing the intervention slightly, so that it involves Grover and—for example—Elmo. If Grover and Elmo were both carrying boxes of coconuts and ran into each other on the screen, scattering the coconuts, and required the participant to place five coconuts in Grover’s box and two coconuts in Elmo’s box, the participant could begin understanding how a single set of seven objects could be broken up into five objects and two objects. Naturally, this recommendation needs empirical testing! Consequently, we go beyond this local recommendation and instead venture to make some recommendations for the field as a whole. The results here indicate that while there were not significant learning gains between the pre- and post-mathematics assessments, our more qualitative analysis reveals intriguing findings that can be explained in part by existing research in mathematics education and cognition. Our ongoing analysis examines the demonstrative behaviors of the study participants as they perform the required activities of the number knowledge assessment items. While this can provide the researchers with a deeper understanding of both participant engagement with a situated learning activity and the nuanced methods in which early learners demonstrate their knowledge of specific content, the suggestions for interactive media development proposed still stand. Interactive media is poised to dramatically change the field of learning, especially when pairing newly emerged technologies like the Kinect with tried-and-true educational interventions like Sesame Street. The results that are most useful for designers and mathematics educators, however, may be hiding behind a simple test that declares discouragingly: “No significant differences.”

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Acknowledgments We would like to thank Alex Games of Microsoft Studios and Rane Johnson of Microsoft Research for supporting the research that led to these findings, and Jordan T. Thevenow-Harrison of the University of Wisconsin-Madison for his invaluable statistical assistance. Finally, thanks to our advisors and mentors who made this research and analysis possible: Drs. Kurt Squire, Constance Steinkuehler, and Amy B. Ellis. Earlier versions of this paper were published in the Games+Learning+Society 9.0 conference proceedings (Rothschild et al. 2013a) and in the Psychology of Mathematics Education—North American chapter proceedings (Rothschild et al. 2013b).

References Alibali, M. W., & Goldin-Meadow, S. (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–523. Ball, S., & Bogatz, G. A. (1970). The first year of Sesame Street: An evaluation. Princeton: Educational Testing Service. Barab, S. A., Gresalfi, M., & Ingram-Goble, A. (2010). Transformational play: Using games to position person, content, and context. Educational Researcher, 39(7), 525–536. doi:10.3102/0 013189X10386593. Baroody, A. J., Lai, M., & Mix, K. S. (2005). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & O. N. Saracho (Eds.), Handbook of research on the education of young children (2nd ed., pp. 187–221). Mahwah: Lawrence Erlbaum Associates. Baroody, A. J., Lai, M.L., & Mix, K. S. (2006). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & N. Olivia (Eds.), Handbook of research on the education of young children (2nd ed.). Mahwah: Lawrence Erlbaum. Barsalou, L. W. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22(4), 577– 609 (disc. 610–660). Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645. Barsalou, L. W., Niedenthal, P. M., Barbey, A., & Ruppert, J. (2003). Social embodiment. In B. Ross (Ed.), The psychology of learning and motivation (Vol. 43, pp. 43–92). San Diego: Academic. Bogatz, G. A., & Ball, S. (1971). The second year of Sesame Street: A continuing evaluation. Princeton: Educational Testing Service. Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds.). (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press. Dogan, M. F., Williams, C. C., Walkington, C., & Nathan, M. (2013). Body-based examples when exploring conjectures: Embodied resources and mathematical proof. Poster presentation conducted at the Research Precession of the 2013 National Council of Teachers of Mathematics Annual Meeting and Exposition, Denver, CO. Fisch, S. M., & Truglio, R. T. (2001). “G” is for growing: Thirty years of research on children and Sesame Street. Mahwah: Erlbaum. Fisch, S. M., Lesh, R., Motoki, E., Crespo, S., & Melfi, V. (2011). Children’s mathematical reasoning in online games: Can data mining reveal strategic thinking? Child Development Perspectives, 5(2), 88–92. Gee, J. P. (2003). What video games have to teach us about learning and literacy. New York: Palgrave MacMillan. Gelman, R., & Meck, E. (1983). Preschoolers’ counting: Principles before skill. Cognition, 13(3), 343–359.

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Glenberg, A., Jaworski, B., Rischal, M., & Levin, J. (2007a). What brains are for: Action, meaning, and reading comprehension. In D. McNamara (Ed.), Reading comprehension strategies: Theories, interventions, and technologies (pp. 221–238). Mahwah: Erlbaum. Glenberg, A. M., Brown, M., & Levin, J. R. (2007b). Enhancing comprehension in small reading groups using a manipulation strategy. Contemporary Educational Psychology, 32, 389–399. Lauricella, A. R., Gola, A. A. H., & Calvert, S. L. (2011). Toddlers’ learning from socially meaningful video characters. Media Psychology, 14, 216–232. Microsoft. (2012). Kinect Sesame Street TV [computer software]. Redmond: Microsoft Studios. Nathan, M., Kintsch, W., & Young, E. (1992). A theory of algebra-word-problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9(4), 329–389. National Association for the Education of Young Children [NAEYC]. (2009). Developmentally appropriate practice in early childhood programs serving children from birth through age 8: Position statement. Washington, DC: NAEYC. www.naeyc.org/files/naeyc/file/positions/position%20statement%20Web.pdf. National Association for the Education of Young Children [NAEYC], & Fred Rogers Center for Early Learning and Children’s Media. (2012). Technology and interactive media as tools in early childhood programs serving children from birth through age 8: Joint position statement. Washington, DC: NAEYC (Latrobe: Fred Rogers Center for Early Learning and Children’s Media at Saint Vincent College). www.naeyc.org/files/naeyc/file/positions/PS_technology_ WEB2.pdf. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: NCTM. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards: Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. http://www. corestandards.org/the-standar ds/mathematics. Rothschild, M., Williams, C. C., & Thevenow-Harrison, J. T. (2013a). Counting apples and coconuts: Young children ‘Kinect-ing’ Sesame Street and mathematics. In C. Williams, A. Ochsner, J. Dietmeier, & C. Steinkuehler (Eds.), Proceedings of the 9th Annual Games+Learning+Society Conference (Vol. 3, pp. 274–280). Pittsburgh: ETC Press. Rothschild, M., Williams, C. C., & Thevenow-Harrison, J. T. (2013b). Performance assessments. In M. V. Martinez & A. C. Superfine (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (p. 1207). Chicago: University of Illinois at Chicago. Shapiro, L. (2011). Embodied cognition. New York: Routledge. Squire, K. (2011). Video games and learning: Teaching and participatory culture in the digital age. Technology, Education–Connections (the TEC Series). New York: Teachers College Press. Steinkuehler, C., & Duncan, S. (2008). Scientific habits of mind in virtual worlds. Journal of Science Education and Technology, 17(6), 530–543. doi:10.1007/s10956-008-9120-8. Walkington, C., Srisurichan, R., Nathan, M., Williams, C., & Alibali, M. (2012). Using the body to build geometry justifications: The link between action and cognition. Paper presented at the 2012 American Educational Research Association Annual Meeting and Exhibition, Vancouver, BC. Walkington, C., Nathan, M., Alibali, M., Pier, L., Boncoddo, R., & Williams, C. (2013). Projection as a mechanism for grounding mathematical justification in embodied action. Paper presented at the 2013 Annual Meeting of the American Educational Research Association, San Francisco, CA. Meagan Rothschild is an Assessment and Design Specialist at WIDA and a PhD candidate at the University of Wisconsin-Madison. She made the courageous leap to the chilly Midwest from balmy Hawaii to pursue a PhD and work with the Games, Learning, and Society Center. Prior to her move, Meagan served as the Instructional Designer for Cosmos Chaos!, an innovative video game designed to support struggling fourth grade readers developed by Pacific Resources for

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Education and Learning (PREL). Her experience at PREL also included the design of a violence and substance abuse prevention curriculum for Native Hawaiian students, using an interdisciplinary approach that merged health and language arts content standards to support literacy-driven prevention activities. Meagan has 6 years of experience in the Hawaii Department of Education system serving in varied roles, including high school classroom teacher, grant writer and manager, technology coordinator, and Magnet E-academy coordinator. Meagan has a BA and MEd from the University of Hawaii at Manoa, with undergraduate studies in Hawaiian Language and special education, and an MEd in Educational Technology. As a PhD candidate in Digital Media and Learning at the University of Wisconsin-Madison, her work now focuses on developing and researching multimedia environments that merge research-based learning principles with interactive/gaming strategies to engage learners. She specifically focuses on the role of play to not only provide opportunities for deeper learning, but to provide relevant contexts for learners to demonstrate content knowledge, challenging traditional views of assessment practices. Caroline C. Williams is a dissertator at the University of Wisconsin-Madison (specializing particularly in Mathematics Education in the Department of Curriculum and Instruction), and affiliated with the Games+Learning+Society group. Her dissertation involves designing a Little Big Planet 2 game to teach fractions and linear functions, and she specializes in research involving middle school students, mathematics, and all things digital media. Caro has presented and published on a wide variety of topics, including building in Little Big Planet, mathematics in World of Warcraft forum posts, using gestures to support mathematical reasoning, example usage in mathematical proof processes, gender and mathematics, and learning trajectories in linear, quadratic, and exponential functions.

Classroom Gaming

Research Brief How Can Teachers Use Video Games to Teach Their Students Mathematics?

A

Pew report published in 2008 reported that 97 percent of teenagers play video games of some type (Lenhart et al., 2008). The U.S. Department of Education’s Transforming American Education: Learning Powered by Technology (2010) suggests video games can support a range of teaching and learning activities in school, from embedded assessment to engagement with locally relevant issues. This recommendation echoes research describing how well-designed video games can simulate professional practices and model real-world problem solving (Gee, 2005, 2007; Salen, 2008; Shaffer, 2006a; Squire, 2006). Recently, a survey of elementary and middle school teachers already using video games in their classrooms found that nearly one-fifth teach with video games every day (Millstone, 2012). Concerning mathematics education, we suggest it is necessary to ask: How can teachers use video games to teach their students mathematics? Our primary concern raises additional questions about how teachers select and evaluate video games, and the various types of video games suitable for mathematics instruction. We address these questions by reviewing current research on video games and mathematics education, and then detailing how to find specific mathematics video games. After highlighting a case of video game play supporting statistical reasoning, we discuss how to adapt video games to support classroom teaching, and we then consider the future of teaching mathematics with video games.

Background: Video Games and Mathematics Education For many teachers, their introduction to teaching mathematics with video games was Math Blaster (Davidson & Associates, 1983), “A standard drill-and-practice-type instructional mechanism . . . within a shooter game idiom” (Ito, 2008, p. 93). Introduced in 1983, Math Blaster was a best-selling piece of software; today, mathematics video games are a broader, dynamic, and profitable enterprise. Video games have been designed across grade levels and to meet Common Core State Standards for Mathematics (2010) and National Council of Teachers of Mathematics (2000) standards. They now range from the mobile drill program Flash Math (Kiger, Herro, &

Prunty, 2012) to narrative-based virtual worlds like Quest Atlantis (Gresalfi, Barab, Siyahhan, & Christensen, 2009). Because definitions of “game” are broad and contested (Salen & Zimmerman, 2003; Schell, 2008), in this brief we consider video games to be “imaginary worlds, hypothetical spaces where players can test ideas and experience their consequences” (Squire & Jenkins, 2003, p. 8). This definition is consistent with a general consensus that video games are more than digital tools; video games are designed environments, or possibility spaces (Squire, 2008), that support learning across multiple social spaces, shared practices, and emergent forms of knowledge (Barab, Gresalfi, Dodge, & Ingram-Goble, 2010; Gee, 2003; Shaffer, 2006b). Our understanding of video games contrasts with digital simulations like Geometer’s Sketchpad (Key Curriculum Press, 1991; e.g., Knuth, 2002; Leong & Lim-Teo, 2003) or “cognitive tutor” systems (e.g. Anderson, Corbett, Koedinger, & Pelletier, 1995), which primarily support consistent and accurate means of interacting with mathematical objects and or notations. Video game play, on the other hand—like play in general—affords positive affect, nonlinearity, intrinsic motivation, process, and free choice (Johnson, Christie, & Wardle, 2005; Vygotsky, 1978). Unlike established research about the impact of video games on students’ science learning (e.g. Clarke & Dede, 2009; Gaydos & Squire, 2012; Squire, 2010), only a handful of empirical investigations have explored how video games influence students’ mathematics experiences and understanding (e.g., Harris, Yuill, & Luckin, 2008; Ke & Grabowski, 2007; Kebritchi, 2008). Young and colleagues (2012) recently examined the pedagogical value of video games in respect to student achievement. Nine studies from the mathematics gaming literature were included in their meta-analysis. Like previous reviews which identified a sparse literature base and insufficient research about instructional gaming among specific age groups (Mitchell & Savill-Smith, 2004), Young et al. (2012) suggest better “correspondence” be developed between a game’s objectives and students’ mathematics learning activities. Teaching mathematics with video games does not invariably equate to students learning mathematics from video games. In other words, the results are mixed. Ke & Grabowski (2007) found that fifth graders who

The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council. Copyright © 2014 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502, Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org.

How Can Teachers Use Video Games to Teach Their Students Mathematics?

played video games outperformed non-gaming peers, yet only those who played games cooperatively (rather than competitively) had effective gains in mathematics understanding and positive changes in attitude (as measured by a modified version of the Attitudes Towards Maths Inventory; Tapia, Marsh, & George, 2004). For high school students, playing DimensionM when aligned with online modules and classroom teaching increased mathematics achievement in comparison to non-gaming students; however, gameplay did not improve motivation to learn mathematics (Kebritchi, 2008). Kebritchi, Hirumi, and Bai (2008) found statistically significant gains for high school students who played video games; their average achievement gain between two district exams was more than double that of non-gaming peers. Further, these students’ teachers reported video games as “effective” learning tools because they were experimental, offered alternative teaching approaches, provided an engaging rationale for learning, and increased time on task. Nonetheless, trends persist across this literature; the success and failure of teaching mathematics with video games reflects many factors, including research design, game mechanics, teaching strategies, learning objectives, and context (Young et al., 2012). The varied influence of video game play in mathematics classrooms is unsurprising: games are no magic bullet. Like Young et al. (2012), we believe that successful learning depends upon teachers developing and supporting “corre-

spondence” between play and instruction. Effectively teaching mathematics with video games requires teachers who can develop, support, and reflect upon how any game corresponds to instructional goals and student learning needs. In sum, mathematically meaningful gameplay is the result of thoughtful and creative teaching. But how does a teacher go about finding mathematics video games to support teaching and learning in his or her classroom?

Finding Mathematics Video Games Imagine a veteran mathematics educator eager to refine her practice. Alternatively, what of the “digitally able” first-year teacher (Starkey, 2010), a novice confident in her ability to adapt a range of new technologies to communicate, problemsolve, play, and share information about teaching and student learning. How might either teacher identify what makes gameplay effective in a mathematics classroom? What will either perceive to be the advantages of video games in contrast to more traditional instructional strategies? Can common pitfalls be avoided, hallmarks of success recognized, and challenges mitigated? Each of these educators can easily visit the “Education” section of Apple’s App Store and search for “math games.” As of May 2014, this search gave 219 results. Some games are free while others are cheap, and the available games cover a range of content areas and grade levels,

Fig. 1. Screenshot of Educade homepage

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

with many claiming to dramatically improve students’ learning. Where might these teachers go to find well-designed video games, related curricular materials, and—perhaps most importantly—a community of professionals who can support implementation and refinement of teaching mathematics with video games? Enthusiasm about video games and learning has resulted in the growth of many online resources and communities, as well as offline conferences and professional learning events. Websites such as GameDesk’s Educade (http://educade.org) and Common Sense Media’s Graphite (http://www.graphite.org) provide free game reviews, lesson plans, teaching tools, and other resources—many of them authored by teachers. Many educators have also created and joined online communities devoted to single games. For example, the game Minecraft (Mojang, 2009) has inspired MinecraftEdu (http://minecraftedu. com), an online community and collaborative effort promoting affordability and school accessibility. Additionally, the Playful Learning Initiative (http://playfullearning.com) supports an online knowledge base of video games, facilitates regional and national partnerships and professional learning summits, and is guided by an advisory board of K–12 educators. As new mathematics video games are designed, so too will the number of websites, online communities, and other resources proliferate. Each website and online community has different characteristics; accordingly, we offer a brief outline

of how to find relevant games using Educade (see fig. 1). A search bar at the top of the website may be used, for example, to find games focused on middle and high school algebra. Seven results are presented. Four are “teaching tools,” or information about specific games and resources for teaching mathematics. The remaining three are “lesson plans” focused on implementing games according to specific learning objectives. One “teaching tool” is Dragonbox (WeWantToKnow, 2012). Accompanied by two lesson plans, Dragonbox is likely a useful resource, and selecting the game leads to additional information. A screenshot from Dragonbox is included as figure 2. Dragonbox is robustly described with text, images, a video, as well as information regarding algebraic content, appropriate grade levels, platforms supporting gameplay, and how teachers can implement the game. There are also two relevant lesson plans, and a link to the official Dragonbox website. While we strongly recommend Dragonbox (particularly for mathematics teachers newly interested in video games), more than a few poorly designed mathematics games are also available. We advise playing a demo before purchasing any mathematics video game. If a game does not have a freely available demo, request one from the company. In our experience, a company unwilling to share a demo may be concerned that they are marketing an inferior product. After downloading the demo, play it! While developing

Fig. 2. Screenshot of a level from Dragonbox

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

ideas about how the game fits with classroom teaching, request funding to purchase the full game for classroom use. Consider how the game aligns with curricula, and the “correspondence” among students’ play, mathematics content, and instruction. Will students play in the classroom, after school, or at home? When playing after school or at home, make sure that a computer lab is available and that the game is properly installed so that all students have access to it. Finally, it is crucial to fully play any game before implementation. While playing, reference desired learning objectives and note which are supported by play and which are not. Consider how the game supports standards and curricula, and what lesson and unit planning could connect student gameplay with learning objectives. Teachers who are eager to consult models of classroom teaching and learning can find online communities of teachers and designers (listed earlier) who openly share tales of triumph and failure. These stories can inform decisions about whether or how a game should be played. Accordingly, we now share a case study that highlights ways of connecting gameplay to classroom teaching.

Case Study: Statistical Reasoning and World of Warcraft Teaching mathematics with video games need not conjure up aged images of Math Blaster; classroom teachers are adapting various gaming environments to support the development of sophisticated mathematical investigations. Consider the case of Scott McClintock, a statistics teacher at West Chester University in West Chester, Pennsylvania. Writing in NCTM’s Mathematics Teacher, McClintock (2011) describes how the game World of Warcraft (Blizzard Entertainment, 2004; fig. 3, below) supported his students’ understanding of statistical reasoning. A primary concern for McClintock had been data collection and analysis; he perceived issues of sample size and survey relevance to be pedagogical challenges for quality teaching and learning. However, with World of Warcraft McClintock realized he could address these challenges by creating assignments that engaged a “wide variety of statistical practices” (p. 215). One key to McClintock’s successes—and his students’ learning—was the opportunities for engagement and reflection found within the game world and also in class after gameplay ended.

Fig. 3. Screenshot of multiplayer battle Al’Akir in World of Warcraft (used with permission from Martin et al., 2012) 4

How Can Teachers Use Video Games to Teach Their Students Mathematics?

One assignment concerns the statistical concept of sampling. McClintock directed his students to survey the race and class of 50 World of Warcraft players in each of three different cities. Knowledgeable about the four different races and four different class categories possible for any given player, students sampled 50 players in the human capital, 50 in the elven capital, and 50 in a racially neutral city. After collecting data, students calculated the percentage of humans in each sample, and then determined which sample proportion most likely represented the true proportion of total human players. Although World of Warcraft was not designed to teach statistics, a number of game features supported students’ statistical reasoning. First, random sampling methods would not work in this virtual context; instead, systematic sampling was possible as the race and class of a given player could be obtained through an unobtrusive mouse click. Second, the design of particular cities biases the sample; sampling in the human capital would likely overestimate the total proportion of humans, whereas sampling in the elven capital would likely underestimate the proportion. Thus, only in a race-neutral city could students obtain the least biased statistical sample. It was because of the game’s contextual features that more sophisticated mathematics, about the probabilities of players’ race and class, became possible. McClintock also addressed the importance of continuing to support students’ investigations beyond gameplay. Reflecting upon his own limitations, he suggests teachers provide written feedback to students about findings, encourage students to ask questions of their data, and conduct iterative rounds of data collection and analysis back in World of Warcraft to test new hypotheses. Ultimately, McClintock argues that both game features and teachers’ pedagogy can appropriately support students’ learning of complex statistics, and that teachers should “consider how we might adapt [video games] to enhance the reach and scope of the classroom” (p. 217).

Discussion: Playing and Adapting Video Games for Teaching Mathematics The intent of this brief is not to support any mathematics teacher in only visiting an online database, downloading a game and related materials, and then having students play “for fun” or as a “reward” after completing another assignment. Rather, we advocate adapting video games to complement the characteristics of a classroom. McCall (2011), writing about his own use of games in the classroom, argues, “Successful game-based lessons are the product of well-designed environments. Teacher/designers must thoughtfully

embed these games in an environment and set of learning activities where students, learning tools, and resources work together in pursuit of the desired outcomes” (p. 61). Robust mathematics education has never been achieved through the blind adoption of new curricula, tools, or methods devoid of teachers’ skillful facilitation; video games are not—and, we believe, will never be—an exception. Teaching mathematics with video games will require that teachers deftly consider trends, limitations, and the insights of case studies such as the one noted above; make professional judgments relevant to local context; and reflect upon the successes and challenges associated with teaching and student learning. We suggest that educators interested in teaching mathematics with video games recognize the substantial effort and intellectual engagement required of such an endeavor. Learning to play any game to support learning – and, in particular, a mathematics video game – entails professional behaviors similar to acquiring any new teaching method: research, planning, implementation, reflection, and iteration. Teachers should investigate games and related teaching resources. Game play should be a central element to any teachers’ personal research and learning. Curricular materials, lesson plans, and assessments should also be prepared before classroom implementation, yet remain flexible enough to be revised iteratively. Furthermore, it may be advantageous for teachers to join online professional-amateur (“pro-am”) gaming communities, like those associated with Minecraft, in order to discuss experiences, share insights, and grow a professional network of teachers-as-players.

The Future of Game-Based Mathematics Education As both literature and popular media indicate, teachers and their students are playing and increasingly designing video games (Squire, 2011). With a growing number of teachers writing about their own experience designing and teaching with video games (e.g. Elford [blog]; McCall, 2011), we believe this is a very exciting time to be a mathematics educator interested in using video games (and other games) for learning. Classroom teachers are beginning to refine how researchers and designers create mathematics video games, so content and gameplay both align to standards and also adapt to individual students’ learning needs (e.g. Riconscente, 2011). What new forms of pedagogical, technological (cf. Mishra & Koehler, 2006), and mathematical knowledge for teaching (cf. Ball, Thames, & Phelps, 2008) might teachers demonstrate when teaching with video games across both formal and informal settings? Might virtual game worlds, like World of Warcraft, become a communal mathematics “third plac-

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

es” (Steinkuehler & D. Williams, 2006) among networked learners around the world? And how might playful pedagogy and game-based curricula alter how researchers and practitioners understand “ambitious” instruction (cf. Lampert et al., 2010)? As Young and colleagues (2012) suggest, the future of game-based education will be shaped by educators and researchers who collaboratively examine “how gaming combined with instructional facilitation by a master teacher affects engagement, student behavior, and overall academic achievement” (p. 83). It is an exciting time to be an educator interested in teaching mathematics with video games.

Acknowledgements We thank our thoughtful advisors and mentors at the University of Wisconsin–Madison and the Games+Learning+Society Center who reviewed this manuscript: Dr. Amy B. Ellis, Dr. Kurt Squire, and Dr. Constance Steinkuehler. We also thank two anonymous reviewers and Series Editor Michael Fish for their useful commentary.

By Jeremiah I. Holden and Caroline C. Williams University of Wisconsin–Madison Michael Fish, Series Editor

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Gee, J.P. (2003). What video games have to teach us about learning and literacy. New York, NY: Palgrave MacMillan. Gee, J. P. (2005). Learning by design: Good video games as learning machines. E-Learning, 2(1), 5–16. Gee, J. P. (2007). What video games have to teach us about learning and literacy. 2nd ed. New York, NY: Palgrave Macmillan. Gresalfi, M., Barab, S., Siyahhan, S., & Christensen, T. (2009). Virtual worlds, conceptual understanding, and me: Designing for consequential engagement. On the Horizon, 17(1), 21–34. Harris, A., Yuill, N., & Luckin, R. (2008). The influence of context-specific and dispositional achievement goals on children’s paired collaborative interaction. British Journal of Educational Psychology, 78, 355–374. Ito, M. (2008). Education vs. entertainment: A cultural history of children’s software. In K. Salen (Ed.), The ecology of games: Connecting youth, games, and learning (pp. 89–116). Cambridge, MA: MIT Press. Johnson, J. E., Christie, J. F., & Wardle, F. (2005). Play, development, and early education. Boston, MA: Pearson Education. Ke, F., & Grabowski, B. (2007). Gameplaying for maths learning: Cooperative or not? British Journal of Educational Technology, 38, 249–259. Kebritchi, M. (2008). Effects of a computer game on mathematics achievement and class motivation: An experimental study. Dissertation Abstracts International Section A: Humanities and Social Sciences, 69(6-A), 2121. Kebritchi, M., Hirumi, A., & Bai, H. (2008). The effects of modern math computer games on learners’ math achievement and math course motivation in a public high school setting. British Journal of Educational Technology, 38(2), 49–259. Key Curriculum Press (1991). Geometer’s sketchpad [computer software]. Berkeley, CA. Kiger, D., Herro, D., & Prunty, D. (2012). Examining the Influence of a Mobile Learning Intervention on Third Grade Math Achievement. Journal of Research on Technology in Education, 45(1). Knuth, E. J. (2002). Fostering mathematical curiosity. Mathematics Teacher, 95(2), 126–130. Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In Instructional explanations in the disciplines (pp. 129–141). New York, NY: Springer. Lenhart, A., Kahne, J., Middaugh, E., Macgill, A. R., Evans, C., & Vitak, J. (2008). Teens, video games, and civics. Washington, D.C.: Pew Internet & American Life Project. Leong, Y. H., & Lim-Teo, S. K. (2003). Effects of geometer’s sketchpad on spatial ability and achievement in transformation geometry among secondary two students in Singapore. The Mathematics Educator, 7(1), 32–48.

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Martin, C., Williams, C., Ochsner, A., Harris, S. King, E., Anton, G., Elmergreen, J. & Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people (pp. 226243). London: Routledge. McCall, J. (2011). Gaming the past: Using video games to teach secondary history. New York, NY: Routledge, Taylor & Francis Group. McClintock, S. (2011). Counting priests, paladins, pets. Mathematics Teacher, 105(3). Millstone, J. (2012). Teacher attitudes about digital games in the classroom. New York, NY: The Joan Ganz Cooney Center at Sesame Workshop. Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A new framework for teacher knowledge. Teachers College Record 108(6), 1017–1054. Mitchell, A., & Savill-Smith, C. (2004). The use of computer and video games for learning: A review of the literature. London, UK: The Learning and Skills Development Agency. Mojang (2009). Minecraft [computer software]. Stockholm, Sweden. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM. National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). (2010). Common core state standards for mathematics. Washington, D.C.: NGA Center and CCSSO. http://www.corestandards.org. Riconscente, M. (2011). Mobile learning game improves 5th graders’ fractions knowledge and attitudes. Los Angeles: GameDesk Institute. Salen, K. (2008). (Ed). The ecology of games: Connecting youth, games, and learning. Cambridge, MA: MIT Press. Salen, K., & Zimmerman, E. (2003). Rules of play. Cambridge, MA: MIT Press. Schell, J. (2008). The art of game design: A book of lenses. Burlington, MA: Morgan Kaufmann Publishers. Shaffer, D. W. (2006a). How computer games help children learn. New York, NY: Palgrave Macmillan. Shaffer, D. W. (2006b). Epistemic frames for epistemic games. Computers & Education, 46(3), 223–234. Squire, K. (2006). From content to context: Video games as designed experience. Educational Researcher, 35(8), 19–29. Squire, K. D. (2008). Video games and education: Designing learning systems for an interactive age. Educational Technology, 48(2), 17. Squire, K. (2010). From information to experience: Place-based augmented reality games as a model for learning in a globally networked society. Teachers College Record, 112(10), 2565– 2602.

Squire, K. (2011). Video games and learning: Teaching and participatory culture in the digital age. New York, NY: Teachers College Press. Squire, K.D. & Jenkins, H. (2003). Harnessing the power of games in education. Insight, 3, 5–33. Starkey, L. (2010). Supporting the digitally able beginning teacher. Teaching and Teacher Education, 26(7), 1429–1438. Steinkuehler, C. A., & Williams, D. (2006). Where everybody knows your (screen) name: Online games as “third places.” Journal of Computer‐Mediated Communication, 11(4), 885–909. Tapia, M., Marsh, I. I., & George, E. (2004). An instrument to measure mathematics attitudes. Academic Exchange Quarterly, 8(2), 16–21. U.S. Department of Education, Office of Educational Technology (2010). Transforming American education: Learning powered by technology. Washington, D.C.: U.S. Department of Education. Vygotsky, L. S. (1978). Mind and society: The development of higher mental processes. Cambridge, MA: Harvard University. WeWantToKnow (2012). Dragonbox [computer software]. Oslo, Norway. Young, M. F., Slota, S., Cutter, A. B., Jalette, G., Mullin, G., Lai, B., Simeoni, Z., Tran, M., & Yukhymenko, M. (2012). Our princess is in another castle: A review of trends in serious gaming for education. Review of Educational Research, 82(1), 61–89.

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Running Head: Quantifying Exponential Growth Ellis, A.B., Ozgur, Z., Kulow, T., Williams, C., & Amidon, J. (2012). Quantifying exponential growth: The case of the Jactus. In R. Mayes & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 93-112). Laramie, WY: University of Wyoming.

Quantifying Exponential Growth: The Case of the Jactus

Amy Ellis Zekiye Ozgur Torrey Kulow Caroline Williams University of Wisconsin-Madison Joel Amidon University of Mississippi

Quantifying Exponential Growth

Abstract This article presents the results of a small-scale teaching experiment with three middle school students who explored exponential growth by reasoning with the co-varying quantities height and time. Three major conceptual shifts occurred during the course of the teaching experiment: a) From repeated multiplication to a coordination of growth in height and time values; b) From coordinating height and time to coordinating constant ratios; and c) Generalizing to non-natural exponents. The details of each of the three shifts is explored, followed by a discussion of the implications of addressing exponential growth from a covariation of quantities perspective.

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The Landscape of Exponential Functions in Common School Treatments Students’ mathematical learning is the reason our profession exists. Everything we do as mathematics educators is, directly or indirectly, to improve the learning attained by anyone who studies mathematics. (Thompson, 2008, p. 45) Exponential functions are an important concept both in school algebra and in higher mathematics. Not only do they play a critical role in college mathematics courses such as calculus, differential equations, and complex analysis (Weber, 2002), they also represent an important transition from middle school mathematics to the more complex ideas students encounter in high school mathematics. A focus on the conceptual underpinnings of exponential growth has increased in recent years; for instance, the Common Core State Standards (CCSS) highlight the need to understand exponential functions in terms of one quantity changing at a constant percent rate per unit interval relative to another. Moreover, these ideas are also being pushed down into middle school mathematics courses, both in terms of national standards such as the CCSS as well as in middle school curricula (e.g., Lappan et al., 2006). The study we report on in this paper is situated in a larger project exploring middleschool students’ understanding when reasoning with co-varying quantities to support their development of function understanding. Our prior studies focused on students’ understanding of linear function and quadratic function as they reasoned with quantitative relationships such as gear ratios, constant speed, and growing rectangles (Ellis, 2007; 2011a; 2011b). A natural extension of this work is to explore how students come to understand exponential growth through reasoning in a similar context, namely, by exploring two co-varying quantities such that one changes exponentially as the other changes linearly. Our approach differed in significant ways from the typical textbook approaches to exponential growth, which we detail below. The Repeated Multiplication Approach

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Quantifying Exponential Growth

A common textbook treatment for introducing the notion of exponential growth is the repeated multiplication approach. For instance, in the middle school curriculum Connected Mathematics Project (2006), students place coins called rubas on a chessboard in a doubling pattern, and then use tables, graphs, and equations to examine the exponential relationship between the number of squares and the number of rubas. These types of tasks require students to perform repeated multiplication to solve a problem and then connect that process to exponential notation (Castillo-Garsow, submitted). A number of researchers have advocated for this approach, suggesting that we define exponentiation as repeated multiplication with natural numbers, and then help students generalize beyond the natural numbers (e.g., Goldin & Herscovics, 1991; Weber, 2002). However, this approach has its limitations. As Davis (2009) noted, generalizing to non-natural exponents may pose difficulties for students; for instance, an 2

expression such as 2 3 can be difficult to understand from a repeated-multiplication perspective. Difficulties in Understanding Exponential Growth

€ literature on students’ and teachers’ understanding of exponential growth is scant, The but the research that does exist supports Davis’ concerns about the difficulties in generalizing one’s understanding of exponentiation as repeated multiplication. For instance, Weber (2002) found that college students struggled to understand or explain the rules of exponentiation and could not connect them to rules for logarithms. Weber described students’ difficulties in explaining what a function such as f (x) = a x meant, as well as in explaining why a function

" 1 %x such as f (x) = $ ' was a decreasing function. Pre-service teachers have not fared much better; # 2& € researchers have identified their struggles not only in understanding exponential functions, but

€ also in recognizing growth as exponential in nature (Davis, 2009; Presmeg & Nenduradu, 2005).

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Quantifying Exponential Growth

Although pre-service teachers appear to have a strong understanding of exponentiation as repeated multiplication, they experience difficulty in connecting this understanding to the closedform equation and in appropriately generalizing rules such as the multiplication and power properties of exponents (Davis, 2009). In general, teachers appear to be able to make some use of graphical, algebraic, and tabular representations, but cannot then leverage their algebraic facility to support their ability to solve exponential problems or to translate from table situations to either recursive representations or correspondence rules (Davis, 2009; Presmeg & Nenduradu, 2005). Research on middle school and high school students reveals difficulties as well; students struggle to transition from linear representations to exponential representations, or to identify what makes data exponential (Alagic & Palenz, 2006). In general, exponential growth appears to be challenging to represent for both students and teachers, and it is difficult for teachers to both anticipate where students might struggle in learning about exponential properties and develop ideas for appropriate contexts that involve exponential growth (Davis, 2009; Weber, 2002). These documented challenges suggest a need for better understanding of how to foster students’ learning about exponential growth, and for identifying more effective modes of instruction on exponential functions. Alternate Approaches to Exponential Growth Repeated multiplication is not the only way to think about exponential growth; one can also approach exponentiation in other ways, for instance, as the relationship between a population of individuals and their collective growth contributions (Castillo-Garsow, submitted), as products of factors (Weber, 2002), or as a multiplicative rate constructed from multiplicative units (Confrey & Smith, 1994; 1995). Weber (2002) offered a theoretical analysis of exponential growth relying on Dubinsky’s (1991) APOS (Action, Process, Object, Schema) theory. Although

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Quantifying Exponential Growth

this approach begins with an action understanding of exponentiation as repeated multiplication, Weber offers a vision of students then transitioning to a process understanding by interiorizing the repeated multiplication action; students would then view exponentiation as a function and be able to reason about its properties. Once students can consider exponential expressions as a result of a process, terms such as 2 3 can be viewed as the product of 3 factors of 2, and ultimately students should then be able to generalize their understanding to view a b as b factors of a.

€ Weber’s analysis offers a vision for moving beyond the repeated multiplication view, but it € these processes. remains an open question how students might actually undergo Confrey and Smith (1994; 1995) introduced an operational basis for multiplication and division called splitting, which is a multiplicative operation that is not repeated addition. A splitting structure is a multiplicative structure in which multiplication and division are inverse operations, such as repeated doubling and repeated halving. Within this model, students also treat the product of a splitting action as the basis for its reapplication; thus, a split can be viewed as a multiplicative unit. Confrey and Smith (1994) assert that “Building concepts of multiplicative rates constructed from multiplicative units should play a central role as students work on understanding how multiplicative worlds generate constant doubling times and constant halflives.” (p. 55) Splitting as an operation can form the basis of a rate of change approach to exponential functions, which we will discuss in further detail below. In Confrey and Smith’s (1994) work, they found a number of different rate-of-change approaches adopted by students making sense of exponential situations, including multiplicative rates of change. Students constructing multiplicative rates of change would interpret a table with, for instance, a growth factor of 9 to be increasing by “a constant rate of nine.” Confrey and Smith suggest that this is an important

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Quantifying Exponential Growth

conception of rate of change, which is found by calculating the ratios between succeeding yvalues for constant unit-change in the x-values. We highlight this conception as an important foundational idea for a rate-of-change approach to exponentiation. The Rate of Change Approach to Exponential Functions: Covariation and Continuous Variation Traditional approaches to function rely on a correspondence view (Smith, 2003), in which a function is seen as the fixed relationship between the members of two sets. From this perspective, y = f (x) represents y as a function of x, in which each value of x is associated with a unique value of y (Farenga & Ness, 2005). This static view underlies the typical treatment of € functions in school mathematics, and it is not difficult to see how students may struggle to

transition from a repeated-multiplication understanding of exponentiation to a correspondence understanding, particularly beyond the domain of natural numbers. Smith and Confrey (Smith, 2003; Smith & Confrey, 1994) offered an alternative to the correspondence view, which they called the covariation approach to functional thinking. Here one examines a function in terms of a coordinated change of x- and y-values: A covariation approach, on the other hand, entails being able to move operationally from ym to ym+1 coordinating with movement from xm to xm+1. For tables, it involves the coordination of the variation in two or more columns as one moves down (or up) the table. (Confrey & Smith, 1994, p. 33) This is what the students described in Confrey and Smith’s 1994 article did when they calculated the ratios between succeeding y-values for a constant unit-change in the x-values in tables of exponential data. Confrey and Smith argue that splitting, juxtaposed with covariation, can

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Quantifying Exponential Growth

provide the basis for the construction of an exponential function. Exponentiation is simply repeated splitting, just as multiplication is repeated addition. Covariation for Exponential Growth Castillow-Garsow (2012, this volume) describes covariation as the imagining of two quantities changing together; students imagine how one variable changes while imagining changes in the other. Relying on situations that involve quantities that students can make sense of, manipulate, experiment with, and investigate can foster their abilities to reason flexibly about dynamically changing events (Carlson & Oehrtman, 2005). An approach that relies on imagining co-varying quantities may be especially useful in helping students understand exponential growth, as this view is strongly tied to how students think about contexts involving multiplicative relationships (Davis, 2009). Thompson (2008) argues that a defining characteristic of exponential functions is the notion that the rate at which an exponential function changes with respect to its argument is proportional to the value of the function at that argument. Approaches that emphasize this concept could help students make strong connections between the change in x-values and the corresponding change in y-values, developing the understanding that the value of f (x + Δx) / f (x) is dependent on Δx (Thompson, 2008). So, for instance, for a repeated doubling

function f (x) = 2 x , a covariation approach could help students coordinate (additive) changes in x

€ values with (multiplicative) changes in y-values to understand that the constant multiplicative

€

€rate of change for Δx = 1 would be 2, for Δx = 2 would be 2 2 , or 4, for Δx = 3 would be 8, and 1 1 for Δx = would be 2 2 . €2

€

€

€

€

One study (Green, 2008) did take a rate-of-change approach to helping students construct € exponential growth and found that expanding the concept of rate of change to include percent 8

Quantifying Exponential Growth

changes helped students understand the meanings of the parameters of exponential functions. In another study with two high-school students, Castillo-Garsow (2012) found that a focus on reasoning covariationally about financial modeling tasks fostered different solutions to a differential equation based on either a discrete or continuous understanding of change. We were interested in developing a situation in which the notion of proportional rate of change would arise naturally. We have found that adopting a rate of change perspective can be accessible even for beginning algebra students in middle school, particularly if they have opportunities to explore situations that encourage students to construct meaningful relationships between quantities (Ellis, 2007; 2011a; 2011b). Given the age group of our students, we aimed to develop a context with co-varying quantities that satisfied three requirements. First, students should be able to visually observe the quantities changing together. Second, students should have a way to easily measure and record the values of both quantities as they covaried. Third, the quantities should vary in a continuous rather than discrete manner. We will describe how we implemented each of these criteria in the sections below. Continuous Variation In order to discuss continuous variation, it is helpful first to address the ideas of chunky reasoning and smooth reasoning (Castillow-Garsow, 2012). Castillow-Garsow describes chunky reasoning in the following manner: a student imagines that a change occurs in completed “chunks”, after a certain amount of time has passed, such as a day or a week. The student does not imagine that change occurs within the chunk unless she can re-conceptualize the change to a smaller chunk size, such as chopping a week into seven days, with each day having its own completed change. Castillow-Garsow explains that “Chunky thinking is inherently discrete. It

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Quantifying Exponential Growth

remains an open question whether or not continuous understanding can be built from pure chunky thinking” (p. 11). In contrast, when reasoning smoothly, a student imagines a quantity changing in the present tense; one can map from one’s own current experiential time to a time period in the mathematical context without needing to resort to convenient units of time. “Smooth thinking, in contrast, is inherently continuous. By imagining change in progress, change is subjected to that person’s understanding of change in the physical universe” (Castillow-Garsow, 2012, p. 11). Castillow-Garsow suggests that the continuous nature of smooth reasoning is critical for understanding exponential growth. Continuous quantitative reasoning then becomes a “repeated process of imagining the smooth change in progress of a quantity over an interval, followed by an actual or imagined numerical measurement of the quantity at the end of each interval” (Castillow-Garsow, 2012, p. 18). Thompson (2011) similarly describes the concept of continuous variation, in which every smooth change in progress is imagined to be composed of smaller chunks (giving numerical values), and every small chunk within the change in progress is thought of as being itself covered by a smooth change in progress. In this manner, a student achieves infinite precision by alternating smooth and chunky thinking by chopping the interval of variation into finer and finer chunks. If continuous reasoning relies on smooth and chunky thinking in this manner, then education targeting continuous quantitative reasoning should focus on developing smooth reasoning skills (Castillow-Garsow, 2012). For this reason we endeavored to develop a scenario in which students would have the opportunity to engage in continuous quantitative reasoning when imagining two quantities co-varying exponentially. The Jactus: Building Exponential Growth by

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Quantifying Exponential Growth

Reasoning with Continuously Covarying Quantities Building on the principles enacted in our linear and quadratic teaching experiments (Ellis, 2007; 2011), we set out to develop a context in which students could explore two continuously co-varying quantities; we wanted to avoid discrete situations such as the chessboard problem. Our intention was to develop a context that would be understandable to a middle-school population. We settled on a scenario in which a plant called the Jactus grew by doubling its initial height every week. As we will discuss later, the choice of a week as the time frame for the plant to double was deliberate. Students explored the growing Jactus plant by comparing its height to time via a specially designed Geogebra script (Figure 1).

Figure 1: Screenshot of the Geogebra Script for a Doubling Jactus Students could manipulate the image of the Jactus plant by dragging its base with the mouse. As they did so, the plant would continuously increase or decrease in height as it moved along the time axis. Over time, we changed the growth factor of the Jactus to values other than 2, the initial height to values other than 1 inch, and the amount of time to double to values other than 1 week.

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Quantifying Exponential Growth

We recognize that situations do not imply reasoning; continuous problem situations do not necessarily mean that continuous reasoning will occur. As Castillo-Garsow (2012) demonstrated in his example with Alice, Bob, and Carol’s solutions to the same problem, a student’s reasoning may not necessarily correspond to the co-varying quantities in the situation. However, we believe that a context with smoothly co-varying quantities could afford the possibility of continuous reasoning in a way that a discrete situation would not. It is possible to imagine a plant that is somewhere between 1 inch tall when it starts growing and 2 inches tall after the first week. It is more difficult to imagine, for instance, a ruba coin that is in the middle of becoming two ruba coins. Encouraging Coordination We hypothesize that significant mathematical learning can take place as a result of students’ engagement in appropriately selected and sequenced mathematical tasks. If individuals have the capacity to learn through their mathematical activity, the possibility exists to engineer a sequence of tasks that promotes the learning of individuals through their engagement with such a task sequence. (Simon et al., 2010, p. 72) We set out to develop a set of sequenced mathematical tasks by first identifying the ways of thinking that we desired for our students as an outcome. Carlson and Oehrtman’s covariation framework (2005) informed our thinking for the initial design of tasks. Their framework emerged to describe the reasoning involved in the meaningful representation and interpretation of graphical models in calculus, but it has still served useful to guide our thinking about what we wanted middle school students to understand about exponential growth. Carlson and Oehrtman described five mental actions, the first four being relevant to middle school students’ reasoning with exponential functions (the fifth addresses instantaneous rates of change). The first mental action identified by Carlson and Oehrtman is to coordinate the dependence of one variable on another variable. In the Jactus context, we developed activities to help students understand that the height of the Jactus plant depends on the amount of time that it 12

Quantifying Exponential Growth

had been growing. Students’ interaction with the Geogebra scripts familiarized them with this dependence relationship. We then designed tasks in which students discussed the variables that could contribute to the Jactus’ growth, drew pictures of the growing Jactus, and devised methods for keeping track of the plant’s growth over time. The second mental action is coordinating the direction of change of one variable with changes in the other variable. We wanted students to understand that as time increases, the plant grows taller, and as time decreases, the plant grows shorter. Our activities with drawing pictures, interacting with the Geogebra script, and identifying relationships between how much time had passed and how tall the plant had grown encouraged this coordination. The third mental action is coordinating the amount of change of one variable with changes in the other variable. Students should initially understand that the growth in height is determined multiplicatively rather than additively. We wanted students to understand that repeated multiplication of the height, such as doubling, occurred every week. Early activities aimed at this idea required students to create drawings. For instance, given a picture of a 1-inch Jactus plant at 0 weeks, students drew the height of the plant at 1, 2, 3, and 4 weeks. Later, students created drawings in which they had to skip weeks, for instance, drawing the height of the plant at week 3 given a picture of the plant at week 0. In order to highlight the fact that the growth rate is the same for any same Δx , we also asked problems such as the following, in which the height value is unknown (imagine that a student has already determined that a

€ each week): “Say you go on vacation for 1 week. How much taller particular Jactus plant triples will the Jactus be when you return?” We combined these tasks with interpolation problems, far prediction problems, and comparison problems across different plants with different growth rates

13

Quantifying Exponential Growth

and initial heights in order to encourage facility with coordinating the amount of change of height with the amount of change of time. Carlson and Oehrtman’s fourth mental action is coordinating the average rate-of-change of the function with uniform increments of change in the input variable. In this case, we focused instead on coordinating multiplicative comparisons of y-values with additive comparisons of xvalues. We wanted students to understand that a) the ratio of f (x 2 ) to f (x1 ) will always be the same for any same Δx = x 2 − x1 , and b) the value of f (x + Δx) / f (x) is dependent on Δx . We € € designed tasks that introduced tables of exponential data in which the week values were not

€ uniformly€distributed and asked students to€find missing week values and missing height values. We also used these table representations when asking students to determine an unknown growth factor. Once students were already coordinating ratios of successive y-values to determine growth factors, we attempted to make the value of Δx explicit by emphasizing different Δx Name ______________________

values, such as in the task in Figure 2: Memo #18

€

Last week each of you divided the inches for successive weeks to find out whether the plant was doubling, tripling, etc: Weeks 0

Inches 1

1

3

2

9

3÷1=3

€

It must be tripling!

9÷3=3

What if you couldn’t do that, because you didn’t have successive weeks in your table? How would you figure out if it was doubling, tripling, or doing something else? Weeks 0

Inches 1

5

1024

18

68,719,476,736

Figure 2: Non-uniform Table Task Encouraging smooth reasoning. Another conceptual goal was that students coordinate the ratio of f (x 2 ) to f (x1 ) for values of Δx that were less than 1, ultimately being able to

€

€

€

14

Quantifying Exponential Growth

imagine this coordination for arbitrarily small increments in x. Weber (2002) argues that students should, in time, be able to generalize their understanding of repeated multiplication to make sense of what it means to be “the product of x factors of a” when x is not a positive integer. The Jactus context could allow for a meaningful interpretation of this idea when students begin to contemplate how much the plant would grow in one day, for instance, given that it doubles in one week. We designed tasks in which students had to determine how tall a plant would be at non whole-number time values such as half a week or two and a half weeks. We then asked students to determine how much a plant would grow for many different time periods, such as for 10 weeks, for one day, or for 1/10th of a week. The Teaching Experiment We recruited three eighth-grade students (ages 13-14), Uditi, Jill, and Laura (all pseudonyms). Jill and Laura were enrolled in eighth-grade mathematics, and Uditi was enrolled in an eighth-grade algebra course. None of the students had encountered exponential functions in their mathematics classes at the time of the study. The students participated in a 12-day teaching experiment (Cobb & Steffe, 1983; Steffe & Thompson, 2000) over the course of three weeks, in which the first author was the teacher-researcher. Two project members familiar with the teaching-experiment methodology observed and videotaped each teaching session. Each session lasted approximately 1 hour. The project team met daily after each session to debrief and discuss the events that transpired during the session. For the purposes of this paper, we present an analysis of the students’ conceptual development over the course of the teaching experiment, with an emphasis on three shifts in understanding. Before embarking on the teaching experiment, the project team developed a progression of tasks according to the design principles described above. We also created a hypothetical

15

Quantifying Exponential Growth

learning trajectory (Simon, 1995), organizing a sequence of tasks that would foster students’ understanding of exponential growth. However, the teaching experiment model demanded a flexibility that meant the initial sequence of tasks served only as a rough model for instruction. We engaged in iterative cycles of intervention, assessment and model building of students’ thinking, and revision of future tasks on an ongoing basis. In this manner we developed, during each teaching session, enhanced hypotheses of the students’ understanding based on the previous cycle (Simon et. al, 2010). Data Analysis We employed retrospective analysis (Simon et. al, 2010) in order to characterize students’ changing conceptions throughout the course of the teaching experiment. We relied on the initial learning trajectory as a source of preliminary codes for the data, coding students’ talk, gestures, actions, and responses to tasks as evidence of understanding at various stages on the trajectory. The act of coding students’ understanding also produced emergent codes, which then altered the initial learning trajectory to account for the events that occurred during the teaching experiment. The codes and the trajectory therefore evolved simultaneously in a cyclical manner until the trajectory stabilized to reflect the final set of codes identifying students’ conceptions. Although the presentation of the final learning trajectory is beyond the scope of this paper, we will present a portion of the trajectory accounting for three major shifts in students’ conceptual understanding of exponential growth. Two members of the research team initially coded the entire data corpus independently. During this process they met weekly with the entire project team in order to discuss boundary cases and clarify and refine uncertain codes. Once this initial phase was complete, the two researchers then recoded the entire data set together by comparing every codes and discussing

16

Quantifying Exponential Growth

any differences until reaching agreement. The iterative process of coding, refining, and recoding continued until no new codes emerged and no more refinement was necessary. From Repeated Multiplication to Constant Multiplicative Rates of Change: Three Conceptual Shifts Early in the teaching experiment, all three students demonstrated an understanding that repeated multiplication of the height determined the manner in which the Jactus plant grew. In this analysis we present evidence of three major conceptual shifts that marked the students’ transitions from understanding exponential growth as repeated multiplication to coordinating multiplicative ratios of height values with additive differences for the time values. We will address each of the shifts (Figure 3) in turn. Conceptual Shift

Definition

Shift 1: From Repeated Multiplication to Coordinating x and y Shift 2: From Coordinating x and y to Coordinated Constant Ratios Shift 3: Generalizing to nonnatural exponents

Students shift from only attending to the repeated multiplication of the y-values to coordinating repeated multiplication with changes in xvalues. Students shift to explicitly coordinating the ratio of f (x 2 ) to f (x1 ) for corresponding x-values for any Δx ≥1. Students shift to explicitly coordinating the ratio of f (x 2 ) to f (x1 ) € for corresponding x-values for any value € of Δx , including values in € which Δx < 1.

Figure 3: Three Conceptual Shifts in Creating Multiplicative Rates of Change € €

€ After exploring the Geogebra script, the students began to quickly identify repeated € multiplication as the mechanism that determined the manner in which the Jactus grew. For instance, the students encountered a situation in which the Jactus was 1 inch tall when it began to grow, and it grew by quadrupling its height every week. Uditi described its growth this way: “They’re all going up by like times 4, like, 16 times 4 is 64 and then 64 times 4 and then 64 times 4…then times 4 that’s 1024.” Explaining how she would determine the height of the plant at 7 weeks, Uditi stated, “Four times 4 is 16, 16 times 4 is 256 then 4…1024 times 4, 4096 times

17

Quantifying Exponential Growth

4 and then it’s 16,384.” Absent from Uditi’s language is an explicit coordination of repeated multiplication of the height of the Jactus with the amount of time it took the Jactus to quadruple. The first shift we address is the one in which the students began to attend to corresponding time values when considering the height of the Jactus. Shift 1: From Repeated Multiplication to Coordinating x and y On Day 2 of the teaching experiment the students encountered a far prediction question: Given a Jactus that doubled every week and began growing when it was 1 inch tall, how tall would it be at 30 weeks? Laura’s response indicated an understanding of repeated doubling, and an attention to the corresponding number of weeks began to make its way into her language: Name

LQuI'UI

“You can do like, so here’s 8, and then the next is 16 and I guess like I said yesterday it goes up, Memo #3

if that’s the rate, and if you minus it you’ll seetables that…for 32showing you would doof 32 timesOne 2, and then Yesterday we saw two on the board the growth the Jactus. had days, and one had weeks:

you have the result for the week, 6 and then youWeeks just keep going like that.” At this Inches Inches Daysfor week 0

7

1 2 4 8 16 32

0 1 2 3 4 5

1 2 4 8 16 32

point, there is little evidence of14explicit coordination of repeated doubling of the height per a 121 28 35

week increase in time. In order to find the value of the plant’s height at week 30, Laura 1. Which do you think is easier to work with, days or weeks?

constructed a table from week 6 all \tJthe e..e-way )L- to S week 30 because she had no way to truncate the repeated multiplication process (Figure 4), which resulted in calculation errors. 2. How tall will the Jactus be after 210 days (30 weeks)?

2-6

--

2-7

q 3 3

7

II 3731-1J 35b

I I

18

Quantifying Exponential Growth

Figure 4: Laura’s table for determining the Jactus’ height at week 30 In an attempt to highlight the coordination of the plant’s height with weeks, the teacherresearcher (TR) introduced a task that required the students to draw the plant’s height after 1 week and after 3 weeks. After the students produced their drawings, the teacher-researcher asked them to think about Week 4: TR:

Uditi: It would be more bigger. Memo #4 TR:

:eP ------,---

So what would happen the next week and week 4? Name

Here's a picture of the Jactus plant. Compared to this picture, draw another

It would be more bigger? picture of what the Jactus will look like after 1 week, and then a picture of what it will look like after 3 weeks.

Jill:

It would be double, it’s…

TR:

Oh, it would be double. Double what?

Jill:

Well, the next week would be double the last week.

Jill’s drawing indicated the beginning of her coordination of the plant’s height with the number of weeks; both are present in her picture (Figure 5). Although it appears that Jill’s image for Week 3 was of a plant twice as tall as Week 1, she labeled the Jactus as 8 inches tall:

i Figure 5: Jill’s drawing of the growing Jactus TR:

So then what would week 5 do?

All:

Double that.

Laura: It would double the week before.

19

Quantifying Exponential Growth

By the fourth day, the students could explicitly coordinate the height of the plant with the week number, but they could not coordinate the growth in height for multiple-week jumps. For instance, Laura worked with a table of exponential values in which she had to find the plant’s height for week 10 and week 13 (Figure 6):

Figure 6: Laura’s table of week / height values Laura was able to coordinate the growth in inches with the number of weeks the plant had been growing by filling in the gaps in the table. Her language reflected an explicit attention to weeks: “For 4 (weeks) I got 48 (inches), for 6 I got 192, for 7 I got 384, for 9 I got 1,536, for 10 I got 3072 and for 13 I got 24,576.” Laura could only double the previous week’s height to find the next week’s height, and she did not yet understand that this process could be truncated, for instance, that she could find the Jactus’ height at week 13 by taking the height at week 10 and multiplying it by 8, or 23 for a jump of 3 weeks. Shift 2: From Coordinating x and y to Coordinated Constant Ratios On the fifth day Jill and Uditi encountered a non-uniform table of exponential data (the function for these data is f (x) = 0.1(2 x ) ). Laura was absent on this day, and her attendance became more sporadic than Uditi and Jill’s for the remainder of the sessions. € 20

Quantifying Exponential Growth

Figure 7: Non-uniform table of exponential data Jill’s response demonstrates the beginning of a coordination of the change in the height of the plant with the change in weeks. She took the ratio of the height at week 8 to the height at week 0, dividing 25.6 inches by 0.1 inches, explaining, “I did 25.6 divided by 0.1 and that was 256.” Jill also knew that she had to account for the 8-week difference. However, rather than recognizing that 256 represented a growth factor repeatedly multiplied 8 times (e.g., b 8 = 256 ), Jill divided by 8: “I just divided that by 8 because there was 8 weeks and I got 32, but I don’t really know why I did that, I just kind of did it.”

€

The teacher-researcher provided Jill with a new problem that contained only two data points (Figure 8), and asked Jill to determine how the plant was growing.

Figure 8: Jill’s recognition that a 2-week jump means the height will grow by 9

21

Quantifying Exponential Growth

At this point Jill began to think about the gap in weeks as representing how many times repeated multiplication must occur: “I tried 3 times 3 and it was the 16 week number, and so then I figured out if I did the 14 week number times 9 it would give me this number.” On the same day, Uditi demonstrated a similar emerging coordination between the Name

multiplicative growth in height and multiple weeks, but did not generalize her understanding. In

1>,

Memo #17

another non-uniform table of exponential data (for which the underlying function was Here's another table for the height of a Jactus in inches after a certain number clf

ratio ofCan theyou first two out height f (x) = 0.1( 4 x ) ), Uditi took theweeks. figure howvalues to find(Figure how tall9): the Jactus will be after any number of weeks?

€

Weeks 0

Inches 0.1

2

1.6.

5

102.4

9

26,214.4

10 12

\

ra

Ih777:Z1 . Co

? 0 \ x Figure 9: Uditi’s work on a non-uniform table of exponential data

Uditi divided 1.6 inches by 0.1 inches to get 16, and she knew that this meant she had to find a number that she could multiply twice to get 16. Given her limited knowledge of algebraic manipulation with exponents, Uditi had to rely on a guess and check method: “I found out that the difference was 16 and then I just tried all these different numbers. I tried to multiply and then multiply it again the same number.” In this manner she discovered that the growth factor was 4. At this stage, however, Uditi was not yet able to generalize her reasoning to any multi-week gap; for instance, she was unable to explain that the plant would grow 4 × 4 × 4 times, or 43 times, between week 2 and week 5 because there was a 3-week gap.

€

It:.

D S 22

I

'"

-r '\

Quantifying Exponential Growth

On Day 7 of the teaching experiment the teacher-researcher gave the students two data points with a 5-week gap in order to encourage the coordination of the multiplicative growth in height with the additive growth in time. For this task, the Jactus was 3,355.443.2 inches tall on week 24, and it was 107,374,182.4 inches tall on week 29. The teacher-researcher asked the students to determine how the Jactus was growing (e.g., whether it was doubling, tripling, etc.) and how tall it was at week 0. Jill took the ratio of the two height values and found it to be 32, and then took the difference of the two week values. She then wrote, “ __⋅ __⋅ __⋅ __⋅ __ ”, searching for a number that she could multiply by itself 5 times in order to yield 32. By guessing € 2 (Figure 10). and checking, Jill determined that the growth factor should be

Figure 10: Jill’s solution to the two-data-point problem Uditi’s response was similar to Jill’s, and once she found that the growth factor was 2, she used this information to determine the height at Week 0. Uditi wrote “ __× 2 24 ” and then guessed and checked in order to find the value that would make her expression equal to the height at 24

€ to be 0.2, so then she produced the weeks, 3,355,443.2 inches. Uditi found the missing value equation “ y = .20 × 2 x ”. Uditi and Jill also appeared to understand that the ratio of height values would remain

€ constant for any same-number increase in weeks. For instance, after determining that one tripling Jactus plant grew 8 times as tall from Week 3 to Week 6, both students immediately predicted that the plant would be 8 times taller at Week 103 than it would be at Week 100.

23

Quantifying Exponential Growth

Shift 3: Generalizing to Non-Natural Exponents Our intention was to foster a coordination between the multiplicative growth of y-values and the additive growth of x-values even for cases in which the difference between x-values was less than 1. Our hope was that students would be able to imagine this coordination for arbitrarily small increments in x, which would provide a way to understand ba even when a is not a whole number. One way to introduce this idea was to ask the students to think about how tall the plant would be in the middle of a week; for instance, if a plant doubles between Week 1 and Week 2, how tall will it be at Week 1.5? Estimation and Reversion to Linear Thinking Early in the teaching experiment, the students struggled to make estimates for how tall the plant would be in between whole week values. For instance, on the third day of the teaching experiment the students worked with a Jactus plant that was 1 inch tall at 0 weeks and doubled each week. The teacher-researcher asked Uditi how tall the plant would be at 1.5 weeks and she guessed that it should be “a little more than three.” Uditi explained, “It increases a little more every time it goes that way [gesturing to successive weeks], so in between them is three so it’s going to increase a little more, like comes to point 5.” Uditi had a qualitative (Behr et al., 1992) understanding of the plant’s growth, in that she could reason about the direction of change in the plant’s height with relying on calculations. Uditi knew that the manner in which the plant increased from one week to the next was not linear. Because the plant was 2 inches tall at 1 week and 4 inches tall at 2 weeks, Uditi guessed that it would be slightly more than 3 inches tall halfway between 1 and 2 weeks, but she was unable to determine the precise amount of growth. Although her estimate is incorrect, it is notable that Uditi did not revert to linear interpolation when trying to determine the plant’s height at 1.5 weeks.

24

Quantifying Exponential Growth

One week into the teaching experiment the students revisited the same doubling Jactus, but this time the teacher-researcher asked the students to think about the plant’s height every day between 0 weeks and 1 week. Laura and Jill both reverted to linear interpolation, stating that the plant would be 1.1 inches tall after 1 day, 1.2 inches tall after 2 days, and so forth. Uditi, however, relied on her ability to coordinate multiplicative growth in the plant’s height with additive growth in weeks. She took the ratio of the plant’s height at 1 week (2 inches) to the plant’s height at 0 weeks (1 inch), and then wrote “ __× __× __× __× __× __× __ = 2 ”. At this point Uditi became stuck, because she was unable to find a number that she could multiply by itself seven times to result in 2.

€

Reliance on Equations to Assist with Interpolation The next day the students visited the same doubling plant with a table that provided its heights for different decimals between Week 0 and Week 2 (Figure 11):

Figure 11: Table of values for doubling Jactus

25

Quantifying Exponential Growth

Laura knew that the ratio of heights for any 0.25-week difference would be the same, so she took the ratio of the plant’s height at week 0.75 and divided it by the plant’s height at week 0.25, which was 1.189. She then wrote “goes up by 1.189207115” and concluded that 1.189 would be the plant’s height at Week 0.25. Although Laura’s “goes up by” language suggests the possibility of an image of an additive difference of 1.189 between successive quarter weeks, later Laura determined the height for Week 3.5 by using appropriate multiplicative reasoning, multiplying the value of the height at Week 3 by 1.189 twice. In contrast, Uditi relied on her ability to create an equation in order to determine the missing height at 0.25 weeks. Uditi wrote “ 1× __^.5 ” and then used a guess and check method to find the missing value that would yield the plant’s height at 0.5 weeks. She easily determined € that the missing value was 2, and then checked the correctness of her equation with other values

in the table, ultimately declaring that the plant doubled each week. Uditi then wrote “Height = 1⋅ 2^ week ” and substituted 0.25 for the exponent in order to find the missing value.

Coordinating the Ratio of Height Values for Δx with bΔx €

In order to encourage Uditi to coordinate the plant’s growth with parts of weeks rather € € than relying on an equation, the teacher-researcher introduced a problem that only had two data

points (Figure 12), asking the students to figure out how the plant was growing.

Figure 12: Two data points from the function f (x) = 3x Jill and Laura both took the ratio of the two height values and stated that the plant would get

€ student could use this information “1.116123172” times as big each tenth of a week, but neither

26

Quantifying Exponential Growth

to determine how much the plant would grow in 1 week. Uditi also took the ratio of the two height values, which she rounded to 1.12. In order to determine how big the plant would grow every week, she then once again relied on an equation, writing “ __× __ 0.1 = 1.12 ”. Uditi indicated that the first blank represented the “starting number”, or the plant’s height at Week 0, and the second blank represented the unknown weekly€growth factor of the plant. Although Uditi’s approach was unexpected, we found it notable because she equated the ratio of the yvalues, 1.12, to abΔx (for a function f (x) = ab x ). This represents a key understanding, that the ratio of height values for a given Δx can be expressed as bΔx . At this point it was not clear how € Uditi might generalize this€understanding, and there was little evidence to determine whether € Uditi also realized that€this ratio will always be the same for any same Δx .

Uditi knew that the plant increased in height by a factor of approximately 1.12 for 1/10th of a week. Rather than taking this value to the tenth power in€order to determine how much the plant would grow in one week, Uditi simply used a guess and check method with her equation to determine that 30.1 ≈ 1.12 . She concluded that the plant tripled every week. The teacherresearcher asked Uditi how the plant grows every 2 weeks and every 3 weeks, and Uditi € concluded that every 2 weeks the plant would grow 9 times as big and every 3 weeks it would

grow 27 times as big. She explained, “Because 3× 3 = 9 and 3× 3× 3 = 27 .” The teacherresearcher decided to ask Uditi about larger numbers: TR:

€ big numbers, € like Week 100 and Week 101? Is What if I gave you really this number, whatever it’s going to be, still going to be 3 times as big the next week?

Uditi: (Nods). TR:

Even all the way up here?

27

Quantifying Exponential Growth

Uditi: (Nods). TR:

How come?

Uditi: Because, the equation says it grows 3 times every week. TR:

Okay, and if it’s going 9 every 2 weeks are you confident that it will be that way for any two weeks?

Uditi: (Nods). TR:

How come?

Uditi: Because, 3 times 3 times 3 is 9. Based on Uditi’s responses, it was unclear what she truly thought; Uditi could have possibly have just been responding positively to the teacher-researcher’s questions. Therefore the teacher-researcher gave her another problem with only two data points. This plant was 64 inches tall at 3 weeks, and 68,719,476,736 inches tall at 18 weeks. The students had to figure out how the plant was growing every week, every 3 weeks, and every ½ week. Both Uditi and Laura determined that the plant grew 4 times as large each week by taking the ratio of the two height values and then determining what number to the 15th power (the difference between 18 weeks and 3 weeks) yielded that result. Both then concluded that the plant would grow 64 times as large every 3 weeks. Uditi then explained that the plant would grow 2 times as large every ½ week, reasoning that at 3 weeks the plant was 64 inches tall, at 3.5 weeks it was double that, 128 inches tall, and at 4 weeks it doubled again to 256 inches tall (Figure 13).

Figure 13: Uditi’s amended table 28

Quantifying Exponential Growth

The teacher-researcher then asked Uditi how the plant would grow every x weeks, and she responded, “It grows by 4 to the x.” Generalizing to Fractional Exponents On the last day of the teaching experiment, the teacher-researcher gave the students a task designed to determine whether they understood that ratio of f (x 2 ) to f (x1 ) will always be the same for any same Δx = x 2 − x1 , even when Δx is a fraction (Figure 14): € €

€

€

Figure 14: Tripling Jactus Table We were curious whether the students would attempt to make a calculation, or whether they would know that the plant would grow 9 times as big regardless of its size at any given week. Laura did not recognize this and attempted to calculate an answer. Jill and Uditi both immediately said the plant would become 9 times as big. Jill explained, “I noticed that 1 times 9 is 9 and 2, or, 9 times 9 is 81, so every 2 weeks it is going up 9 times.” The teacher-researcher then asked the students how much the plant would grow from Week 155 to Week 160, and Uditi wrote “ 35 = 243 ”, explaining that this was a way to represent “three times three times three times three times three.” Laura and Jill were then able to use Uditi’s reasoning to correctly determine

€ how much larger the plant would grow in 10 weeks. The teacher-researcher then asked the students to determine how much the plant would grow in 1 day, or 1/7th of a week. Uditi wrote “ 3.14 = 1.17 ”, explaining, “It’s 1 divided by 7 because it only shows the result for 1 week on the table, and there are 7 days in a week. So I € 29

Quantifying Exponential Growth

divided 1 week into 7 parts, which represents 1 day each and it’s .14 of a week.” In this manner Uditi was able to make sense of a non-whole number exponent and generalize her coordination of the ratio of height values with growth in time to Δx values less than 1. Discussion

€ The students began the teaching experiment with an understanding of exponential growth as repeated multiplication, but they did not explicitly coordinate repeated multiplication of height with growth in time. Over time, the students began to coordinate the growth of the height of the Jactus with time, first implicitly by thinking about the plant as doubling “every time”, and then explicitly as they had to negotiate non-uniform tables of values. The students then began to truncate the process of repeated multiplication, coordinating increases in height with multipleweek time spans. Initially they did this linearly, by dividing the ratio of height values by Δx , but Δx ultimately they shifted to coordinating the ratio of height values with b for the growth factor b.

€ The students realized that this relationship would hold for any same Δx ; at first, they understood € this only for Δx values greater than 1, representing multi-week jumps, but ultimately one

€ 1, making use of fractional student, Uditi, was able to coordinate for Δx values less than € exponents. € The students’ ability to coordinate the ratio of height values with the additive difference in time values played a significant role in supporting their ability to develop algebraic representations of the plant’s growth. This was evident, for example, in Uditi’s development of the equation 0.2 × 2 x , in which she had determined the growth factor 2 by taking the ratio of two height values 5 weeks apart and then determining the number b such that b5 was equal to that € ratio. In general, the students’ covariational thinking preceded their ability to develop

correspondence rules of the form y = f(x), which reflects Smith and Confrey’s (Smith, 2003;

30

Quantifying Exponential Growth

Smith & Confrey, 1994) assertion that students typically approach functional relationships from a covariational perspective first. One reason why we focused on the middle school population for this study was because students in our participating schools encounter exponential growth formally in their mathematics classrooms in eighth grade. We were interested in exploring students’ evolving conceptual development as they encountered exponential situations for the first time, which necessitated a younger participant group. However, this also resulted in some challenges and constraints in the types of problems and tasks we were able to design. Enabling the students to physically manipulate and visually observe the growing plant with Geogebra supported a qualitative experience of the nature of exponential growth, but it limited the growth factors we could use. A growth factor larger than 4 resulted in numbers too large for Geogebra and the students’ scientific calculators, so they only had opportunities to explore plants that doubled, tripled, or quadrupled. In addition, we typically constrained the growth factor to whole numbers because our participants did not have access to sophisticated algebraic manipulation abilities or the notion of logarithms. This meant that they were limited to guess and check methods for determining the growth factor for mystery plants. For instance, the students often determined the growth factor by taking the ratio of two height values a certain number of weeks apart. Imagine a situation in which the growth factor is approximately 97.66 for a time period of 5 weeks. A high school or 1 5

college student could write the equation b = 97.66 and then solve for b, calculating 97.66 in 5

order to determine that the mystery growth factor is 2.5. Because the middle school students did € not possess this degree of facility with algebraic manipulation, they instead€had to guess and

check to determine a number they could multiply by itself 5 times that would equal 97.66. If the

31

Quantifying Exponential Growth

growth factor was something other than a whole number, the guess and check method was typically too time consuming to allow for any significant progress within the constraints of the teaching experiment. Despite these limitations, however, the results presented in this paper offer a proof of concept that even with their relative lack of algebraic sophistication, middle school students can engage in an impressive degree of coordination of co-varying quantities when exploring exponential growth. In addition, we have presented evidence that students can generalize their a understanding of exponential growth to view b as a factors of b for non-natural values of a, as

suggested in theory by Weber (2002). We contend that reasoning with continuously co-varying

€ quantities was a critical element in constructing this particular understanding of exponentiation. The Jactus context offered a scenario in which students could begin to make meaningful sense of non-natural exponents as they imagined the height of the plant smoothly growing over time. Although we hesitate to make the definite claim that Uditi employed smooth thinking, we suspect that contexts such as the Jactus situation can support this development because they provide opportunities for students to re-imagine the nature of the plant’s growth for varying units of time.

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References Alagic, M., & Palenz, D. (2006). Teachers explore linear and exponential growth: Spreadsheets as cognitive tools. Journal of Technology and Teacher Education, 14(3), 633 – 649. Behr, M.J., Harel, G., Post., T., & Lesh, R. (1992). Rational number, ratio, and proportion. In Grouws, D.A. (Ed.), Handbook of Research on Mathematics Teaching and Learning, (pp. 296 – 333). New York: Macmillan, 1992. Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler. Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. Mayes & L. Hatfield (Eds.), Quantitative Reasoning and Mathematical Modeling: A Driver for STEM Integrated Education and Teaching in Context (WISDOMe Monograph Volume 2). Castillo-Garsow, C. (Submitted). The role of various modeling perspectives in students’ learning of exponential growth. Davis, J. (2009). Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12, 365 – 389. Cobb, P., & Steffe, L.P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 28, 258 – 277. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2), 135–164. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.),

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Advanced mathematical thinking (pp. 95 – 123). Dordrecht, The Netherlands: Kluwer Academic Publishers. Ellis, A.B. (2011a). Generalizing promoting actions: How classroom collaborations can support students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308 – 345. Ellis, A.B. (2011b). Algebra in the middle school: Developing functional relationships through quantitative reasoning. In J. Cai & E. Knuth (Eds.), Early Algebraization: A Global Dialogue from Multiple Perspectives (pp. 215 – 235). New York: Springer. Ellis, A.B. (2007). Connections between generalizing and justifying: Students’ reasoning with – linear relationships. Journal for Research in Mathematics Education, 38(3), 194 – 229. Farenga, S.J., & Ness, D. (2005). Algebraic thinking part II: The use of functions in scientific inquiry. Science Scope, 29(1), 62 – 64. Goldin, G., & Herscovics, N. (1991). Towards a conceptual-representational analysis of the exponential function. In F. Furinghetti (Ed.), Proceedings of the Fifteenth Annual Conference for the Psychology of Mathematics Education (PME) (Vol 2, pp. 64-71). Genoa, Italy: Dipartimento di Matematica dell’Universita di Geneva. Green, K. (2008). Using spreadsheepts to discover meaning for parameters in nonlinear models. The Journal of Computers in Mathematics and Science Teaching, 27(4), 423 – 441. Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Phillips, E.D. (2006). Connected Mathematics 2. Hilldale, NJ: Pearson Prentice Hall. Presmeg, N., & Nenduardu, R. (2005) An investigation of a pre-service teacher’s use of

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representations in solving algebraic problems involving exponential relationships. In H. L. Chick & J. K. Vincent (Eds). Proceedings of the 29th PMEInternational Conference, 4, pp. 105-112. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 347 – 395. Simon, M., Saldanha, L., McClintock, E., Akar, G., Watanabe, T., & Zembat, I. (2010). A Developing Approach to Studying Students' Learning through Their Mathematical Activity. Cognition and Instruction, 28(1), 70-112 Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. (pp. 136 – 150). New York: Erlbaum. Smith, E., & Confrey, J. (1994). Multiplicative structures and the development of logarithms: What was lost by the invention of function. In Harel, G. & Confrey, J. (Eds.), The development of multiplicative reasoning in the learning of mathematics. Albany, NY: State University of New York Press, pp. 333 – 364. Steffe, L.P., & Thompson, P.W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum Associates. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education. Laramie, WY: University of Wyoming.

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Thompson, P.W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. In O Figueras, J L Cortina, S Alatorre, T Rojano, and A Sepulveda, Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education, volume 1, pages 45–64, Morelia, Mexico. PME. Weber, K. (2002). Students' understanding of exponential and logarithmic functions. Second International Conference on the Teaching of Mathematics (pp. 1-10). Crete, Greece: University of Crete.

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14 Playing Together Separately Mapping Out Literacy and Social Synchronicity Crystle Caroline C. Amanda Ochsner, Shannon Elizabeth Gabriella Anton, Jonathon and Constance Steinkuehler In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people. London: Routledge.

INTRODUCTION Jaea sits at a desk with a laptop on each side of him and a keyboard and large computer monitor directly in front of him; he has three different keyboards and three different screens. He also wears headphones for both listening and speaking. Eyes shifting rapidly, he sometimes focuses on a quickly changing scene on a single screen and sometimes glances across multiple screens each offer different information. At the same time, he receives feedback from voices coming through on the speakers, leading him to shift his attention elsewhere on one of the screens, and speaking into the microphone he returns information to the voices. With so much equipment and so much rapid movement and communication, one might guess that he is doing very important and very complicated work. Perhaps he is saving lives. In a way, he is. In the scene described earlier, Jaea is playing the massively multiplayer online role-playing game (MMORPG) World ofWare raft (WoW), and he is playing it well. Very well. His role in the group is to act as a healer, keeping the rest of his raid members alive while they take large amounts of damage from intimidating and powerful enemies. The other group members are playing ftom other rooms, similarly equipped with screens, keyboards, and headphones, but located throughout the country, perhaps even across the world. Their work is just as complicated and is equally demanding. What Jaea is really doing is reading. He is reading text conversations with his fellow players on the screen in front of him; he is reading the graphics he sees on the screen; he is reading the actions of other players; he is reading the needs and abilities of his own character; and he is reading his physical environment as well. After all, the girlfriend and three very large, very fuzzy cats that he shares this space with require attention sometimes too. (The cats, in fact, demand it!) This reading occurs in his physical environment as in the game world, where he maneuvers his avatar.

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His temporal existence is also full of multiplicity: he simultaneously has an understanding of when it is in the room he plays in, where his raid group is in terms of progressing through their raid dungeon, and the time at which his character exists in the game's fantasy universe. This chapter will explore the multiplicity of presence that is exhibited within gameplay in World of Wareraft, and demonstrate how leveraging multiple presences is essential to successful high-end virtualliteracies like MMOs.

EXAMINING ATTENTION From the phenomena described in the introduction, many may think of existing explanations to describe what we are seeing. Terms like split attention, multitasking, and constant partial attention are often used to describe situations in which a person undertakes multiple tasks at once. These terms often carry a negative connotation. In theories of split attention, the research focuses on "the limitations of working memory" and the idea that multiple tasks "overburden working memory" (Kalyuga, Chandler, & Sweller, 1999). The theory of split attention views attention as something that is allocated, and when divided up it weakens the person's ability to accomplish tasks (Awh & Pashler, 2000; Tabbers, Martens, & van Merrienboer, 2000; Mayer & Moreno, 1998). Multitasking is another explanation for attention paid to several tasks at once, and although multitasking is seen by many as a valuable skill that makes their world function, it is seen by others as a less than positive structure. Gonzalez and Mark (2004, 2005) represented multitasking as an issue to be remedied. They used as an illustration the everyday example of working in an office and having to deal with multiple conversations, telephones ringing, people walking into other cubicles (2004: 115). The issue of multitasking was mitigated through working spheres, which they defined as "a set of interrelated events, which share a common motive" Gonzalez and Mark's (2005) suggestions turned to system design, namely that system design needed to take into account multitasking and help to reinforce workers' working spheres. In the context of this paper, the multiple activities that are undertaken simultaneously could be considered split attention or multitasking; however, these descriptions in the existing research are narrow in focus, negatively valenced at the outset, and do not capture the subtleties of the game experience on its own terms. To help describe the nuances of the game experience, we explore a variety of research to illustrate the complexities inherent to the activity. Understanding these subtleties is important because it allows a clearer picture of the actions and information management engaged in by players. The erature will be framed in three contexts of gaming-that of literacy, time, and place-and will finish with new explanations for attending to several tasks as once.

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GAMING AS VIRTUAL LITERACY Gaming as a literacy has been studied by many researchers. Gee (2003), in his seminal book What Videogames Have to Teach Us about Learning and Literacy, described 36 learning principles that are manifested in good games. He claimed that in order to participate in a game the player must understand the literacy of the game and that playing was participating in a semiotic domain in which the player must understand rules, symbols, social interactions, and discourse to be successful. In addition to games, research has been conducted on online communities, which are centered on literacy practices (e.g., FacFiction.net (Black, 2007a, 2007b, 2008)). Black described the literacy habits of adolescents in online fan fiction communities, detailing their level of community interaction, their editing abilities, the use of the space by second language learners to develop their English skills, and their development of 'metavocabulary of editors.' Black and Steinkuehler (2009) stated that participants in virtual worlds engage in reading and writing in a variety of formats, which they found to align with national educational standards (Standards for the English Language Arts, as cited in Black & Steinkuehler, 2009). Steinkuehler (2006, 2007, 2008) also studies literacy in massively multiplayer online games (MMOs) as both a discourse (which is both languagein-use and language-in-action (see Gee, 1999) and as a constellation of Iiteracies, which includes the game and all of the resources (e.g., wikis, hlogs, forums, videos, etc.) that are used as information sources by the game community. The constellation of Iiteracies is the 'space' in which the information needed for success in a game or other affinity space exists. Gee's (2004, 2005) semiotic social and affinity spaces describe the 'place' of MMOs. The affordances of these perspectives offer flexibility, focusing on space and how it functions to bring people with common interests together. Steinkuehler and King (2009) used play within the place of an MMO to bolster the literacy practices of disengaged adolescent boys using the space of the constellation of literacies that surround the game to help them to become re-engaged. Martin and Steinkuehler (2010) explore another form of literacy that functions within the information constellation of an MMO, that of information literacy. By observing practices in the naturalistic setting of the game, Martin and Steinkuehler (2010) examine what and how resources are utilized to maintain success in a game-that is, how players traverse the space of the constellation of literacies in order to successfully retrieve information and resources they need. Time can be viewed in a multitude of ways. Gelliooks at the keeping of time in two ways (1992). He divided time up into what he termed A-series time, or standardized time as measured by a clock, and B-series time, or time that is run by the punctuation of activity (similar to Erickson's (2004) characterization of kronos and kairos). We appropriate Gell's classification and apply it to the distinction between a person's time spent in and

Playing Together Separately 229 out of games. Gamers live a hybrid existence of A-series and B-series time (Martin, in progress) where A- and B-series time becomes intertwined and entangled in the gaming activities. This is true in many genres of games but it is especially true in relation to MMOs. In an MMO, a player follows B-series time by focusing on the game world where much activity is punctuated by a cycle of activities rather than by the time increments of a clock. However, the player may at the same time keep track of when a certain time in the physical world comes-for example, when they need to eat dinner or when they need to make a phone call. Here, A-series time is intruding on the B-series time of the game. Participation in group activities in MMOs, although possibly starting at an A-series time, is punctuated by phases of the activity in B-series time. Although states of engagement and flow can be found in other activities, games offer an always available environment to delve into and MMOs offer the added benefit of constant social interaction in the virtual space, although the players are generally separated in the physical space. Situating the activity of in-game activities within literacy, time, and space gives a basis for understanding the activities that take place within the game. Through this lens of activities a different approach of cognition could be taken. Distributed cognition (Hutchins, 1995) is an approach that emphasizes the use of resources expanding cognition beyond the individual's mind. This theory focuses on the coordination of enaction among agents within the community, which can include individuals, artifacts, and the environment itself (Hollan, Hutchins, & Kirsh, 2000), and here, the environment includes the game environment. This notion of the distribution of cognitive processes among the community, in this case the players of the MMO World of Warcraft, can be seen in Martin and Steinkuehler's (2010) idea of collective information literacy, also termed distributed information literacy.

METHODS To comprehend the visual and information literacy experiences and practices of an expert-level World of Warcraft player, the researchers involved in this investigation used an instrumental case study model in which the actor being studied was selected for purposes of better understanding the surrounding problem space of one expert player during normative gameplay (Stake, 1995). Our goal is to identify the ways in which literacies are practiced and constructed in a complex, navigationally demanding, fastpaced, and visually elaborate digital environment in which successful literacy practices and problem solving efforts are essential to success. The goal of this case study was to develop a hermeneutic understanding of the practices common to expert-level WoW play in a holistic, empirical, interpretive, and an empathetic manner (Stake, 1995).

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Selecting an expert-level player was imperative to understanding the advanced literacy practices and concomitant problem representations evidenced by many participants of this highly effective upper echelon of gameplay. Problem representations, such as determining the most effective and information literacy practices in order to achieve advanced game world success, were inherently designed by the participant and were grounded in his domain-related knowledge and subsequent semantic organization. Furthermore, experts solve problems differently than do novices (Chi, Feltovich, & Glaser, 1981) and we expected this difference to manifest evcnor, perhaps, especially-in very complex problems spaces like WoW. For this study, a 25-year-old, Caucasian, male, expert-level World of Warcraft player was observed playing WoW (Stevens, Satwicz, & McCarthy, 2(08) on a weekly basis from January of 2011 until April of 2011. The researchers are familiar with WoW gameplay and are garners themselves, although they do not play at the expert level of the case study participant. The expert-level player, Jaea (a pseudonym), typically plays WoW for about two hours each weekday, in addition to logging in about 8-12 hours of playtime over the weekends. He is a successful raider, and his guild is consistently first-ranked server-side (on their server) in ten-person are collaborative and complex in-game actions to defeat a series of level enemies usually called bosses. The guild has even been first on their server to complete some ten-person raids, a prized achievement. Data was collected through the use of detailed and structured handwritten field notes taken by One case researcher present during gameplay. In

Figure 14.1

jaea's physical context.

Figure 14.1 Jaea's physical context.

addition to constructing presence flow charts Figures 14.2 and 14.3) capturing Jaea's foci of attention, the case researcher also took abundant notes used to reconstruct each of the ten fast-paced and complex sessions. The case researcher mainly observed Jaea's actions, both person) and on-screen, during gameplay, and occasionally asked questions when it was not obvious what was happening-on average about once or twice during each gameplay session. To assist with data collection, the case study participant, Jaea, frequently thought aloud as he played, meaning he talked through the problems or experiences he encountered during gameplay. In addition, he modified his typical gameplay habits to play without headphones. This allowed the case researcher to hear utterances from Mumble, a voice-over IP application that he uses to communicate with other players, as well as contextualize his responses to Mumble. Having the case researcher carefully observe and note Jaea's head movement captured shifts in player focus (see Figure 14.1 for Jaea's physical context). A movement of Jaea's head to the left indicated focus on a second laptop, in which he would Twitter or Reddit forums a and rated news links forum, while playing. The second was positioned far enough to the left that Jaea couldn't read screen merely by shifting his eyes' focus, but had to turn his head slightly, which let the case researcher follow his presence more closely. The case researcher initially asked about Jaea's practices when he shifted focus to his information feed laptop, but he quickly developed a habit of automatically describing what he was doing. Jaea's engagement with the actual ingame WoW chat was also recorded whenever he typed a response. The case researchers also learned what the user interface (UI) customizations Jaea used looked like, so that they could identify times at which he tweaked or changed his UI without interrupting gameplay. A screenshot was taken of Jaea's screen at the end of gameplay so that the level of visual information captured on-screen could be paired with the other data collected bv the case researchers. For each observation a presence was taking place. This visualization included graphic representations of where presence-operationalized as eye gaze, verbosity, and physical actions (including the physical actions necessary to create digital movement)-was most focused, as well as descriptors for the activities taking place within each delineation of time. Our operationalization of presence relied upon visual and audio indicators: Jaea's movement in physical space could highlight his physical presence (such as going to the bathroom), or it may indicate a shift from in-game WoW to information constellation (such as shifting his gaze from his primary computer screen, loaded with WoW, to his secondary laptop to the left, loaded with Twitter or WoW forums). Additionally, audio input from raid team members on Mumble would be represented in the information constellation layer, while audio WoW feedback from the game itself would appear in the

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Martin, C., Williams, C., Ochsner, A., Harris, S. King, E., Anton, G., Elmergreen, J. & Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people (pp. 226243). London: Routledge.

Figure 14.3 Presence flow chart 2.

Martin, C., Williams, C., Ochsner, A., Harris, S. King, E., Anton, G., Elmergreen, J. & Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people (pp. 226243). London: Routledge.

Martin et al.

The player's ability to navigate within these different spaces and be successful in the game demonstrates a level of literacy, as the player must possess considerable information literacy as well as the traditional literacy of reading in order to interact with the information available around the game. Along with the literacy space players need to navigate for successful WoW play, our method is designed to capture both the physical and virtual spaces that Jaea regularly travels within. It is through the simultaneous traversal of such disparate places that makes expert-and even novice!-WoW play so difficult to understand and analyze. Our conceptualization of space is quite broad, capturing both digital and physical measurable space, as well as pursuit of information and the development of mental maps and plans. time, while Figure 14.3 Figure 14.2 represents 41 minutes of represents 38 minutes; the former is nine chunks of B-series time, while the latter is ten. From the data collected, Figures 14.2 and 14.3 were created for visualization and analysis. Each of the follows a format. Across the top are A-series time stamps that mark the demarcations section of time has a between B-series chunks of time (Gell, 1992). clear beginning and ending with 'fight begins' and 'wipe' appearing as the most common periods in these two presence visualizations. The former indicates that the raid group engaged in battle, while the latter indicates that all ten raid members died during battle and were unable to complete their goal. The emergent rule for breaking raiding up into B-series time is consequently 'fight begins' and 'wipe' (or 'win'), although in non-raid activities the time can be sectioned quite differently. The framework for analysis developed for this study was originally inspired by Lemke's (2000) notion of timescales. Lemke focuses on activity in time and how timescales interact with one another. His view of the interactions of people within different timescales caused us to consider the presence or focus of an individual when engaging in a multifaceted activity like that of playing an MMO. The distribution of attention across multiple layers looks much more like Hutchin's (1995) distributed cognition than the description of split attention that plagues research on digital spaces (Kalyuga et aI., 1999; Mayer & Moreno, 1998: 318; Tabbers et Awh & Pashler, 2000). Distributed cognition is an active process of engaging in multiple activities simultaneously and successfully. Successful here entails being able to learn from mistakes, being able to find needed inforand being able to cognitively manage multiple inputs from multiple presences at once. months of observations, two contrary examples are reprein Figure 14.2 and Figure 14.3. The first example (Figure 14.2) was a raid that Jaea's guild is very familiar with and that they were helping a new guild member through. Figure 14.3 is a raid that they had never attempted before, although the raiders had prepared beforehand by watching videos and reading guides. These two examples were chosen to illustrate a known activity and an unknown activity both in the context of a ten-person raid.

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The digital contexts of Figure 14.2 and Figure 14.3 vary in important ways, even though both represent the presence flow of Jaea engaging in tenperson raids with his guild. Figure 14.2 was a familiar raid for Jaea and the guild, and one that they had successfully completed many times before. However, a new guild member had never encountered this particular content before and was being carefully guided by the nine other guild members. Figure 14.3, on the other hand, represents a new raid boss that none of the ten players, including Jaea, had attempted before. And while all the players had researched the raid information closely beforehand, strategies and coordination were developed by the guild in situ. Researching raids before attempting them is a common practice of raiding guilds. The reason for this is to familiarize the members with the layout of the environment, the types of enemies that they will be facing, and anything that is specific to a boss that they may is used for preparation is community-creresources like written walk-throughs or Information literacy skills are needed in order players to locate and understand the information provided in these resources as well as to evaluate the reliability of each resource. In terms of time, the two presence visualizations are quite similar. The following is a description of the digital contexts for Figures 14.2 and 14.3. In order to maintain accessibility for readers who are not WoW players, the following accounts are considerably simplified. This simplified version is to make the description accessible for all readers regardless of whether they are familiar with WoW. As stated in the introduction, WoW is an online game played by millions of people across the world. Smaller groups within the game, called 'guilds,' play and plan together, often in order to complete 'raids' (that is, incredibly complicated and challenging battles with powerful boss monsters that require a group of people to defeat). During this playtime, the guild groups tend to communicate verbally and via text in order to coordinate their attacks, and teamwork, prior knowledge, and careful attention are necessary in order to succeed. Figure 14.2 represents the first phase of the ten-person AI'Akir Raids may be played in either the raid bosses are even more powertu is a three-phase raid, and in the first phase, final boss of the Throne of Four Winds raid) stands in the center of a floating circular platform. Around AI'Akir spins a wall of damage-dealing whirlwinds that require players to either move close to the boss or to the very edge of the platform, requiring constant attention by the players. A general WoW strategy is to distract the boss into attacking a high-defense player (otherwise known as a tank) while other players deal damage to him (otherwise known as DPS-ers, named after the phrase Damage {Jer Second). All at the same time, other players are healing the tank and DPSers to ensure they stay alive. As this strategy plays out, AI'Akir casts chain lightning, the damage of which is multiplied by the number of players near

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the afflicted players-so the players must maintain a carefully coordinated distance from each other. In addition, Al'Akir will occasionally cast a speU that pushes players a certain distance from him, requiring that players stay close enough to him so that they don't fall off the platform when pushed back. In order to prevent the push back spell from being cast, the tank can run away from Al'Akir, but Al'Akir punishes the fleeing tank by doing extra damage to him or her, and consequently the healers need to be prepared to cast extra healing spells on the tank. There are more spells that Al'Akir casts during the battle (too many to describe here), each of which requires specific strategies that can sometimes conflict with one another. Figure 14.3 represents the first phase of the ten-person heroic mode Val. iona and Theralion raid, which-at the time of writing-Jaea's guild just successfully completed. During the observation that led to Figure 14.3, the guild was unsuccessfully attempting this raid for the first time. This raid has two bosses, both dragons, which periodically switch off between the roles of fighting on the ground and flying while attacking the raid from the air. During the first phase, Valiona is on the ground, casting proximitybased flame attacks while the tank tries to keep Valiona's attacks focused on him or her. The DPS-ers try to bring Valiona's health down (both dragons share a common hea Ith pool), and healers work to keep everyone alive. Meanwhile, Valiona has another attack that requires players to stack other words, get as close as possible to each other) in order to minimize damage, while Theralion has an attack that requires players to spread out as far away as possible, forcing the raid team to be constantly aware of their relative positioning and to be prepared to move as needed.

FINDINGS Despite differences in the raid itself, as well as the group's familiarity with each raid, both show a strikingly similar pattern. During fights, the in-game WoW presence predominates, but the between-battle chunks show Jaea shifting presence heavily towards information flow. This strong pattern is, in retrospect, the very pattern that led to our B-series time demarcations. While fighting and not fighting may seem like simple activity changes that occur only within the virtual world that the in-game WoW self inhabits, the influence on presence flow in all three layers is sharp and clear. Furthermore, Jaea's smooth yet rapid movement from one distribution of presence to another is practiced and familiar and presents no evidence of cognitive confusion or adaptive difficulties, which might be seen if split attention and multitasking were an issue. On the left of the figures are three labels, each of which represents a 'layer' of presence. At the top, the information constellation represents all the information accessed by laea that is not included by default in WoW. This information includes Mumble, the heavily customized user interface

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that Jaea relies upon, and any online resources or activities that he engages in, such as Twitter or email. In the middle, the Physical self represents Jaea's physical body, located within the physical world and subject to various distractions like bathroom breaks, cell phones calls, and aggressively playful cats. At the bottom, the in-game WoW represents the virtual body and the default WoW information, which includes virtual cues from fellow raid members and bosses, mana (magic points) and health information, equipment durability, and bag space. The separation of the information constellation layer from the in-game WoW layer and the physical self layer is a complex one. While information comes from the physical world (such as in-body signals that say, "I'm hungry") and the in-game WoW layer (virtual indicators that say, "My avatar needs to eat"), this information is part and parcel of living in and being able to perceive and respond to those worlds. The information constellation layer includes only the information that is separate from the default physical messages (the hardness of the chair against your legs) and the default WoW messages (the attack upon your avatar by a monster). In other words, the information constellation includes the additional information that laea (and other players) choose to receive or not receive, to seek out or not seek out. Within each chunk, text summarizes laea's activities by presence type. The order is not chronological because the actions overlap one another, and the text merely indicates what happened during that time frame. Each item represents an activity or exchange of some sort, ranging from going to the bathroom (physical self), listening to raid chatter via Mumble (information flow), to specific movements present only during boss engagements (in-game WoW). Repetitions of activities or exchanges are condensed and represented with only one label. for example, Healing teammates (in-game WoW) is an action Jaea engaged in repeatedly in each of the chunks; however, that entry is included only once, and there is no indication of frequency except indirectly by the height of the in-game WoW layer. Time progresses from left to right, and the three layers co-occur within each B-series time column. The height of each layer within blocks indicates the proportion of presence: in Figure 14.2, for example, Jaea's presence in the first block is mOst strongly in information constellation, followed by in-game WoW presence, with the least amount of presence in physical self. In the following block, a distinct shift occurs, as information constellation and physical self become shorter, and in-game WoW nearly doubles. The dramatic shifts of presence accompanying events in WoW dictated the B-series time distinctions and help to foreground and background different elements of the data. Specific actions taken by Jaea are noted within the blocks by text but are otherwise backgrounded. Instead, we can clearly see the flow of activity and presence by backgrounding the details and foregrounding the descriptors of actions, which is key to our analysis. The graphs emphasize the proportion of overall presence allocated to each layer and the changes of those proportions based on activity.

There are two particular codes that are worth explaining briefly here. The descriptor Minimal presence is used to indicate that Jaea is barely present in that layer during a time chunk. It is used only in the physical self category to represent that some presence is unavoidable. Another descriptor, Customized UI (see Figure 14.4), is ever-present in the information constellation category, and represents Jaea's modified user interface. His reorga. nization of WoW information flow is so drastic that his every engagement with the virtual world is mediated by his unique design. Jaea's constant tweaking of his customized UI contributes to his level of expertise because his ability to process information included in the customized UI at a glance allows him to have faster reaction times and more precision when playing. A constant striving to improve the customized UI shows a perpetual striving to improve his gameplay. Just as in situations of other forms of fandom (Black & Stein kuehler, 2009) the player's ability to 'read' the text of the situation is important to success. Coiro and Dobler (2007) found that reading online required different skills than reading in traditional print format. The reader needed to understand more than the words on the screen-also the context clues and structure of the digital materials. The situation is similar for Jaea in-game; as Coiro and Dobler found for websites, Jaea must be able to interpret symbols, and understand the context clues and structure of the game interface. Modifying the interface could be seen as similar to writing fan fiction-tweaking the story and changing parts to make the readability more customized to his needs. While writing fan fiction and maximizing healing performance may seem quite different, both are examples of the same phenomena: participants acting upon their digital environments in order to rewrite their own stories. Jaea's UI modification could be conceived as intentionally modifying the spatial arrangement (Hollan et aI., 2000) in order to simplify choice, perception, and internal computation. One difference between the two presence visualizations is particularly clear. The physical self presence is similarly minimal during fights, but Figure 14.2 shows a shift towards a stronger physical self presence between battles, while the Figure 14.3 physical self presence stays quite minimal both in and out of fights. The differences in raid familiarity provide a possible rationale: Figure 14.2, as a familiar raid, required less attention from Jaea during non-battle times, as he was already comfortable with the actions required of him during battle, and didn't need to devote resources towards planning for the restart of battle. Figure 14.3, representing a neverbefore-attempted raid, required a consistently high presence in information constellation, thus limiting the physical self presence even during non-battle time chunks. Note that although there is still a high information flow in Figure 14.2 during non-battle times, it contains some considerably different items that indicate a shift away from the in-game information. Instead of just raiding team discussions and user interface tweaking, Jaea is reading various forums and looking through Twitter, activities that are sometimes unrelated

Figure 14.4

Jaea’s customized UI before Al’Akir.

Figure 14.4 Jaea's customized

or before AI'Akir.

to the game or his current activity within the game. This divergence in activity pairs with the increased presence in physical self, as Jaea stops devoting his presence to the game-related activities, and shifts into a more relaxed state.

VISUALIZATION OF EXPERTISE

Our expert's success at play, relying on more than just his ability to play his character well, requires fluid and rapid shifts of presence. As can be seen in the presence visualizations, Jaea's ability to distribute attention to different layers means that he can balance the influx of information and information needs that arise, as well simultaneously engage within the physical space and within the game. Jaea's expertise moves beyond his ability to control his character and relies on his ability to interact with his surrounding virtual and physical spaces, as well as informational resources. Jaea's layers of presence vary in proportion with the activity he is currently engaged in during game play. The more temporally rapid the activity in the game, the larger the proportion of the game layer presence, whereas between activities the information layer or the physical self layer increases in proportion in the presence visualization. The information constellation layer always fills a larger proportion than the physical self layer. In Figure 14.3, the physical self layer is a smaller proportion because of the intensity of information flow both during fighting and non-fighting periods due to the fact that the raid is unfamiliar, which requires high levels of

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verbal coordination and information seeking even between fights. Despite the fact that all raid members studied the raid before they attempted it, the attempts were not going well and required just-in-time (Gee, 2003) information seeking in order to help them re-strategize. The information flow of a player is contingent upon the player's information literacy skills, and in this case the raid group's information literacy skills. Collective information literacy (Martin & Steinkuehler, 2010) is implemented within the raid group in order to pull from the group's collective knowledge of the game-at-large, raid strategy, player roles, specific raid layouts, boss phases, etc. Using this collective, connected cloud of information, the raid guild is able to be more effective and better informed together than they would be individually. This method allows for the capture of more than just Jaea's movements in the space of the game, the information constellation, and his physical location. It captures social interactions, use of resources, and focus in correspondence with the activities in the B-series timescale. This connection of activities to time with reference to presence offers a deeper look into the intricate functioning of expertise in the game, and demonstrates the importance of distributed presence. From this conceptualization of distributed presence, we can extend the concept of distributed cognition as discussed by Hutchins (1995) and Hollan et al. (2000). Presence illustrates distributed cognition in several ways. First, it has a unit of analysis, in this case the larger unit of a coherent WoW activity and the smaller units of partitioned activity as demarcated by B-series time. Second, there are multiple actors in the cognitive process-in this example, members of a social group, as well as a variety of resources. Finally, the actions and process are distributed across time and previous actions affect the nature of later actions. Crucially, our examination of presence is based on the fact that "people form a tightly coupled system with their environments" (Hollan et aI., 2000: 192), even as we extend the definition of environment to include the digital context of WoW and the complex constellation of information. At the same time the actions we see here are reminiscent of collective information literacy (Martin & Steinkuehler, 2010), which could be viewed as distributed information literacy, in view of Hutchins' definition of distributed cognition. Collective information literacy occurs when multiple people in a group or affinity space work together to solve a problem. This happens in both examples given earlier but most notably in the example with the group facing an unfamiliar raid. In Figure 14.3, the strategy and problem solving sessions carried out at the end of each wipe are a perfect example of collective information literacy: the raid team always used that time to determine what needed to be changed from the prior strategy, and what information players were missing in order to accomplish the raid. The use of distributed cognition and collective information literacy is not surprising given the social nature of games. However, the evidence of these two models functioning in a punctuated way within the B-series time allows for

a useful way to track the intellectual work as well as the action that goes into success in an affinity space or other collaborative spaces that requires joint activity. We offer Jaea as an exemplar of how expertise functions in a space like this, surrounded by other experts and experts-in-training working towards a common challenging goal. THE SHIFTING NATURE OF EXPERTISE Throughout his play, Jaea consistently engages in multiple simultaneous actions and thought processes. The number of actions he juggles at once combined with the speed with which he shifts between these actions is likely to seem overwhelming to a novice or non-player. This shifting distribution of presence should not be confused for partial or even split attention, but rather for the necessarily flexible nature of expertise. Jaea's focus is not splintered; rather, he is unquestioningly focused on playing his character well. The expert's shifting distributions of presence can be clearly distinguished from the metaphor of split attention: Jaea's increase in focus in one layer occurs because of activities in the other layers. Instead of 'losing' cognitive power (as in split attention), the expert's shift occurs because of a holistic understanding of the context and attentional demands. In fact, play at such an expert level cannot take place without such shifts in presence. Players who focus their attention on only one component of the game at a time cannot possibly play at the same level as Jaea. Only players who can engage in action with distributed attention and distributed information literacy across a group of players can successfully participate in highlevel raiding activities. This ability demonstrates the literacy of the player to navigate the space of the game, the physical world, and the information constellation. These literacy practices, because of their ability to help the player succeed in the game, also affect the social identity of the player within the community. This method helps to visually identify the literacy practices that take place in the virtual space, the physical space, and the information constellation, all of which are contained in the constellation of literacies that a player or person uses and experiences. Through this visualization process we can observe the literacy practices as they take place and see what practices are layered together. Expertise in World of Warcraft and other games requires masterful coordination of information resources across multiple timescales and spaces, and is the same no matter the physical age of the player. Jaea has to be able to sustain his attention across multiple spaces in his immediate vicinity, in the virtual game world, and with fellow players who are participating from other locations throughout the world. Considering that this delicate balance of shifting distributions of presence is necessary for expert-level play, we suggest that one has to consider the timescales and spaces involved all as equally valid.

Martin et al. CONCLUSIONS

This study of a single expert World of Warcraft player revealed important insights about activity as it OCCurs across multiple timescales and in multiple spaces, both real and virtual. For future studies that hope to continue with similar kinds of analyses, we recommend observing players with varied amounts of expertise. The methodology of presence that the research_ ers of this study utilize could be used to study the gameplay of any level of player. Novice players are not likely to engage these acts of distributed and collective cognition with the same ease with which laea is able to transition across times and spaces using both his physical self and in-game WoW self. Additional research could allow researchers to map out a trajectory of expertise development in terms of time, space, and literacy. We anticipate that such a trajectory will provide illumination into the development of facile shifts in particular layers of presence. We also suggest that studying an entire raid group might reveal more about how attention is divided across multiple activities and events during high-level play of World of Warcraft. A sudden shift from in-game Wo W self to physical self because of some event occurring in their physical space could have substantive effects on the attention of the rest of the players in the raid group even if they are distributed across multiple locations hundreds, if not thousands, of miles apart. As raiding players tend to operate at a fairly expert level, however, they may have means and strategies for coping with such interruptions and shifts such that it does not dramatically disrupt their play. Knowing more about how the players' manage their distributed attention to maximize their collective efforts could be useful for an array of contexts, both for study of games and beyond. Understanding how groups manage their resources across multiple spaces and times has a variety of implications in a globally connected world where teams of problem solvers are often located throughout the world as they work on shared problems.

REFERENCES Awh, E. & Pashler, H. (2000). Evidence for split attention a I foci. Journal of Experimental Psychology Human Perception and Performance 26(2): 834-846. Black, R. W. (2008). Adolescents and Online rl1/l Fiction. New York: Peter Lang. Black, R. W. (2007a). Digital design: English language learners and reader reviews in online fiction. In C. Lankshear & M. Knobel (eds.), New Literacies Sampler. New York: Peter Lang. Black, R. W. (2007b). Fanfiction writing and the construction of space. e-Learning 4(4): 384-397. Black, R. W. & Steinkuehler, C. (2009). Literacy in virtual worlds. In 1.. Christenbury, R. Bomer, & P. Smagorinsky (eds.), Handbook of Adolescent Literacy Research. New York: Guilford. Bourdieu, P. (1977). Outline ofa Theory of Practice (R. Nice, Trans.). Cambridge, MA: Cambridge University Press. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science 5: 121-1S2.

Coiro, J., & Dobler, E. (2007). Exploring the online reading comprehension strategies used by sixth-grade skilled readers to search for and locate information on the Internet. Reading Research Quarterly 42(2): 214-257. Erickson, F. (2004). Talk and Social Theory: Ecologies of Speaking and Listening in Everyday Life. Cambridge: Polity Press. Gee, J. P. (2005). Semiotic social spaces and affinity spaces: From The Age of Mythology to today's schools. In D. Barton & K. Tusting (eds.), Beyond Communities of Practice: Language, Power, and Social Context. New York: Cambridge University Press. Gee, J. P. (2004). Situated Language and Learning. New York: Routledge. Gee, J. (2003). What Videogames Have to Teach Us about Learning and Literacy. New York: Palgrave Macmillan. Gee, J. P. (1999). An Introduction to Discourse Analysis: Theory and Method. New York: Routledge. Gell, A. (1992). A-series:B-series::Gemeinschaft:Gesellschaft::Them:Us. The Anthropology of Time: Culture Construction of Temporal Maps and Images. Oxford: Berg. Gonzalez, V. M. & Mark, G. (2005). Managing currents of work: Multi-tasking among multiple collaborations. In H. Gellersen et al. (eds.), ECSCW 20005: Proceedings of the Ninth European Conference 011 Computer-Supported Cooperative Work. Paris: Springer. Gonzalez, V. M. & Mark, G. (2004). "Constant, constant, multi-tasking craziness": Managing multiple working spheres. CHI 2004 6(1): 113-120. Hollan, J., Hutchins, E., & Kirsh, D. (2000). Distributed cognition: Toward a new foundation for human-computer interaction research. ACM TrallSactions on Computer- Human Interaction 7(2): 174-196. Hutchins, E. (1995). Cognition in the Wild. Cambridge, MA: MIT Press. Kalyuga, S., Chandler, P., & Sweller,]. (1999). Making split-attention and redundancy in multimedia instruction. Applied Cognitive Psychology 13: 351-371. Lemke, J. (2000). Across the scales of time: Artifacts, activities, and meanings in ecosocial systems. Mind, Culture, and Activity 7(4): 273-290. Martin, C. (in progress). A-series and B-series Time Maps in World of Warcraft. Martin, C. & Steinkuehler, C. (2010). Collective information literacy in massively multiplayer online games. e-Learning and Digital Media 7(4): 355-365. Mayer, R. E. & Moreno, R. (1998). A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory. Journal of Education Psychology 90(2): 312-320. Stake, R. (1995). The Art of Case Study Research. Thousand Oaks, CA: Sage. Steinkuehler, C. (2008). Cognition and literacy in massively multiplayer online games. In J. Coiro et al. (eds.), Handbook of Research on New Literacies. New York: Routledge. Steinkuehler, C. (2007). Massively multiplayer online games as a constellation of literacy practices. e-Learnillg and Digital Media 4(3): 297-318. Steinkuehler, C. (2006). Massively multiplayer online video gaming as participation in a discourse. Mind, Culture, and Activity 13(1): 38-52. Steinkuehler, C. & King, B. (2009). Digitalliteracies for the disengaged: Creating after school contexts to support boys' game-based literacy skills. On the Horizon 17(1): 47-59. Stevens, R., Satwicz, T., & McCarthy, 1.. (2008). In-game, in-room, in-world: Reconnecting video game play to the rest of kids' lives. In K. Salen (ed.), The Ecology or Games: Connecting Youth. Games, and Learning. Cambridge, MA: MIT Press. Tabhers, H., Martens, R., & van Merrienboer, J. (2000, February). Multimedia instructions and cognitive load theory: Split-attention and modality effects. Paper presented at the AECT 2000, Long Beach, CA.

Super Meat Boy Special Bonus Pack Caro Williams The experience of failure is a beautiful one. Granted, failure is generally only beautiful when viewed from the other side of the currently insurmountable obstacle—but without failure, where is the glory in success? When Team Meat was designing their love letter to classic platformers (Payne & Cambell) and their experience turned from joy to horror (Wolfenstein), they pushed on. And Super Meat Boy is the beautiful reward that Team Meat—and we—received from their brutal experience. The following odes to failure and Super Meat Boy share the beauty and difficulty in pushing through aspects of life—in games and out— that cause suffering and frustration. What both Wolfenstein and Payne & Campbell tell us repeatedly, despite their very different approaches to the subject, is this: The only real failure is when you put down the controller and never return. Team Meat made the world a better place by not succumbing to their suffering and frustration—and Super Meat Boy players every day testify against failure by picking up the controller again and again. “We can do this,” they say, “even as the world transforms from pastoral woodlands to hellish nightscapes— give me just one more try…” Team Meat has left Meat Boy and Bandage Girl in our hands—fail well, and play on.

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Knuth, E., Kalish, C., Ellis, A., Williams, C., & Felton, M. (2011). Adolescent reasoning in mathematical and non-mathematical domains: Exploring the paradox. In V. Reyna, S. Chapman, M. Dougherty, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making (pp. 183-210). Washington, DC: American Psychological Association.

Adolescent Reasoning in Mathematical and Non-Mathematical Domains: Exploring the Paradox

Eric Knuth Charles Kalish Amy Ellis Caroline Williams University of Wisconsin Matthew Felton University of Arizona

The research is supported in part by the National Science Foundation under grant DRL-0814710. The opinions expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Abstract Mathematics education and cognitive science research paint differing portrayals of adolescents’ reasoning. A perennial concern in mathematics education is that students fail to understand the nature of evidence and justification in mathematics. In particular, students rely overwhelming on examples-based (inductive) reasoning to justify the truth of mathematical statements, and often fail to successfully navigate the transition from inductive to deductive reasoning. In contrast, cognitive science research has demonstrated that children often rely quite successfully on inductive inference strategies to make sense of the natural world. In fact, by the time children reach middle school, they have had countless experiences successfully employing empirical, inductive reasoning in domains outside of mathematics. In this chapter, we explore this seeming paradox and, in particular, explore the question of whether the skills or knowledge that underlie adolescents’ abilities to reason in non-mathematical domains can be leveraged to foster the development of increasingly more sophisticated ways of reasoning in mathematical domains.

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A perennial concern in mathematics education is that students fail to understand the nature of evidence and justification in mathematics (Kloosterman & Lester, 2004). Consequently, mathematical reasoning—proof, in particular—has been receiving increased attention in the mathematics education community with researchers and reform initiatives alike advocating that proof should play a central role in the mathematics education of students at all grade levels (e.g., Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; National Council of Teachers of Mathematics, 2000; Knuth, 2002a, 2002b; RAND Mathematics Study Panel, 2002; Sowder & Harel, 1998; Yackel & Hanna, 2003). Proof plays a critical role in promoting deep learning in mathematics (Hanna, 2000); as Stylianides (2007) noted, “proof and proving are fundamental to doing and knowing mathematics; they are the basis of mathematical understanding and essential in developing, establishing, and communicating mathematical knowledge” (p. 289). Yet, despite its importance to learning as well as the growing emphasis being placed on proof in school mathematics, research continues to paint a bleak picture of students’ abilities to reason mathematically (e.g., Dreyfus, 1999; Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Martin et al., 2005). In contrast, cognitive science research has revealed surprising strengths in children’s abilities to reason inferentially in non-mathematical domains (e.g., Gelman & Kalish, 2006; Gopnik, et al., 2004). Although more traditional (Piagetian) views posit children as limited to understanding obvious relations among observable properties, there is growing evidence that children are capable of developing sophisticated causal theories, and of using powerful strategies of inductive inference when reasoning about the natural world (for review, see Gelman & Kalish, 2006). In the former case, for example, children can integrate statistical patterns to form

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representations of underlying causal mechanisms (Gopnik & Schulz, 2007). In the latter case, for example, children often organize their knowledge of living things in ways that reflect theoretical principles rather than superficial appearances (Gelman, 2003). Thus, this raises something of a paradox: Why do children appear so capable when reasoning in non-mathematical domains, yet seemingly appear so incapable when reasoning in mathematical domains? In this chapter, we explore the paradox by considering the research on adolescents’ reasoning capabilities within mathematics education as well as within cognitive science. In particular, we briefly consider research that provides a portrayal of adolescents’ reasoning in mathematical and non-mathematical domains. Next, we present preliminary results from the first phase of our multi-year research effort to better understand the relationships between adolescents’ reasoning in mathematical and non-mathematical domains. We view such relationships as a means for potentially leveraging the strengths adolescents demonstrate when reasoning in non-mathematical domains to foster the development of their mathematical ways of reasoning. Finally, we discuss the implications of our research as well as its future directions. Situating the Paradox Mathematics education and cognitive science research paint differing portrayals of adolescents’ reasoning, particularly with respect to the nature of their reasoning strategies. In the world outside the mathematics classroom, children typically rely quite successfully on inductive inference strategies—empirical generalizations and causal theories—to make sense of the natural world. For example, preschool-aged children are able to interpret and construct interventions to identify causal mechanisms in simple systems (Gopnik, et al., 2004). Young children also have rich knowledge structures supporting explanation and predictions of physical, biological, and social phenomena (Gelman & Kalish, 2006). In fact, by the time children reach middle school,

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they have had countless experiences successfully employing empirical, inductive reasoning in

domains outside of mathematics.1 Not surprisingly, many students also employ similar reasoning strategies as they encounter ideas and problems in mathematics (Recio & Godino, 2001); however, they often fail to successfully navigate the transition from inductive to deductive reasoning—the latter being the essence of reasoning in mathematics. As Bretscher (2003) noted, “Proof in everyday life tends to take the form of evidence used to back up a statement. Mathematical proof is something quite distinct: evidence alone might support a conjecture but would not be sufficient to be called a proof” (p. 3). Adolescent Reasoning in Mathematical Domains It is generally accepted that students’ understandings of mathematical justification are “likely to proceed from inductive toward deductive and toward greater generality” (Simon & Blume, 1996, p. 9). Indeed, various mathematical reasoning hierarchies have been proposed that reflect this expected progression (e.g., Balacheff, 1987; Bell, 1976; van Dormolen, 1977; Waring, 2000); yet, research continues to show that many students fail to successfully make the transition from inductive to deductive reasoning.2 One of the primary challenges students face in developing an understanding of deductive proof is overcoming their reliance on empirical evidence (Fischbein, 1982). In fact, the wealth of studies investigating students’ proving

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Unfortunately, children’s experiences successfully employing empirical, inductive reasoning in elementary school also tend to engender the belief that such reasoning suffices as proof in mathematical domains. 2 The hierarchies that have been proposed, although based on empirical data, do not provide accounts regarding the actual transition from inductive to deductive reasoning. Rather, the hierarchies primarily note differences in the nature of students’ inductive reasoning (e.g., justifications that rely on several “typical” cases versus those that rely on “extreme” cases) with deductive reasoning being at the “upper end” of the hierarchies, and not how (or if) such inductive reasoning strategies can develop into deductive reasoning strategies.

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competencies demonstrates that students overwhelmingly rely on examples to justify the truth of statements (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Porteous, 1990).3 As a means of illustrating adolescent reasoning in mathematics, we briefly present results from our prior work concerning middle school students’ proving and justifying competencies. The following longitudinal data are from 78 middle school students who completed a written assessment at the beginning of Grades 6 and 7, and at the end of Grade 8; the assessment focused on students’ production of justifications as well as on their comprehension of justifications.4 In the narrative that follows, we present a representative sample of the assessment items and corresponding student responses. Justifications in which examples were used to support the truth of a statement were categorized as empirical, justifications in which there was an attempt to treat the general case (i.e., demonstrate that the statement is true for all members of the set) were categorized as general, and justifications that did not fit either of these two aforementioned categories were categorized as other.5 As an example, students were asked to provide a justification to the following item: If you add any three odd numbers together, is your answer always odd? The following two student responses are representative of empirical justifications: Yes because 7+7+7=21; 3+3+3=9; 13+13+13=39. Those problems are proof that it is true. (Grade 6 student)

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For the purposes of this chapter, we define inductive reasoning to be reasoning that is based on the use of empirical evidence, and by empirical evidence we mean the use of examples to justify statements or conjectures. Moreover, inductive reasoning is not to be confused with mathematical induction—a mathematically valid method of proving. 4 The assessment items presented below were the same for each administration of the assessment, and the same group of students completed the assessment at all three time points. 5 We have simplified the categorizations described in this chapter as we are primarily interested in highlighting the differences between empirical-based justifications and more general, deductive justifications. See Knuth, Choppin, and Bieda (2009) and Knuth, Bieda, and Choppin (forthcoming) for more detail about the study’s results.

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1+3+3=7. 3+11+1=15. Yes it would be but you will have to do it a 100 times just to make sure. (Grade 7 student)

In contrast, the following two responses are representative of general justifications: If you add two odds, the result is even. An even plus one more odd is odd. So three odds added together always results in odd. (Grade 6 student)

We know that odd and odd equals even. So even (2 odds) added together with odd equals odd. This shows us that no matter what three odd numbers you add together, the sum will always be an odd number. (Grade 8 student)

Responses categorized as other (see Footnote 2) were often restatements of the question (without further justification) or nonsensical responses (e.g., “It is not always odd because some problems are even like 1+2+3=6, 3+4+5=12”). Table 1 displays the overall results of students’ justifications for this item. As the table illustrates, a significant number of students relied on examples as their means of justification, with very little change occurring across the middle grades. We also see an increase in the number of students attempting to produce more general, deductive justifications from Grade 6 to Grade 8; yet still less than half the students produce such justifications even by the end of their middle school mathematics education. As a second example, consider students’ responses to the following assessment item: Sarah discovers a cool number trick. She thinks of a number between 1 and 10, she adds 3 to the number, doubles the result, and then she writes this answer down. She goes back to the number she first thought of, she doubles it, she adds 6 to the result, and then she writes this answer

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down. [A worked-out example, including the computations, followed the preceding text.] Will Sarah’s two answers always be equal to each other for any number between 1 and 10? In this

case it is also worth noting that students could use examples to prove that the statement is always true by testing the entire set of possible numbers (i.e., numbers between 1 and 10). Table 2 presents the results for this item; justifications based on proof-by-exhaustion are also included in the general category (only approximately 5% of students in each grade level used this method). Given the significant proportion of students whose responses were categorized as other, it is worth briefly discussing potential reasons underlying their responses. The majority of these responses were either a result of (i) students misinterpreting the problem, thinking that the end result must always be twenty (the result that was provided in the worked-out example that accompanied the problem); or (ii) students not being able to articulate a general argument— students could “see” what was going on but were unable to provide an adequate justification. In the former case, the following response is representative: “No, because the number comes out differently if you chose a number like 11. It does not come out as 20” (Grade 8 student). In the latter case, the following response is representative: “The answers will always be equal because you’re just doing the same thing” (Grade 7 student). Although the percentage of students who relied on empirical-based justifications decreased relative to the previous example, the gradelevel trend regarding the number of students providing general, deductive justifications remained about the same (again, less than 50% of the students at any grade level provided this type of justification). As a final example, consider the following item in which students were asked to compare an empirical-based justification with a general, deductive justification: The teacher says the following is a mathematical fact: When you add any two consecutive numbers, the answer is

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always odd. Two students offer their explanations to show that this fact is true [Note: an empirical-based justification and a general, deductive justification are then presented for the students]. Whose response proves that if we were to add any two consecutive numbers we would get an answer that is an odd number? As Table 3 suggests, for many middle school students, an empirical-based justification seems to suffice as proof. We see a slight grade level increase in viewing the general, deductive justification as proving the claim and, interestingly, we see a substantial decrease by Grade 7 of students who think both justifications prove the claim. In summary, the snapshot of adolescents’ mathematical reasoning illustrated above is quite typical of the findings from much of the research: adolescents are limited in their

understanding of what constitutes evidence and justification in mathematics and, moreover, they demonstrate a proclivity for empirical-based, inductive reasoning rather than more general, deductive reasoning. Although most studies have focused on adolescents in a particular grade level or across grade levels (i.e., cross-sectional studies), the research discussed above provides a longitudinal view into the development of adolescents’ mathematical reasoning. And given this longitudinal view, we see very little development as adolescents progress through their middle school years, and what development we do see falls far short of desired outcomes. Adolescent Reasoning in Non-mathematical Domains The difficulties that adolescents show with regard to mathematical reasoning, including the apparent lack of development as they progress through middle school, raise the question of whether there is some developmental constraint that limits adolescents’ mathematical reasoning. The most likely candidate would be abilities to do and understand deductive inference. The emergence of deductive inference has been a central focus of research on adolescent cognitive development, spurred in part by Piaget’s theory of formal operations. Although there is

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considerable debate within this literature, a plausible reading suggests there is nothing special about adolescence in terms of acquiring deduction. Younger children, for example, have been shown to appreciate that deductive inference leads to certain conclusions, and is stronger than inductive inference (Pillow, 2002). At the same time, however, even adults struggle to reason formally and deductively.6 Thus, deductive inference seems neither impossible before adolescence, nor guaranteed after. Rather than review the literature on the development of

deductive inference in non-mathematical domains (see Falmange & Gonsalves, 1995), we take a slightly different approach here. Similar to the mathematics education research on proof discussed above, researchers in psychology have often argued that people rely on empirical solutions to deductive problems. An interesting difference with the literature in mathematics education, however, is that these empirical-based solutions are typically evaluated quite positively in non-mathematical domains. That is, the kind of performance that makes people look like poor deductive reasoners is actually consistent with their being quite good inductive reasoners. Deductive and inductive arguments have very different qualities. On the one hand, in making a deductive argument, one endeavors to show that the hypothesized conjecture must be true as a logical consequence of the premises (i.e., axioms, theorems). On the other hand, in making an inductive argument, one seeks supporting evidence as the means for justifying that the conjecture is likely to be true. We will refer to arguments based on accumulation of evidence as “empirical.” Often times the empirical support in inductive arguments is provided by examples. The conclusion “All ravens are black” is supported by encounters with black ravens (and the

6

Note that in this literature, as in almost all work in psychology, “adult” means college-aged.

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absence of no black ones).7 The critical point is that deductive arguments prove their conclusions though logic, while inductive arguments provide evidence that conclusions are likely. Again, though there is strong debate, one influential view is that people often seek evidence (make inductive arguments based on examples from a class) when asked to evaluate logical validity (make deductive arguments). One of the clearest examples of inductive approaches to a deductive problem is Oaksford and Chater’s (1994) analysis of performance on the Wason selection task. The selection task is a classic test of logical argumentation. A participant is asked to evaluate a conjecture about a conditional relation, such as “If there is a p on one side of a card, then there is a q on the other side.” The participant is presented with four cards: one each showing p, not-p, q, and not-q. The task is to select just the cards necessary to validate the conjecture. The logical solution is to ensure that the cards are consistent with the conjecture: that there are no p and not-q cards. To confirm this involves checking that there is a q on the back of the p card, and checking that there is a not-p on the back of the not-q card. In practice, most people do check the p card, but very few examine the not-q card. Rather, most people opt to explore the q card, which is logically irrelevant (both p and q and not-p and not-q are consistent with the conjecture). This behavior is often interpreted as akin to the logical fallacy of affirming the consequent (if p then q, q, therefore p). Oaksford and Chater argue that selecting the p and the q cards is actually a reasonable strategy for assessing the evidential support for the conjecture. Though the details are quite complex, they show that given reasonable assumptions about the relative frequencies of p and q, the cards selected are the most informative tests. That is, people’s behavior conforms to a normative standard of hypothesis testing (e.g., optimal experiment design). 7

There are many other sources of inductive support. For example, that one’s teacher says, “All ravens are black.” provides some reason for adopting the belief.

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The distinction turns on two different ways of construing the task. Interpreting the problem as a deductive one (the way experimenters’ intend it) can be glossed something like this: Is the statement logically consistent with the features of the four cards on the table? The

inductive construal is something like the following: Is the statement likely to be true of cards in general? Although the deductive problem can be solved conclusively, it is not really that interesting or important (who cares about these four cards?) The inductive problem can never be truly solved (absent investigation of every card in the world), however, it is just the kind of problem that people really care about and face in their everyday lives. How can past experience (the four cards) help in the future (expectations about new cards)? The idea that people often employ inductive, evidential support, strategies to solve deductive problems is part of a general approach to cognition and cognitive development that emphasizes probabilistic reasoning (Chater & Oaksford, 2008). From this perspective, most of the cognitive challenges people face involve estimating probabilities from evidence. This is straightforward for processes of categorization and property projection. Learning that barking things tend to be dogs, and that dogs tend to bark, seems to involve learning some conditional probabilities. Influential accounts of language acquisition suggest that children are not learning formal grammars (deductive re-write rules) but rather patterns of probabilities in word cooccurrences and transitions. Even vision has been analyzed as Bayesian inference about structures likely to have generated a given perceptual experience. The general perspective is that inductive inference is ubiquitous; we are continually engaged in the task of evaluating and seeking evidential support. Given the centrality of inductive inference, it should not be surprising that many psychologists argue that we are surprisingly good at it and, good at it from a surprisingly young age (Xu & Tenenbaum, 2007a; 2007b).

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There are several lines of research potentially relevant to understanding adolescents’ reasoning strategies in mathematics. Perhaps the most direct connection is with research on

evaluations of inductive arguments. In contrast to deductive arguments, which are either valid or invalid, inductive arguments can vary in strength. If people are good at reasoning inductively, then they ought to be able to distinguish better and poorer arguments, stronger and weaker evidence, for conclusions. This work has both a descriptive focus—how do people distinguish stronger and weaker evidence—and a normative focus—do people’s strategies conform to normative standards for evidence evaluation? Most work on adolescent inductive inference has focused on the problem of identifying causal relations in multivariate domains (see Kuhn, 2002, for a review). Questions center on children’s abilities to construct and recognize unconfounded experiments, distinguish between hypotheses and evidence, and generally to adopt systematic investigation strategies. Similar to the literature on deductive inference, the conclusions are generally that young children show some important abilities but are quite limited; adults are better, but far from perfect; and adolescents are somewhere in the middle. Other forms or elements of inductive reasoning show a significantly different profile: Even young children are skilled at inductive inference (see Gopnik & Schulz, 2007). Adolescents have not been the direct focus of research, but there seems no reason to believe that inductive inference skills should decline from early childhood to adolescence. Research on evidential support explores how people respond to or generate evidence. Evidence in this work consists of different kinds of examples or instances.8 The task is to make or evaluate a conclusion based on that evidence. For example, given that robins are known to 8

In much of this work, the “examples” are categories of animals. It is unclear whether category-to-category inferences (“robins” to “owls”) is different than individual-to-individual inferences (“these 3 robins” to “these 3 owls”).

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have a certain property (e.g., hollow bones), how likely is it that owls also have the property?

The evidence consists of examples known to have the property in question. These examples can be understood as premises in an argument about the conclusion: The strength of the argument is confidence in the conclusion conditional on the evidence. Osherson and colleagues (Osherson, et al., 1990) developed one of the first models and described several criteria for evidential strength. Subsequent research has explored the development and application of these and other criteria (Lopez, Gelman, Gutheil, & Smith, 1992; Heit & Hahn, 2001; Rhodes, Gelman, & Brickman, in press). Table 4 provides a list of proposed criteria; note that some of these criteria are more normatively defensible than others. Although there remains some debate about preschool-aged children, most researchers would agree that by middle childhood children use the criteria in Table 4 to evaluate examples as evidence. Thus, children judge that many examples are more convincing than are fewer, that a diverse set of examples is better than a set of very similar examples, and that an argument based on a typical example is stronger than an argument based on an atypical example. Research continues on other principles of example-based arguments, such as the role of contrasting cases (e.g., non-birds that do not have hollow-bones; Kalish & Lawson, 2007) and children’s appreciation of the importance of sampling. The general conclusion is that children, including adolescents, are similar to adults in their evaluations of evidence. Moreover, children’s evaluations accord quite well with normative standards of evidence. Adolescent Reasoning in Mathematical and Non-mathematical Domains The preceding discussion highlights some important differences between adolescent reasoning in mathematical and non-mathematical domains. In mathematics education, inductive strategies are typically treated as a stumbling block to overcome rather than as an object of study.

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Moreover, mathematics education research has focused primarily on distinctions between empirical/inductive and formal/deductive justifications, and questions such as what makes one empirical justification better than another or what constitutes better/stronger evidence has not

been well addressed.9 In a recent paper Christou and Papageorgiou (2007) argued that the skills involved in induction, such as “comparing” or “distinguishing,” were similar in mathematical and non-mathematical domains. Christou and Papageorgiou’s work showed that students can identify similarities among numbers, distinguish non-conforming examples, and extend a pattern to include new instances. Thus, identifying how adolescents use such abilities to evaluate both mathematical and non-mathematical conjectures and how they think about the nature of evidence used to support conjectures may suggest a means for leveraging their inductive reasoning skills to foster the development of more sophisticated (deductive) ways of reasoning in mathematical domains. Exploring the Paradox What kinds of skills or knowledge underlie adolescents’ abilities to reason in nonmathematical domains, and might such skills or knowledge have any relevance to reasoning in mathematical domains? There are many different accounts of inductive inference, but one fairly consistent component is a representation of relevant similarity in the domain. To make or evaluate empirical-based, inductive inferences one must have a sense of the significant relations among the examples or objects. For example, if the task is to decide whether birds have hemoglobin in their blood or not, the most informative examples will be objects similar to birds. The argument that since spiders lack hemoglobin, birds must lack it as well is not particularly convincing because spiders and birds seem very different. In contrast, knowing that reptiles have 9

Although mathematics education researchers have noted differences in the nature of empirical justifications— checking a few “random” cases, systematically checking a few cases (e.g., even and odd numbers), and checking extreme cases (e.g., Balacheff, 1987)—they have not engaged in any deeper study of empirical justifications.

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hemoglobin seems quite relevant if we believe birds and reptiles are relevantly similar. The critical question, then, is “what makes two things relevantly similar?” Other principles of inductive inference described earlier also depend on similarity relations (e.g., typical examples

are better because they are similar to many other examples). The relevant similarity relations are, in part, knowledge dependent. Airplanes are similar to birds and may be useful examples to use when making inferences about aerodynamics, however, the question about hemoglobin calls for a biological sense of similarity. Getting the right similarity relations is a critical part of expertise. For example, experts tend to see “deep” similarities (e.g., evolutionary history), while novices often rely on shallow, domain-general similarities (e.g., appearances; Bedard & Chi, 1992). Put another way, reasoning from similar cases will only be successful if one’s representation of “similar” really does capture important relations in the domain. A considerable amount of research and debate in the cognitive developmental literature involves just what kinds of similarity relations children recognize and how such relations are acquired. Some argue, for example, that evolution has equipped us to be sensitive to significant similarities (Spelke, 2000; Quine, 1969). Others argue that domain general learning principles allow children to hone in on the important relations (Rogers & McClelland, 2004). Regardless, the general finding is that quite young children seem to display useful and productive intuitions about similarity in the empirical domains studied. Even preschoolers recognize that reptiles are more like birds than are airplanes when biological questions are involved, but that airplanes may be more informative about birds when the questions involve aerodynamics (for example, Kalish & Gelman, 1992). Unfortunately, the mechanisms that have been hypothesized to underlie the development of similarity in empirical domains may fail to support a sense of mathematical similarity useful for evaluating mathematical conjectures.

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The nativist view of similarity suggests that evolutionary pressures have shaped the human cognitive system to focus on important relations. For example, snakes often look like sticks, but an organism that focused on these similarities would find itself in significant peril. Clearly, quantitative relations have adaptive significance, and a long enough history that our species could have evolved specific cognitive dispositions to represent such relations. Indeed, there are important claims of just such a “number sense” involving representations of

approximate magnitude (see Dahaene, 1999). Beyond the early grades, however, such relations are generally not important parts of mathematical thinking or reasoning. A sense of numerical similarity based on approximate magnitude is a limited basis for evaluating or making inferences about mathematical relations. The principles of mathematical relations depend on a formal system, which is too recent an invention to have had any significant selective pressure on the human cognitive system (see Geary, 1995). In geometry, basic mechanisms for representing shape provide a natural organization to the domain. It seems possible that this sense of similarity may be more productive, more related to mathematically significant properties, than representations of number. Empiricist views of the development of similarity also suggest pessimism about a mathematical sense of similarity. The empiricist idea is that children form representations in a domain by tracking statistical patterns. In the natural world, objects tend to form clusters: There are natural discontinuities (Rosch, et al., 1976). The features that are important for representing animals tend to come in groups, with high intra-group correlations among features and low intergroup correlations. For example, birds tend to fly, have wings, and have feathers. These features co-occur and tend to be distinctive from the features of mammals that walk, have legs, and have fur. These patterns in the distributions of observed features allow people to pick out informative

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features and represent kinds or categories that reflect those distributions. A sense of mathematical similarity will also be dependent on the kinds of relations and distributions of properties observed in experience. Again, it seems likely that many of the most significant

relations among mathematical objects in children’s regular as well as school experiences may not be particularly well correlated with mathematically significant relations. A tendency to notice similarity in appearance does lead one toward a fairly useful notion of similarity among animals, because biological properties tend to be correlated with appearance. In contrast, a tendency to notice frequency or magnitude among numbers does not typically lead to a mathematically useful notion of similarity of numbers. Moreover, mathematical objects, at least numbers, have a network organization: There are many cross-cutting dimensions of similarity. In contrast, there is one, taxonomic, way of representing similarity relations among living things that seems primary (though see Ross, Medin, & Cox, 2007 on significance of ecological relations). Again, geometric objects, with a strong hierarchical organization, and a closer tie to psychological mechanisms of shape perception, may be somewhat different than numbers in this regard. Thus, an important first step toward developing a deeper understanding of students’ inductive reasoning in the domain of mathematics is to explore their representations of similarity relations among mathematical objects.10 Successful inductive reasoning depends on seeing objects as similar to the degree they really do share important features or characteristics. How do students make judgments about whether two numbers or two geometric shapes are similar? What features or characteristics do students attend to when considering the similarity of numbers or geometric shapes? How do students’ similarity judgments compare with experts’ similarity judgments? Answers to such questions may provide insight into students’ choices for the 10

Note that our use of similarity refers to conceptual similarity unless we explicitly write mathematical similarity.

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empirical evidence they use to justify mathematical conjectures, which, in turn, may provide insight into means to foster their transition to more deductive ways of reasoning.11 Assessing Similarity in Mathematical Domains The first phase of our current research was to determine which features and characteristics individuals attend to when considering whole numbers and common geometric

shapes; in particular, on what features might individuals base their decisions when determining whether a particular number or shape is typical? We conducted semi-structured interviews with 14 middle school students, 14 undergraduates, and 14 doctoral students in mathematics and engineering fields (hereinafter, STEM experts). Participants examined various numbers and shapes on individual cards and then sorted and re-sorted them into groups according to whatever principles they chose (Medin et al., 1997). The numbers and shapes presented to participants for inclusion are shown in Figures 2 and 3. Participants engaged in three types of sorts: an open sort, a prompted sort, and a constrained sort. For the open sort, participants grouped and re-grouped numbers or shapes into categories of their own choosing until they had exhausted the types of categories they deemed relevant. For the prompted sort, the interviewer grouped some numbers (or shapes) according to a characteristic and asked participants to place additional numbers (or shapes) into the group. For instance, the interviewer might place the numbers 4, 25, and 81 (all perfect squares) into a group and ask the participant to include other numbers in the group. For the constrained sort, the interviewer provided a group of numbers (such as 4, 25, and 81) and then included an additional set of numbers (such as 23, 36, 51, and 100) and asked the participants which, if any, of the 11

Mathematics education research has revealed very little insight into students’ thinking regarding their choices of empirical evidence, yet such insight is critical in helping students develop more sophisticated ways of reasoning. For example, selecting examples that provide insight into the structure underlying why a conjecture is true can offer a potential means for generating a general, deductive justification (e.g., Yopp, 2009).

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additional numbers should be included in the group. The prompted and constrained sorts allowed us to determine whether participants would sort by particular features deemed mathematically interesting, such as a number being a perfect square, versus other noticeable, yet mathematically uninteresting features, such as the number of digits of a number or the value of one of its digits. The participants’ responses to the sorting and tree building interview yielded 13 number categories and 13 shape categories denoting features deemed relevant in each domain. Table 5 presents the number categories and their meanings. One of the more salient results from the number-sorting task was the number of similarities between the middle school students and the STEM experts in terms of which features they noticed. For instance, consider the parity category. Figure 4 shows the percentage of participants from each group who sorted according to parity in the three sorts. The “Parity 1st” part of the graph shows the percentage of participants who sorted by parity as their first-choice sort in the open sort. The “Parity Open” part shows the percentage of participants who sorted by parity in the open sort, but not as their first sort, and the “Parity Prompt” section shows the percentage of participants who were able to sort according to parity only in the prompted or constrained sorts. In this case it is clear that parity was a particularly salient feature for all three groups. The similarities between the middle school students and the STEM experts led us to wonder, were there any features that one group attended to but the other did not? The factors category was the only category that appeared to be salient to the middle school students but not to the undergraduates or the STEM experts. Just over 20% of the middle school students sorted according to factors in the open sort, whereas none of the undergraduates or STEM experts sorted according to factors. There was also just one category that the STEM experts could sort by more readily than the middle school students, and this was the squared category, referring to

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numbers that are perfect squares. Figure 5 shows the percentage of participants in each group who were able to sort by perfect squares. The “prime” category was the other category of number we expected to be more salient

to STEM experts, but this turned out not to be the case. Almost 90% of the STEM experts could sort according to primes, but 70% of the middle school students and the undergraduates could sort according to primes as well. Additionally, there were some mathematically uninteresting features that we expected the middle school students to attend to more than the experts, such as intervals, value of the digits, number of digits, contains a digit, and arithmetic. But of those five categories, differences only emerged for contains a digit and number of digits, and the differences were not large: 36% of the middle school students versus 21% of the experts sorted according to contains a digit, and 43% and 29% respectively sorted according to number of digits. Table 6 presents the categories for the number sort organized by the features to which each group of participants attended, in order from the most salient to the least. The middle school students and undergraduates were somewhat more attentive to common digits and number of digits than the STEM experts, whereas the STEM experts were more attentive to perfect squares and arithmetic relationships. Table 7 presents the categories of shape that the participants identified in the sorting task. The most salient category across all three groups was the number of sides, which all of the participants in each group used for grouping in either the first sort or the open sort. The other two categories that were also salient across all three groups were shape and size. The only category that was more salient for the STEM experts than for the other participants was the regular category. Forty-three percent of the STEM experts grouped shapes

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according to whether they were regular, but only 14% of the middle school students sorted according to this principle, and those students did so only in a prompted sort. None of the undergraduates sorted according to regularity. There were three more categories that we

anticipated would be more salient for the STEM experts: similar, symmetry, and tessellate. This turned out to be the case only for symmetry (see Figure 6): STEM experts sorted according to symmetry more often than middle school students (47% versus 21% for both the middle school students and the undergraduate students). Contrary to our expectations, as shown in Figure 6, middle school students attended to similarity slightly more than STEM experts did (50% versus 40%). Only one participant across the three groups attended to tessellations, and this participant was a middle school student. We anticipated that categories such as size, familiar, and orientation would be ones that would be more salient for middle school students, particularly because we consider these categories to denote principles of shape that are not mathematically important. However, orientation and size were actually slightly more salient for the STEM experts, and the familiar category was equally salient across all three groups (36% of each group sorted according familiarity of shape). Twenty-seven percent of the STEM experts grouped according to orientation, while only 14% of the middle school students and 7% of the undergraduates sorted by orientation. All of the STEM experts sorted according size, versus 78% of middle school students and 85% of the undergraduates. Table 8 presents the categories of shape sorted by which features each group of participants attended to in order from the most salient to the least. We found that STEM experts noticed symmetry and regularity more than did middle school students, whereas middle school students attended to similarity and equal sides more readily.

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Findings from the sorting and tree building study showed that in general, there were not many differences between the middle school students, the undergraduates, and the STEM experts in terms of the characteristics of number and shape that they attended to. Furthermore, we found that some of the most salient features of number included multiples, parity, primes, and intervals (i.e., the relative size of numbers). Several of the more salient features of shape included the number of sides, the shape’s size, recognizable features of a shape, and the size of its angles. These findings are important in that they reveal particular characteristics that participants find noticeable, such as a number’s relative size or a shape’s size, that matter to students but are not mathematically important from our perspective. Discussion and Concluding Remarks The results from our initial study suggest that adolescents’ and experts have very similar representations of similarity among (some) mathematical objects. In particular, adolescents did notice mathematically significant relations among the objects; part of what makes two numbers or shapes similar is that they share properties relevant to mathematical theorems and conjectures. Of course, participants did notice less significant properties as well, for example, shared digits of numbers, and shared orientation of shapes. There was some evidence that these less significant properties played a larger role in adolescents’ representations of number and shape, however, such properties also showed up in the STEM experts’ sorts. One possible explanation for this result is due in part to the extremely open-ended, unconstrained, nature of our similarity measures. Participants, for example, were not instructed to focus on “mathematical” similarity. As noted above, part of expertise consists of being able to select an appropriate similarity metric for the task at hand. We suspect that experts would tend to ignore irrelevant features (e.g., orientation) in the context of evaluating mathematical conjectures. It is less clear whether

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adolescents would show the same selectivity—exploring the significance of contextual variations (e.g., mathematics class) is one aspect included in the next steps for this research program. The importance of the current findings, though, is that adolescents do represent mathematically significant similarity relations. The pressing question for future research, however, is how they use such relations to evaluate conjectures. In this chapter we have taken a relatively new perspective (in mathematics education research) on adolescents’ use of empirical strategies for evaluating mathematical conjectures. Rather than seeing such strategies as limited or as failures to adopt deductive strategies, we suggest that there may be value in such inductive strategies. The argument so far has been that inductive inference is a powerful and useful form of reasoning, and one that people (especially adolescents) seem both disposed to use and use relatively successfully. Our proposal is to consider inductive inference about mathematical conjectures as an object of study in and of itself. To that end, we seek to better understand the adolescents’ inductive reasoning in the domain of mathematics. The empirical work presented in this chapter is a first step in a larger project of exploring just how adolescents use empirical examples and inductive methods to reason about mathematical objects. In short, we believe inductive inference strategies should play an important role in mathematics, and understanding adolescents’ inductive reasoning may provide important insight into helping adolescents transition to more sophisticated, deductive ways of reasoning in mathematics. In closing we want to make a more extended argument in favor of our perspective that inductive inference can and should play a productive role in school mathematics? Can this kind of reasoning support the transition to more general, deductive ways of reasoning? We began the chapter by noting that inductive arguments are commonplace in mathematics classrooms among

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middle school adolescents, and that more general, deductive reasoning is relatively rare. In contrast, we also noted that inductive arguments are important outside of mathematics and that

adolescents often employ quite sophisticated inductive strategies on many tasks that seem to call for deductive inference. The perspective from cognitive science is that people are not so much poor deductive reasoners as they are reluctant deductive reasoners. One reason for this reluctance may be that (outside of mathematics) we rarely care about the deductive implications of some set of facts or propositions. Invariably, it is the empirical significance that people seem to care about in most aspects of their lives. Yet, mathematics is different, precisely because of the demand to attend to deductive relations. To the question of the proper place of inductive inference in mathematics education we offer three responses. Although induction is not the accepted form of mathematical inference, it is a form of inference. Inductive reasoning can help students develop a feel for a mathematical situation and can aid in the formation of conjectures (Polya, 1954). It also provides a means of testing the validity of a general proof, especially where students are uncertain about the scope and logic of their argument (Jahnke, 2005). A major challenge in mathematics education, however, lies in moving students from reasoning based on empirical cases to making inferences and deductions from a basis of mathematical structures. By using more accessible inductive inference strategies, at least as an intermediate step, students may begin to appreciate that mathematics is a body of knowledge that can be reasoned about, explained, and justified. A concern with justification and explanation, even if inductively based, may support, rather than undermine, acquisition of more formal proof strategies. Inductive inferences are important mathematical strategies in their own right. Mathematical problems do not always demand formal solution approaches. This point is very

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much akin to the value of estimation in relation to exact computation. For example, it is very

useful to be able to guess whether a novel problem will be like some familiar problem; perhaps the same solution strategies will work in both cases. One does not need a formal proof that the two problems are isomorphic. Of course, the induction may be false and the apparent similarities misleading. However, developing better inductive strategies, such as recognizing which dimensions of similarity are important, is an important mathematical skill. Even in the context of theorem proving, inductive strategies are invaluable as they can often be used to provide evidence that suggests a conjecture may be true (or false). Second, the process of producing a proof depends on intuitions about the likely value of different steps or transformations. Intuitions that certain problems are related, or that some problems are more difficult, are expectations derived from experience. The critical point is that some intuitions and perceptions of similarities will be more useful than others. If students employ inappropriate inductive strategies they will not develop adequate mathematical reasoning skills. Although empirical induction is not an accepted form of proof within mathematics, it is a form of justification (and as previously discussed, a very common form among students). If students are encouraged to reason in more familiar ways, inductively, they may come to recognize the limitations of such reasoning with regard to proof as well as the power (in terms of proving) of deductive methods. Moreover, reflecting on the strengths and limitations of inductive argumentation may be an excellent bridge to introduce deductive methods. For example, empirical methods cannot conclusively prove conjectures, but they can conclusively disprove them (by exposing counter-examples). The idea of proof by contradiction could flow naturally from discussion of this feature of inductive reasoning. A similar trajectory might work for introducing mathematical induction as a kind of systemization or grounding of empirical

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induction. For example, students could be prompted to consider the limits of empirical induction and challenged to identify how (or if) mathematical induction overcomes those limits. We argue that inductive strategies are an important and valuable part of mathematical reasoning. Yet, even from the perspective that inductive strategies are shortcomings in the long run, there is overwhelming evidence that students do rely on them. Understanding inductive strategies is critical to understanding what students are taking from mathematics instruction. For example, teachers illustrate mathematical concepts with specific instances, but what do students infer from these particular instances? In such cases, are students led to believe that examples suffice as proof? The overwhelming message from mathematics education and cognitive science is that students do use empirical, inductive, strategies to reason about their world (including mathematics). Mathematics education can either ignore such strategies by treating them as “errors” to be overcome, or it can ask whether there is some value, as instructional tools, or as important mathematical content, to supporting inductive approaches. The perspective from cognitive science emphasizes the value of induction; to be a good reasoner is largely to be a good inductive reasoner. Mathematics may be different, but that difference does not obviate the need for or value of inductive reasoning. The study of adolescents’ inductive reasoning in the domain of mathematics is at a very early stage. If the literature on non-mathematical domains is any guide, we should expect to see powerful and sophisticated strategies of inference in the domain of mathematics. Inductive inference is likely a real source of strength upon which mathematics education can build.

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mathematics assessments of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics. Knuth, E. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405. Knuth, E. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61–88. Knuth, E., Bieda, K., & Choppin, J. (Forthcoming). The longitudinal development of middle school students’ justifying and proving competencies. Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). New York, NY: Routledge. Kuhn, D. (2002). What is scientific thinking, and how does it develop? In U. Goswami (Ed.), Blackwell handbook of childhood cognitive development. (pp. 371): Blackwell Publishing. Lopez, A., Gelman, S. A., Gutheil, G., & Smith, E. E. (1992). The development of categorybased induction. Child Development, 63, 171-190. Martin, T. S., McCrone, S. M. S., Bower, M. L., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof. Educational Studies in Mathematics, 60(1), 95–124. Medin, D. L., Lynch, E. B., Coley, J. D., & Atran, S. (1997). Categorization and reasoning among tree experts: Do all roads lead to Rome? Cognitive Psychology, 32, 49. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

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Pillow, B. H. (2002). Children’s and adults’ evaluation of the certainty of deductive inferences, inductive inferences, and guesses. Child Development, 73, 779-792. Polya, G. (1954). Induction and analogy in mathematics. Princeton, NJ: Princeton University Press. Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589–598. Quine, W. (1969). Natural kinds. In N. Rescher et al (Eds.), Essays in Honor of Carl G. Hempel, pp. 5-23, Dordrecht: Reidel RAND Mathematics Study Panel Report. (2002). Mathematical proficiency for all students: A strategic research and development program in mathematics education. Washington, DC: U.S. Department of Education. Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational Studies in Mathematics, 48(1), 83–99. Rhodes, M., Gelman, S. A., & Brickman, D. (In press). Children’s attention to sample composition in learning, teaching, and discovery. Developmental Science. Rogers, T. T., & McClelland, J. L. (2004). Semantic cognition: A parallel distributed processing approach. Cambridge, MA: MIT Press. Rosch, E., Mervis, C. B., Gray, W. D., Johnson, D. M., & Boyes-Braem, P. (1976). Basic objects in natural categories. Cognitive Psychology, 8, 382-439.

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Ross, N., Medin, D., & Cox, D. (2007). Epistemological models and culture conflict: Menominee and Euro-American hunters in Wisconsin. Ethos, 35, 478-515. Simon, M. & Blume, G. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31. Sowder, L. & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher, 91(8), 670–675. Spelke, E. (2000). Core knowledge. American Psychologist, 55, 1233-1243. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

van Dormolen, J. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8, 27–34. Waring, S. (2000). Can you prove it? Developing concepts of proof in primary and secondary schools. Leicester, UK: The Mathematical Association. Xu, F., & Tenenbaum, J. B. (2007a). Word learning as Bayesian inference. Psychological review, 114, 245-272. Xu, F., & Tenenbaum, J. B. (2007b). Sensitivity to sampling in Bayesian word learning. Developmental Science, 10, 288. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to the Principles and Standards for School Mathematics (pp. 227–236). Reston, VA: NCTM. Yopp, D. (2009). From inductive reasoning to proof. Mathematics Teaching in the Middle School, 15(5), 286-291.

33

Grade

Empirical

General

Other

6

37%

21%

42%

7

42%

36%

22%

8

40%

46%

14%

Table 1. Categories of Student Justifications to the Three Odd Numbers Sum item.

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Grade

Empirical

General

Other

6

30%

28%

42%

7

28%

32%

40%

8

22%

42%

36%

Table 2. Categories of Student Justifications to the Number Trick item.

35

Grade

Empirical

General

Both

Other

6

37%

32%

20%

11%

7

40%

39%

3%

18%

8

36%

49%

7%

8%

Table 3. Categories of Student Responses to the Consecutive Numbers Sum item.

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Principle

Amount

Diversity

Typicality

Description: Arguments

Example for conclusions concerning

with…

“all birds” hve X

More examples are stronger

(robins, sparrows, & cardinals have X) >

than fewer

(robins have X)

Dissimilar examples are

(robins, hawks, & penguins have X) >

stronger than similar

(robins, sparrows, & cardinals have X)

Typical examples are

(robins have X) > (penguins have X)

stronger than atypical Contrast

Negative examples are

(robins have X, cats lack X) > (robins

stronger than those without

have X)

Table 4. Some Examples of Criteria for Evidential Strength.

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Category

Meaning

Parity

Even vs. odd

Multiples

3, 9, 36, 81, 90 are all multiples of 3

Factors

3, 9, 15, 30, and 90 all go into 90 evenly

Prime

2, 5, 11, 17 go together because they’re prime

Composite

25, 30, and 60 are all composite numbers

Squared

4, 25, and 81 are perfect squares

Sequence

3, 9, and 15 go together because they go up by 6

Value of digit

1, 11, 21, and 81 all have a “1” at the same spot

Intervals

11, 14, 15, 17 are all between 10 and 20

Number of digits

1, 2, 3, 5, and 9 are all one-digit numbers

Contains a digit

5, 15, and 51 all contain a 5 so they belong together

Arithmetic

2, 3, and 5 are a group because 2 + 3 = 5

Relational

100, 25, 36, & 9 because 100/25 = 4 and 36/9 = 4

Table 5: Number categories and their meanings.

38

Principle

Middle

Undergrad

STEM

School Multiples

86% (1)

93% (2)

93% (1)

Parity

79% (2)

93% (1)

87% (2)

Prime

50% (3)

71% (3)

80% (3)

Vale of Digit

50% (3)

29% (9)

47% (6)

# of Digits

43% (5)

36% (6)

20% (8)

Intervals

36% (6)

50% (4)

53% (4)

Contains Digit

29% (7)

43% (5)

13%

Squared

21% (8)

36% (6)

53% (11) (4)

Sequence

21% (8)

14% (11)

20% (8)

Factors

21% (8)

0% (12)

0% (13)

Arithmetic

14% (11)

36% (6)

40% (7)

Composite

14% (11)

21% (10)

20% (8)

Relational

7% (13)

0% (12)

13% (8)

Table 6: Number categories sorted according to salience.

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Category

What it Means

# of Sides

Grouping shapes with the same number of sides

Angles

Grouping shapes on angle size (all have an obtuse)

Equal sides

Two or more equal sides

Regular

Grouping regular shapes together

Shape

Resemblance to a shape (e.g., “arrows”, “sharp”)

Familiar

Common shapes you see in school

Size

Grouping large or small shapes together

Orientation

Grouping according to orientation on paper

Compose

A group of shapes that can be made from others

Tessellate

Grouping shapes that would tessellate

Similar

Grouping similar shapes together

Symmetry

Grouping symmetric shapes together

Convex/Concave

Grouping according to concavity

Table 7: Shape categories and their meanings.

40

Principle

Middle

Undergrad

STEM

School Size

78% (2)

85% (3)

100% (3)

Shape

64% (2)

78% (4)

93% (2)

Angles

64% (2)

85% (2)

80% (4)

Similar

50% (5)

43% (8)

40% (11)

Concavity

43% (5)

14% (8)

40% (7)

Equal Sides

36% (5)

21% (6)

26% (11)

Composition

36% (10)

43% (9)

27% (8)

Familiar

35% (5)

43% (5)

34% (8)

Symmetry

21% (8)

21% (6)

47% (5)

Orientation

14% (8)

7% (9)

27% (8)

Regular

14% (11)

0% (11)

53% (6)

Tessellate

7% (11)

0% (11)

0% (12)

# of Sides

100% (1)

100% (1)

100% (1)

Table 8: Shape categories sorted according to salience.

WELL PLAYED 3.0

The pleasures of WoW as a player are not at odds

CRYSTLE MARTIN, SARAH CHU, DEE JOHNSON, AMANDA OCHSNER, CARO WILLIAMS, & CONSTANCE STEINKUEHLER

252

DING! WORLD OF WARCRAFT WELL PLAYED, WELL RESEARCHED

CRYSTLE MARTIN, SARAH CHU, DEE JOHNSON, AMANDA OCHSNER, CARO WILLIAMS, & CONSTANCE STEINKUEHLER

World of Warcraft (WoW) is a massively multiplayer online (MMO) role-playing game that takes place in the fantasy realm of Azeroth and boasts over ten million players. WoW was vast in scope when originally released, and has since added on more territories and character customization choices. Originally consisting of two continents, Kalimdor and the Eastern Kingdoms, two expansion packs added the realm of Outland and the continent of Northrend to the map, and, have also expanded the content of the game with the addition of new races, lands, quests, etc., and raising the level cap (highest attainable level). When designing a character, the player is offered a variety of choices, such as selecting a character’s faction (Horde or Alliance), race (10 playable races which include Night Elf, Troll, and Undead), and class (9 different classes which include Made, Paladin, and Druid). The player is able to further specialize their character by selecting two professions, as well as spending points to develop talent builds. In addition, during gameplay, decisions must be made about what armor to wear, what weapons to wield, and in what order to cast spells. The choices are vast and are able to be molded to fit a variety of playing styles, especially considering the social interactions that raids and guilds support, the wide variety of gear choices that are made and re-made regularly, the complex role-play opportunities, and the various patterns that different players develop (and swear by). As such, WoW offers a tremendous number of avenues for the player’s enjoyment and just as many avenues of study for researchers interested in informal learning, especially in online collaborative spaces. We all play WoW but our backgrounds vary beyond that, ranging from a professor who researches informal learning and MMOs, four graduate students with a variety of research interests (including math, visual studies, and literacy), and a high school senior who is an avid gamer. As players and researchers of WoW, we could ramble endlessly about the game but we have instead decided to talk about nine of our most loved things. In this essay, we delve into how WoW has redefined gaming, narrative and raid centric playing styles, as well as the multi-level social interactions in WoW, as an exploration of what we love about playing the game. Then, we explore character aesthetics and player-produced visual models, the use of math and information literacy, and finally a player’s experience with time in WoW, for our more research-oriented pursuits. Though we cannot go into much depth for each section in this short essay, we do cover a breadth of topics to offer the reader an overview of the wide variety of perspectives from which to think about WoW as players and as researchers.

253

WELL PLAYED 3.0

WoW

JMEXXX10.1177/1052562916633867Journal of Management EducationWilliams-Pierce

Rejoinder

On Reading and Digital Media: Rejoinder to “Digital Technology and Student Cognitive Development: The Neuroscience of the University Classroom”

Journal of Management Education 2016, Vol. 40(4) 398–404 © The Author(s) 2016 Reprints and permissions: sagepub.com/journalsPermissions.nav DOI: 10.1177/1052562916633867 jme.sagepub.com

Caroline Williams-Pierce1 The article “Digital Technology and Student Cognitive Development: The Neuroscience of the University Classroom,” by Cavanaugh, Giapponi, and Golden (2015), succeeded tremendously in its goal to provoke conversation, and I enjoyed the opportunity to be one of the first respondents! My expertise lies particularly in how digital media can be used productively in formal and informal contexts, and how designing with different representations and interactions can increase learning and interest. I have also conducted research on the power of interest-driven play in commercial games—that is, games not designed with academic learning goals in mind—and the communities that emerge online to discuss and create around those game worlds. Consequently, the following commentary serves as an introduction to multiple scholarly fields about the value of digital media for providing contexts for—and provoking—learning.

On Reading Cavanaugh et al. (2015) set digital reading and book reading as opposed activities, with the former aligned with quick sips of information in a constant 1University

at Albany, State University of New York, Albany, NY, USA

Corresponding Author: Caroline Williams-Pierce, University at Albany, State University of New York, ED 127A, 1400 Washington Ave, Albany, NY 12222, USA. Email: [email protected]

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waterfall of information overload, and the latter aligned with the familiar and comforting image of sitting cuddled in an oversized chair, surrounded by the smells of a crackling fire and hard-bound books. While the latter image is deeply welcoming to me personally, I cannot help but think: What then are the cognitive implications of sitting in that oversized chair with a Kindle in my hands, reading Fahrenheit 451? I am using a technological object to read a digital book about physical books being destroyed such that their only trace is left within the heads of individuals who have memorized their content. This illustration pushes back against the distinction between digital media and reading by Cavanaugh et al., but perhaps this Kindle-as-digital-media fruit is too low-hanging to make my point. Another digital media project, Bedtime Math, takes the form of free mobile apps with word problems that are designed for parents to read—mathematize?—with their young children at night, and were shown to have a significant influence on their children’s mathematical achievement, with the most significant improvements with parents who suffered from math anxiety (Berkowitz et al., 2015). David Landy (2015), a cognitive scientist at Indiana University, cheerfully warns other parents to avoid this product because his “n = 2 field study” (para. 2) has shown that children enjoy the math activities too much, and his do not always fall asleep on time when Bedtime Math is in the picture. But perhaps this is still too close to what physical books can provide, so I will instead veer into a high school English classroom led by Paul Darvasi. Darvasi, who has designed and implemented augmented reality games in his classroom for books such as One Flew Over the Cuckoo’s Nest (Darvasi, 2016, in press-a), discovered a short Indie videogame, Gone Home (Darvasi, 2014, in press-b), that was universally loved by game critics. Curiosity piqued, he played the game, which involves exploring an abandoned house and seeking to find out what had happened to the family that lived there. Darvasi realized that the nonlinear form of player-driven exploration meant that “this type of dynamic would not work as well in a novel or a film. This video game had staked out narrative territory where its traditional forerunners could not follow” (Darvasi, 2014, para. 7), and he promptly decided to use reframe the game as a text for use in his senior English class. He had his students play the game and analyze it using the traditional strategies of annotation and close reading, while expanding the normal analytic evidence of cited quotes with screenshots and video clips as a way to more deeply express the experience of playing Gone Home, gently merging traditional analytic methods with ones more suited to this new form. In other words, Darvasi used this videogame as a substitute for a more traditional text, supporting widely accepted methods of deep reading and analysis, while expanding the

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methods to align more clearly with the new digital media we are surrounded by. He concludes that English class can become a sort of theater where everyone pretends the texts are being read. . . . Our duty as educators is to design our courses to prepare students to think critically and succeed in their current communication context, as that is the environment where they must survive and, hopefully, prosper. (Darvasi, in press-b)

Naturally, Darvasi is not saying that we should get rid of books, but rather that as new types of texts come into existence, we can—and should—support students in using powerful reading strategies for both books and digital media. Consequently, I propose that instead of considering a dichotomy between reading physical books and reading digital media, as encouraged by Cavanaugh et al. (2015), we consider a scale of sorts between reading as pertaining to linear written endeavors and reading as situated experiences within designed spaces. Then, reading Fahrenheit 451 on a Kindle is fully placed within the former, while reading Bedtime Math is closer to the middle of the scale, as it is a written product that does not require linear engagement, since parents and their children can flip through the word problems in any order. Gone Home, however, is designed to be a situated (digital) experience that readers can apply respected, new, and adapted analytic methods to understand and deconstruct, enriching their ability to apply deep reading methods to both the old and new experiences that we are surrounded by.

Beyond Reading Numerous online communities actively engage in reading, reviewing, creating, and analyzing texts from this proposed spectrum. For example, the Hunger Games and Harry Potter books have powerful online presences, where readers voluntarily engage in a wide variety of practices that we value in the classroom, such as reading, writing, reviewing, arguing, discussing, and so on (Curwood, 2012; Jenkins, 2004). On the other end of the spectrum, the same practices are engaged in by game players who were inspired by game worlds, characters, or experiences. For example, the epic fantasy game World of Warcraft involves considerable reading within the game by even the most casual player, both in traditional text-based form and in terms of the situated reading of complex symbols and indicators within the virtual world (what Gee [2003] would call the game’s semiotic domain). The incredible popularity of the game, combined with its complexity and difficulty, has led to considerable production of text- and video-based information resources by

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players, and Steinkuehler has termed the game and its attendant player-created resources a constellation of literacies (Steinkuehler, 2007, 2008). How players navigate this constellation relates to various factors, such their identity (Martin, 2012) or level of expertise (Martin et al., 2012),1 but engagement in literacy practices are an integral part of the game experience and surrounding community. One small part of this constellation is termed theorycrafting, which involves collecting data during gameplay in order to further understand the game mechanics, and it “is a rich, compelling intellectual activity involving hypothesis generation, testing, numerical analysis, logical argumentation, rhetoric and writing” (Choontanom & Nardi, 2012, p. 187). In fact, theorycrafting has a lot in common with scientific fields in our physical worlds: Unknown laws and patterns undergird our natural or virtual world experiences, and we can further understand those laws and patterns by interrogating collected data. Unsurprisingly, then, when Steinkuehler and Duncan (2008) examined randomly selected posts on World of Warcraft forums, they determined that the vast majority of posts were engaging in scientific reasoning about the game world. They concluded that “forms of inquiry within play contexts such as these are authentic although synthetic: even though the worlds themselves are fantasy, the knowledge building communities around them are quite real” (p. 541). One particularly mathematically rich post from that data corpus was written by an author who combined a theorycrafting argument with a narrative structure to emphasize his/her argument, transforming mathematical algorithms into characters within a story (Steinkuehler & Williams, 2009; Williams, 2011). Such forum posters and theorycrafters provide examples of voluntary reading, writing, creating, and analyzing that we most desire from our students—provoked specifically by digital media experiences and presented in online spaces that we may have no awareness of. In short, our students may be walking into their first class on their first day with some powerful and relevant, albeit unrecognized, skills, thanks to the very technology that we often denigrate! Online reading-related activities, some very few of which I highlighted above, often take place in what James Paul Gee calls affinity groups or spaces (Gee, 2003, 2005). These spaces are where people with common endeavors work together to discuss, create, argue, invent, share, review, and so on. Gee and Hayes (2012) identified some important features of affinity spaces, but particularly relevant here is one that can occur specifically thanks to the anonymity available online—participants are not segregated by age, race, gender, expertise, and so on. Regardless of these characteristics of their human bodies or past history, they can create and critique content, develop further general and specialized knowledge to contribute to the group, gain status

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through multiple routes, earn the role of a respected leader in various ways, and so on—all important experiences that can lend themselves to online or offline interactions. I highlight this point, and these spaces, particularly to emphasize to the reader one of the distinct differences between traditional education contexts and digital media contexts; in the latter spaces, students may already be deeply engaged in activities that we are planning to teach them. In fact, they may already be considered respected leaders, despite their “student” label in our classroom.

Conclusion There are numerous points relating to learning, in general, and management education, in particular, in this commentary. However, I want to push back specifically against Cavanaugh et al.’s (2015) claim that today’s students will struggle with effective experiential learning exercises. I have presented considerable evidence that digital media can provide situated experiences that provoke reading, creating, evaluating, and so on through a variety of forms, which means that we cannot blame the students alone if they struggle with our educational versions of experiential learning. Rather, we need to provide sufficient autonomy such that students can engage in deep interest-driven learning, bring their own knowledge to the table, and contribute unique content and vision to the class. This is what our students do outside of class on a regular basis, for no reason rather than to create and contribute to their affinity spaces, so we must learn to see—and respect—how their new approaches to learning are enacted and displayed. Give the younger generation some space, they may surprise us! Declaration of Conflicting Interests The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author received no financial support for the research, authorship, and/or publication of this article.

Note 1.

As an aside about multitasking, Martin et al. (2012) found in their case study of a World of Warcraft player that his ability to multitask “across multiple timescales and in multiple spaces, real and virtual” (p. 242) is what enabled him to engage in advanced play at an expert level. In other words, what would traditionally be seen as drawback to efficiency was instead a necessary component of his successful engagement in a complex activity.

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Williams-Pierce References

Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement in school. Science, 350, 196-198. Cavanaugh, J. M., Giapponi, C. C., & Golden, T. D. (2015). Digital technology and student cognitive development: The neuroscience of the university classroom. Journal of Management Education, 40(4), 374-397. doi:10.1177/1052562915614051 Choontanom, T., & Nardi, B. (2012). Theorycrafting: The art and science of using numbers to interpret the world. In C. Steinkuehler, K. Squire, & S. Barab (Eds.), Games, learning, and society: Learning and meaning in the digital age (pp. 185-209). Cambridge, England: Cambridge University Press. Curwood, J. S. (2012). Cultural shifts, multimodal representations, and assessment practices: A case study. E-Learning and Digital Media, 9, 232-244. Darvasi, P. (2014, March 5). Prologue: A video game’s epic-ish journey to a high school English class. Retrieved from http://www.ludiclearning.org/2014/03/05/ gone-home-in-education/ Darvasi, P. (2016, January 17). The Ward game: How McMurphy, McLuhan and MacGyver might help free us from McEducation. Retrieved from http://www. ludiclearning.org/2016/01/17/the-ward-game-how-mcmurphy-mcluhan-andmacgyver-might-help-free-us-from-mceducation/ Darvasi, P. (in press-a). The Ward game: How McMurphy, McLuhan and MacGyver might free us from McEducation. In C. Williams (Ed.), Teacher pioneers: Visions from the edge of the map. Pittsburgh, PA: ETC Press. Darvasi, P. (in press-b). Gone home and the apocalypse of high school English. In C. Williams (Ed.), Teacher pioneers: Visions from the edge of the map. Pittsburgh, PA: ETC Press. Gee, J. P. (2003). What video games have to teach us about learning and literacy. Computers in Entertainment (CIE), 1(1), 20-20. Gee, J. P. (2005). Learning by design: Good video games as learning machines. E-Learning and Digital Media, 2(1), 5-16. http://doi.org/10.2304/elea.2005.2.1.5 Gee, J. P., & Hayes, E. (2012). Nurturing affinity spaces and game-based learning. In C. Steinkuehler & K. Squire (Eds.), Games, learning, and society: Learning and meaning in the digital age (pp. 123-153). Cambridge, England: Cambridge University Press. Jenkins, H. (2004, February 6). Why heather can write. Retrieved from http://www. technologyreview.com/news/402471/why-heather-can-write/?p=1 Landy, D. (2015). What is bedtime math really good for. Retrieved from https://davidlandy. net/the-dangerous-lie-of-bedtime-math/ Martin, C. (2012). Video games, identity, and the constellation of information. Bulletin of Science, Technology & Society, 32, 384-392. doi:10.1177/0270467612463797 Martin, C., Williams, C., Ochsner, A., Harris, S., King, E., Anton, G., . . .Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh, & J. Davies (Eds.), Virtual

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literacies: Interactive spaces for children and young people (pp. 226-243). London, England: Routledge. Steinkuehler, C. (2007). Massively multiplayer online gaming as a constellation of literacy practices. eLearning, 4, 297-318. Steinkuehler, C. (2008). Cognition and literacy in massively multiplayer online games. In J. Coiro, M. Knobel, C. Lankshear, & D. Leu (Eds.), Handbook of research on new literacies (pp. 611-634). Hillsdale, NJ: Erlbaum. Steinkuehler, C., & Duncan, S. (2008). Scientific habits of mind in virtual worlds. Journal of Science Education and Technology, 17, 530-543. doi:10.1007/s10956008-9120-8 Steinkuehler, C., & Williams, C. (2009). Math as narrative in WoW forum discussions. International Journal of Learning and Media, 1(3). Retrieved from http:// www.mitpressjournals.org/doi/abs/10.1162/ijlm_a_00028 Stone, J. C. (2007). Popular websites in adolescents’ out-of-school lives: Critical lessons on literacy. In C. Lankshear & M. Knobel (Eds.), A new literacies sampler (pp. 49-65). New York, NY: Peter Lang. Williams, C. (2011, April). “Shadow has crap scaling—FACT”: The intertwining of mathematics and narrative on a game forum. Paper presented at the 2011 American Educational Research Association Annual Meeting and Exhibition, New Orleans, LA.

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Apples and Coconuts: Young Children ‘Kinect-ing’ with Mathematics and Sesame Street Meagan Rothschild and Caroline C. Williams

Abstract The ability to count objects is a crucial skill for young children. We report on an experimental study that utilized a Kinect Sesame Street TV intervention designed to support two types of counting activities. We conducted quantitative as well as open-coding based analyses, on video data with 3- and 4-year-olds. The complexity of interactive digital media contexts for mathematical learning is unpacked with the assistance of literature from the fields of mathematics education and cognitive science. We conclude by making recommendations for interactive educational design in general. Keywords Common Core Standards for mathematics · Kindergarten · Microsoft · Kinect · Sesame Street · Embodied cognition · Early education · Number knowledge · Interactive television · Informal learning · Xbox

Introduction A foundational skill that young children need to develop for mathematics learning is counting. The United States Common Core State Standards for Mathematics Kindergarten standards state that students should learn the number names and count sequence, and be able to count objects (National Governors Association Center for Best Practices 2010). The National Council of Teachers of Mathematics (2000) include in the pre-kindergarten to second grade-band the requirement that all students learn to count with understanding, be able to determine the size of sets of objects, and use numbers to count quantities. Being able to count and connect the counting specifically to specific objects is a crucial part of learning how to mathematize M. Rothschild ( ) WIDA, Wisconsin Center for Education and Research, University of Wisconsin-Madison, Madison, WI, USA e-mail: [email protected] C. C. Williams Department of Curriculum and Instruction, Mathematics Education, University of WisconsinMadison, Madison, WI, USA e-mail: [email protected] © Springer Science+Business Media Dordrecht 2015 T. Lowrie, R. Jorgensen (Zevenbergen) (eds.), Digital Games and Mathematics Learning, Mathematics Education in the Digital Era 4, DOI 10.1007/978-94-017-9517-3_8

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the world, as well as to continue in further mathematics learning trajectories. This project focuses specifically on supporting 3- and 4-year-old children in counting by using a Sesame Street episode made interactive through the medium of the Microsoft Kinect 2012, a motion capture device for the Xbox console. In the following section, we briefly review literature on video games and learning, and on embodied cognition and mathematics. We then further describe the relationship between the Kinect and Sesame Street, before transitioning to describe our study design, implementation, and analysis. Considerably varied research indicates that video games can be powerful vessels for learning (Barab et al. 2010; Fisch et al. 2011; Gee 2003; Squire 2011; Steinkuehler and Duncan 2008). By leveraging some elements of video game design and transforming the traditionally televised one-way information flow into an interactive learning experience, the Sesame Street Kinect series has the potential to increase the engagement and learning of its participants. In particular, this multimodal design aligns with embodied cognition research that suggests that cognition and action are intertwined (Shapiro 2011). Theories of embodied cognition contend that thinking and learning are not based on amodal symbol systems, but rather are inextricably woven into action and perception systems (Barsalou 1999, 2008). Researchers examining the relationship of action and gesture to mathematics learning have found promising results (Alibali and Goldin-Meadow 1993; Glenberg et al. 2007a; Nathan et al. 1992), including interventions in which actions and gestures are designed to be related to successful solving of specific conjectures (Dogan et al. 2013; Walkington et al. 2012, 2013). In summary, physical action can influence mathematical cognition and, consequently, using the Kinect in conjunction with episodes designed to support mathematical learning may leverage action as a way to support cognition. The questions about the nature of learning with Kinect Sesame Street TV led to a research project conducted at Microsoft Studios in which the first author began to investigate the nature of participant experiences in two-way episodes and traditional television episode viewing. Two interconnected questions formed the focus of the project: How are mathematical concepts learned in each context, and how may interactivity relate to concept learning? The episode follows Sesame Street’s emphasis on literacy and STEM; it includes a word of the day, a number of the day, and—to connect to the interactive elements—a move of the day. The internal white paper produced as a result of the initial analysis of the study (Rothschild, internal Microsoft white paper 2012) presents preliminary results showing all students that watched the episode (both experimental and comparison groups) showed statistically significant learning gains when all the tests were collapsed. This included assessment items related to letter recognition, relational concepts, and number knowledge. Initial analysis of the assessment total did not, however, demonstrate a statistically significant difference by condition or gender. A review of observational notes and engagement data indicates that there may be more nuanced issues to explore within the data set in order to more deeply understand the experiences of participants engaging with the episode and related assessment. This paper uses the data collected in the earlier study conducted at Microsoft, and goes deeper into a quantitative and

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qualitative analysis of the questions specifically related to number knowledge, and presents the investigation of the number knowledge component of the episode studied within the frames of current math education and cognition research.

Sesame Street and Kinect Television Research and position papers by leading early education organizations recognize that varied media use is becoming ubiquitous in early childhood, and when used within developmentally appropriate frameworks, can effectively promote learning and development for young children (National Association for the Education of Young Children [NAEYC] and Fred Rogers Center for Early Learning and Children’s Media 2012). Sesame Street is a proven television format with an extended media legacy of success. The format has been shown to produce learning gains in younger viewers across studies over the last 43 years, including a longitudinal study that supports the findings of learning gains (Ball and Bogatz 1970; Bogatz and Ball 1971; Fisch and Truglio 2001). Additional studies suggest that as children form parasocial relationships with the characters in the Sesame Street narrative world, they are more apt to learn targeted video content (Lauricella et al. 2011). The Sesame Street Workshop leverages multiple media to extend educational content and play-based connections to the Sesame Street narrative world, including web-based games and resources, character toys, and video game console products. In 2010, Microsoft Studios released the Kinect, an Xbox peripheral device for motion-sensing input. Since the release, Microsoft has worked on ways to engage audiences beyond their traditional core gamer, producing titles like Dance Central, Kinect Sports, Disneyland Adventures, and Nike+ Kinect Training in order to engage kids and families. Using the Kinect, participants are not bound in their play experience by holding a controller, as the Kinect peripheral device uses gesture, facial, and voice recognition that turns the player’s body and physical participation into the controlling agent. Among the products that Microsoft has released to push the boundaries of traditional gaming and television viewing experiences is Kinect TV (2012), featuring initial product lines that include a uniquely developed set of Sesame Street and National Geographic interactive television episodes. For the developers of Kinect Sesame Street TV, the goal was to extend an already successful media property and viewing format. The designers wanted to design their products based on firm research, in order to make sure that the added Kinect interactivity would not adversely disrupt the potential for learning gains found in the linear television format (Rothschild, internal Microsoft white paper 2012). This included understanding situated learning theory and the role of learning in the context of relevant activity (Gee 2003; Barsalou et al. 2003), as well as scrutinizing the potential learning through a lens of embodied cognition, connecting concepts to a learner’s own perceptions—which includes relationships between the content and themselves/their own bodies (Glenberg et al. 2007a, b).

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Informal Mathematical Learning Cross et al. (2009) note that while young children are capable of becoming competent in mathematics, a lack of appropriate formal instruction or informal opportunities in the home or community often prevents the learning of foundational concepts. The authors go on to recommend two areas of focus, Number as well as Geometry and Measurement, and further suggest that one way to remedy suboptimal learning situations is to provide various informal opportunities for learning mathematics outside of school. Baroody et al. (2005) further suggest that informal mathematical learning is a key part of successful trajectories in learning mathematics, specifically for developing number sense in young children. Consequently, the Kinect Sesame Street TV episode format is poised to fit this gap by providing content-driven informal opportunities to engage in mathematical learning.

The Number Core The mathematics included in the study design and assessment involve what Cross et al. (2009) call the ‘number core’, in this case by modeling and asking participants to coordinate cardinality, the number word list, and one-to-one counting correspondences. They define each as following: cardinality involves perceptual or conceptual subitizing; the number word list involves knowing the order of number words (i.e., 1, 2, 3, …); and one-to-one correspondences requires matching the two such that, for example, each object being counted requires one and only one number word in the appropriate list order. Cross et al. (2009) note that practicing all three of these activities, as well as coordinating between them, will improve the ability of young children to be successful—for example, 2- and 3-year-old children are considerably more likely to be able to count five objects successfully if they have had repeated practice at this task. The methodology, reported in the next section, was designed specifically to support repeated practice, and the open-coding analysis, reported later in this paper, found subitizing, the number word list, and one-to-one correspondences to be an integral part of understanding the results.

Methods This chapter analyzes data that was collected in an earlier study that took place at Microsoft, led by the first author. Forty-two 3- and 4-year-olds participated in the study. The group was composed of a mix of boys and girls from Seattle and its surrounding areas. The requirements for participant families were that they needed to have regular access to an Xbox 360 and Kinect in their home, that they had not previously viewed the episodes, and that the child was proficient in English. Data was

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collected at the Microsoft User Research Labs, and consisted of video footage, observation notes, pre-tests and post-tests, and parent surveys (including demographic data). Participants were divided into two groups of 21 by a process of stratified random sampling, accounting for gender and known family annual income. One group of participants was designated as the KINECT group, in which Kinect Sesame Street TV experiences took place as designed with all interactions on. The other group was the TRADITIONAL group, in which interactions were still elicited but the episode did not require interactions to progress. In other words, the participant experienced the same content but edited to a non-interactive, linear format. Participants came in to the research lab with a parent or caregiver, and participated in a pre-test, watched the episode, then completed a mid-test. The child and guardian left the lab with a copy of the episode in the format that they viewed (KINECT or TRADITIONAL) and then played the same episode at home over the next couple weeks. Parents logged their child’s play and made observations. The child and a parent or guardian returned to the lab one more time to view the episode and then participate in a post-test. For the purposes of this paper, analysis is specifically targeting the questions regarding the number five (the number of the day for the episode), and comparing pre- and post-test scores, with mid-test scores used to interpret the open-coding analysis. The nature of these tests will be discussed in detail in a later section.

Number Knowledge in the Episode In the episode, the scene opens with Cookie Monster dropping a banana peel on the ground, which a bustling Grover then slips on, dropping his delivery of five coconuts. Grover then asks the audience member to please help him collect his five coconuts by throwing them into his box. For each throw, an image of the box is displayed with a visual of how many coconuts are now in the box. The number of coconuts in the box is displayed in the lower right corner of the box (see Fig. 1). Grover states, “Now I have ( number) coconuts in the box.” At the end, the box with five coconuts and the number five in the bottom right corner is displayed as Grover cheers, “Hooray! Now I have FIVE coconuts!” In the KINECT group, when the participants threw, the Kinect motion sensor would respond to their movement in the system, and the coconut would fly into the screen and into Grover’s box, sometimes in silly and surprising ways (see Fig. 2). If the child did not throw the coconut, Cookie Monster would come into the scene having ‘found’ one, and drop it into Grover’s box. Grover would then ask the audience member to try throwing the next one. The TRADITIONAL group would get the verbal prompts from Grover to throw the coconut, however, their activity did not affect the way the show progressed, and for each coconut, the show would progress as if the child had made a successful throw.

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Fig. 1 Throwing coconuts into Grover’s box

The Performance Assessment Games, interactive media, and playful learning are taking a prominent role in educational dialogue. Consequently, the issue of assessing these media must be raised. For early learners, design foundations should meet Developmentally Appropriate Practice (DAP), articulated by the National Association for the Education of Young Children (NAEYC). The 2009 policy statement describes the ways that knowledge Fig. 2 Participant throwing coconut

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of child development and learning and methods that are adaptive and responsive to individual children should be linked to social and cultural contexts of a child’s life. DAP contends that the domains of development and learning are interrelated. Understanding child development, trajectories of learning, and the contexts of media and play for learning became paramount in the development of the assessment activities for this research. The assessment activities were designed to feel playful and both match the spirit of the episode and align with the sorts of performance elicited in the show. Because the study that took place at Microsoft was a pilot study for Kinect Two-Way TV, the net cast in the research was wide and would encompass a broad variety of participants with a multitude of media and learning contexts, from home languages other than English to specific behavioral needs, to experiences with media and the narrative world of Sesame Street. This meant that the assessment tasks needed to be designed to allow a variety of levels of conceptual knowledge to be demonstrated. The protocol needed to remain reflexive to the behaviors and abilities of an individual child, particularly given the long time period of each study session (a 40-min episode and time for assessment activities). In addition, the move from a visual TV format of participation to a paper and analog manipulative format of assessment represented a shift in modality. Thus the characters and playful nature of episode activity were important for connecting the episode’s learning stimulus to the assessment performance activities. The activities related to letter recognition, number knowledge, and relational concepts were designed to include participant feedback and decision-making in the hopes of increasing participant agency in the activities without detracting from the ability to elicit specific modes of content knowledge demonstration. Because this new analysis focuses on the nuances of the mathematical activities, this chapter will describe number knowledge assessment items in details. The researcher began by asking the participant to pretend with her, imagining that they had been walking through an apple farm together (situating the activity). The researcher said, “Oh look! We found some apples on the ground!” and displayed a page with five apples on it (see Fig. 3). The researcher then asked, “Can you count how many apples we found?” and, if needed, prompted with “Point to and count each apple that you see”. If the child counted to five, it was initially coded as correct; anything other than counting exactly to five was initially coded as incorrect. Immediately following the Enumeration activity, the researcher segued into the Number Application activity by telling the child that Cookie Monster loves apples, and that today they were going to help him cook. The participant helped decide what should be cooked (e.g., apple cookies, apple cake, applesauce), further situating the activity with recognizable characters from the episode and providing the participant with an opportunity to determine elements of the assessment narrative. The researcher brought out a bowl, seven foam core apples, and an image of Cookie Monster, placing them in front of the child (see Fig. 4). The researcher then told the participant that Cookie Monster needed exactly five apples to make his recipe, and asked, “Can you put five apples in Cookie Monster’s

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Fig. 3 Enumeration activity

bowl?” If the participant placed five apples in the bowl, it was initially coded as correct. Anything other than five apples in the bowl was initially coded as incorrect. The goal of these two activities was to provide the child with avenues to convey a range of number knowledge abilities. Enumeration was most directly modeled in the episode, and matched a developmentally appropriate benchmark for 3- and 4-year-olds. The second activity, counting apples for Cookie Monster’s recipe, deviated from the episode. In the episode, there was no way in the interactive system to miscount—the activity was physically tied to throwing five and only five coconuts. The designed interactive system limited the participant to throwing or not throwing, and did not support actively applying number knowledge to a situation and receiving corresponding feedback. However, understanding the depth of the participants’ understanding of the number five required an assessment inquiry of more than enumeration ability.

Fig. 4 Number Application activity

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Results In this section, we share the results we anticipated, and then share the actual empiric results. We then explore the data using qualitative open-coding in order to further understand the quantitative results. We conclude by making recommendations to the researchers and designers interested in interactive digital learning experiences, in order to give insight into designs that more deeply support the desired types of learning.

Expected Results As a consequence of playing the Sesame Street episode, and due to the general benefit of this intervention discovered in Rothschild’s internal white paper (internal Microsoft communication 2012), we expected the children to improve in their ability to count to five, which requires attending to cardinality, the number word list, and one-to-one correspondences. Based on the existing success of the Sesame Street platform and the earlier preliminary results of the overall assessment (Rothschild, internal Microsoft communication 2012), we theorized that both groups would show learning gains, with the possibility of the KINECT group showing greater gain due to increased activity and engagement.

Empiric Results The actual results did not unilaterally fulfill our expectations. Regarding our hypothesis that the KINECT group would perform better than the TRADITIONAL group, no significant difference was found between the two conditions according to Fisher’s Exact Test for the Enumeration ( p > 0.05) or the Number Application ( p > 0.05) tasks. Furthermore, no significant difference was found when the conditions were collapsed ( p > 0.05). However, the results of our qualitative analyses align quite well with literature on child development and mathematics learning, and suggest that the lack of significance is due to considerably different reasons for each test. (Because of the lack of statistical differences between the conditions for this intervention, we collapse the conditions for the remainder of this chapter). For the Enumeration test, 38 children contributed complete data to our analysis. Of those 38 children, 28 were successful in enumerating five apples during the pre-test, indicating that counting to five was a skill that these participants were already quite competent at. At post-test, 32 participants were successful (which included all 28 who replied accurately during the pre-test). Given that nearly 75 % of participants came into the study with the target skill, it is hardly unexpected that a ceiling effect occurred.

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The Number Application test, on the other hand, suffered from no ceiling effect but similarly demonstrated few gains. Sixteen of the 38 participants were successful during the pre-test, and only 20 were successful during the post-test (again, all the participants who performed correctly during the pre-test continued to be correct in their post-test). Intriguingly, as an exact but nonverbal task, this performance assessment appears to be quite achievable, even for participants of this age (e.g., Baroody et al. 2006), so the study did not accidentally include a task with achievable content but overly challenging performance demands (as Gelman and Meck 1983 so eloquently warn us about). Consequently, we more deeply examined the data qualitatively to determine exactly why there were no significant results.

Open-Coding Analysis We analyzed the video data of each participant by coding each action undertaken. Since each participant was assessed three times (before encountering the intervention, after encountering the intervention once, and after encountering the intervention multiple times), we analyzed each of the two tasks during each of the three performance assessments per participant. If three or more of the six data points per participant were absent, the participant was dropped from this analysis, leaving a total of 35 participants from the original 43. Our coding schemes were emergent, achieved through open-coding the participants’ actions and merging similar codes into several core codes, presented in Tables 1 and 2. The schemes are structurally quite different, as the Enumeration coding scheme is built of categories, while the Number Application coding scheme has multiple codes which are applied in a concatenated fashion. The codes for the Enumeration task (Table 1) are applied singly, except for the ( circuitous) code, which is appended to the primary code when appropriate. The codes for the application task (Table 2) are broken down further than the codes for the enumerating task (Table 1), due to the increased complexity of the concept under examination. The physical process of moving the apples to the bowl is labeled as conducted in a sequential or grouping fashion, and appended to this code is whether the participant counted aloud (verbal or nonverbal) and whether the participant is correct or incorrect. The NOTHING code, used in both the Enumeration and Number Application tasks, is used during analysis as though data were absent. Following the completion of coding, an additional round of open-coding was conducted that built upon the patterns observed by each participant. For example, the majority of participants were already adept at the Enumeration task at pre-test (25 of the 35 in this analysis), and those who demonstrated their proficiency at all three assessments were coding more broadly as “All Correct—No Change.” A similar code was also applied to the smaller number of participants who were always able to perform correctly during the Number Application task (13 out of the 35)— and in both cases, the method with which each participant showed their knowledge was irrelevant to our needs. They came to the intervention knowing how to count,

Enumeration Code Definition The participant sequentially touches the apples (or points at them) while verbally following the appropriate number word list, and concludes with the accurate count of ‘five’ The participant verbally follows the appropriate number word list, but does not physically touch nor gesture towards the apples The participant sequentially touches the apples while verbally following the appropriate number word list, but over- or under-counts to x The participant sequentially touches the apples while verbally following the appropriate number word list, but divides the set into two subsets of x and y while counting. The most frequent occurrence of this code included the subsets of 2 and 3, likely due to the visual stimuli layout (see Fig. 3) Participant does not verbally enumerate, but concludes by stating the number x The participant follows an incorrect number word list, and ends the count with the number x, after speaking y number words in some order The participant follows the correct number word list, but ends the count with the number word of x, putting y objects under a single number word This is a subcode, appended to the previous codes, merely to indicate that the participant does not follow a path that makes it easy to remember which numbers have already been counted. For example, most correct participants follow a ‘loop’ while they’re counting; but participants who received this subcode may have followed, for example, a ‘W’ shape on the stimuli (see Fig. 3) Participant says and does nothing, or refuses to cooperate

Enumeration Code

One-to-one counting correspondence

One-to-one counting correspondence—no movement

One-to-one counting correspondence over-counts to x

One-to-one counting correspondence Subsets ( x, y)

Nonverbal enumeration to x

Without accurate number word list ( x, y)

With accurate number word list ( x, y)

(Circuitous)

NOTHING

Table 1 Enumeration codes and definitions

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Table 2 Number Application codes, types, and definitions Application Code

Code Type

Application Code Definition

Sequential

Apple Movement

The participant placed the apples in the bowl in a sequential fashion, one by one

Grouping

Apple Movement

The participant grouped the apples in some fashion before placing them in the bowl. Primary types of grouping included moving apples into clumps on the ground, and placing more than one apple in their hand at once

Verbal( x)

Communication

The participant speaks the number word list out loud, correctly or incorrectly. The x refers to how many number words were spoken in total

Nonverbal

Communication

The participant does not speak while completing the assessment

_Correct

Solution

Participant accurately placed exactly five apples in the bowl

_x

Solution

Participant placed a number of apples other than five in the bowl, with a total of x placed

NOTHING

Participant says and does nothing, or refuses to cooperate

and they left in the same condition. (And, unsurprisingly, the same 13 who were consistently successful at the Number Application task were also always accurate at the Enumeration task.) Once the “All Correct—No Change” participants in the Enumeration task were removed, additional patterns emerged, but with the weakness one expects when over 70 % of the participants are absent. Of the remaining nine participants, three showed no consistent improvement, four improved in their ability to perform oneto-one correspondences, one became more accurate in her use of the number word list and one-to-one correspondences, and two appeared to become worse. Thankfully, once the “All Correct—No Change” participants in the Number Application task were removed, a more interesting pattern emerged among the 22 participants remaining. A full 14 of those participants (64 %; 40 % of total) did not deviate from their initial response, doing the exact same action during the pre-test, mid-test, and post-test: placing all seven of the foam core apples in the bowl. We termed this the “All the Objects” rule, and unlike the “All Correct—No Change” participants, the “All the Objects” participants showed a wide variety of abilities, strategies, and trajectories in their Enumeration task performances, perhaps signifying that the “All the Objects” solution method is a particularly sticky one that requires deeper understanding than Cross et al.’s (2009) number core. Lastly, of the remaining eight participants in this grouping, two showed no improvement (but did

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not consistently put all seven apples in the bowl), two appeared to worsen, and the remaining four improved.

Discussion As 40 % of the participants used the “All the Apples” strategy to complete the Number Application task, and varied widely in their ability to succeed in the Enumeration task, some skill or level of conceptual understanding appears to be missing. Upon first glance, the two tasks seem quite similar: in both, you are asked to count five apples. Upon further reflection, however, the Number Application task requires that the participant carve out a subset of five apples from a set of seven—which was only partially modeled by the intervention. For example, the intervention modeled that every time a coconut is added to the box, the number of coconuts in the box goes up by one (e.g., the intervention gently focuses on the impact of the addition to the group). However, the intervention does not model the division of a set into subsets (e.g., how the whole can be separated into new wholes). Consequently, whether the 40 % of participants are not grasping the mathematical concept or whether they are merely following the social training of the episode (i.e., putting all the coconuts in Grover’s box), is difficult to tell. Given that carving out five apples from seven appears to be so difficult, we became interested in the few participants who used a grouping strategy prior to putting the apples into the bowl. (However, as the number of participants using grouping is so small, we include this in the Discussion section as a thought-provoking mention instead of in Analysis as a more significant finding.) Since some participants would use a grouping strategy during one assessment and a sequential strategy during the next, we broke up the assessments into individual ‘clips’, so that they stood alone (for example, Participant T001 is now broken up into T001Pretest, T001Midtest, and T001Postest, and grain size is now the clip). This resulted in a total of 105 clips, and of those, 87 were coded as sequential, while merely 18 were coded as grouping. The sequential clips had approximately a 40 % chance of being correct—while the grouping clips, remarkably, had approximately an 82 % of being correct! We do not have the data to conclude whether children who grouped were more successful because of the strategy, or because they knew more (and consequently knew to use the grouping strategy), but a possible next direction for teaching young children mathematics in this multimodal context would be to have the grouping strategy specifically modeled for them on screen in some fashion. As a brief note prior to concluding the Discussion section, the children who became worse are particularly interesting, but unfortunately too few in number to glean much from. We tentatively hypothesize that they were attempting to adjust their strategies, not knowing why their strategies were wrong but knowing that they were, and adjusted them in the incorrect direction.

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Conclusion The interactive media industry is saturated with products and applications targeting basic math and literacy skills for early childhood. A strong conceptual foundation requires that children have the ability to move from basic knowledge to content application. This analysis shows that for an older preschool target audience, interactive media developers would be well advised to move beyond enumeration activities and look into supporting the transition from enumeration to number application. Additionally, this analysis shows that what may appear (particularly to adults) to be a simple cognitive progression may be riddled with complexities for a young child who is learning higher order number sense. Interactive media tools hold promise for providing meaningful learning experiences for children, but the complex nuances of learning, particularly in mathematics education, may require specific forms of scaffolding, like that suggested above. While it is quite simple to merely discard results that, like ours, show no significant difference between pre- and post-tests, it is through more qualitative analyses that we—as members of many fields interested in similar design and research—can unpack the complications of learning and design more powerful interactive educational opportunities. Our design recommendations are broad and go beyond the scope of this particular study. It is quite easy to examine the findings of the second performance assessment and make particular design recommendations. For example, based on the literature cited above, the finding that participants struggled to count five apples into the bowl is not surprising—and fixing it may be as simple as re-designing the intervention slightly, so that it involves Grover and—for example—Elmo. If Grover and Elmo were both carrying boxes of coconuts and ran into each other on the screen, scattering the coconuts, and required the participant to place five coconuts in Grover’s box and two coconuts in Elmo’s box, the participant could begin understanding how a single set of seven objects could be broken up into five objects and two objects. Naturally, this recommendation needs empirical testing! Consequently, we go beyond this local recommendation and instead venture to make some recommendations for the field as a whole. The results here indicate that while there were not significant learning gains between the pre- and post-mathematics assessments, our more qualitative analysis reveals intriguing findings that can be explained in part by existing research in mathematics education and cognition. Our ongoing analysis examines the demonstrative behaviors of the study participants as they perform the required activities of the number knowledge assessment items. While this can provide the researchers with a deeper understanding of both participant engagement with a situated learning activity and the nuanced methods in which early learners demonstrate their knowledge of specific content, the suggestions for interactive media development proposed still stand. Interactive media is poised to dramatically change the field of learning, especially when pairing newly emerged technologies like the Kinect with tried-and-true educational interventions like Sesame Street. The results that are most useful for designers and mathematics educators, however, may be hiding behind a simple test that declares discouragingly: “No significant differences.”

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Acknowledgments We would like to thank Alex Games of Microsoft Studios and Rane Johnson of Microsoft Research for supporting the research that led to these findings, and Jordan T. Thevenow-Harrison of the University of Wisconsin-Madison for his invaluable statistical assistance. Finally, thanks to our advisors and mentors who made this research and analysis possible: Drs. Kurt Squire, Constance Steinkuehler, and Amy B. Ellis. Earlier versions of this paper were published in the Games+Learning+Society 9.0 conference proceedings (Rothschild et al. 2013a) and in the Psychology of Mathematics Education—North American chapter proceedings (Rothschild et al. 2013b).

References Alibali, M. W., & Goldin-Meadow, S. (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–523. Ball, S., & Bogatz, G. A. (1970). The first year of Sesame Street: An evaluation. Princeton: Educational Testing Service. Barab, S. A., Gresalfi, M., & Ingram-Goble, A. (2010). Transformational play: Using games to position person, content, and context. Educational Researcher, 39(7), 525–536. doi:10.3102/0 013189X10386593. Baroody, A. J., Lai, M., & Mix, K. S. (2005). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & O. N. Saracho (Eds.), Handbook of research on the education of young children (2nd ed., pp. 187–221). Mahwah: Lawrence Erlbaum Associates. Baroody, A. J., Lai, M.L., & Mix, K. S. (2006). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & N. Olivia (Eds.), Handbook of research on the education of young children (2nd ed.). Mahwah: Lawrence Erlbaum. Barsalou, L. W. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22(4), 577– 609 (disc. 610–660). Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645. Barsalou, L. W., Niedenthal, P. M., Barbey, A., & Ruppert, J. (2003). Social embodiment. In B. Ross (Ed.), The psychology of learning and motivation (Vol. 43, pp. 43–92). San Diego: Academic. Bogatz, G. A., & Ball, S. (1971). The second year of Sesame Street: A continuing evaluation. Princeton: Educational Testing Service. Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds.). (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press. Dogan, M. F., Williams, C. C., Walkington, C., & Nathan, M. (2013). Body-based examples when exploring conjectures: Embodied resources and mathematical proof. Poster presentation conducted at the Research Precession of the 2013 National Council of Teachers of Mathematics Annual Meeting and Exposition, Denver, CO. Fisch, S. M., & Truglio, R. T. (2001). “G” is for growing: Thirty years of research on children and Sesame Street. Mahwah: Erlbaum. Fisch, S. M., Lesh, R., Motoki, E., Crespo, S., & Melfi, V. (2011). Children’s mathematical reasoning in online games: Can data mining reveal strategic thinking? Child Development Perspectives, 5(2), 88–92. Gee, J. P. (2003). What video games have to teach us about learning and literacy. New York: Palgrave MacMillan. Gelman, R., & Meck, E. (1983). Preschoolers’ counting: Principles before skill. Cognition, 13(3), 343–359.

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Glenberg, A., Jaworski, B., Rischal, M., & Levin, J. (2007a). What brains are for: Action, meaning, and reading comprehension. In D. McNamara (Ed.), Reading comprehension strategies: Theories, interventions, and technologies (pp. 221–238). Mahwah: Erlbaum. Glenberg, A. M., Brown, M., & Levin, J. R. (2007b). Enhancing comprehension in small reading groups using a manipulation strategy. Contemporary Educational Psychology, 32, 389–399. Lauricella, A. R., Gola, A. A. H., & Calvert, S. L. (2011). Toddlers’ learning from socially meaningful video characters. Media Psychology, 14, 216–232. Microsoft. (2012). Kinect Sesame Street TV [computer software]. Redmond: Microsoft Studios. Nathan, M., Kintsch, W., & Young, E. (1992). A theory of algebra-word-problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9(4), 329–389. National Association for the Education of Young Children [NAEYC]. (2009). Developmentally appropriate practice in early childhood programs serving children from birth through age 8: Position statement. Washington, DC: NAEYC. www.naeyc.org/files/naeyc/file/positions/position%20statement%20Web.pdf. National Association for the Education of Young Children [NAEYC], & Fred Rogers Center for Early Learning and Children’s Media. (2012). Technology and interactive media as tools in early childhood programs serving children from birth through age 8: Joint position statement. Washington, DC: NAEYC (Latrobe: Fred Rogers Center for Early Learning and Children’s Media at Saint Vincent College). www.naeyc.org/files/naeyc/file/positions/PS_technology_ WEB2.pdf. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: NCTM. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards: Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. http://www. corestandards.org/the-standar ds/mathematics. Rothschild, M., Williams, C. C., & Thevenow-Harrison, J. T. (2013a). Counting apples and coconuts: Young children ‘Kinect-ing’ Sesame Street and mathematics. In C. Williams, A. Ochsner, J. Dietmeier, & C. Steinkuehler (Eds.), Proceedings of the 9th Annual Games+Learning+Society Conference (Vol. 3, pp. 274–280). Pittsburgh: ETC Press. Rothschild, M., Williams, C. C., & Thevenow-Harrison, J. T. (2013b). Performance assessments. In M. V. Martinez & A. C. Superfine (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (p. 1207). Chicago: University of Illinois at Chicago. Shapiro, L. (2011). Embodied cognition. New York: Routledge. Squire, K. (2011). Video games and learning: Teaching and participatory culture in the digital age. Technology, Education–Connections (the TEC Series). New York: Teachers College Press. Steinkuehler, C., & Duncan, S. (2008). Scientific habits of mind in virtual worlds. Journal of Science Education and Technology, 17(6), 530–543. doi:10.1007/s10956-008-9120-8. Walkington, C., Srisurichan, R., Nathan, M., Williams, C., & Alibali, M. (2012). Using the body to build geometry justifications: The link between action and cognition. Paper presented at the 2012 American Educational Research Association Annual Meeting and Exhibition, Vancouver, BC. Walkington, C., Nathan, M., Alibali, M., Pier, L., Boncoddo, R., & Williams, C. (2013). Projection as a mechanism for grounding mathematical justification in embodied action. Paper presented at the 2013 Annual Meeting of the American Educational Research Association, San Francisco, CA. Meagan Rothschild is an Assessment and Design Specialist at WIDA and a PhD candidate at the University of Wisconsin-Madison. She made the courageous leap to the chilly Midwest from balmy Hawaii to pursue a PhD and work with the Games, Learning, and Society Center. Prior to her move, Meagan served as the Instructional Designer for Cosmos Chaos!, an innovative video game designed to support struggling fourth grade readers developed by Pacific Resources for

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Education and Learning (PREL). Her experience at PREL also included the design of a violence and substance abuse prevention curriculum for Native Hawaiian students, using an interdisciplinary approach that merged health and language arts content standards to support literacy-driven prevention activities. Meagan has 6 years of experience in the Hawaii Department of Education system serving in varied roles, including high school classroom teacher, grant writer and manager, technology coordinator, and Magnet E-academy coordinator. Meagan has a BA and MEd from the University of Hawaii at Manoa, with undergraduate studies in Hawaiian Language and special education, and an MEd in Educational Technology. As a PhD candidate in Digital Media and Learning at the University of Wisconsin-Madison, her work now focuses on developing and researching multimedia environments that merge research-based learning principles with interactive/gaming strategies to engage learners. She specifically focuses on the role of play to not only provide opportunities for deeper learning, but to provide relevant contexts for learners to demonstrate content knowledge, challenging traditional views of assessment practices. Caroline C. Williams is a dissertator at the University of Wisconsin-Madison (specializing particularly in Mathematics Education in the Department of Curriculum and Instruction), and affiliated with the Games+Learning+Society group. Her dissertation involves designing a Little Big Planet 2 game to teach fractions and linear functions, and she specializes in research involving middle school students, mathematics, and all things digital media. Caro has presented and published on a wide variety of topics, including building in Little Big Planet, mathematics in World of Warcraft forum posts, using gestures to support mathematical reasoning, example usage in mathematical proof processes, gender and mathematics, and learning trajectories in linear, quadratic, and exponential functions.

Classroom Gaming

Research Brief How Can Teachers Use Video Games to Teach Their Students Mathematics?

A

Pew report published in 2008 reported that 97 percent of teenagers play video games of some type (Lenhart et al., 2008). The U.S. Department of Education’s Transforming American Education: Learning Powered by Technology (2010) suggests video games can support a range of teaching and learning activities in school, from embedded assessment to engagement with locally relevant issues. This recommendation echoes research describing how well-designed video games can simulate professional practices and model real-world problem solving (Gee, 2005, 2007; Salen, 2008; Shaffer, 2006a; Squire, 2006). Recently, a survey of elementary and middle school teachers already using video games in their classrooms found that nearly one-fifth teach with video games every day (Millstone, 2012). Concerning mathematics education, we suggest it is necessary to ask: How can teachers use video games to teach their students mathematics? Our primary concern raises additional questions about how teachers select and evaluate video games, and the various types of video games suitable for mathematics instruction. We address these questions by reviewing current research on video games and mathematics education, and then detailing how to find specific mathematics video games. After highlighting a case of video game play supporting statistical reasoning, we discuss how to adapt video games to support classroom teaching, and we then consider the future of teaching mathematics with video games.

Background: Video Games and Mathematics Education For many teachers, their introduction to teaching mathematics with video games was Math Blaster (Davidson & Associates, 1983), “A standard drill-and-practice-type instructional mechanism . . . within a shooter game idiom” (Ito, 2008, p. 93). Introduced in 1983, Math Blaster was a best-selling piece of software; today, mathematics video games are a broader, dynamic, and profitable enterprise. Video games have been designed across grade levels and to meet Common Core State Standards for Mathematics (2010) and National Council of Teachers of Mathematics (2000) standards. They now range from the mobile drill program Flash Math (Kiger, Herro, &

Prunty, 2012) to narrative-based virtual worlds like Quest Atlantis (Gresalfi, Barab, Siyahhan, & Christensen, 2009). Because definitions of “game” are broad and contested (Salen & Zimmerman, 2003; Schell, 2008), in this brief we consider video games to be “imaginary worlds, hypothetical spaces where players can test ideas and experience their consequences” (Squire & Jenkins, 2003, p. 8). This definition is consistent with a general consensus that video games are more than digital tools; video games are designed environments, or possibility spaces (Squire, 2008), that support learning across multiple social spaces, shared practices, and emergent forms of knowledge (Barab, Gresalfi, Dodge, & Ingram-Goble, 2010; Gee, 2003; Shaffer, 2006b). Our understanding of video games contrasts with digital simulations like Geometer’s Sketchpad (Key Curriculum Press, 1991; e.g., Knuth, 2002; Leong & Lim-Teo, 2003) or “cognitive tutor” systems (e.g. Anderson, Corbett, Koedinger, & Pelletier, 1995), which primarily support consistent and accurate means of interacting with mathematical objects and or notations. Video game play, on the other hand—like play in general—affords positive affect, nonlinearity, intrinsic motivation, process, and free choice (Johnson, Christie, & Wardle, 2005; Vygotsky, 1978). Unlike established research about the impact of video games on students’ science learning (e.g. Clarke & Dede, 2009; Gaydos & Squire, 2012; Squire, 2010), only a handful of empirical investigations have explored how video games influence students’ mathematics experiences and understanding (e.g., Harris, Yuill, & Luckin, 2008; Ke & Grabowski, 2007; Kebritchi, 2008). Young and colleagues (2012) recently examined the pedagogical value of video games in respect to student achievement. Nine studies from the mathematics gaming literature were included in their meta-analysis. Like previous reviews which identified a sparse literature base and insufficient research about instructional gaming among specific age groups (Mitchell & Savill-Smith, 2004), Young et al. (2012) suggest better “correspondence” be developed between a game’s objectives and students’ mathematics learning activities. Teaching mathematics with video games does not invariably equate to students learning mathematics from video games. In other words, the results are mixed. Ke & Grabowski (2007) found that fifth graders who

The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council. Copyright © 2014 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502, Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org.

How Can Teachers Use Video Games to Teach Their Students Mathematics?

played video games outperformed non-gaming peers, yet only those who played games cooperatively (rather than competitively) had effective gains in mathematics understanding and positive changes in attitude (as measured by a modified version of the Attitudes Towards Maths Inventory; Tapia, Marsh, & George, 2004). For high school students, playing DimensionM when aligned with online modules and classroom teaching increased mathematics achievement in comparison to non-gaming students; however, gameplay did not improve motivation to learn mathematics (Kebritchi, 2008). Kebritchi, Hirumi, and Bai (2008) found statistically significant gains for high school students who played video games; their average achievement gain between two district exams was more than double that of non-gaming peers. Further, these students’ teachers reported video games as “effective” learning tools because they were experimental, offered alternative teaching approaches, provided an engaging rationale for learning, and increased time on task. Nonetheless, trends persist across this literature; the success and failure of teaching mathematics with video games reflects many factors, including research design, game mechanics, teaching strategies, learning objectives, and context (Young et al., 2012). The varied influence of video game play in mathematics classrooms is unsurprising: games are no magic bullet. Like Young et al. (2012), we believe that successful learning depends upon teachers developing and supporting “corre-

spondence” between play and instruction. Effectively teaching mathematics with video games requires teachers who can develop, support, and reflect upon how any game corresponds to instructional goals and student learning needs. In sum, mathematically meaningful gameplay is the result of thoughtful and creative teaching. But how does a teacher go about finding mathematics video games to support teaching and learning in his or her classroom?

Finding Mathematics Video Games Imagine a veteran mathematics educator eager to refine her practice. Alternatively, what of the “digitally able” first-year teacher (Starkey, 2010), a novice confident in her ability to adapt a range of new technologies to communicate, problemsolve, play, and share information about teaching and student learning. How might either teacher identify what makes gameplay effective in a mathematics classroom? What will either perceive to be the advantages of video games in contrast to more traditional instructional strategies? Can common pitfalls be avoided, hallmarks of success recognized, and challenges mitigated? Each of these educators can easily visit the “Education” section of Apple’s App Store and search for “math games.” As of May 2014, this search gave 219 results. Some games are free while others are cheap, and the available games cover a range of content areas and grade levels,

Fig. 1. Screenshot of Educade homepage

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

with many claiming to dramatically improve students’ learning. Where might these teachers go to find well-designed video games, related curricular materials, and—perhaps most importantly—a community of professionals who can support implementation and refinement of teaching mathematics with video games? Enthusiasm about video games and learning has resulted in the growth of many online resources and communities, as well as offline conferences and professional learning events. Websites such as GameDesk’s Educade (http://educade.org) and Common Sense Media’s Graphite (http://www.graphite.org) provide free game reviews, lesson plans, teaching tools, and other resources—many of them authored by teachers. Many educators have also created and joined online communities devoted to single games. For example, the game Minecraft (Mojang, 2009) has inspired MinecraftEdu (http://minecraftedu. com), an online community and collaborative effort promoting affordability and school accessibility. Additionally, the Playful Learning Initiative (http://playfullearning.com) supports an online knowledge base of video games, facilitates regional and national partnerships and professional learning summits, and is guided by an advisory board of K–12 educators. As new mathematics video games are designed, so too will the number of websites, online communities, and other resources proliferate. Each website and online community has different characteristics; accordingly, we offer a brief outline

of how to find relevant games using Educade (see fig. 1). A search bar at the top of the website may be used, for example, to find games focused on middle and high school algebra. Seven results are presented. Four are “teaching tools,” or information about specific games and resources for teaching mathematics. The remaining three are “lesson plans” focused on implementing games according to specific learning objectives. One “teaching tool” is Dragonbox (WeWantToKnow, 2012). Accompanied by two lesson plans, Dragonbox is likely a useful resource, and selecting the game leads to additional information. A screenshot from Dragonbox is included as figure 2. Dragonbox is robustly described with text, images, a video, as well as information regarding algebraic content, appropriate grade levels, platforms supporting gameplay, and how teachers can implement the game. There are also two relevant lesson plans, and a link to the official Dragonbox website. While we strongly recommend Dragonbox (particularly for mathematics teachers newly interested in video games), more than a few poorly designed mathematics games are also available. We advise playing a demo before purchasing any mathematics video game. If a game does not have a freely available demo, request one from the company. In our experience, a company unwilling to share a demo may be concerned that they are marketing an inferior product. After downloading the demo, play it! While developing

Fig. 2. Screenshot of a level from Dragonbox

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

ideas about how the game fits with classroom teaching, request funding to purchase the full game for classroom use. Consider how the game aligns with curricula, and the “correspondence” among students’ play, mathematics content, and instruction. Will students play in the classroom, after school, or at home? When playing after school or at home, make sure that a computer lab is available and that the game is properly installed so that all students have access to it. Finally, it is crucial to fully play any game before implementation. While playing, reference desired learning objectives and note which are supported by play and which are not. Consider how the game supports standards and curricula, and what lesson and unit planning could connect student gameplay with learning objectives. Teachers who are eager to consult models of classroom teaching and learning can find online communities of teachers and designers (listed earlier) who openly share tales of triumph and failure. These stories can inform decisions about whether or how a game should be played. Accordingly, we now share a case study that highlights ways of connecting gameplay to classroom teaching.

Case Study: Statistical Reasoning and World of Warcraft Teaching mathematics with video games need not conjure up aged images of Math Blaster; classroom teachers are adapting various gaming environments to support the development of sophisticated mathematical investigations. Consider the case of Scott McClintock, a statistics teacher at West Chester University in West Chester, Pennsylvania. Writing in NCTM’s Mathematics Teacher, McClintock (2011) describes how the game World of Warcraft (Blizzard Entertainment, 2004; fig. 3, below) supported his students’ understanding of statistical reasoning. A primary concern for McClintock had been data collection and analysis; he perceived issues of sample size and survey relevance to be pedagogical challenges for quality teaching and learning. However, with World of Warcraft McClintock realized he could address these challenges by creating assignments that engaged a “wide variety of statistical practices” (p. 215). One key to McClintock’s successes—and his students’ learning—was the opportunities for engagement and reflection found within the game world and also in class after gameplay ended.

Fig. 3. Screenshot of multiplayer battle Al’Akir in World of Warcraft (used with permission from Martin et al., 2012) 4

How Can Teachers Use Video Games to Teach Their Students Mathematics?

One assignment concerns the statistical concept of sampling. McClintock directed his students to survey the race and class of 50 World of Warcraft players in each of three different cities. Knowledgeable about the four different races and four different class categories possible for any given player, students sampled 50 players in the human capital, 50 in the elven capital, and 50 in a racially neutral city. After collecting data, students calculated the percentage of humans in each sample, and then determined which sample proportion most likely represented the true proportion of total human players. Although World of Warcraft was not designed to teach statistics, a number of game features supported students’ statistical reasoning. First, random sampling methods would not work in this virtual context; instead, systematic sampling was possible as the race and class of a given player could be obtained through an unobtrusive mouse click. Second, the design of particular cities biases the sample; sampling in the human capital would likely overestimate the total proportion of humans, whereas sampling in the elven capital would likely underestimate the proportion. Thus, only in a race-neutral city could students obtain the least biased statistical sample. It was because of the game’s contextual features that more sophisticated mathematics, about the probabilities of players’ race and class, became possible. McClintock also addressed the importance of continuing to support students’ investigations beyond gameplay. Reflecting upon his own limitations, he suggests teachers provide written feedback to students about findings, encourage students to ask questions of their data, and conduct iterative rounds of data collection and analysis back in World of Warcraft to test new hypotheses. Ultimately, McClintock argues that both game features and teachers’ pedagogy can appropriately support students’ learning of complex statistics, and that teachers should “consider how we might adapt [video games] to enhance the reach and scope of the classroom” (p. 217).

Discussion: Playing and Adapting Video Games for Teaching Mathematics The intent of this brief is not to support any mathematics teacher in only visiting an online database, downloading a game and related materials, and then having students play “for fun” or as a “reward” after completing another assignment. Rather, we advocate adapting video games to complement the characteristics of a classroom. McCall (2011), writing about his own use of games in the classroom, argues, “Successful game-based lessons are the product of well-designed environments. Teacher/designers must thoughtfully

embed these games in an environment and set of learning activities where students, learning tools, and resources work together in pursuit of the desired outcomes” (p. 61). Robust mathematics education has never been achieved through the blind adoption of new curricula, tools, or methods devoid of teachers’ skillful facilitation; video games are not—and, we believe, will never be—an exception. Teaching mathematics with video games will require that teachers deftly consider trends, limitations, and the insights of case studies such as the one noted above; make professional judgments relevant to local context; and reflect upon the successes and challenges associated with teaching and student learning. We suggest that educators interested in teaching mathematics with video games recognize the substantial effort and intellectual engagement required of such an endeavor. Learning to play any game to support learning – and, in particular, a mathematics video game – entails professional behaviors similar to acquiring any new teaching method: research, planning, implementation, reflection, and iteration. Teachers should investigate games and related teaching resources. Game play should be a central element to any teachers’ personal research and learning. Curricular materials, lesson plans, and assessments should also be prepared before classroom implementation, yet remain flexible enough to be revised iteratively. Furthermore, it may be advantageous for teachers to join online professional-amateur (“pro-am”) gaming communities, like those associated with Minecraft, in order to discuss experiences, share insights, and grow a professional network of teachers-as-players.

The Future of Game-Based Mathematics Education As both literature and popular media indicate, teachers and their students are playing and increasingly designing video games (Squire, 2011). With a growing number of teachers writing about their own experience designing and teaching with video games (e.g. Elford [blog]; McCall, 2011), we believe this is a very exciting time to be a mathematics educator interested in using video games (and other games) for learning. Classroom teachers are beginning to refine how researchers and designers create mathematics video games, so content and gameplay both align to standards and also adapt to individual students’ learning needs (e.g. Riconscente, 2011). What new forms of pedagogical, technological (cf. Mishra & Koehler, 2006), and mathematical knowledge for teaching (cf. Ball, Thames, & Phelps, 2008) might teachers demonstrate when teaching with video games across both formal and informal settings? Might virtual game worlds, like World of Warcraft, become a communal mathematics “third plac-

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es” (Steinkuehler & D. Williams, 2006) among networked learners around the world? And how might playful pedagogy and game-based curricula alter how researchers and practitioners understand “ambitious” instruction (cf. Lampert et al., 2010)? As Young and colleagues (2012) suggest, the future of game-based education will be shaped by educators and researchers who collaboratively examine “how gaming combined with instructional facilitation by a master teacher affects engagement, student behavior, and overall academic achievement” (p. 83). It is an exciting time to be an educator interested in teaching mathematics with video games.

Acknowledgements We thank our thoughtful advisors and mentors at the University of Wisconsin–Madison and the Games+Learning+Society Center who reviewed this manuscript: Dr. Amy B. Ellis, Dr. Kurt Squire, and Dr. Constance Steinkuehler. We also thank two anonymous reviewers and Series Editor Michael Fish for their useful commentary.

By Jeremiah I. Holden and Caroline C. Williams University of Wisconsin–Madison Michael Fish, Series Editor

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How Can Teachers Use Video Games to Teach Their Students Mathematics?

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Squire, K. (2011). Video games and learning: Teaching and participatory culture in the digital age. New York, NY: Teachers College Press. Squire, K.D. & Jenkins, H. (2003). Harnessing the power of games in education. Insight, 3, 5–33. Starkey, L. (2010). Supporting the digitally able beginning teacher. Teaching and Teacher Education, 26(7), 1429–1438. Steinkuehler, C. A., & Williams, D. (2006). Where everybody knows your (screen) name: Online games as “third places.” Journal of Computer‐Mediated Communication, 11(4), 885–909. Tapia, M., Marsh, I. I., & George, E. (2004). An instrument to measure mathematics attitudes. Academic Exchange Quarterly, 8(2), 16–21. U.S. Department of Education, Office of Educational Technology (2010). Transforming American education: Learning powered by technology. Washington, D.C.: U.S. Department of Education. Vygotsky, L. S. (1978). Mind and society: The development of higher mental processes. Cambridge, MA: Harvard University. WeWantToKnow (2012). Dragonbox [computer software]. Oslo, Norway. Young, M. F., Slota, S., Cutter, A. B., Jalette, G., Mullin, G., Lai, B., Simeoni, Z., Tran, M., & Yukhymenko, M. (2012). Our princess is in another castle: A review of trends in serious gaming for education. Review of Educational Research, 82(1), 61–89.

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Running Head: Quantifying Exponential Growth Ellis, A.B., Ozgur, Z., Kulow, T., Williams, C., & Amidon, J. (2012). Quantifying exponential growth: The case of the Jactus. In R. Mayes & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 93-112). Laramie, WY: University of Wyoming.

Quantifying Exponential Growth: The Case of the Jactus

Amy Ellis Zekiye Ozgur Torrey Kulow Caroline Williams University of Wisconsin-Madison Joel Amidon University of Mississippi

Quantifying Exponential Growth

Abstract This article presents the results of a small-scale teaching experiment with three middle school students who explored exponential growth by reasoning with the co-varying quantities height and time. Three major conceptual shifts occurred during the course of the teaching experiment: a) From repeated multiplication to a coordination of growth in height and time values; b) From coordinating height and time to coordinating constant ratios; and c) Generalizing to non-natural exponents. The details of each of the three shifts is explored, followed by a discussion of the implications of addressing exponential growth from a covariation of quantities perspective.

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Quantifying Exponential Growth

The Landscape of Exponential Functions in Common School Treatments Students’ mathematical learning is the reason our profession exists. Everything we do as mathematics educators is, directly or indirectly, to improve the learning attained by anyone who studies mathematics. (Thompson, 2008, p. 45) Exponential functions are an important concept both in school algebra and in higher mathematics. Not only do they play a critical role in college mathematics courses such as calculus, differential equations, and complex analysis (Weber, 2002), they also represent an important transition from middle school mathematics to the more complex ideas students encounter in high school mathematics. A focus on the conceptual underpinnings of exponential growth has increased in recent years; for instance, the Common Core State Standards (CCSS) highlight the need to understand exponential functions in terms of one quantity changing at a constant percent rate per unit interval relative to another. Moreover, these ideas are also being pushed down into middle school mathematics courses, both in terms of national standards such as the CCSS as well as in middle school curricula (e.g., Lappan et al., 2006). The study we report on in this paper is situated in a larger project exploring middleschool students’ understanding when reasoning with co-varying quantities to support their development of function understanding. Our prior studies focused on students’ understanding of linear function and quadratic function as they reasoned with quantitative relationships such as gear ratios, constant speed, and growing rectangles (Ellis, 2007; 2011a; 2011b). A natural extension of this work is to explore how students come to understand exponential growth through reasoning in a similar context, namely, by exploring two co-varying quantities such that one changes exponentially as the other changes linearly. Our approach differed in significant ways from the typical textbook approaches to exponential growth, which we detail below. The Repeated Multiplication Approach

3

Quantifying Exponential Growth

A common textbook treatment for introducing the notion of exponential growth is the repeated multiplication approach. For instance, in the middle school curriculum Connected Mathematics Project (2006), students place coins called rubas on a chessboard in a doubling pattern, and then use tables, graphs, and equations to examine the exponential relationship between the number of squares and the number of rubas. These types of tasks require students to perform repeated multiplication to solve a problem and then connect that process to exponential notation (Castillo-Garsow, submitted). A number of researchers have advocated for this approach, suggesting that we define exponentiation as repeated multiplication with natural numbers, and then help students generalize beyond the natural numbers (e.g., Goldin & Herscovics, 1991; Weber, 2002). However, this approach has its limitations. As Davis (2009) noted, generalizing to non-natural exponents may pose difficulties for students; for instance, an 2

expression such as 2 3 can be difficult to understand from a repeated-multiplication perspective. Difficulties in Understanding Exponential Growth

€ literature on students’ and teachers’ understanding of exponential growth is scant, The but the research that does exist supports Davis’ concerns about the difficulties in generalizing one’s understanding of exponentiation as repeated multiplication. For instance, Weber (2002) found that college students struggled to understand or explain the rules of exponentiation and could not connect them to rules for logarithms. Weber described students’ difficulties in explaining what a function such as f (x) = a x meant, as well as in explaining why a function

" 1 %x such as f (x) = $ ' was a decreasing function. Pre-service teachers have not fared much better; # 2& € researchers have identified their struggles not only in understanding exponential functions, but

€ also in recognizing growth as exponential in nature (Davis, 2009; Presmeg & Nenduradu, 2005).

4

Quantifying Exponential Growth

Although pre-service teachers appear to have a strong understanding of exponentiation as repeated multiplication, they experience difficulty in connecting this understanding to the closedform equation and in appropriately generalizing rules such as the multiplication and power properties of exponents (Davis, 2009). In general, teachers appear to be able to make some use of graphical, algebraic, and tabular representations, but cannot then leverage their algebraic facility to support their ability to solve exponential problems or to translate from table situations to either recursive representations or correspondence rules (Davis, 2009; Presmeg & Nenduradu, 2005). Research on middle school and high school students reveals difficulties as well; students struggle to transition from linear representations to exponential representations, or to identify what makes data exponential (Alagic & Palenz, 2006). In general, exponential growth appears to be challenging to represent for both students and teachers, and it is difficult for teachers to both anticipate where students might struggle in learning about exponential properties and develop ideas for appropriate contexts that involve exponential growth (Davis, 2009; Weber, 2002). These documented challenges suggest a need for better understanding of how to foster students’ learning about exponential growth, and for identifying more effective modes of instruction on exponential functions. Alternate Approaches to Exponential Growth Repeated multiplication is not the only way to think about exponential growth; one can also approach exponentiation in other ways, for instance, as the relationship between a population of individuals and their collective growth contributions (Castillo-Garsow, submitted), as products of factors (Weber, 2002), or as a multiplicative rate constructed from multiplicative units (Confrey & Smith, 1994; 1995). Weber (2002) offered a theoretical analysis of exponential growth relying on Dubinsky’s (1991) APOS (Action, Process, Object, Schema) theory. Although

5

Quantifying Exponential Growth

this approach begins with an action understanding of exponentiation as repeated multiplication, Weber offers a vision of students then transitioning to a process understanding by interiorizing the repeated multiplication action; students would then view exponentiation as a function and be able to reason about its properties. Once students can consider exponential expressions as a result of a process, terms such as 2 3 can be viewed as the product of 3 factors of 2, and ultimately students should then be able to generalize their understanding to view a b as b factors of a.

€ Weber’s analysis offers a vision for moving beyond the repeated multiplication view, but it € these processes. remains an open question how students might actually undergo Confrey and Smith (1994; 1995) introduced an operational basis for multiplication and division called splitting, which is a multiplicative operation that is not repeated addition. A splitting structure is a multiplicative structure in which multiplication and division are inverse operations, such as repeated doubling and repeated halving. Within this model, students also treat the product of a splitting action as the basis for its reapplication; thus, a split can be viewed as a multiplicative unit. Confrey and Smith (1994) assert that “Building concepts of multiplicative rates constructed from multiplicative units should play a central role as students work on understanding how multiplicative worlds generate constant doubling times and constant halflives.” (p. 55) Splitting as an operation can form the basis of a rate of change approach to exponential functions, which we will discuss in further detail below. In Confrey and Smith’s (1994) work, they found a number of different rate-of-change approaches adopted by students making sense of exponential situations, including multiplicative rates of change. Students constructing multiplicative rates of change would interpret a table with, for instance, a growth factor of 9 to be increasing by “a constant rate of nine.” Confrey and Smith suggest that this is an important

6

Quantifying Exponential Growth

conception of rate of change, which is found by calculating the ratios between succeeding yvalues for constant unit-change in the x-values. We highlight this conception as an important foundational idea for a rate-of-change approach to exponentiation. The Rate of Change Approach to Exponential Functions: Covariation and Continuous Variation Traditional approaches to function rely on a correspondence view (Smith, 2003), in which a function is seen as the fixed relationship between the members of two sets. From this perspective, y = f (x) represents y as a function of x, in which each value of x is associated with a unique value of y (Farenga & Ness, 2005). This static view underlies the typical treatment of € functions in school mathematics, and it is not difficult to see how students may struggle to

transition from a repeated-multiplication understanding of exponentiation to a correspondence understanding, particularly beyond the domain of natural numbers. Smith and Confrey (Smith, 2003; Smith & Confrey, 1994) offered an alternative to the correspondence view, which they called the covariation approach to functional thinking. Here one examines a function in terms of a coordinated change of x- and y-values: A covariation approach, on the other hand, entails being able to move operationally from ym to ym+1 coordinating with movement from xm to xm+1. For tables, it involves the coordination of the variation in two or more columns as one moves down (or up) the table. (Confrey & Smith, 1994, p. 33) This is what the students described in Confrey and Smith’s 1994 article did when they calculated the ratios between succeeding y-values for a constant unit-change in the x-values in tables of exponential data. Confrey and Smith argue that splitting, juxtaposed with covariation, can

7

Quantifying Exponential Growth

provide the basis for the construction of an exponential function. Exponentiation is simply repeated splitting, just as multiplication is repeated addition. Covariation for Exponential Growth Castillow-Garsow (2012, this volume) describes covariation as the imagining of two quantities changing together; students imagine how one variable changes while imagining changes in the other. Relying on situations that involve quantities that students can make sense of, manipulate, experiment with, and investigate can foster their abilities to reason flexibly about dynamically changing events (Carlson & Oehrtman, 2005). An approach that relies on imagining co-varying quantities may be especially useful in helping students understand exponential growth, as this view is strongly tied to how students think about contexts involving multiplicative relationships (Davis, 2009). Thompson (2008) argues that a defining characteristic of exponential functions is the notion that the rate at which an exponential function changes with respect to its argument is proportional to the value of the function at that argument. Approaches that emphasize this concept could help students make strong connections between the change in x-values and the corresponding change in y-values, developing the understanding that the value of f (x + Δx) / f (x) is dependent on Δx (Thompson, 2008). So, for instance, for a repeated doubling

function f (x) = 2 x , a covariation approach could help students coordinate (additive) changes in x

€ values with (multiplicative) changes in y-values to understand that the constant multiplicative

€

€rate of change for Δx = 1 would be 2, for Δx = 2 would be 2 2 , or 4, for Δx = 3 would be 8, and 1 1 for Δx = would be 2 2 . €2

€

€

€

€

One study (Green, 2008) did take a rate-of-change approach to helping students construct € exponential growth and found that expanding the concept of rate of change to include percent 8

Quantifying Exponential Growth

changes helped students understand the meanings of the parameters of exponential functions. In another study with two high-school students, Castillo-Garsow (2012) found that a focus on reasoning covariationally about financial modeling tasks fostered different solutions to a differential equation based on either a discrete or continuous understanding of change. We were interested in developing a situation in which the notion of proportional rate of change would arise naturally. We have found that adopting a rate of change perspective can be accessible even for beginning algebra students in middle school, particularly if they have opportunities to explore situations that encourage students to construct meaningful relationships between quantities (Ellis, 2007; 2011a; 2011b). Given the age group of our students, we aimed to develop a context with co-varying quantities that satisfied three requirements. First, students should be able to visually observe the quantities changing together. Second, students should have a way to easily measure and record the values of both quantities as they covaried. Third, the quantities should vary in a continuous rather than discrete manner. We will describe how we implemented each of these criteria in the sections below. Continuous Variation In order to discuss continuous variation, it is helpful first to address the ideas of chunky reasoning and smooth reasoning (Castillow-Garsow, 2012). Castillow-Garsow describes chunky reasoning in the following manner: a student imagines that a change occurs in completed “chunks”, after a certain amount of time has passed, such as a day or a week. The student does not imagine that change occurs within the chunk unless she can re-conceptualize the change to a smaller chunk size, such as chopping a week into seven days, with each day having its own completed change. Castillow-Garsow explains that “Chunky thinking is inherently discrete. It

9

Quantifying Exponential Growth

remains an open question whether or not continuous understanding can be built from pure chunky thinking” (p. 11). In contrast, when reasoning smoothly, a student imagines a quantity changing in the present tense; one can map from one’s own current experiential time to a time period in the mathematical context without needing to resort to convenient units of time. “Smooth thinking, in contrast, is inherently continuous. By imagining change in progress, change is subjected to that person’s understanding of change in the physical universe” (Castillow-Garsow, 2012, p. 11). Castillow-Garsow suggests that the continuous nature of smooth reasoning is critical for understanding exponential growth. Continuous quantitative reasoning then becomes a “repeated process of imagining the smooth change in progress of a quantity over an interval, followed by an actual or imagined numerical measurement of the quantity at the end of each interval” (Castillow-Garsow, 2012, p. 18). Thompson (2011) similarly describes the concept of continuous variation, in which every smooth change in progress is imagined to be composed of smaller chunks (giving numerical values), and every small chunk within the change in progress is thought of as being itself covered by a smooth change in progress. In this manner, a student achieves infinite precision by alternating smooth and chunky thinking by chopping the interval of variation into finer and finer chunks. If continuous reasoning relies on smooth and chunky thinking in this manner, then education targeting continuous quantitative reasoning should focus on developing smooth reasoning skills (Castillow-Garsow, 2012). For this reason we endeavored to develop a scenario in which students would have the opportunity to engage in continuous quantitative reasoning when imagining two quantities co-varying exponentially. The Jactus: Building Exponential Growth by

10

Quantifying Exponential Growth

Reasoning with Continuously Covarying Quantities Building on the principles enacted in our linear and quadratic teaching experiments (Ellis, 2007; 2011), we set out to develop a context in which students could explore two continuously co-varying quantities; we wanted to avoid discrete situations such as the chessboard problem. Our intention was to develop a context that would be understandable to a middle-school population. We settled on a scenario in which a plant called the Jactus grew by doubling its initial height every week. As we will discuss later, the choice of a week as the time frame for the plant to double was deliberate. Students explored the growing Jactus plant by comparing its height to time via a specially designed Geogebra script (Figure 1).

Figure 1: Screenshot of the Geogebra Script for a Doubling Jactus Students could manipulate the image of the Jactus plant by dragging its base with the mouse. As they did so, the plant would continuously increase or decrease in height as it moved along the time axis. Over time, we changed the growth factor of the Jactus to values other than 2, the initial height to values other than 1 inch, and the amount of time to double to values other than 1 week.

11

Quantifying Exponential Growth

We recognize that situations do not imply reasoning; continuous problem situations do not necessarily mean that continuous reasoning will occur. As Castillo-Garsow (2012) demonstrated in his example with Alice, Bob, and Carol’s solutions to the same problem, a student’s reasoning may not necessarily correspond to the co-varying quantities in the situation. However, we believe that a context with smoothly co-varying quantities could afford the possibility of continuous reasoning in a way that a discrete situation would not. It is possible to imagine a plant that is somewhere between 1 inch tall when it starts growing and 2 inches tall after the first week. It is more difficult to imagine, for instance, a ruba coin that is in the middle of becoming two ruba coins. Encouraging Coordination We hypothesize that significant mathematical learning can take place as a result of students’ engagement in appropriately selected and sequenced mathematical tasks. If individuals have the capacity to learn through their mathematical activity, the possibility exists to engineer a sequence of tasks that promotes the learning of individuals through their engagement with such a task sequence. (Simon et al., 2010, p. 72) We set out to develop a set of sequenced mathematical tasks by first identifying the ways of thinking that we desired for our students as an outcome. Carlson and Oehrtman’s covariation framework (2005) informed our thinking for the initial design of tasks. Their framework emerged to describe the reasoning involved in the meaningful representation and interpretation of graphical models in calculus, but it has still served useful to guide our thinking about what we wanted middle school students to understand about exponential growth. Carlson and Oehrtman described five mental actions, the first four being relevant to middle school students’ reasoning with exponential functions (the fifth addresses instantaneous rates of change). The first mental action identified by Carlson and Oehrtman is to coordinate the dependence of one variable on another variable. In the Jactus context, we developed activities to help students understand that the height of the Jactus plant depends on the amount of time that it 12

Quantifying Exponential Growth

had been growing. Students’ interaction with the Geogebra scripts familiarized them with this dependence relationship. We then designed tasks in which students discussed the variables that could contribute to the Jactus’ growth, drew pictures of the growing Jactus, and devised methods for keeping track of the plant’s growth over time. The second mental action is coordinating the direction of change of one variable with changes in the other variable. We wanted students to understand that as time increases, the plant grows taller, and as time decreases, the plant grows shorter. Our activities with drawing pictures, interacting with the Geogebra script, and identifying relationships between how much time had passed and how tall the plant had grown encouraged this coordination. The third mental action is coordinating the amount of change of one variable with changes in the other variable. Students should initially understand that the growth in height is determined multiplicatively rather than additively. We wanted students to understand that repeated multiplication of the height, such as doubling, occurred every week. Early activities aimed at this idea required students to create drawings. For instance, given a picture of a 1-inch Jactus plant at 0 weeks, students drew the height of the plant at 1, 2, 3, and 4 weeks. Later, students created drawings in which they had to skip weeks, for instance, drawing the height of the plant at week 3 given a picture of the plant at week 0. In order to highlight the fact that the growth rate is the same for any same Δx , we also asked problems such as the following, in which the height value is unknown (imagine that a student has already determined that a

€ each week): “Say you go on vacation for 1 week. How much taller particular Jactus plant triples will the Jactus be when you return?” We combined these tasks with interpolation problems, far prediction problems, and comparison problems across different plants with different growth rates

13

Quantifying Exponential Growth

and initial heights in order to encourage facility with coordinating the amount of change of height with the amount of change of time. Carlson and Oehrtman’s fourth mental action is coordinating the average rate-of-change of the function with uniform increments of change in the input variable. In this case, we focused instead on coordinating multiplicative comparisons of y-values with additive comparisons of xvalues. We wanted students to understand that a) the ratio of f (x 2 ) to f (x1 ) will always be the same for any same Δx = x 2 − x1 , and b) the value of f (x + Δx) / f (x) is dependent on Δx . We € € designed tasks that introduced tables of exponential data in which the week values were not

€ uniformly€distributed and asked students to€find missing week values and missing height values. We also used these table representations when asking students to determine an unknown growth factor. Once students were already coordinating ratios of successive y-values to determine growth factors, we attempted to make the value of Δx explicit by emphasizing different Δx Name ______________________

values, such as in the task in Figure 2: Memo #18

€

Last week each of you divided the inches for successive weeks to find out whether the plant was doubling, tripling, etc: Weeks 0

Inches 1

1

3

2

9

3÷1=3

€

It must be tripling!

9÷3=3

What if you couldn’t do that, because you didn’t have successive weeks in your table? How would you figure out if it was doubling, tripling, or doing something else? Weeks 0

Inches 1

5

1024

18

68,719,476,736

Figure 2: Non-uniform Table Task Encouraging smooth reasoning. Another conceptual goal was that students coordinate the ratio of f (x 2 ) to f (x1 ) for values of Δx that were less than 1, ultimately being able to

€

€

€

14

Quantifying Exponential Growth

imagine this coordination for arbitrarily small increments in x. Weber (2002) argues that students should, in time, be able to generalize their understanding of repeated multiplication to make sense of what it means to be “the product of x factors of a” when x is not a positive integer. The Jactus context could allow for a meaningful interpretation of this idea when students begin to contemplate how much the plant would grow in one day, for instance, given that it doubles in one week. We designed tasks in which students had to determine how tall a plant would be at non whole-number time values such as half a week or two and a half weeks. We then asked students to determine how much a plant would grow for many different time periods, such as for 10 weeks, for one day, or for 1/10th of a week. The Teaching Experiment We recruited three eighth-grade students (ages 13-14), Uditi, Jill, and Laura (all pseudonyms). Jill and Laura were enrolled in eighth-grade mathematics, and Uditi was enrolled in an eighth-grade algebra course. None of the students had encountered exponential functions in their mathematics classes at the time of the study. The students participated in a 12-day teaching experiment (Cobb & Steffe, 1983; Steffe & Thompson, 2000) over the course of three weeks, in which the first author was the teacher-researcher. Two project members familiar with the teaching-experiment methodology observed and videotaped each teaching session. Each session lasted approximately 1 hour. The project team met daily after each session to debrief and discuss the events that transpired during the session. For the purposes of this paper, we present an analysis of the students’ conceptual development over the course of the teaching experiment, with an emphasis on three shifts in understanding. Before embarking on the teaching experiment, the project team developed a progression of tasks according to the design principles described above. We also created a hypothetical

15

Quantifying Exponential Growth

learning trajectory (Simon, 1995), organizing a sequence of tasks that would foster students’ understanding of exponential growth. However, the teaching experiment model demanded a flexibility that meant the initial sequence of tasks served only as a rough model for instruction. We engaged in iterative cycles of intervention, assessment and model building of students’ thinking, and revision of future tasks on an ongoing basis. In this manner we developed, during each teaching session, enhanced hypotheses of the students’ understanding based on the previous cycle (Simon et. al, 2010). Data Analysis We employed retrospective analysis (Simon et. al, 2010) in order to characterize students’ changing conceptions throughout the course of the teaching experiment. We relied on the initial learning trajectory as a source of preliminary codes for the data, coding students’ talk, gestures, actions, and responses to tasks as evidence of understanding at various stages on the trajectory. The act of coding students’ understanding also produced emergent codes, which then altered the initial learning trajectory to account for the events that occurred during the teaching experiment. The codes and the trajectory therefore evolved simultaneously in a cyclical manner until the trajectory stabilized to reflect the final set of codes identifying students’ conceptions. Although the presentation of the final learning trajectory is beyond the scope of this paper, we will present a portion of the trajectory accounting for three major shifts in students’ conceptual understanding of exponential growth. Two members of the research team initially coded the entire data corpus independently. During this process they met weekly with the entire project team in order to discuss boundary cases and clarify and refine uncertain codes. Once this initial phase was complete, the two researchers then recoded the entire data set together by comparing every codes and discussing

16

Quantifying Exponential Growth

any differences until reaching agreement. The iterative process of coding, refining, and recoding continued until no new codes emerged and no more refinement was necessary. From Repeated Multiplication to Constant Multiplicative Rates of Change: Three Conceptual Shifts Early in the teaching experiment, all three students demonstrated an understanding that repeated multiplication of the height determined the manner in which the Jactus plant grew. In this analysis we present evidence of three major conceptual shifts that marked the students’ transitions from understanding exponential growth as repeated multiplication to coordinating multiplicative ratios of height values with additive differences for the time values. We will address each of the shifts (Figure 3) in turn. Conceptual Shift

Definition

Shift 1: From Repeated Multiplication to Coordinating x and y Shift 2: From Coordinating x and y to Coordinated Constant Ratios Shift 3: Generalizing to nonnatural exponents

Students shift from only attending to the repeated multiplication of the y-values to coordinating repeated multiplication with changes in xvalues. Students shift to explicitly coordinating the ratio of f (x 2 ) to f (x1 ) for corresponding x-values for any Δx ≥1. Students shift to explicitly coordinating the ratio of f (x 2 ) to f (x1 ) € for corresponding x-values for any value € of Δx , including values in € which Δx < 1.

Figure 3: Three Conceptual Shifts in Creating Multiplicative Rates of Change € €

€ After exploring the Geogebra script, the students began to quickly identify repeated € multiplication as the mechanism that determined the manner in which the Jactus grew. For instance, the students encountered a situation in which the Jactus was 1 inch tall when it began to grow, and it grew by quadrupling its height every week. Uditi described its growth this way: “They’re all going up by like times 4, like, 16 times 4 is 64 and then 64 times 4 and then 64 times 4…then times 4 that’s 1024.” Explaining how she would determine the height of the plant at 7 weeks, Uditi stated, “Four times 4 is 16, 16 times 4 is 256 then 4…1024 times 4, 4096 times

17

Quantifying Exponential Growth

4 and then it’s 16,384.” Absent from Uditi’s language is an explicit coordination of repeated multiplication of the height of the Jactus with the amount of time it took the Jactus to quadruple. The first shift we address is the one in which the students began to attend to corresponding time values when considering the height of the Jactus. Shift 1: From Repeated Multiplication to Coordinating x and y On Day 2 of the teaching experiment the students encountered a far prediction question: Given a Jactus that doubled every week and began growing when it was 1 inch tall, how tall would it be at 30 weeks? Laura’s response indicated an understanding of repeated doubling, and an attention to the corresponding number of weeks began to make its way into her language: Name

LQuI'UI

“You can do like, so here’s 8, and then the next is 16 and I guess like I said yesterday it goes up, Memo #3

if that’s the rate, and if you minus it you’ll seetables that…for 32showing you would doof 32 timesOne 2, and then Yesterday we saw two on the board the growth the Jactus. had days, and one had weeks:

you have the result for the week, 6 and then youWeeks just keep going like that.” At this Inches Inches Daysfor week 0

7

1 2 4 8 16 32

0 1 2 3 4 5

1 2 4 8 16 32

point, there is little evidence of14explicit coordination of repeated doubling of the height per a 121 28 35

week increase in time. In order to find the value of the plant’s height at week 30, Laura 1. Which do you think is easier to work with, days or weeks?

constructed a table from week 6 all \tJthe e..e-way )L- to S week 30 because she had no way to truncate the repeated multiplication process (Figure 4), which resulted in calculation errors. 2. How tall will the Jactus be after 210 days (30 weeks)?

2-6

--

2-7

q 3 3

7

II 3731-1J 35b

I I

18

Quantifying Exponential Growth

Figure 4: Laura’s table for determining the Jactus’ height at week 30 In an attempt to highlight the coordination of the plant’s height with weeks, the teacherresearcher (TR) introduced a task that required the students to draw the plant’s height after 1 week and after 3 weeks. After the students produced their drawings, the teacher-researcher asked them to think about Week 4: TR:

Uditi: It would be more bigger. Memo #4 TR:

:eP ------,---

So what would happen the next week and week 4? Name

Here's a picture of the Jactus plant. Compared to this picture, draw another

It would be more bigger? picture of what the Jactus will look like after 1 week, and then a picture of what it will look like after 3 weeks.

Jill:

It would be double, it’s…

TR:

Oh, it would be double. Double what?

Jill:

Well, the next week would be double the last week.

Jill’s drawing indicated the beginning of her coordination of the plant’s height with the number of weeks; both are present in her picture (Figure 5). Although it appears that Jill’s image for Week 3 was of a plant twice as tall as Week 1, she labeled the Jactus as 8 inches tall:

i Figure 5: Jill’s drawing of the growing Jactus TR:

So then what would week 5 do?

All:

Double that.

Laura: It would double the week before.

19

Quantifying Exponential Growth

By the fourth day, the students could explicitly coordinate the height of the plant with the week number, but they could not coordinate the growth in height for multiple-week jumps. For instance, Laura worked with a table of exponential values in which she had to find the plant’s height for week 10 and week 13 (Figure 6):

Figure 6: Laura’s table of week / height values Laura was able to coordinate the growth in inches with the number of weeks the plant had been growing by filling in the gaps in the table. Her language reflected an explicit attention to weeks: “For 4 (weeks) I got 48 (inches), for 6 I got 192, for 7 I got 384, for 9 I got 1,536, for 10 I got 3072 and for 13 I got 24,576.” Laura could only double the previous week’s height to find the next week’s height, and she did not yet understand that this process could be truncated, for instance, that she could find the Jactus’ height at week 13 by taking the height at week 10 and multiplying it by 8, or 23 for a jump of 3 weeks. Shift 2: From Coordinating x and y to Coordinated Constant Ratios On the fifth day Jill and Uditi encountered a non-uniform table of exponential data (the function for these data is f (x) = 0.1(2 x ) ). Laura was absent on this day, and her attendance became more sporadic than Uditi and Jill’s for the remainder of the sessions. € 20

Quantifying Exponential Growth

Figure 7: Non-uniform table of exponential data Jill’s response demonstrates the beginning of a coordination of the change in the height of the plant with the change in weeks. She took the ratio of the height at week 8 to the height at week 0, dividing 25.6 inches by 0.1 inches, explaining, “I did 25.6 divided by 0.1 and that was 256.” Jill also knew that she had to account for the 8-week difference. However, rather than recognizing that 256 represented a growth factor repeatedly multiplied 8 times (e.g., b 8 = 256 ), Jill divided by 8: “I just divided that by 8 because there was 8 weeks and I got 32, but I don’t really know why I did that, I just kind of did it.”

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The teacher-researcher provided Jill with a new problem that contained only two data points (Figure 8), and asked Jill to determine how the plant was growing.

Figure 8: Jill’s recognition that a 2-week jump means the height will grow by 9

21

Quantifying Exponential Growth

At this point Jill began to think about the gap in weeks as representing how many times repeated multiplication must occur: “I tried 3 times 3 and it was the 16 week number, and so then I figured out if I did the 14 week number times 9 it would give me this number.” On the same day, Uditi demonstrated a similar emerging coordination between the Name

multiplicative growth in height and multiple weeks, but did not generalize her understanding. In

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another non-uniform table of exponential data (for which the underlying function was Here's another table for the height of a Jactus in inches after a certain number clf

ratio ofCan theyou first two out height f (x) = 0.1( 4 x ) ), Uditi took theweeks. figure howvalues to find(Figure how tall9): the Jactus will be after any number of weeks?

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Uditi divided 1.6 inches by 0.1 inches to get 16, and she knew that this meant she had to find a number that she could multiply twice to get 16. Given her limited knowledge of algebraic manipulation with exponents, Uditi had to rely on a guess and check method: “I found out that the difference was 16 and then I just tried all these different numbers. I tried to multiply and then multiply it again the same number.” In this manner she discovered that the growth factor was 4. At this stage, however, Uditi was not yet able to generalize her reasoning to any multi-week gap; for instance, she was unable to explain that the plant would grow 4 × 4 × 4 times, or 43 times, between week 2 and week 5 because there was a 3-week gap.

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Quantifying Exponential Growth

On Day 7 of the teaching experiment the teacher-researcher gave the students two data points with a 5-week gap in order to encourage the coordination of the multiplicative growth in height with the additive growth in time. For this task, the Jactus was 3,355.443.2 inches tall on week 24, and it was 107,374,182.4 inches tall on week 29. The teacher-researcher asked the students to determine how the Jactus was growing (e.g., whether it was doubling, tripling, etc.) and how tall it was at week 0. Jill took the ratio of the two height values and found it to be 32, and then took the difference of the two week values. She then wrote, “ __⋅ __⋅ __⋅ __⋅ __ ”, searching for a number that she could multiply by itself 5 times in order to yield 32. By guessing € 2 (Figure 10). and checking, Jill determined that the growth factor should be

Figure 10: Jill’s solution to the two-data-point problem Uditi’s response was similar to Jill’s, and once she found that the growth factor was 2, she used this information to determine the height at Week 0. Uditi wrote “ __× 2 24 ” and then guessed and checked in order to find the value that would make her expression equal to the height at 24

€ to be 0.2, so then she produced the weeks, 3,355,443.2 inches. Uditi found the missing value equation “ y = .20 × 2 x ”. Uditi and Jill also appeared to understand that the ratio of height values would remain

€ constant for any same-number increase in weeks. For instance, after determining that one tripling Jactus plant grew 8 times as tall from Week 3 to Week 6, both students immediately predicted that the plant would be 8 times taller at Week 103 than it would be at Week 100.

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Quantifying Exponential Growth

Shift 3: Generalizing to Non-Natural Exponents Our intention was to foster a coordination between the multiplicative growth of y-values and the additive growth of x-values even for cases in which the difference between x-values was less than 1. Our hope was that students would be able to imagine this coordination for arbitrarily small increments in x, which would provide a way to understand ba even when a is not a whole number. One way to introduce this idea was to ask the students to think about how tall the plant would be in the middle of a week; for instance, if a plant doubles between Week 1 and Week 2, how tall will it be at Week 1.5? Estimation and Reversion to Linear Thinking Early in the teaching experiment, the students struggled to make estimates for how tall the plant would be in between whole week values. For instance, on the third day of the teaching experiment the students worked with a Jactus plant that was 1 inch tall at 0 weeks and doubled each week. The teacher-researcher asked Uditi how tall the plant would be at 1.5 weeks and she guessed that it should be “a little more than three.” Uditi explained, “It increases a little more every time it goes that way [gesturing to successive weeks], so in between them is three so it’s going to increase a little more, like comes to point 5.” Uditi had a qualitative (Behr et al., 1992) understanding of the plant’s growth, in that she could reason about the direction of change in the plant’s height with relying on calculations. Uditi knew that the manner in which the plant increased from one week to the next was not linear. Because the plant was 2 inches tall at 1 week and 4 inches tall at 2 weeks, Uditi guessed that it would be slightly more than 3 inches tall halfway between 1 and 2 weeks, but she was unable to determine the precise amount of growth. Although her estimate is incorrect, it is notable that Uditi did not revert to linear interpolation when trying to determine the plant’s height at 1.5 weeks.

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Quantifying Exponential Growth

One week into the teaching experiment the students revisited the same doubling Jactus, but this time the teacher-researcher asked the students to think about the plant’s height every day between 0 weeks and 1 week. Laura and Jill both reverted to linear interpolation, stating that the plant would be 1.1 inches tall after 1 day, 1.2 inches tall after 2 days, and so forth. Uditi, however, relied on her ability to coordinate multiplicative growth in the plant’s height with additive growth in weeks. She took the ratio of the plant’s height at 1 week (2 inches) to the plant’s height at 0 weeks (1 inch), and then wrote “ __× __× __× __× __× __× __ = 2 ”. At this point Uditi became stuck, because she was unable to find a number that she could multiply by itself seven times to result in 2.

€

Reliance on Equations to Assist with Interpolation The next day the students visited the same doubling plant with a table that provided its heights for different decimals between Week 0 and Week 2 (Figure 11):

Figure 11: Table of values for doubling Jactus

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Quantifying Exponential Growth

Laura knew that the ratio of heights for any 0.25-week difference would be the same, so she took the ratio of the plant’s height at week 0.75 and divided it by the plant’s height at week 0.25, which was 1.189. She then wrote “goes up by 1.189207115” and concluded that 1.189 would be the plant’s height at Week 0.25. Although Laura’s “goes up by” language suggests the possibility of an image of an additive difference of 1.189 between successive quarter weeks, later Laura determined the height for Week 3.5 by using appropriate multiplicative reasoning, multiplying the value of the height at Week 3 by 1.189 twice. In contrast, Uditi relied on her ability to create an equation in order to determine the missing height at 0.25 weeks. Uditi wrote “ 1× __^.5 ” and then used a guess and check method to find the missing value that would yield the plant’s height at 0.5 weeks. She easily determined € that the missing value was 2, and then checked the correctness of her equation with other values

in the table, ultimately declaring that the plant doubled each week. Uditi then wrote “Height = 1⋅ 2^ week ” and substituted 0.25 for the exponent in order to find the missing value.

Coordinating the Ratio of Height Values for Δx with bΔx €

In order to encourage Uditi to coordinate the plant’s growth with parts of weeks rather € € than relying on an equation, the teacher-researcher introduced a problem that only had two data

points (Figure 12), asking the students to figure out how the plant was growing.

Figure 12: Two data points from the function f (x) = 3x Jill and Laura both took the ratio of the two height values and stated that the plant would get

€ student could use this information “1.116123172” times as big each tenth of a week, but neither

26

Quantifying Exponential Growth

to determine how much the plant would grow in 1 week. Uditi also took the ratio of the two height values, which she rounded to 1.12. In order to determine how big the plant would grow every week, she then once again relied on an equation, writing “ __× __ 0.1 = 1.12 ”. Uditi indicated that the first blank represented the “starting number”, or the plant’s height at Week 0, and the second blank represented the unknown weekly€growth factor of the plant. Although Uditi’s approach was unexpected, we found it notable because she equated the ratio of the yvalues, 1.12, to abΔx (for a function f (x) = ab x ). This represents a key understanding, that the ratio of height values for a given Δx can be expressed as bΔx . At this point it was not clear how € Uditi might generalize this€understanding, and there was little evidence to determine whether € Uditi also realized that€this ratio will always be the same for any same Δx .

Uditi knew that the plant increased in height by a factor of approximately 1.12 for 1/10th of a week. Rather than taking this value to the tenth power in€order to determine how much the plant would grow in one week, Uditi simply used a guess and check method with her equation to determine that 30.1 ≈ 1.12 . She concluded that the plant tripled every week. The teacherresearcher asked Uditi how the plant grows every 2 weeks and every 3 weeks, and Uditi € concluded that every 2 weeks the plant would grow 9 times as big and every 3 weeks it would

grow 27 times as big. She explained, “Because 3× 3 = 9 and 3× 3× 3 = 27 .” The teacherresearcher decided to ask Uditi about larger numbers: TR:

€ big numbers, € like Week 100 and Week 101? Is What if I gave you really this number, whatever it’s going to be, still going to be 3 times as big the next week?

Uditi: (Nods). TR:

Even all the way up here?

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Quantifying Exponential Growth

Uditi: (Nods). TR:

How come?

Uditi: Because, the equation says it grows 3 times every week. TR:

Okay, and if it’s going 9 every 2 weeks are you confident that it will be that way for any two weeks?

Uditi: (Nods). TR:

How come?

Uditi: Because, 3 times 3 times 3 is 9. Based on Uditi’s responses, it was unclear what she truly thought; Uditi could have possibly have just been responding positively to the teacher-researcher’s questions. Therefore the teacher-researcher gave her another problem with only two data points. This plant was 64 inches tall at 3 weeks, and 68,719,476,736 inches tall at 18 weeks. The students had to figure out how the plant was growing every week, every 3 weeks, and every ½ week. Both Uditi and Laura determined that the plant grew 4 times as large each week by taking the ratio of the two height values and then determining what number to the 15th power (the difference between 18 weeks and 3 weeks) yielded that result. Both then concluded that the plant would grow 64 times as large every 3 weeks. Uditi then explained that the plant would grow 2 times as large every ½ week, reasoning that at 3 weeks the plant was 64 inches tall, at 3.5 weeks it was double that, 128 inches tall, and at 4 weeks it doubled again to 256 inches tall (Figure 13).

Figure 13: Uditi’s amended table 28

Quantifying Exponential Growth

The teacher-researcher then asked Uditi how the plant would grow every x weeks, and she responded, “It grows by 4 to the x.” Generalizing to Fractional Exponents On the last day of the teaching experiment, the teacher-researcher gave the students a task designed to determine whether they understood that ratio of f (x 2 ) to f (x1 ) will always be the same for any same Δx = x 2 − x1 , even when Δx is a fraction (Figure 14): € €

€

€

Figure 14: Tripling Jactus Table We were curious whether the students would attempt to make a calculation, or whether they would know that the plant would grow 9 times as big regardless of its size at any given week. Laura did not recognize this and attempted to calculate an answer. Jill and Uditi both immediately said the plant would become 9 times as big. Jill explained, “I noticed that 1 times 9 is 9 and 2, or, 9 times 9 is 81, so every 2 weeks it is going up 9 times.” The teacher-researcher then asked the students how much the plant would grow from Week 155 to Week 160, and Uditi wrote “ 35 = 243 ”, explaining that this was a way to represent “three times three times three times three times three.” Laura and Jill were then able to use Uditi’s reasoning to correctly determine

€ how much larger the plant would grow in 10 weeks. The teacher-researcher then asked the students to determine how much the plant would grow in 1 day, or 1/7th of a week. Uditi wrote “ 3.14 = 1.17 ”, explaining, “It’s 1 divided by 7 because it only shows the result for 1 week on the table, and there are 7 days in a week. So I € 29

Quantifying Exponential Growth

divided 1 week into 7 parts, which represents 1 day each and it’s .14 of a week.” In this manner Uditi was able to make sense of a non-whole number exponent and generalize her coordination of the ratio of height values with growth in time to Δx values less than 1. Discussion

€ The students began the teaching experiment with an understanding of exponential growth as repeated multiplication, but they did not explicitly coordinate repeated multiplication of height with growth in time. Over time, the students began to coordinate the growth of the height of the Jactus with time, first implicitly by thinking about the plant as doubling “every time”, and then explicitly as they had to negotiate non-uniform tables of values. The students then began to truncate the process of repeated multiplication, coordinating increases in height with multipleweek time spans. Initially they did this linearly, by dividing the ratio of height values by Δx , but Δx ultimately they shifted to coordinating the ratio of height values with b for the growth factor b.

€ The students realized that this relationship would hold for any same Δx ; at first, they understood € this only for Δx values greater than 1, representing multi-week jumps, but ultimately one

€ 1, making use of fractional student, Uditi, was able to coordinate for Δx values less than € exponents. € The students’ ability to coordinate the ratio of height values with the additive difference in time values played a significant role in supporting their ability to develop algebraic representations of the plant’s growth. This was evident, for example, in Uditi’s development of the equation 0.2 × 2 x , in which she had determined the growth factor 2 by taking the ratio of two height values 5 weeks apart and then determining the number b such that b5 was equal to that € ratio. In general, the students’ covariational thinking preceded their ability to develop

correspondence rules of the form y = f(x), which reflects Smith and Confrey’s (Smith, 2003;

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Quantifying Exponential Growth

Smith & Confrey, 1994) assertion that students typically approach functional relationships from a covariational perspective first. One reason why we focused on the middle school population for this study was because students in our participating schools encounter exponential growth formally in their mathematics classrooms in eighth grade. We were interested in exploring students’ evolving conceptual development as they encountered exponential situations for the first time, which necessitated a younger participant group. However, this also resulted in some challenges and constraints in the types of problems and tasks we were able to design. Enabling the students to physically manipulate and visually observe the growing plant with Geogebra supported a qualitative experience of the nature of exponential growth, but it limited the growth factors we could use. A growth factor larger than 4 resulted in numbers too large for Geogebra and the students’ scientific calculators, so they only had opportunities to explore plants that doubled, tripled, or quadrupled. In addition, we typically constrained the growth factor to whole numbers because our participants did not have access to sophisticated algebraic manipulation abilities or the notion of logarithms. This meant that they were limited to guess and check methods for determining the growth factor for mystery plants. For instance, the students often determined the growth factor by taking the ratio of two height values a certain number of weeks apart. Imagine a situation in which the growth factor is approximately 97.66 for a time period of 5 weeks. A high school or 1 5

college student could write the equation b = 97.66 and then solve for b, calculating 97.66 in 5

order to determine that the mystery growth factor is 2.5. Because the middle school students did € not possess this degree of facility with algebraic manipulation, they instead€had to guess and

check to determine a number they could multiply by itself 5 times that would equal 97.66. If the

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Quantifying Exponential Growth

growth factor was something other than a whole number, the guess and check method was typically too time consuming to allow for any significant progress within the constraints of the teaching experiment. Despite these limitations, however, the results presented in this paper offer a proof of concept that even with their relative lack of algebraic sophistication, middle school students can engage in an impressive degree of coordination of co-varying quantities when exploring exponential growth. In addition, we have presented evidence that students can generalize their a understanding of exponential growth to view b as a factors of b for non-natural values of a, as

suggested in theory by Weber (2002). We contend that reasoning with continuously co-varying

€ quantities was a critical element in constructing this particular understanding of exponentiation. The Jactus context offered a scenario in which students could begin to make meaningful sense of non-natural exponents as they imagined the height of the plant smoothly growing over time. Although we hesitate to make the definite claim that Uditi employed smooth thinking, we suspect that contexts such as the Jactus situation can support this development because they provide opportunities for students to re-imagine the nature of the plant’s growth for varying units of time.

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Quantifying Exponential Growth

References Alagic, M., & Palenz, D. (2006). Teachers explore linear and exponential growth: Spreadsheets as cognitive tools. Journal of Technology and Teacher Education, 14(3), 633 – 649. Behr, M.J., Harel, G., Post., T., & Lesh, R. (1992). Rational number, ratio, and proportion. In Grouws, D.A. (Ed.), Handbook of Research on Mathematics Teaching and Learning, (pp. 296 – 333). New York: Macmillan, 1992. Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler. Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. Mayes & L. Hatfield (Eds.), Quantitative Reasoning and Mathematical Modeling: A Driver for STEM Integrated Education and Teaching in Context (WISDOMe Monograph Volume 2). Castillo-Garsow, C. (Submitted). The role of various modeling perspectives in students’ learning of exponential growth. Davis, J. (2009). Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12, 365 – 389. Cobb, P., & Steffe, L.P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 28, 258 – 277. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2), 135–164. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.),

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Advanced mathematical thinking (pp. 95 – 123). Dordrecht, The Netherlands: Kluwer Academic Publishers. Ellis, A.B. (2011a). Generalizing promoting actions: How classroom collaborations can support students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308 – 345. Ellis, A.B. (2011b). Algebra in the middle school: Developing functional relationships through quantitative reasoning. In J. Cai & E. Knuth (Eds.), Early Algebraization: A Global Dialogue from Multiple Perspectives (pp. 215 – 235). New York: Springer. Ellis, A.B. (2007). Connections between generalizing and justifying: Students’ reasoning with – linear relationships. Journal for Research in Mathematics Education, 38(3), 194 – 229. Farenga, S.J., & Ness, D. (2005). Algebraic thinking part II: The use of functions in scientific inquiry. Science Scope, 29(1), 62 – 64. Goldin, G., & Herscovics, N. (1991). Towards a conceptual-representational analysis of the exponential function. In F. Furinghetti (Ed.), Proceedings of the Fifteenth Annual Conference for the Psychology of Mathematics Education (PME) (Vol 2, pp. 64-71). Genoa, Italy: Dipartimento di Matematica dell’Universita di Geneva. Green, K. (2008). Using spreadsheepts to discover meaning for parameters in nonlinear models. The Journal of Computers in Mathematics and Science Teaching, 27(4), 423 – 441. Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Phillips, E.D. (2006). Connected Mathematics 2. Hilldale, NJ: Pearson Prentice Hall. Presmeg, N., & Nenduardu, R. (2005) An investigation of a pre-service teacher’s use of

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representations in solving algebraic problems involving exponential relationships. In H. L. Chick & J. K. Vincent (Eds). Proceedings of the 29th PMEInternational Conference, 4, pp. 105-112. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 347 – 395. Simon, M., Saldanha, L., McClintock, E., Akar, G., Watanabe, T., & Zembat, I. (2010). A Developing Approach to Studying Students' Learning through Their Mathematical Activity. Cognition and Instruction, 28(1), 70-112 Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. (pp. 136 – 150). New York: Erlbaum. Smith, E., & Confrey, J. (1994). Multiplicative structures and the development of logarithms: What was lost by the invention of function. In Harel, G. & Confrey, J. (Eds.), The development of multiplicative reasoning in the learning of mathematics. Albany, NY: State University of New York Press, pp. 333 – 364. Steffe, L.P., & Thompson, P.W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum Associates. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education. Laramie, WY: University of Wyoming.

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Thompson, P.W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. In O Figueras, J L Cortina, S Alatorre, T Rojano, and A Sepulveda, Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education, volume 1, pages 45–64, Morelia, Mexico. PME. Weber, K. (2002). Students' understanding of exponential and logarithmic functions. Second International Conference on the Teaching of Mathematics (pp. 1-10). Crete, Greece: University of Crete.

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Playing Together Separately

14 Playing Together Separately Mapping Out Literacy and Social Synchronicity Crystle Caroline C. Amanda Ochsner, Shannon Elizabeth Gabriella Anton, Jonathon and Constance Steinkuehler In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people. London: Routledge.

INTRODUCTION Jaea sits at a desk with a laptop on each side of him and a keyboard and large computer monitor directly in front of him; he has three different keyboards and three different screens. He also wears headphones for both listening and speaking. Eyes shifting rapidly, he sometimes focuses on a quickly changing scene on a single screen and sometimes glances across multiple screens each offer different information. At the same time, he receives feedback from voices coming through on the speakers, leading him to shift his attention elsewhere on one of the screens, and speaking into the microphone he returns information to the voices. With so much equipment and so much rapid movement and communication, one might guess that he is doing very important and very complicated work. Perhaps he is saving lives. In a way, he is. In the scene described earlier, Jaea is playing the massively multiplayer online role-playing game (MMORPG) World ofWare raft (WoW), and he is playing it well. Very well. His role in the group is to act as a healer, keeping the rest of his raid members alive while they take large amounts of damage from intimidating and powerful enemies. The other group members are playing ftom other rooms, similarly equipped with screens, keyboards, and headphones, but located throughout the country, perhaps even across the world. Their work is just as complicated and is equally demanding. What Jaea is really doing is reading. He is reading text conversations with his fellow players on the screen in front of him; he is reading the graphics he sees on the screen; he is reading the actions of other players; he is reading the needs and abilities of his own character; and he is reading his physical environment as well. After all, the girlfriend and three very large, very fuzzy cats that he shares this space with require attention sometimes too. (The cats, in fact, demand it!) This reading occurs in his physical environment as in the game world, where he maneuvers his avatar.

227

His temporal existence is also full of multiplicity: he simultaneously has an understanding of when it is in the room he plays in, where his raid group is in terms of progressing through their raid dungeon, and the time at which his character exists in the game's fantasy universe. This chapter will explore the multiplicity of presence that is exhibited within gameplay in World of Wareraft, and demonstrate how leveraging multiple presences is essential to successful high-end virtualliteracies like MMOs.

EXAMINING ATTENTION From the phenomena described in the introduction, many may think of existing explanations to describe what we are seeing. Terms like split attention, multitasking, and constant partial attention are often used to describe situations in which a person undertakes multiple tasks at once. These terms often carry a negative connotation. In theories of split attention, the research focuses on "the limitations of working memory" and the idea that multiple tasks "overburden working memory" (Kalyuga, Chandler, & Sweller, 1999). The theory of split attention views attention as something that is allocated, and when divided up it weakens the person's ability to accomplish tasks (Awh & Pashler, 2000; Tabbers, Martens, & van Merrienboer, 2000; Mayer & Moreno, 1998). Multitasking is another explanation for attention paid to several tasks at once, and although multitasking is seen by many as a valuable skill that makes their world function, it is seen by others as a less than positive structure. Gonzalez and Mark (2004, 2005) represented multitasking as an issue to be remedied. They used as an illustration the everyday example of working in an office and having to deal with multiple conversations, telephones ringing, people walking into other cubicles (2004: 115). The issue of multitasking was mitigated through working spheres, which they defined as "a set of interrelated events, which share a common motive" Gonzalez and Mark's (2005) suggestions turned to system design, namely that system design needed to take into account multitasking and help to reinforce workers' working spheres. In the context of this paper, the multiple activities that are undertaken simultaneously could be considered split attention or multitasking; however, these descriptions in the existing research are narrow in focus, negatively valenced at the outset, and do not capture the subtleties of the game experience on its own terms. To help describe the nuances of the game experience, we explore a variety of research to illustrate the complexities inherent to the activity. Understanding these subtleties is important because it allows a clearer picture of the actions and information management engaged in by players. The erature will be framed in three contexts of gaming-that of literacy, time, and place-and will finish with new explanations for attending to several tasks as once.

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GAMING AS VIRTUAL LITERACY Gaming as a literacy has been studied by many researchers. Gee (2003), in his seminal book What Videogames Have to Teach Us about Learning and Literacy, described 36 learning principles that are manifested in good games. He claimed that in order to participate in a game the player must understand the literacy of the game and that playing was participating in a semiotic domain in which the player must understand rules, symbols, social interactions, and discourse to be successful. In addition to games, research has been conducted on online communities, which are centered on literacy practices (e.g., FacFiction.net (Black, 2007a, 2007b, 2008)). Black described the literacy habits of adolescents in online fan fiction communities, detailing their level of community interaction, their editing abilities, the use of the space by second language learners to develop their English skills, and their development of 'metavocabulary of editors.' Black and Steinkuehler (2009) stated that participants in virtual worlds engage in reading and writing in a variety of formats, which they found to align with national educational standards (Standards for the English Language Arts, as cited in Black & Steinkuehler, 2009). Steinkuehler (2006, 2007, 2008) also studies literacy in massively multiplayer online games (MMOs) as both a discourse (which is both languagein-use and language-in-action (see Gee, 1999) and as a constellation of Iiteracies, which includes the game and all of the resources (e.g., wikis, hlogs, forums, videos, etc.) that are used as information sources by the game community. The constellation of Iiteracies is the 'space' in which the information needed for success in a game or other affinity space exists. Gee's (2004, 2005) semiotic social and affinity spaces describe the 'place' of MMOs. The affordances of these perspectives offer flexibility, focusing on space and how it functions to bring people with common interests together. Steinkuehler and King (2009) used play within the place of an MMO to bolster the literacy practices of disengaged adolescent boys using the space of the constellation of literacies that surround the game to help them to become re-engaged. Martin and Steinkuehler (2010) explore another form of literacy that functions within the information constellation of an MMO, that of information literacy. By observing practices in the naturalistic setting of the game, Martin and Steinkuehler (2010) examine what and how resources are utilized to maintain success in a game-that is, how players traverse the space of the constellation of literacies in order to successfully retrieve information and resources they need. Time can be viewed in a multitude of ways. Gelliooks at the keeping of time in two ways (1992). He divided time up into what he termed A-series time, or standardized time as measured by a clock, and B-series time, or time that is run by the punctuation of activity (similar to Erickson's (2004) characterization of kronos and kairos). We appropriate Gell's classification and apply it to the distinction between a person's time spent in and

Playing Together Separately 229 out of games. Gamers live a hybrid existence of A-series and B-series time (Martin, in progress) where A- and B-series time becomes intertwined and entangled in the gaming activities. This is true in many genres of games but it is especially true in relation to MMOs. In an MMO, a player follows B-series time by focusing on the game world where much activity is punctuated by a cycle of activities rather than by the time increments of a clock. However, the player may at the same time keep track of when a certain time in the physical world comes-for example, when they need to eat dinner or when they need to make a phone call. Here, A-series time is intruding on the B-series time of the game. Participation in group activities in MMOs, although possibly starting at an A-series time, is punctuated by phases of the activity in B-series time. Although states of engagement and flow can be found in other activities, games offer an always available environment to delve into and MMOs offer the added benefit of constant social interaction in the virtual space, although the players are generally separated in the physical space. Situating the activity of in-game activities within literacy, time, and space gives a basis for understanding the activities that take place within the game. Through this lens of activities a different approach of cognition could be taken. Distributed cognition (Hutchins, 1995) is an approach that emphasizes the use of resources expanding cognition beyond the individual's mind. This theory focuses on the coordination of enaction among agents within the community, which can include individuals, artifacts, and the environment itself (Hollan, Hutchins, & Kirsh, 2000), and here, the environment includes the game environment. This notion of the distribution of cognitive processes among the community, in this case the players of the MMO World of Warcraft, can be seen in Martin and Steinkuehler's (2010) idea of collective information literacy, also termed distributed information literacy.

METHODS To comprehend the visual and information literacy experiences and practices of an expert-level World of Warcraft player, the researchers involved in this investigation used an instrumental case study model in which the actor being studied was selected for purposes of better understanding the surrounding problem space of one expert player during normative gameplay (Stake, 1995). Our goal is to identify the ways in which literacies are practiced and constructed in a complex, navigationally demanding, fastpaced, and visually elaborate digital environment in which successful literacy practices and problem solving efforts are essential to success. The goal of this case study was to develop a hermeneutic understanding of the practices common to expert-level WoW play in a holistic, empirical, interpretive, and an empathetic manner (Stake, 1995).

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Selecting an expert-level player was imperative to understanding the advanced literacy practices and concomitant problem representations evidenced by many participants of this highly effective upper echelon of gameplay. Problem representations, such as determining the most effective and information literacy practices in order to achieve advanced game world success, were inherently designed by the participant and were grounded in his domain-related knowledge and subsequent semantic organization. Furthermore, experts solve problems differently than do novices (Chi, Feltovich, & Glaser, 1981) and we expected this difference to manifest evcnor, perhaps, especially-in very complex problems spaces like WoW. For this study, a 25-year-old, Caucasian, male, expert-level World of Warcraft player was observed playing WoW (Stevens, Satwicz, & McCarthy, 2(08) on a weekly basis from January of 2011 until April of 2011. The researchers are familiar with WoW gameplay and are garners themselves, although they do not play at the expert level of the case study participant. The expert-level player, Jaea (a pseudonym), typically plays WoW for about two hours each weekday, in addition to logging in about 8-12 hours of playtime over the weekends. He is a successful raider, and his guild is consistently first-ranked server-side (on their server) in ten-person are collaborative and complex in-game actions to defeat a series of level enemies usually called bosses. The guild has even been first on their server to complete some ten-person raids, a prized achievement. Data was collected through the use of detailed and structured handwritten field notes taken by One case researcher present during gameplay. In

Figure 14.1

jaea's physical context.

Figure 14.1 Jaea's physical context.

addition to constructing presence flow charts Figures 14.2 and 14.3) capturing Jaea's foci of attention, the case researcher also took abundant notes used to reconstruct each of the ten fast-paced and complex sessions. The case researcher mainly observed Jaea's actions, both person) and on-screen, during gameplay, and occasionally asked questions when it was not obvious what was happening-on average about once or twice during each gameplay session. To assist with data collection, the case study participant, Jaea, frequently thought aloud as he played, meaning he talked through the problems or experiences he encountered during gameplay. In addition, he modified his typical gameplay habits to play without headphones. This allowed the case researcher to hear utterances from Mumble, a voice-over IP application that he uses to communicate with other players, as well as contextualize his responses to Mumble. Having the case researcher carefully observe and note Jaea's head movement captured shifts in player focus (see Figure 14.1 for Jaea's physical context). A movement of Jaea's head to the left indicated focus on a second laptop, in which he would Twitter or Reddit forums a and rated news links forum, while playing. The second was positioned far enough to the left that Jaea couldn't read screen merely by shifting his eyes' focus, but had to turn his head slightly, which let the case researcher follow his presence more closely. The case researcher initially asked about Jaea's practices when he shifted focus to his information feed laptop, but he quickly developed a habit of automatically describing what he was doing. Jaea's engagement with the actual ingame WoW chat was also recorded whenever he typed a response. The case researchers also learned what the user interface (UI) customizations Jaea used looked like, so that they could identify times at which he tweaked or changed his UI without interrupting gameplay. A screenshot was taken of Jaea's screen at the end of gameplay so that the level of visual information captured on-screen could be paired with the other data collected bv the case researchers. For each observation a presence was taking place. This visualization included graphic representations of where presence-operationalized as eye gaze, verbosity, and physical actions (including the physical actions necessary to create digital movement)-was most focused, as well as descriptors for the activities taking place within each delineation of time. Our operationalization of presence relied upon visual and audio indicators: Jaea's movement in physical space could highlight his physical presence (such as going to the bathroom), or it may indicate a shift from in-game WoW to information constellation (such as shifting his gaze from his primary computer screen, loaded with WoW, to his secondary laptop to the left, loaded with Twitter or WoW forums). Additionally, audio input from raid team members on Mumble would be represented in the information constellation layer, while audio WoW feedback from the game itself would appear in the

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Martin, C., Williams, C., Ochsner, A., Harris, S. King, E., Anton, G., Elmergreen, J. & Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people (pp. 226243). London: Routledge.

Figure 14.3 Presence flow chart 2.

Martin, C., Williams, C., Ochsner, A., Harris, S. King, E., Anton, G., Elmergreen, J. & Steinkuehler, C. (2012). Playing together separately: Mapping out literacy and social synchronicity. In G. Merchant, J. Gillen, J. Marsh & J. Davies (Eds.), Virtual literacies: Interactive spaces for children and young people (pp. 226243). London: Routledge.

Martin et al.

The player's ability to navigate within these different spaces and be successful in the game demonstrates a level of literacy, as the player must possess considerable information literacy as well as the traditional literacy of reading in order to interact with the information available around the game. Along with the literacy space players need to navigate for successful WoW play, our method is designed to capture both the physical and virtual spaces that Jaea regularly travels within. It is through the simultaneous traversal of such disparate places that makes expert-and even novice!-WoW play so difficult to understand and analyze. Our conceptualization of space is quite broad, capturing both digital and physical measurable space, as well as pursuit of information and the development of mental maps and plans. time, while Figure 14.3 Figure 14.2 represents 41 minutes of represents 38 minutes; the former is nine chunks of B-series time, while the latter is ten. From the data collected, Figures 14.2 and 14.3 were created for visualization and analysis. Each of the follows a format. Across the top are A-series time stamps that mark the demarcations section of time has a between B-series chunks of time (Gell, 1992). clear beginning and ending with 'fight begins' and 'wipe' appearing as the most common periods in these two presence visualizations. The former indicates that the raid group engaged in battle, while the latter indicates that all ten raid members died during battle and were unable to complete their goal. The emergent rule for breaking raiding up into B-series time is consequently 'fight begins' and 'wipe' (or 'win'), although in non-raid activities the time can be sectioned quite differently. The framework for analysis developed for this study was originally inspired by Lemke's (2000) notion of timescales. Lemke focuses on activity in time and how timescales interact with one another. His view of the interactions of people within different timescales caused us to consider the presence or focus of an individual when engaging in a multifaceted activity like that of playing an MMO. The distribution of attention across multiple layers looks much more like Hutchin's (1995) distributed cognition than the description of split attention that plagues research on digital spaces (Kalyuga et aI., 1999; Mayer & Moreno, 1998: 318; Tabbers et Awh & Pashler, 2000). Distributed cognition is an active process of engaging in multiple activities simultaneously and successfully. Successful here entails being able to learn from mistakes, being able to find needed inforand being able to cognitively manage multiple inputs from multiple presences at once. months of observations, two contrary examples are reprein Figure 14.2 and Figure 14.3. The first example (Figure 14.2) was a raid that Jaea's guild is very familiar with and that they were helping a new guild member through. Figure 14.3 is a raid that they had never attempted before, although the raiders had prepared beforehand by watching videos and reading guides. These two examples were chosen to illustrate a known activity and an unknown activity both in the context of a ten-person raid.

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The digital contexts of Figure 14.2 and Figure 14.3 vary in important ways, even though both represent the presence flow of Jaea engaging in tenperson raids with his guild. Figure 14.2 was a familiar raid for Jaea and the guild, and one that they had successfully completed many times before. However, a new guild member had never encountered this particular content before and was being carefully guided by the nine other guild members. Figure 14.3, on the other hand, represents a new raid boss that none of the ten players, including Jaea, had attempted before. And while all the players had researched the raid information closely beforehand, strategies and coordination were developed by the guild in situ. Researching raids before attempting them is a common practice of raiding guilds. The reason for this is to familiarize the members with the layout of the environment, the types of enemies that they will be facing, and anything that is specific to a boss that they may is used for preparation is community-creresources like written walk-throughs or Information literacy skills are needed in order players to locate and understand the information provided in these resources as well as to evaluate the reliability of each resource. In terms of time, the two presence visualizations are quite similar. The following is a description of the digital contexts for Figures 14.2 and 14.3. In order to maintain accessibility for readers who are not WoW players, the following accounts are considerably simplified. This simplified version is to make the description accessible for all readers regardless of whether they are familiar with WoW. As stated in the introduction, WoW is an online game played by millions of people across the world. Smaller groups within the game, called 'guilds,' play and plan together, often in order to complete 'raids' (that is, incredibly complicated and challenging battles with powerful boss monsters that require a group of people to defeat). During this playtime, the guild groups tend to communicate verbally and via text in order to coordinate their attacks, and teamwork, prior knowledge, and careful attention are necessary in order to succeed. Figure 14.2 represents the first phase of the ten-person AI'Akir Raids may be played in either the raid bosses are even more powertu is a three-phase raid, and in the first phase, final boss of the Throne of Four Winds raid) stands in the center of a floating circular platform. Around AI'Akir spins a wall of damage-dealing whirlwinds that require players to either move close to the boss or to the very edge of the platform, requiring constant attention by the players. A general WoW strategy is to distract the boss into attacking a high-defense player (otherwise known as a tank) while other players deal damage to him (otherwise known as DPS-ers, named after the phrase Damage {Jer Second). All at the same time, other players are healing the tank and DPSers to ensure they stay alive. As this strategy plays out, AI'Akir casts chain lightning, the damage of which is multiplied by the number of players near

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the afflicted players-so the players must maintain a carefully coordinated distance from each other. In addition, Al'Akir will occasionally cast a speU that pushes players a certain distance from him, requiring that players stay close enough to him so that they don't fall off the platform when pushed back. In order to prevent the push back spell from being cast, the tank can run away from Al'Akir, but Al'Akir punishes the fleeing tank by doing extra damage to him or her, and consequently the healers need to be prepared to cast extra healing spells on the tank. There are more spells that Al'Akir casts during the battle (too many to describe here), each of which requires specific strategies that can sometimes conflict with one another. Figure 14.3 represents the first phase of the ten-person heroic mode Val. iona and Theralion raid, which-at the time of writing-Jaea's guild just successfully completed. During the observation that led to Figure 14.3, the guild was unsuccessfully attempting this raid for the first time. This raid has two bosses, both dragons, which periodically switch off between the roles of fighting on the ground and flying while attacking the raid from the air. During the first phase, Valiona is on the ground, casting proximitybased flame attacks while the tank tries to keep Valiona's attacks focused on him or her. The DPS-ers try to bring Valiona's health down (both dragons share a common hea Ith pool), and healers work to keep everyone alive. Meanwhile, Valiona has another attack that requires players to stack other words, get as close as possible to each other) in order to minimize damage, while Theralion has an attack that requires players to spread out as far away as possible, forcing the raid team to be constantly aware of their relative positioning and to be prepared to move as needed.

FINDINGS Despite differences in the raid itself, as well as the group's familiarity with each raid, both show a strikingly similar pattern. During fights, the in-game WoW presence predominates, but the between-battle chunks show Jaea shifting presence heavily towards information flow. This strong pattern is, in retrospect, the very pattern that led to our B-series time demarcations. While fighting and not fighting may seem like simple activity changes that occur only within the virtual world that the in-game WoW self inhabits, the influence on presence flow in all three layers is sharp and clear. Furthermore, Jaea's smooth yet rapid movement from one distribution of presence to another is practiced and familiar and presents no evidence of cognitive confusion or adaptive difficulties, which might be seen if split attention and multitasking were an issue. On the left of the figures are three labels, each of which represents a 'layer' of presence. At the top, the information constellation represents all the information accessed by laea that is not included by default in WoW. This information includes Mumble, the heavily customized user interface

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that Jaea relies upon, and any online resources or activities that he engages in, such as Twitter or email. In the middle, the Physical self represents Jaea's physical body, located within the physical world and subject to various distractions like bathroom breaks, cell phones calls, and aggressively playful cats. At the bottom, the in-game WoW represents the virtual body and the default WoW information, which includes virtual cues from fellow raid members and bosses, mana (magic points) and health information, equipment durability, and bag space. The separation of the information constellation layer from the in-game WoW layer and the physical self layer is a complex one. While information comes from the physical world (such as in-body signals that say, "I'm hungry") and the in-game WoW layer (virtual indicators that say, "My avatar needs to eat"), this information is part and parcel of living in and being able to perceive and respond to those worlds. The information constellation layer includes only the information that is separate from the default physical messages (the hardness of the chair against your legs) and the default WoW messages (the attack upon your avatar by a monster). In other words, the information constellation includes the additional information that laea (and other players) choose to receive or not receive, to seek out or not seek out. Within each chunk, text summarizes laea's activities by presence type. The order is not chronological because the actions overlap one another, and the text merely indicates what happened during that time frame. Each item represents an activity or exchange of some sort, ranging from going to the bathroom (physical self), listening to raid chatter via Mumble (information flow), to specific movements present only during boss engagements (in-game WoW). Repetitions of activities or exchanges are condensed and represented with only one label. for example, Healing teammates (in-game WoW) is an action Jaea engaged in repeatedly in each of the chunks; however, that entry is included only once, and there is no indication of frequency except indirectly by the height of the in-game WoW layer. Time progresses from left to right, and the three layers co-occur within each B-series time column. The height of each layer within blocks indicates the proportion of presence: in Figure 14.2, for example, Jaea's presence in the first block is mOst strongly in information constellation, followed by in-game WoW presence, with the least amount of presence in physical self. In the following block, a distinct shift occurs, as information constellation and physical self become shorter, and in-game WoW nearly doubles. The dramatic shifts of presence accompanying events in WoW dictated the B-series time distinctions and help to foreground and background different elements of the data. Specific actions taken by Jaea are noted within the blocks by text but are otherwise backgrounded. Instead, we can clearly see the flow of activity and presence by backgrounding the details and foregrounding the descriptors of actions, which is key to our analysis. The graphs emphasize the proportion of overall presence allocated to each layer and the changes of those proportions based on activity.

There are two particular codes that are worth explaining briefly here. The descriptor Minimal presence is used to indicate that Jaea is barely present in that layer during a time chunk. It is used only in the physical self category to represent that some presence is unavoidable. Another descriptor, Customized UI (see Figure 14.4), is ever-present in the information constellation category, and represents Jaea's modified user interface. His reorga. nization of WoW information flow is so drastic that his every engagement with the virtual world is mediated by his unique design. Jaea's constant tweaking of his customized UI contributes to his level of expertise because his ability to process information included in the customized UI at a glance allows him to have faster reaction times and more precision when playing. A constant striving to improve the customized UI shows a perpetual striving to improve his gameplay. Just as in situations of other forms of fandom (Black & Stein kuehler, 2009) the player's ability to 'read' the text of the situation is important to success. Coiro and Dobler (2007) found that reading online required different skills than reading in traditional print format. The reader needed to understand more than the words on the screen-also the context clues and structure of the digital materials. The situation is similar for Jaea in-game; as Coiro and Dobler found for websites, Jaea must be able to interpret symbols, and understand the context clues and structure of the game interface. Modifying the interface could be seen as similar to writing fan fiction-tweaking the story and changing parts to make the readability more customized to his needs. While writing fan fiction and maximizing healing performance may seem quite different, both are examples of the same phenomena: participants acting upon their digital environments in order to rewrite their own stories. Jaea's UI modification could be conceived as intentionally modifying the spatial arrangement (Hollan et aI., 2000) in order to simplify choice, perception, and internal computation. One difference between the two presence visualizations is particularly clear. The physical self presence is similarly minimal during fights, but Figure 14.2 shows a shift towards a stronger physical self presence between battles, while the Figure 14.3 physical self presence stays quite minimal both in and out of fights. The differences in raid familiarity provide a possible rationale: Figure 14.2, as a familiar raid, required less attention from Jaea during non-battle times, as he was already comfortable with the actions required of him during battle, and didn't need to devote resources towards planning for the restart of battle. Figure 14.3, representing a neverbefore-attempted raid, required a consistently high presence in information constellation, thus limiting the physical self presence even during non-battle time chunks. Note that although there is still a high information flow in Figure 14.2 during non-battle times, it contains some considerably different items that indicate a shift away from the in-game information. Instead of just raiding team discussions and user interface tweaking, Jaea is reading various forums and looking through Twitter, activities that are sometimes unrelated

Figure 14.4

Jaea’s customized UI before Al’Akir.

Figure 14.4 Jaea's customized

or before AI'Akir.

to the game or his current activity within the game. This divergence in activity pairs with the increased presence in physical self, as Jaea stops devoting his presence to the game-related activities, and shifts into a more relaxed state.

VISUALIZATION OF EXPERTISE

Our expert's success at play, relying on more than just his ability to play his character well, requires fluid and rapid shifts of presence. As can be seen in the presence visualizations, Jaea's ability to distribute attention to different layers means that he can balance the influx of information and information needs that arise, as well simultaneously engage within the physical space and within the game. Jaea's expertise moves beyond his ability to control his character and relies on his ability to interact with his surrounding virtual and physical spaces, as well as informational resources. Jaea's layers of presence vary in proportion with the activity he is currently engaged in during game play. The more temporally rapid the activity in the game, the larger the proportion of the game layer presence, whereas between activities the information layer or the physical self layer increases in proportion in the presence visualization. The information constellation layer always fills a larger proportion than the physical self layer. In Figure 14.3, the physical self layer is a smaller proportion because of the intensity of information flow both during fighting and non-fighting periods due to the fact that the raid is unfamiliar, which requires high levels of

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verbal coordination and information seeking even between fights. Despite the fact that all raid members studied the raid before they attempted it, the attempts were not going well and required just-in-time (Gee, 2003) information seeking in order to help them re-strategize. The information flow of a player is contingent upon the player's information literacy skills, and in this case the raid group's information literacy skills. Collective information literacy (Martin & Steinkuehler, 2010) is implemented within the raid group in order to pull from the group's collective knowledge of the game-at-large, raid strategy, player roles, specific raid layouts, boss phases, etc. Using this collective, connected cloud of information, the raid guild is able to be more effective and better informed together than they would be individually. This method allows for the capture of more than just Jaea's movements in the space of the game, the information constellation, and his physical location. It captures social interactions, use of resources, and focus in correspondence with the activities in the B-series timescale. This connection of activities to time with reference to presence offers a deeper look into the intricate functioning of expertise in the game, and demonstrates the importance of distributed presence. From this conceptualization of distributed presence, we can extend the concept of distributed cognition as discussed by Hutchins (1995) and Hollan et al. (2000). Presence illustrates distributed cognition in several ways. First, it has a unit of analysis, in this case the larger unit of a coherent WoW activity and the smaller units of partitioned activity as demarcated by B-series time. Second, there are multiple actors in the cognitive process-in this example, members of a social group, as well as a variety of resources. Finally, the actions and process are distributed across time and previous actions affect the nature of later actions. Crucially, our examination of presence is based on the fact that "people form a tightly coupled system with their environments" (Hollan et aI., 2000: 192), even as we extend the definition of environment to include the digital context of WoW and the complex constellation of information. At the same time the actions we see here are reminiscent of collective information literacy (Martin & Steinkuehler, 2010), which could be viewed as distributed information literacy, in view of Hutchins' definition of distributed cognition. Collective information literacy occurs when multiple people in a group or affinity space work together to solve a problem. This happens in both examples given earlier but most notably in the example with the group facing an unfamiliar raid. In Figure 14.3, the strategy and problem solving sessions carried out at the end of each wipe are a perfect example of collective information literacy: the raid team always used that time to determine what needed to be changed from the prior strategy, and what information players were missing in order to accomplish the raid. The use of distributed cognition and collective information literacy is not surprising given the social nature of games. However, the evidence of these two models functioning in a punctuated way within the B-series time allows for

a useful way to track the intellectual work as well as the action that goes into success in an affinity space or other collaborative spaces that requires joint activity. We offer Jaea as an exemplar of how expertise functions in a space like this, surrounded by other experts and experts-in-training working towards a common challenging goal. THE SHIFTING NATURE OF EXPERTISE Throughout his play, Jaea consistently engages in multiple simultaneous actions and thought processes. The number of actions he juggles at once combined with the speed with which he shifts between these actions is likely to seem overwhelming to a novice or non-player. This shifting distribution of presence should not be confused for partial or even split attention, but rather for the necessarily flexible nature of expertise. Jaea's focus is not splintered; rather, he is unquestioningly focused on playing his character well. The expert's shifting distributions of presence can be clearly distinguished from the metaphor of split attention: Jaea's increase in focus in one layer occurs because of activities in the other layers. Instead of 'losing' cognitive power (as in split attention), the expert's shift occurs because of a holistic understanding of the context and attentional demands. In fact, play at such an expert level cannot take place without such shifts in presence. Players who focus their attention on only one component of the game at a time cannot possibly play at the same level as Jaea. Only players who can engage in action with distributed attention and distributed information literacy across a group of players can successfully participate in highlevel raiding activities. This ability demonstrates the literacy of the player to navigate the space of the game, the physical world, and the information constellation. These literacy practices, because of their ability to help the player succeed in the game, also affect the social identity of the player within the community. This method helps to visually identify the literacy practices that take place in the virtual space, the physical space, and the information constellation, all of which are contained in the constellation of literacies that a player or person uses and experiences. Through this visualization process we can observe the literacy practices as they take place and see what practices are layered together. Expertise in World of Warcraft and other games requires masterful coordination of information resources across multiple timescales and spaces, and is the same no matter the physical age of the player. Jaea has to be able to sustain his attention across multiple spaces in his immediate vicinity, in the virtual game world, and with fellow players who are participating from other locations throughout the world. Considering that this delicate balance of shifting distributions of presence is necessary for expert-level play, we suggest that one has to consider the timescales and spaces involved all as equally valid.

Martin et al. CONCLUSIONS

This study of a single expert World of Warcraft player revealed important insights about activity as it OCCurs across multiple timescales and in multiple spaces, both real and virtual. For future studies that hope to continue with similar kinds of analyses, we recommend observing players with varied amounts of expertise. The methodology of presence that the research_ ers of this study utilize could be used to study the gameplay of any level of player. Novice players are not likely to engage these acts of distributed and collective cognition with the same ease with which laea is able to transition across times and spaces using both his physical self and in-game WoW self. Additional research could allow researchers to map out a trajectory of expertise development in terms of time, space, and literacy. We anticipate that such a trajectory will provide illumination into the development of facile shifts in particular layers of presence. We also suggest that studying an entire raid group might reveal more about how attention is divided across multiple activities and events during high-level play of World of Warcraft. A sudden shift from in-game Wo W self to physical self because of some event occurring in their physical space could have substantive effects on the attention of the rest of the players in the raid group even if they are distributed across multiple locations hundreds, if not thousands, of miles apart. As raiding players tend to operate at a fairly expert level, however, they may have means and strategies for coping with such interruptions and shifts such that it does not dramatically disrupt their play. Knowing more about how the players' manage their distributed attention to maximize their collective efforts could be useful for an array of contexts, both for study of games and beyond. Understanding how groups manage their resources across multiple spaces and times has a variety of implications in a globally connected world where teams of problem solvers are often located throughout the world as they work on shared problems.

REFERENCES Awh, E. & Pashler, H. (2000). Evidence for split attention a I foci. Journal of Experimental Psychology Human Perception and Performance 26(2): 834-846. Black, R. W. (2008). Adolescents and Online rl1/l Fiction. New York: Peter Lang. Black, R. W. (2007a). Digital design: English language learners and reader reviews in online fiction. In C. Lankshear & M. Knobel (eds.), New Literacies Sampler. New York: Peter Lang. Black, R. W. (2007b). Fanfiction writing and the construction of space. e-Learning 4(4): 384-397. Black, R. W. & Steinkuehler, C. (2009). Literacy in virtual worlds. In 1.. Christenbury, R. Bomer, & P. Smagorinsky (eds.), Handbook of Adolescent Literacy Research. New York: Guilford. Bourdieu, P. (1977). Outline ofa Theory of Practice (R. Nice, Trans.). Cambridge, MA: Cambridge University Press. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science 5: 121-1S2.

Coiro, J., & Dobler, E. (2007). Exploring the online reading comprehension strategies used by sixth-grade skilled readers to search for and locate information on the Internet. Reading Research Quarterly 42(2): 214-257. Erickson, F. (2004). Talk and Social Theory: Ecologies of Speaking and Listening in Everyday Life. Cambridge: Polity Press. Gee, J. P. (2005). Semiotic social spaces and affinity spaces: From The Age of Mythology to today's schools. In D. Barton & K. Tusting (eds.), Beyond Communities of Practice: Language, Power, and Social Context. New York: Cambridge University Press. Gee, J. P. (2004). Situated Language and Learning. New York: Routledge. Gee, J. (2003). What Videogames Have to Teach Us about Learning and Literacy. New York: Palgrave Macmillan. Gee, J. P. (1999). An Introduction to Discourse Analysis: Theory and Method. New York: Routledge. Gell, A. (1992). A-series:B-series::Gemeinschaft:Gesellschaft::Them:Us. The Anthropology of Time: Culture Construction of Temporal Maps and Images. Oxford: Berg. Gonzalez, V. M. & Mark, G. (2005). Managing currents of work: Multi-tasking among multiple collaborations. In H. Gellersen et al. (eds.), ECSCW 20005: Proceedings of the Ninth European Conference 011 Computer-Supported Cooperative Work. Paris: Springer. Gonzalez, V. M. & Mark, G. (2004). "Constant, constant, multi-tasking craziness": Managing multiple working spheres. CHI 2004 6(1): 113-120. Hollan, J., Hutchins, E., & Kirsh, D. (2000). Distributed cognition: Toward a new foundation for human-computer interaction research. ACM TrallSactions on Computer- Human Interaction 7(2): 174-196. Hutchins, E. (1995). Cognition in the Wild. Cambridge, MA: MIT Press. Kalyuga, S., Chandler, P., & Sweller,]. (1999). Making split-attention and redundancy in multimedia instruction. Applied Cognitive Psychology 13: 351-371. Lemke, J. (2000). Across the scales of time: Artifacts, activities, and meanings in ecosocial systems. Mind, Culture, and Activity 7(4): 273-290. Martin, C. (in progress). A-series and B-series Time Maps in World of Warcraft. Martin, C. & Steinkuehler, C. (2010). Collective information literacy in massively multiplayer online games. e-Learning and Digital Media 7(4): 355-365. Mayer, R. E. & Moreno, R. (1998). A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory. Journal of Education Psychology 90(2): 312-320. Stake, R. (1995). The Art of Case Study Research. Thousand Oaks, CA: Sage. Steinkuehler, C. (2008). Cognition and literacy in massively multiplayer online games. In J. Coiro et al. (eds.), Handbook of Research on New Literacies. New York: Routledge. Steinkuehler, C. (2007). Massively multiplayer online games as a constellation of literacy practices. e-Learnillg and Digital Media 4(3): 297-318. Steinkuehler, C. (2006). Massively multiplayer online video gaming as participation in a discourse. Mind, Culture, and Activity 13(1): 38-52. Steinkuehler, C. & King, B. (2009). Digitalliteracies for the disengaged: Creating after school contexts to support boys' game-based literacy skills. On the Horizon 17(1): 47-59. Stevens, R., Satwicz, T., & McCarthy, 1.. (2008). In-game, in-room, in-world: Reconnecting video game play to the rest of kids' lives. In K. Salen (ed.), The Ecology or Games: Connecting Youth. Games, and Learning. Cambridge, MA: MIT Press. Tabhers, H., Martens, R., & van Merrienboer, J. (2000, February). Multimedia instructions and cognitive load theory: Split-attention and modality effects. Paper presented at the AECT 2000, Long Beach, CA.

Super Meat Boy Special Bonus Pack Caro Williams The experience of failure is a beautiful one. Granted, failure is generally only beautiful when viewed from the other side of the currently insurmountable obstacle—but without failure, where is the glory in success? When Team Meat was designing their love letter to classic platformers (Payne & Cambell) and their experience turned from joy to horror (Wolfenstein), they pushed on. And Super Meat Boy is the beautiful reward that Team Meat—and we—received from their brutal experience. The following odes to failure and Super Meat Boy share the beauty and difficulty in pushing through aspects of life—in games and out— that cause suffering and frustration. What both Wolfenstein and Payne & Campbell tell us repeatedly, despite their very different approaches to the subject, is this: The only real failure is when you put down the controller and never return. Team Meat made the world a better place by not succumbing to their suffering and frustration—and Super Meat Boy players every day testify against failure by picking up the controller again and again. “We can do this,” they say, “even as the world transforms from pastoral woodlands to hellish nightscapes— give me just one more try…” Team Meat has left Meat Boy and Bandage Girl in our hands—fail well, and play on.

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Knuth, E., Kalish, C., Ellis, A., Williams, C., & Felton, M. (2011). Adolescent reasoning in mathematical and non-mathematical domains: Exploring the paradox. In V. Reyna, S. Chapman, M. Dougherty, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making (pp. 183-210). Washington, DC: American Psychological Association.

Adolescent Reasoning in Mathematical and Non-Mathematical Domains: Exploring the Paradox

Eric Knuth Charles Kalish Amy Ellis Caroline Williams University of Wisconsin Matthew Felton University of Arizona

The research is supported in part by the National Science Foundation under grant DRL-0814710. The opinions expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Abstract Mathematics education and cognitive science research paint differing portrayals of adolescents’ reasoning. A perennial concern in mathematics education is that students fail to understand the nature of evidence and justification in mathematics. In particular, students rely overwhelming on examples-based (inductive) reasoning to justify the truth of mathematical statements, and often fail to successfully navigate the transition from inductive to deductive reasoning. In contrast, cognitive science research has demonstrated that children often rely quite successfully on inductive inference strategies to make sense of the natural world. In fact, by the time children reach middle school, they have had countless experiences successfully employing empirical, inductive reasoning in domains outside of mathematics. In this chapter, we explore this seeming paradox and, in particular, explore the question of whether the skills or knowledge that underlie adolescents’ abilities to reason in non-mathematical domains can be leveraged to foster the development of increasingly more sophisticated ways of reasoning in mathematical domains.

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A perennial concern in mathematics education is that students fail to understand the nature of evidence and justification in mathematics (Kloosterman & Lester, 2004). Consequently, mathematical reasoning—proof, in particular—has been receiving increased attention in the mathematics education community with researchers and reform initiatives alike advocating that proof should play a central role in the mathematics education of students at all grade levels (e.g., Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; National Council of Teachers of Mathematics, 2000; Knuth, 2002a, 2002b; RAND Mathematics Study Panel, 2002; Sowder & Harel, 1998; Yackel & Hanna, 2003). Proof plays a critical role in promoting deep learning in mathematics (Hanna, 2000); as Stylianides (2007) noted, “proof and proving are fundamental to doing and knowing mathematics; they are the basis of mathematical understanding and essential in developing, establishing, and communicating mathematical knowledge” (p. 289). Yet, despite its importance to learning as well as the growing emphasis being placed on proof in school mathematics, research continues to paint a bleak picture of students’ abilities to reason mathematically (e.g., Dreyfus, 1999; Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Martin et al., 2005). In contrast, cognitive science research has revealed surprising strengths in children’s abilities to reason inferentially in non-mathematical domains (e.g., Gelman & Kalish, 2006; Gopnik, et al., 2004). Although more traditional (Piagetian) views posit children as limited to understanding obvious relations among observable properties, there is growing evidence that children are capable of developing sophisticated causal theories, and of using powerful strategies of inductive inference when reasoning about the natural world (for review, see Gelman & Kalish, 2006). In the former case, for example, children can integrate statistical patterns to form

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representations of underlying causal mechanisms (Gopnik & Schulz, 2007). In the latter case, for example, children often organize their knowledge of living things in ways that reflect theoretical principles rather than superficial appearances (Gelman, 2003). Thus, this raises something of a paradox: Why do children appear so capable when reasoning in non-mathematical domains, yet seemingly appear so incapable when reasoning in mathematical domains? In this chapter, we explore the paradox by considering the research on adolescents’ reasoning capabilities within mathematics education as well as within cognitive science. In particular, we briefly consider research that provides a portrayal of adolescents’ reasoning in mathematical and non-mathematical domains. Next, we present preliminary results from the first phase of our multi-year research effort to better understand the relationships between adolescents’ reasoning in mathematical and non-mathematical domains. We view such relationships as a means for potentially leveraging the strengths adolescents demonstrate when reasoning in non-mathematical domains to foster the development of their mathematical ways of reasoning. Finally, we discuss the implications of our research as well as its future directions. Situating the Paradox Mathematics education and cognitive science research paint differing portrayals of adolescents’ reasoning, particularly with respect to the nature of their reasoning strategies. In the world outside the mathematics classroom, children typically rely quite successfully on inductive inference strategies—empirical generalizations and causal theories—to make sense of the natural world. For example, preschool-aged children are able to interpret and construct interventions to identify causal mechanisms in simple systems (Gopnik, et al., 2004). Young children also have rich knowledge structures supporting explanation and predictions of physical, biological, and social phenomena (Gelman & Kalish, 2006). In fact, by the time children reach middle school,

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they have had countless experiences successfully employing empirical, inductive reasoning in

domains outside of mathematics.1 Not surprisingly, many students also employ similar reasoning strategies as they encounter ideas and problems in mathematics (Recio & Godino, 2001); however, they often fail to successfully navigate the transition from inductive to deductive reasoning—the latter being the essence of reasoning in mathematics. As Bretscher (2003) noted, “Proof in everyday life tends to take the form of evidence used to back up a statement. Mathematical proof is something quite distinct: evidence alone might support a conjecture but would not be sufficient to be called a proof” (p. 3). Adolescent Reasoning in Mathematical Domains It is generally accepted that students’ understandings of mathematical justification are “likely to proceed from inductive toward deductive and toward greater generality” (Simon & Blume, 1996, p. 9). Indeed, various mathematical reasoning hierarchies have been proposed that reflect this expected progression (e.g., Balacheff, 1987; Bell, 1976; van Dormolen, 1977; Waring, 2000); yet, research continues to show that many students fail to successfully make the transition from inductive to deductive reasoning.2 One of the primary challenges students face in developing an understanding of deductive proof is overcoming their reliance on empirical evidence (Fischbein, 1982). In fact, the wealth of studies investigating students’ proving

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Unfortunately, children’s experiences successfully employing empirical, inductive reasoning in elementary school also tend to engender the belief that such reasoning suffices as proof in mathematical domains. 2 The hierarchies that have been proposed, although based on empirical data, do not provide accounts regarding the actual transition from inductive to deductive reasoning. Rather, the hierarchies primarily note differences in the nature of students’ inductive reasoning (e.g., justifications that rely on several “typical” cases versus those that rely on “extreme” cases) with deductive reasoning being at the “upper end” of the hierarchies, and not how (or if) such inductive reasoning strategies can develop into deductive reasoning strategies.

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competencies demonstrates that students overwhelmingly rely on examples to justify the truth of statements (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Porteous, 1990).3 As a means of illustrating adolescent reasoning in mathematics, we briefly present results from our prior work concerning middle school students’ proving and justifying competencies. The following longitudinal data are from 78 middle school students who completed a written assessment at the beginning of Grades 6 and 7, and at the end of Grade 8; the assessment focused on students’ production of justifications as well as on their comprehension of justifications.4 In the narrative that follows, we present a representative sample of the assessment items and corresponding student responses. Justifications in which examples were used to support the truth of a statement were categorized as empirical, justifications in which there was an attempt to treat the general case (i.e., demonstrate that the statement is true for all members of the set) were categorized as general, and justifications that did not fit either of these two aforementioned categories were categorized as other.5 As an example, students were asked to provide a justification to the following item: If you add any three odd numbers together, is your answer always odd? The following two student responses are representative of empirical justifications: Yes because 7+7+7=21; 3+3+3=9; 13+13+13=39. Those problems are proof that it is true. (Grade 6 student)

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For the purposes of this chapter, we define inductive reasoning to be reasoning that is based on the use of empirical evidence, and by empirical evidence we mean the use of examples to justify statements or conjectures. Moreover, inductive reasoning is not to be confused with mathematical induction—a mathematically valid method of proving. 4 The assessment items presented below were the same for each administration of the assessment, and the same group of students completed the assessment at all three time points. 5 We have simplified the categorizations described in this chapter as we are primarily interested in highlighting the differences between empirical-based justifications and more general, deductive justifications. See Knuth, Choppin, and Bieda (2009) and Knuth, Bieda, and Choppin (forthcoming) for more detail about the study’s results.

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1+3+3=7. 3+11+1=15. Yes it would be but you will have to do it a 100 times just to make sure. (Grade 7 student)

In contrast, the following two responses are representative of general justifications: If you add two odds, the result is even. An even plus one more odd is odd. So three odds added together always results in odd. (Grade 6 student)

We know that odd and odd equals even. So even (2 odds) added together with odd equals odd. This shows us that no matter what three odd numbers you add together, the sum will always be an odd number. (Grade 8 student)

Responses categorized as other (see Footnote 2) were often restatements of the question (without further justification) or nonsensical responses (e.g., “It is not always odd because some problems are even like 1+2+3=6, 3+4+5=12”). Table 1 displays the overall results of students’ justifications for this item. As the table illustrates, a significant number of students relied on examples as their means of justification, with very little change occurring across the middle grades. We also see an increase in the number of students attempting to produce more general, deductive justifications from Grade 6 to Grade 8; yet still less than half the students produce such justifications even by the end of their middle school mathematics education. As a second example, consider students’ responses to the following assessment item: Sarah discovers a cool number trick. She thinks of a number between 1 and 10, she adds 3 to the number, doubles the result, and then she writes this answer down. She goes back to the number she first thought of, she doubles it, she adds 6 to the result, and then she writes this answer

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down. [A worked-out example, including the computations, followed the preceding text.] Will Sarah’s two answers always be equal to each other for any number between 1 and 10? In this

case it is also worth noting that students could use examples to prove that the statement is always true by testing the entire set of possible numbers (i.e., numbers between 1 and 10). Table 2 presents the results for this item; justifications based on proof-by-exhaustion are also included in the general category (only approximately 5% of students in each grade level used this method). Given the significant proportion of students whose responses were categorized as other, it is worth briefly discussing potential reasons underlying their responses. The majority of these responses were either a result of (i) students misinterpreting the problem, thinking that the end result must always be twenty (the result that was provided in the worked-out example that accompanied the problem); or (ii) students not being able to articulate a general argument— students could “see” what was going on but were unable to provide an adequate justification. In the former case, the following response is representative: “No, because the number comes out differently if you chose a number like 11. It does not come out as 20” (Grade 8 student). In the latter case, the following response is representative: “The answers will always be equal because you’re just doing the same thing” (Grade 7 student). Although the percentage of students who relied on empirical-based justifications decreased relative to the previous example, the gradelevel trend regarding the number of students providing general, deductive justifications remained about the same (again, less than 50% of the students at any grade level provided this type of justification). As a final example, consider the following item in which students were asked to compare an empirical-based justification with a general, deductive justification: The teacher says the following is a mathematical fact: When you add any two consecutive numbers, the answer is

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always odd. Two students offer their explanations to show that this fact is true [Note: an empirical-based justification and a general, deductive justification are then presented for the students]. Whose response proves that if we were to add any two consecutive numbers we would get an answer that is an odd number? As Table 3 suggests, for many middle school students, an empirical-based justification seems to suffice as proof. We see a slight grade level increase in viewing the general, deductive justification as proving the claim and, interestingly, we see a substantial decrease by Grade 7 of students who think both justifications prove the claim. In summary, the snapshot of adolescents’ mathematical reasoning illustrated above is quite typical of the findings from much of the research: adolescents are limited in their

understanding of what constitutes evidence and justification in mathematics and, moreover, they demonstrate a proclivity for empirical-based, inductive reasoning rather than more general, deductive reasoning. Although most studies have focused on adolescents in a particular grade level or across grade levels (i.e., cross-sectional studies), the research discussed above provides a longitudinal view into the development of adolescents’ mathematical reasoning. And given this longitudinal view, we see very little development as adolescents progress through their middle school years, and what development we do see falls far short of desired outcomes. Adolescent Reasoning in Non-mathematical Domains The difficulties that adolescents show with regard to mathematical reasoning, including the apparent lack of development as they progress through middle school, raise the question of whether there is some developmental constraint that limits adolescents’ mathematical reasoning. The most likely candidate would be abilities to do and understand deductive inference. The emergence of deductive inference has been a central focus of research on adolescent cognitive development, spurred in part by Piaget’s theory of formal operations. Although there is

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considerable debate within this literature, a plausible reading suggests there is nothing special about adolescence in terms of acquiring deduction. Younger children, for example, have been shown to appreciate that deductive inference leads to certain conclusions, and is stronger than inductive inference (Pillow, 2002). At the same time, however, even adults struggle to reason formally and deductively.6 Thus, deductive inference seems neither impossible before adolescence, nor guaranteed after. Rather than review the literature on the development of

deductive inference in non-mathematical domains (see Falmange & Gonsalves, 1995), we take a slightly different approach here. Similar to the mathematics education research on proof discussed above, researchers in psychology have often argued that people rely on empirical solutions to deductive problems. An interesting difference with the literature in mathematics education, however, is that these empirical-based solutions are typically evaluated quite positively in non-mathematical domains. That is, the kind of performance that makes people look like poor deductive reasoners is actually consistent with their being quite good inductive reasoners. Deductive and inductive arguments have very different qualities. On the one hand, in making a deductive argument, one endeavors to show that the hypothesized conjecture must be true as a logical consequence of the premises (i.e., axioms, theorems). On the other hand, in making an inductive argument, one seeks supporting evidence as the means for justifying that the conjecture is likely to be true. We will refer to arguments based on accumulation of evidence as “empirical.” Often times the empirical support in inductive arguments is provided by examples. The conclusion “All ravens are black” is supported by encounters with black ravens (and the

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Note that in this literature, as in almost all work in psychology, “adult” means college-aged.

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absence of no black ones).7 The critical point is that deductive arguments prove their conclusions though logic, while inductive arguments provide evidence that conclusions are likely. Again, though there is strong debate, one influential view is that people often seek evidence (make inductive arguments based on examples from a class) when asked to evaluate logical validity (make deductive arguments). One of the clearest examples of inductive approaches to a deductive problem is Oaksford and Chater’s (1994) analysis of performance on the Wason selection task. The selection task is a classic test of logical argumentation. A participant is asked to evaluate a conjecture about a conditional relation, such as “If there is a p on one side of a card, then there is a q on the other side.” The participant is presented with four cards: one each showing p, not-p, q, and not-q. The task is to select just the cards necessary to validate the conjecture. The logical solution is to ensure that the cards are consistent with the conjecture: that there are no p and not-q cards. To confirm this involves checking that there is a q on the back of the p card, and checking that there is a not-p on the back of the not-q card. In practice, most people do check the p card, but very few examine the not-q card. Rather, most people opt to explore the q card, which is logically irrelevant (both p and q and not-p and not-q are consistent with the conjecture). This behavior is often interpreted as akin to the logical fallacy of affirming the consequent (if p then q, q, therefore p). Oaksford and Chater argue that selecting the p and the q cards is actually a reasonable strategy for assessing the evidential support for the conjecture. Though the details are quite complex, they show that given reasonable assumptions about the relative frequencies of p and q, the cards selected are the most informative tests. That is, people’s behavior conforms to a normative standard of hypothesis testing (e.g., optimal experiment design). 7

There are many other sources of inductive support. For example, that one’s teacher says, “All ravens are black.” provides some reason for adopting the belief.

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The distinction turns on two different ways of construing the task. Interpreting the problem as a deductive one (the way experimenters’ intend it) can be glossed something like this: Is the statement logically consistent with the features of the four cards on the table? The

inductive construal is something like the following: Is the statement likely to be true of cards in general? Although the deductive problem can be solved conclusively, it is not really that interesting or important (who cares about these four cards?) The inductive problem can never be truly solved (absent investigation of every card in the world), however, it is just the kind of problem that people really care about and face in their everyday lives. How can past experience (the four cards) help in the future (expectations about new cards)? The idea that people often employ inductive, evidential support, strategies to solve deductive problems is part of a general approach to cognition and cognitive development that emphasizes probabilistic reasoning (Chater & Oaksford, 2008). From this perspective, most of the cognitive challenges people face involve estimating probabilities from evidence. This is straightforward for processes of categorization and property projection. Learning that barking things tend to be dogs, and that dogs tend to bark, seems to involve learning some conditional probabilities. Influential accounts of language acquisition suggest that children are not learning formal grammars (deductive re-write rules) but rather patterns of probabilities in word cooccurrences and transitions. Even vision has been analyzed as Bayesian inference about structures likely to have generated a given perceptual experience. The general perspective is that inductive inference is ubiquitous; we are continually engaged in the task of evaluating and seeking evidential support. Given the centrality of inductive inference, it should not be surprising that many psychologists argue that we are surprisingly good at it and, good at it from a surprisingly young age (Xu & Tenenbaum, 2007a; 2007b).

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There are several lines of research potentially relevant to understanding adolescents’ reasoning strategies in mathematics. Perhaps the most direct connection is with research on

evaluations of inductive arguments. In contrast to deductive arguments, which are either valid or invalid, inductive arguments can vary in strength. If people are good at reasoning inductively, then they ought to be able to distinguish better and poorer arguments, stronger and weaker evidence, for conclusions. This work has both a descriptive focus—how do people distinguish stronger and weaker evidence—and a normative focus—do people’s strategies conform to normative standards for evidence evaluation? Most work on adolescent inductive inference has focused on the problem of identifying causal relations in multivariate domains (see Kuhn, 2002, for a review). Questions center on children’s abilities to construct and recognize unconfounded experiments, distinguish between hypotheses and evidence, and generally to adopt systematic investigation strategies. Similar to the literature on deductive inference, the conclusions are generally that young children show some important abilities but are quite limited; adults are better, but far from perfect; and adolescents are somewhere in the middle. Other forms or elements of inductive reasoning show a significantly different profile: Even young children are skilled at inductive inference (see Gopnik & Schulz, 2007). Adolescents have not been the direct focus of research, but there seems no reason to believe that inductive inference skills should decline from early childhood to adolescence. Research on evidential support explores how people respond to or generate evidence. Evidence in this work consists of different kinds of examples or instances.8 The task is to make or evaluate a conclusion based on that evidence. For example, given that robins are known to 8

In much of this work, the “examples” are categories of animals. It is unclear whether category-to-category inferences (“robins” to “owls”) is different than individual-to-individual inferences (“these 3 robins” to “these 3 owls”).

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have a certain property (e.g., hollow bones), how likely is it that owls also have the property?

The evidence consists of examples known to have the property in question. These examples can be understood as premises in an argument about the conclusion: The strength of the argument is confidence in the conclusion conditional on the evidence. Osherson and colleagues (Osherson, et al., 1990) developed one of the first models and described several criteria for evidential strength. Subsequent research has explored the development and application of these and other criteria (Lopez, Gelman, Gutheil, & Smith, 1992; Heit & Hahn, 2001; Rhodes, Gelman, & Brickman, in press). Table 4 provides a list of proposed criteria; note that some of these criteria are more normatively defensible than others. Although there remains some debate about preschool-aged children, most researchers would agree that by middle childhood children use the criteria in Table 4 to evaluate examples as evidence. Thus, children judge that many examples are more convincing than are fewer, that a diverse set of examples is better than a set of very similar examples, and that an argument based on a typical example is stronger than an argument based on an atypical example. Research continues on other principles of example-based arguments, such as the role of contrasting cases (e.g., non-birds that do not have hollow-bones; Kalish & Lawson, 2007) and children’s appreciation of the importance of sampling. The general conclusion is that children, including adolescents, are similar to adults in their evaluations of evidence. Moreover, children’s evaluations accord quite well with normative standards of evidence. Adolescent Reasoning in Mathematical and Non-mathematical Domains The preceding discussion highlights some important differences between adolescent reasoning in mathematical and non-mathematical domains. In mathematics education, inductive strategies are typically treated as a stumbling block to overcome rather than as an object of study.

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Moreover, mathematics education research has focused primarily on distinctions between empirical/inductive and formal/deductive justifications, and questions such as what makes one empirical justification better than another or what constitutes better/stronger evidence has not

been well addressed.9 In a recent paper Christou and Papageorgiou (2007) argued that the skills involved in induction, such as “comparing” or “distinguishing,” were similar in mathematical and non-mathematical domains. Christou and Papageorgiou’s work showed that students can identify similarities among numbers, distinguish non-conforming examples, and extend a pattern to include new instances. Thus, identifying how adolescents use such abilities to evaluate both mathematical and non-mathematical conjectures and how they think about the nature of evidence used to support conjectures may suggest a means for leveraging their inductive reasoning skills to foster the development of more sophisticated (deductive) ways of reasoning in mathematical domains. Exploring the Paradox What kinds of skills or knowledge underlie adolescents’ abilities to reason in nonmathematical domains, and might such skills or knowledge have any relevance to reasoning in mathematical domains? There are many different accounts of inductive inference, but one fairly consistent component is a representation of relevant similarity in the domain. To make or evaluate empirical-based, inductive inferences one must have a sense of the significant relations among the examples or objects. For example, if the task is to decide whether birds have hemoglobin in their blood or not, the most informative examples will be objects similar to birds. The argument that since spiders lack hemoglobin, birds must lack it as well is not particularly convincing because spiders and birds seem very different. In contrast, knowing that reptiles have 9

Although mathematics education researchers have noted differences in the nature of empirical justifications— checking a few “random” cases, systematically checking a few cases (e.g., even and odd numbers), and checking extreme cases (e.g., Balacheff, 1987)—they have not engaged in any deeper study of empirical justifications.

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hemoglobin seems quite relevant if we believe birds and reptiles are relevantly similar. The critical question, then, is “what makes two things relevantly similar?” Other principles of inductive inference described earlier also depend on similarity relations (e.g., typical examples

are better because they are similar to many other examples). The relevant similarity relations are, in part, knowledge dependent. Airplanes are similar to birds and may be useful examples to use when making inferences about aerodynamics, however, the question about hemoglobin calls for a biological sense of similarity. Getting the right similarity relations is a critical part of expertise. For example, experts tend to see “deep” similarities (e.g., evolutionary history), while novices often rely on shallow, domain-general similarities (e.g., appearances; Bedard & Chi, 1992). Put another way, reasoning from similar cases will only be successful if one’s representation of “similar” really does capture important relations in the domain. A considerable amount of research and debate in the cognitive developmental literature involves just what kinds of similarity relations children recognize and how such relations are acquired. Some argue, for example, that evolution has equipped us to be sensitive to significant similarities (Spelke, 2000; Quine, 1969). Others argue that domain general learning principles allow children to hone in on the important relations (Rogers & McClelland, 2004). Regardless, the general finding is that quite young children seem to display useful and productive intuitions about similarity in the empirical domains studied. Even preschoolers recognize that reptiles are more like birds than are airplanes when biological questions are involved, but that airplanes may be more informative about birds when the questions involve aerodynamics (for example, Kalish & Gelman, 1992). Unfortunately, the mechanisms that have been hypothesized to underlie the development of similarity in empirical domains may fail to support a sense of mathematical similarity useful for evaluating mathematical conjectures.

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The nativist view of similarity suggests that evolutionary pressures have shaped the human cognitive system to focus on important relations. For example, snakes often look like sticks, but an organism that focused on these similarities would find itself in significant peril. Clearly, quantitative relations have adaptive significance, and a long enough history that our species could have evolved specific cognitive dispositions to represent such relations. Indeed, there are important claims of just such a “number sense” involving representations of

approximate magnitude (see Dahaene, 1999). Beyond the early grades, however, such relations are generally not important parts of mathematical thinking or reasoning. A sense of numerical similarity based on approximate magnitude is a limited basis for evaluating or making inferences about mathematical relations. The principles of mathematical relations depend on a formal system, which is too recent an invention to have had any significant selective pressure on the human cognitive system (see Geary, 1995). In geometry, basic mechanisms for representing shape provide a natural organization to the domain. It seems possible that this sense of similarity may be more productive, more related to mathematically significant properties, than representations of number. Empiricist views of the development of similarity also suggest pessimism about a mathematical sense of similarity. The empiricist idea is that children form representations in a domain by tracking statistical patterns. In the natural world, objects tend to form clusters: There are natural discontinuities (Rosch, et al., 1976). The features that are important for representing animals tend to come in groups, with high intra-group correlations among features and low intergroup correlations. For example, birds tend to fly, have wings, and have feathers. These features co-occur and tend to be distinctive from the features of mammals that walk, have legs, and have fur. These patterns in the distributions of observed features allow people to pick out informative

18

features and represent kinds or categories that reflect those distributions. A sense of mathematical similarity will also be dependent on the kinds of relations and distributions of properties observed in experience. Again, it seems likely that many of the most significant

relations among mathematical objects in children’s regular as well as school experiences may not be particularly well correlated with mathematically significant relations. A tendency to notice similarity in appearance does lead one toward a fairly useful notion of similarity among animals, because biological properties tend to be correlated with appearance. In contrast, a tendency to notice frequency or magnitude among numbers does not typically lead to a mathematically useful notion of similarity of numbers. Moreover, mathematical objects, at least numbers, have a network organization: There are many cross-cutting dimensions of similarity. In contrast, there is one, taxonomic, way of representing similarity relations among living things that seems primary (though see Ross, Medin, & Cox, 2007 on significance of ecological relations). Again, geometric objects, with a strong hierarchical organization, and a closer tie to psychological mechanisms of shape perception, may be somewhat different than numbers in this regard. Thus, an important first step toward developing a deeper understanding of students’ inductive reasoning in the domain of mathematics is to explore their representations of similarity relations among mathematical objects.10 Successful inductive reasoning depends on seeing objects as similar to the degree they really do share important features or characteristics. How do students make judgments about whether two numbers or two geometric shapes are similar? What features or characteristics do students attend to when considering the similarity of numbers or geometric shapes? How do students’ similarity judgments compare with experts’ similarity judgments? Answers to such questions may provide insight into students’ choices for the 10

Note that our use of similarity refers to conceptual similarity unless we explicitly write mathematical similarity.

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empirical evidence they use to justify mathematical conjectures, which, in turn, may provide insight into means to foster their transition to more deductive ways of reasoning.11 Assessing Similarity in Mathematical Domains The first phase of our current research was to determine which features and characteristics individuals attend to when considering whole numbers and common geometric

shapes; in particular, on what features might individuals base their decisions when determining whether a particular number or shape is typical? We conducted semi-structured interviews with 14 middle school students, 14 undergraduates, and 14 doctoral students in mathematics and engineering fields (hereinafter, STEM experts). Participants examined various numbers and shapes on individual cards and then sorted and re-sorted them into groups according to whatever principles they chose (Medin et al., 1997). The numbers and shapes presented to participants for inclusion are shown in Figures 2 and 3. Participants engaged in three types of sorts: an open sort, a prompted sort, and a constrained sort. For the open sort, participants grouped and re-grouped numbers or shapes into categories of their own choosing until they had exhausted the types of categories they deemed relevant. For the prompted sort, the interviewer grouped some numbers (or shapes) according to a characteristic and asked participants to place additional numbers (or shapes) into the group. For instance, the interviewer might place the numbers 4, 25, and 81 (all perfect squares) into a group and ask the participant to include other numbers in the group. For the constrained sort, the interviewer provided a group of numbers (such as 4, 25, and 81) and then included an additional set of numbers (such as 23, 36, 51, and 100) and asked the participants which, if any, of the 11

Mathematics education research has revealed very little insight into students’ thinking regarding their choices of empirical evidence, yet such insight is critical in helping students develop more sophisticated ways of reasoning. For example, selecting examples that provide insight into the structure underlying why a conjecture is true can offer a potential means for generating a general, deductive justification (e.g., Yopp, 2009).

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additional numbers should be included in the group. The prompted and constrained sorts allowed us to determine whether participants would sort by particular features deemed mathematically interesting, such as a number being a perfect square, versus other noticeable, yet mathematically uninteresting features, such as the number of digits of a number or the value of one of its digits. The participants’ responses to the sorting and tree building interview yielded 13 number categories and 13 shape categories denoting features deemed relevant in each domain. Table 5 presents the number categories and their meanings. One of the more salient results from the number-sorting task was the number of similarities between the middle school students and the STEM experts in terms of which features they noticed. For instance, consider the parity category. Figure 4 shows the percentage of participants from each group who sorted according to parity in the three sorts. The “Parity 1st” part of the graph shows the percentage of participants who sorted by parity as their first-choice sort in the open sort. The “Parity Open” part shows the percentage of participants who sorted by parity in the open sort, but not as their first sort, and the “Parity Prompt” section shows the percentage of participants who were able to sort according to parity only in the prompted or constrained sorts. In this case it is clear that parity was a particularly salient feature for all three groups. The similarities between the middle school students and the STEM experts led us to wonder, were there any features that one group attended to but the other did not? The factors category was the only category that appeared to be salient to the middle school students but not to the undergraduates or the STEM experts. Just over 20% of the middle school students sorted according to factors in the open sort, whereas none of the undergraduates or STEM experts sorted according to factors. There was also just one category that the STEM experts could sort by more readily than the middle school students, and this was the squared category, referring to

21

numbers that are perfect squares. Figure 5 shows the percentage of participants in each group who were able to sort by perfect squares. The “prime” category was the other category of number we expected to be more salient

to STEM experts, but this turned out not to be the case. Almost 90% of the STEM experts could sort according to primes, but 70% of the middle school students and the undergraduates could sort according to primes as well. Additionally, there were some mathematically uninteresting features that we expected the middle school students to attend to more than the experts, such as intervals, value of the digits, number of digits, contains a digit, and arithmetic. But of those five categories, differences only emerged for contains a digit and number of digits, and the differences were not large: 36% of the middle school students versus 21% of the experts sorted according to contains a digit, and 43% and 29% respectively sorted according to number of digits. Table 6 presents the categories for the number sort organized by the features to which each group of participants attended, in order from the most salient to the least. The middle school students and undergraduates were somewhat more attentive to common digits and number of digits than the STEM experts, whereas the STEM experts were more attentive to perfect squares and arithmetic relationships. Table 7 presents the categories of shape that the participants identified in the sorting task. The most salient category across all three groups was the number of sides, which all of the participants in each group used for grouping in either the first sort or the open sort. The other two categories that were also salient across all three groups were shape and size. The only category that was more salient for the STEM experts than for the other participants was the regular category. Forty-three percent of the STEM experts grouped shapes

22

according to whether they were regular, but only 14% of the middle school students sorted according to this principle, and those students did so only in a prompted sort. None of the undergraduates sorted according to regularity. There were three more categories that we

anticipated would be more salient for the STEM experts: similar, symmetry, and tessellate. This turned out to be the case only for symmetry (see Figure 6): STEM experts sorted according to symmetry more often than middle school students (47% versus 21% for both the middle school students and the undergraduate students). Contrary to our expectations, as shown in Figure 6, middle school students attended to similarity slightly more than STEM experts did (50% versus 40%). Only one participant across the three groups attended to tessellations, and this participant was a middle school student. We anticipated that categories such as size, familiar, and orientation would be ones that would be more salient for middle school students, particularly because we consider these categories to denote principles of shape that are not mathematically important. However, orientation and size were actually slightly more salient for the STEM experts, and the familiar category was equally salient across all three groups (36% of each group sorted according familiarity of shape). Twenty-seven percent of the STEM experts grouped according to orientation, while only 14% of the middle school students and 7% of the undergraduates sorted by orientation. All of the STEM experts sorted according size, versus 78% of middle school students and 85% of the undergraduates. Table 8 presents the categories of shape sorted by which features each group of participants attended to in order from the most salient to the least. We found that STEM experts noticed symmetry and regularity more than did middle school students, whereas middle school students attended to similarity and equal sides more readily.

23

Findings from the sorting and tree building study showed that in general, there were not many differences between the middle school students, the undergraduates, and the STEM experts in terms of the characteristics of number and shape that they attended to. Furthermore, we found that some of the most salient features of number included multiples, parity, primes, and intervals (i.e., the relative size of numbers). Several of the more salient features of shape included the number of sides, the shape’s size, recognizable features of a shape, and the size of its angles. These findings are important in that they reveal particular characteristics that participants find noticeable, such as a number’s relative size or a shape’s size, that matter to students but are not mathematically important from our perspective. Discussion and Concluding Remarks The results from our initial study suggest that adolescents’ and experts have very similar representations of similarity among (some) mathematical objects. In particular, adolescents did notice mathematically significant relations among the objects; part of what makes two numbers or shapes similar is that they share properties relevant to mathematical theorems and conjectures. Of course, participants did notice less significant properties as well, for example, shared digits of numbers, and shared orientation of shapes. There was some evidence that these less significant properties played a larger role in adolescents’ representations of number and shape, however, such properties also showed up in the STEM experts’ sorts. One possible explanation for this result is due in part to the extremely open-ended, unconstrained, nature of our similarity measures. Participants, for example, were not instructed to focus on “mathematical” similarity. As noted above, part of expertise consists of being able to select an appropriate similarity metric for the task at hand. We suspect that experts would tend to ignore irrelevant features (e.g., orientation) in the context of evaluating mathematical conjectures. It is less clear whether

24

adolescents would show the same selectivity—exploring the significance of contextual variations (e.g., mathematics class) is one aspect included in the next steps for this research program. The importance of the current findings, though, is that adolescents do represent mathematically significant similarity relations. The pressing question for future research, however, is how they use such relations to evaluate conjectures. In this chapter we have taken a relatively new perspective (in mathematics education research) on adolescents’ use of empirical strategies for evaluating mathematical conjectures. Rather than seeing such strategies as limited or as failures to adopt deductive strategies, we suggest that there may be value in such inductive strategies. The argument so far has been that inductive inference is a powerful and useful form of reasoning, and one that people (especially adolescents) seem both disposed to use and use relatively successfully. Our proposal is to consider inductive inference about mathematical conjectures as an object of study in and of itself. To that end, we seek to better understand the adolescents’ inductive reasoning in the domain of mathematics. The empirical work presented in this chapter is a first step in a larger project of exploring just how adolescents use empirical examples and inductive methods to reason about mathematical objects. In short, we believe inductive inference strategies should play an important role in mathematics, and understanding adolescents’ inductive reasoning may provide important insight into helping adolescents transition to more sophisticated, deductive ways of reasoning in mathematics. In closing we want to make a more extended argument in favor of our perspective that inductive inference can and should play a productive role in school mathematics? Can this kind of reasoning support the transition to more general, deductive ways of reasoning? We began the chapter by noting that inductive arguments are commonplace in mathematics classrooms among

25

middle school adolescents, and that more general, deductive reasoning is relatively rare. In contrast, we also noted that inductive arguments are important outside of mathematics and that

adolescents often employ quite sophisticated inductive strategies on many tasks that seem to call for deductive inference. The perspective from cognitive science is that people are not so much poor deductive reasoners as they are reluctant deductive reasoners. One reason for this reluctance may be that (outside of mathematics) we rarely care about the deductive implications of some set of facts or propositions. Invariably, it is the empirical significance that people seem to care about in most aspects of their lives. Yet, mathematics is different, precisely because of the demand to attend to deductive relations. To the question of the proper place of inductive inference in mathematics education we offer three responses. Although induction is not the accepted form of mathematical inference, it is a form of inference. Inductive reasoning can help students develop a feel for a mathematical situation and can aid in the formation of conjectures (Polya, 1954). It also provides a means of testing the validity of a general proof, especially where students are uncertain about the scope and logic of their argument (Jahnke, 2005). A major challenge in mathematics education, however, lies in moving students from reasoning based on empirical cases to making inferences and deductions from a basis of mathematical structures. By using more accessible inductive inference strategies, at least as an intermediate step, students may begin to appreciate that mathematics is a body of knowledge that can be reasoned about, explained, and justified. A concern with justification and explanation, even if inductively based, may support, rather than undermine, acquisition of more formal proof strategies. Inductive inferences are important mathematical strategies in their own right. Mathematical problems do not always demand formal solution approaches. This point is very

26

much akin to the value of estimation in relation to exact computation. For example, it is very

useful to be able to guess whether a novel problem will be like some familiar problem; perhaps the same solution strategies will work in both cases. One does not need a formal proof that the two problems are isomorphic. Of course, the induction may be false and the apparent similarities misleading. However, developing better inductive strategies, such as recognizing which dimensions of similarity are important, is an important mathematical skill. Even in the context of theorem proving, inductive strategies are invaluable as they can often be used to provide evidence that suggests a conjecture may be true (or false). Second, the process of producing a proof depends on intuitions about the likely value of different steps or transformations. Intuitions that certain problems are related, or that some problems are more difficult, are expectations derived from experience. The critical point is that some intuitions and perceptions of similarities will be more useful than others. If students employ inappropriate inductive strategies they will not develop adequate mathematical reasoning skills. Although empirical induction is not an accepted form of proof within mathematics, it is a form of justification (and as previously discussed, a very common form among students). If students are encouraged to reason in more familiar ways, inductively, they may come to recognize the limitations of such reasoning with regard to proof as well as the power (in terms of proving) of deductive methods. Moreover, reflecting on the strengths and limitations of inductive argumentation may be an excellent bridge to introduce deductive methods. For example, empirical methods cannot conclusively prove conjectures, but they can conclusively disprove them (by exposing counter-examples). The idea of proof by contradiction could flow naturally from discussion of this feature of inductive reasoning. A similar trajectory might work for introducing mathematical induction as a kind of systemization or grounding of empirical

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induction. For example, students could be prompted to consider the limits of empirical induction and challenged to identify how (or if) mathematical induction overcomes those limits. We argue that inductive strategies are an important and valuable part of mathematical reasoning. Yet, even from the perspective that inductive strategies are shortcomings in the long run, there is overwhelming evidence that students do rely on them. Understanding inductive strategies is critical to understanding what students are taking from mathematics instruction. For example, teachers illustrate mathematical concepts with specific instances, but what do students infer from these particular instances? In such cases, are students led to believe that examples suffice as proof? The overwhelming message from mathematics education and cognitive science is that students do use empirical, inductive, strategies to reason about their world (including mathematics). Mathematics education can either ignore such strategies by treating them as “errors” to be overcome, or it can ask whether there is some value, as instructional tools, or as important mathematical content, to supporting inductive approaches. The perspective from cognitive science emphasizes the value of induction; to be a good reasoner is largely to be a good inductive reasoner. Mathematics may be different, but that difference does not obviate the need for or value of inductive reasoning. The study of adolescents’ inductive reasoning in the domain of mathematics is at a very early stage. If the literature on non-mathematical domains is any guide, we should expect to see powerful and sophisticated strategies of inference in the domain of mathematics. Inductive inference is likely a real source of strength upon which mathematics education can build.

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Ross, N., Medin, D., & Cox, D. (2007). Epistemological models and culture conflict: Menominee and Euro-American hunters in Wisconsin. Ethos, 35, 478-515. Simon, M. & Blume, G. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31. Sowder, L. & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher, 91(8), 670–675. Spelke, E. (2000). Core knowledge. American Psychologist, 55, 1233-1243. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

van Dormolen, J. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8, 27–34. Waring, S. (2000). Can you prove it? Developing concepts of proof in primary and secondary schools. Leicester, UK: The Mathematical Association. Xu, F., & Tenenbaum, J. B. (2007a). Word learning as Bayesian inference. Psychological review, 114, 245-272. Xu, F., & Tenenbaum, J. B. (2007b). Sensitivity to sampling in Bayesian word learning. Developmental Science, 10, 288. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to the Principles and Standards for School Mathematics (pp. 227–236). Reston, VA: NCTM. Yopp, D. (2009). From inductive reasoning to proof. Mathematics Teaching in the Middle School, 15(5), 286-291.

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Grade

Empirical

General

Other

6

37%

21%

42%

7

42%

36%

22%

8

40%

46%

14%

Table 1. Categories of Student Justifications to the Three Odd Numbers Sum item.

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Grade

Empirical

General

Other

6

30%

28%

42%

7

28%

32%

40%

8

22%

42%

36%

Table 2. Categories of Student Justifications to the Number Trick item.

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Grade

Empirical

General

Both

Other

6

37%

32%

20%

11%

7

40%

39%

3%

18%

8

36%

49%

7%

8%

Table 3. Categories of Student Responses to the Consecutive Numbers Sum item.

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Principle

Amount

Diversity

Typicality

Description: Arguments

Example for conclusions concerning

with…

“all birds” hve X

More examples are stronger

(robins, sparrows, & cardinals have X) >

than fewer

(robins have X)

Dissimilar examples are

(robins, hawks, & penguins have X) >

stronger than similar

(robins, sparrows, & cardinals have X)

Typical examples are

(robins have X) > (penguins have X)

stronger than atypical Contrast

Negative examples are

(robins have X, cats lack X) > (robins

stronger than those without

have X)

Table 4. Some Examples of Criteria for Evidential Strength.

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Category

Meaning

Parity

Even vs. odd

Multiples

3, 9, 36, 81, 90 are all multiples of 3

Factors

3, 9, 15, 30, and 90 all go into 90 evenly

Prime

2, 5, 11, 17 go together because they’re prime

Composite

25, 30, and 60 are all composite numbers

Squared

4, 25, and 81 are perfect squares

Sequence

3, 9, and 15 go together because they go up by 6

Value of digit

1, 11, 21, and 81 all have a “1” at the same spot

Intervals

11, 14, 15, 17 are all between 10 and 20

Number of digits

1, 2, 3, 5, and 9 are all one-digit numbers

Contains a digit

5, 15, and 51 all contain a 5 so they belong together

Arithmetic

2, 3, and 5 are a group because 2 + 3 = 5

Relational

100, 25, 36, & 9 because 100/25 = 4 and 36/9 = 4

Table 5: Number categories and their meanings.

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Principle

Middle

Undergrad

STEM

School Multiples

86% (1)

93% (2)

93% (1)

Parity

79% (2)

93% (1)

87% (2)

Prime

50% (3)

71% (3)

80% (3)

Vale of Digit

50% (3)

29% (9)

47% (6)

# of Digits

43% (5)

36% (6)

20% (8)

Intervals

36% (6)

50% (4)

53% (4)

Contains Digit

29% (7)

43% (5)

13%

Squared

21% (8)

36% (6)

53% (11) (4)

Sequence

21% (8)

14% (11)

20% (8)

Factors

21% (8)

0% (12)

0% (13)

Arithmetic

14% (11)

36% (6)

40% (7)

Composite

14% (11)

21% (10)

20% (8)

Relational

7% (13)

0% (12)

13% (8)

Table 6: Number categories sorted according to salience.

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Category

What it Means

# of Sides

Grouping shapes with the same number of sides

Angles

Grouping shapes on angle size (all have an obtuse)

Equal sides

Two or more equal sides

Regular

Grouping regular shapes together

Shape

Resemblance to a shape (e.g., “arrows”, “sharp”)

Familiar

Common shapes you see in school

Size

Grouping large or small shapes together

Orientation

Grouping according to orientation on paper

Compose

A group of shapes that can be made from others

Tessellate

Grouping shapes that would tessellate

Similar

Grouping similar shapes together

Symmetry

Grouping symmetric shapes together

Convex/Concave

Grouping according to concavity

Table 7: Shape categories and their meanings.

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Principle

Middle

Undergrad

STEM

School Size

78% (2)

85% (3)

100% (3)

Shape

64% (2)

78% (4)

93% (2)

Angles

64% (2)

85% (2)

80% (4)

Similar

50% (5)

43% (8)

40% (11)

Concavity

43% (5)

14% (8)

40% (7)

Equal Sides

36% (5)

21% (6)

26% (11)

Composition

36% (10)

43% (9)

27% (8)

Familiar

35% (5)

43% (5)

34% (8)

Symmetry

21% (8)

21% (6)

47% (5)

Orientation

14% (8)

7% (9)

27% (8)

Regular

14% (11)

0% (11)

53% (6)

Tessellate

7% (11)

0% (11)

0% (12)

# of Sides

100% (1)

100% (1)

100% (1)

Table 8: Shape categories sorted according to salience.

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DING! WORLD OF WARCRAFT WELL PLAYED, WELL RESEARCHED

CRYSTLE MARTIN, SARAH CHU, DEE JOHNSON, AMANDA OCHSNER, CARO WILLIAMS, & CONSTANCE STEINKUEHLER

World of Warcraft (WoW) is a massively multiplayer online (MMO) role-playing game that takes place in the fantasy realm of Azeroth and boasts over ten million players. WoW was vast in scope when originally released, and has since added on more territories and character customization choices. Originally consisting of two continents, Kalimdor and the Eastern Kingdoms, two expansion packs added the realm of Outland and the continent of Northrend to the map, and, have also expanded the content of the game with the addition of new races, lands, quests, etc., and raising the level cap (highest attainable level). When designing a character, the player is offered a variety of choices, such as selecting a character’s faction (Horde or Alliance), race (10 playable races which include Night Elf, Troll, and Undead), and class (9 different classes which include Made, Paladin, and Druid). The player is able to further specialize their character by selecting two professions, as well as spending points to develop talent builds. In addition, during gameplay, decisions must be made about what armor to wear, what weapons to wield, and in what order to cast spells. The choices are vast and are able to be molded to fit a variety of playing styles, especially considering the social interactions that raids and guilds support, the wide variety of gear choices that are made and re-made regularly, the complex role-play opportunities, and the various patterns that different players develop (and swear by). As such, WoW offers a tremendous number of avenues for the player’s enjoyment and just as many avenues of study for researchers interested in informal learning, especially in online collaborative spaces. We all play WoW but our backgrounds vary beyond that, ranging from a professor who researches informal learning and MMOs, four graduate students with a variety of research interests (including math, visual studies, and literacy), and a high school senior who is an avid gamer. As players and researchers of WoW, we could ramble endlessly about the game but we have instead decided to talk about nine of our most loved things. In this essay, we delve into how WoW has redefined gaming, narrative and raid centric playing styles, as well as the multi-level social interactions in WoW, as an exploration of what we love about playing the game. Then, we explore character aesthetics and player-produced visual models, the use of math and information literacy, and finally a player’s experience with time in WoW, for our more research-oriented pursuits. Though we cannot go into much depth for each section in this short essay, we do cover a breadth of topics to offer the reader an overview of the wide variety of perspectives from which to think about WoW as players and as researchers.

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WELL PLAYED 3.0

WoW