Supporting Students' Ability to Reason about Data - Springer Link

KAY MCCLAIN and PAUL COBB

SUPPORTING STUDENTS’ ABILITY TO REASON ABOUT DATA

ABSTRACT. The purpose of this paper is to describe the role of an instructional sequence and two accompanying computer-based tools in supporting students’ developing understandings of statistical data analysis. In doing so, we also take account of the role of the data creation process in supporting students’ ability to engage in genuine data analysis. Data is taken from two classroom teaching experiments conducted with middle-grades students (ages twelve and thirteen) in the fall semester of 1998 and 1999. Through analysis of two classroom episodes we document 1) the emergence of the sociomathematical norm of what counts as a mathematical argument in the context of data analysis, and 2) the importance of the data creation process in grounding the students’ activity in the context of a problem or question under investigation. These claims are grounded in students’ ways of reasoning about data as they made arguments in the course of their analyses.

I NTRODUCTION In the analysis presented in this paper, we describe the role of an instructional sequence and two accompanying computer-based tools in supporting students’ developing understandings of statistical data analysis. In our analysis we will also take account of the role of the data creation process in supporting students’ ability to engage in genuine data analysis. The data reported in this paper are taken from two classroom teaching experiments. The first was conducted in the fall semester of 1998 with a group of twenty-nine American seventh-grade students (age twelve) and the second was conducted in the fall semester of 1999 with some of the same students as eighth-graders (age thirteen). During the twelve weeks of each teaching experiment, the research team1 assumed total responsibility for the class sessions, including teaching.2 Our primary goal for the teaching experiments was to investigate ways to proactively support middle-school students’ ability to reason about data while developing statistical understandings related to exploratory data analysis. An integral aspect of that understanding entailed students coming to view data sets as distributions. The research reported in this paper was supported by the National Science Foundation under grant no. REC-9814898 and by the Office of Educational Research and Improvement through the National Center under grant no. R305A60007. The opinions do not necessarily reflect the views of either the Foundation or the Center.

Educational Studies in Mathematics 45: 103–129, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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KAY MCCLAIN AND PAUL COBB

The instructional activities developed for the seventh-grade teaching experiment focused on univariate data sets. We then planned to introduce bivariate data sets in the course of the eighth-grade teaching experiment. Our interest in statistics was motivated by current debates about its role in school curricula (cf. Burrill, 1996; Burrill and Romberg, in press; Cobb, 1997; Lajoie, in press; Lajoie and Romberg, in press; Lipson and Jones, 1996; National Council of Teachers of Mathematics, 1989, 1991, 2000; Shaughnessy, 1992; Shaughnessy, et al., 1996). The guiding image that emerged for us as we read and synthesized the literature was that of students engaging in instructional activities in which they both developed and critiqued data-based arguments (cf. Wilensky, 1997). In the following sections of this paper, we begin by outlining the theoretical framework and methodology that guided our analyses. We then describe the instructional sequence and computer-based analysis tools that we developed while planning for and conducting the seventh-grade teaching experiment. Against this background, we present an analysis of a classroom episode that occurred in the first weeks of the seventh-grade teaching experiment. The purpose of this analysis is two-fold. First, it highlights starting points in the instructional sequence as they related to students’ current understandings. Second, it provides the reader with an understanding of the role of the data creation process in supporting the students’ understandings of what it means to engage in genuine data analysis. We follow the analysis of this episode with a brief summary of the seventhgrade teaching experiment in order to describe the students’ learning that occurred over the course of the twelve weeks of the teaching experiment. We then present an analysis of a second episode that occurred in the first weeks of the eighth-grade teaching experiment. The instructional activity that is the focus of the second episode was one of four univariate data sets given to the students as pre-assessments before introducing bivariate data sets. Our goal was to determine if there had been any regression in the students’ understandings in the intervening nine months since the completion of the seventh-grade teaching experiment. The purpose of the analysis is then to document the shifts that occurred in the students’ ways of reasoning while highlighting the evolution of the role of the norms for argumentation and the data creation process. The two activities analyzed in this paper therefore focus on univariate data sets and the computer tools developed in the course of the seventh-grade teaching experiment. It is important to note that detailed retrospective analyses have been conducted of both the seventh- and eighth-grade teaching experiments (cf. Cobb, 1999; McClain, Cobb and Gravemeijer, 2000; Cobb, McClain and Gravemeijer, in press). These analyses document student growth by ac-


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counting for the classroom mathematical practices that emerged over the course of each teaching experiment. The purpose of this paper is not to provide a summary of these analyses. It is instead to draw on these analyses through the use of representative episodes to clarify the role of both (1) the instructional sequence and computer-based tools, and (2) the data creation process in supporting students’ understandings. It is also important to clarify that we do not want to be perceived as giving agency to the instructional sequence and computer tools. Instead, we would argue that it was the use of the activities and the tools as they became constituted in the classroom that offered the means of support to advance our mathematical agenda. Inherent in this argument is an emphasis on a classroom participation structure that values the active participation of the students as they make reasoned arguments during deliberately facilitated discussions.

M ETHODOLOGY Theoretical framework The emergent perspective is the theoretical framework that guided the detailed longitudinal analyses of the classroom teaching experiments. It explicitly coordinates sociological and psychological perspectives. The psychological perspective is constructivist and treats mathematical development as a process of self-organization in which the learner reorganizes his or her activity in an attempt to achieve purposes or goals (von Glaserfeld, 1995). The sociological perspective is interactionist and views communication as a process of mutual adaptation wherein individuals negotiate mathematical meaning (Bauersfeld, Krummheuer and Voigt, 1988). From this latter perspective, learning is characterized as the personal reconstruction of societal means and models through negotiation in interaction. Together, the two perspectives treat mathematical learning as both a process of active individual construction and a process of enculturation into the mathematical practices of wider society. Individual and collective processes are viewed as reflexively related in that one does not exist without the other (Cobb, Perlwitz and Underwood, 1994). This coordination of the social and psychological perspectives within the emergent perspective is highlighted by the two columns shown in Figure 1 (cf. Cobb and Yackel, 1996). The entries in the column headed ‘social perspective’ indicate three aspects of the classroom microculture that we have found useful to differentiate when conducting analyses of the classroom community. The three corresponding entries in the column

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Social perspective

Psychological perspective

Classroom social norms

Beliefs about own role, others’ role, and the general nature of mathematical activity

Sociomathematical norms

Mathematical beliefs and values

Classroom mathematical practices

Mathematical interpretations and reasoning

Figure 1. The Theoretical Framework.

headed ‘psychological perspective’ indicate three related aspects of individual students’ reasoning as they participate in communal classroom processes. For the purposes of this paper, we will focus on the aspect of the framework that deals with sociomathematical norms (cf. Yackel and Cobb, 1996). (Analyses which draw on the mathematical practices can be found in Cobb, 1999; and Cobb, McClain and Gravemeijer, in press. We have also described and illustrated the criteria for delineating classroom mathematical practices in some detail in Cobb, Stephan, McClain and Gravemeijer, 2001.) Sociomathematical norms include what counts as a different mathematical solution, a sophisticated mathematical solution, an efficient mathematical solution, and an acceptable mathematical explanation and justification (cf. Herschowitz and Schwartz, 1999; Lampert, 1990; Simon and Blume, 1996; Voigt, 1995). In contrast, social norms can be characterized as general norms that are necessary for engaging in classroom discussions and can apply to any subject matter. Such norms include explaining and justifying solutions, attempting to make sense of explanations given by others, and challenging others’ thinking (Cobb and Yackel, 1996). As indicated in the figure, the psychological correlates of the sociomathematical norms are taken to be the specifically mathematical beliefs and values that constitute what might be termed students’ mathematical dispositions. In the perspective that we take, students are seen to reorganize their understandings of what counts as different, sophisticated, efficient, or acceptable solutions and explanations as they participate in the negotiation of sociomathematical norms (cf. McClain and Cobb, 2001). The analyses presented in this paper will focus on the sociomathematical norm of what counts as an acceptable mathematical argument and the norms for justification for these arguments. We will use sociomathematical norms to frame our analysis and to provide evidence of growth in the students’ ways of understanding. However, it is important to note that a focus on the sociomathematical norm of what counts as an acceptable mathematical argument makes assumptions about the establishment of cer-


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tain social norms. It is therefore important to note that both the social and sociomathematical norms were a focus of attention by the research team throughout both teaching experiments. Method of analysis The approach we took when conducting retrospective analyses of the data generated during the teaching experiments involves a method described by Cobb and Whitenack (1996) for analyzing large sets of classroom data. This method is an adaptation of Glaser and Strauss’ (1967) constant comparative method. The initial orientation for a retrospective analysis is provided by the tentative and eminently revisable conjectures that the research team developed while actually conducting the teaching experiment. The method involves continually testing and revising conjectures while working through the data chronologically episode by episode. These conjectures might focus on such issues as the evolution of classroom mathematical practices or the means by which individual students’ learning was supported and organized. This constant comparison of conjectures with data results in the formulation of claims or assertions that span the entire data set but yet remain empirically grounded in the details of specific episodes. The reader is referred to Cobb et al. (2001) for an illustration of this process of testing and revising conjectures about both communal classroom processes and individual students’ mathematical learning. Data sources The data sources for the analyses include videorecordings of each class session from two cameras, copies of the students’ work, two sets of field notes and the teacher’s daily journal. In addition, we conducted and videorecorded individual interviews with all students prior to the seventh-grade experiment and at the end of both teaching experiments.

I NSTRUCTIONAL SEQUENCE Identifying the ‘big ideas’ As part of the process of designing the instructional sequences to be used in the classroom teaching experiments, we attempted to identify the ‘big ideas’ in statistics. Our goal was to develop coherent sequences that would tie together the separate, loosely related topics that typically characterize American middle-school statistics curricula. The notion that emerged as central from our synthesis of the literature was that of distribution. We therefore wanted students to come to view data sets as entities that are

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distributed within a space of values (Hancock, in press; Konold, Pollatsek, Well and Gagnon, in press; Wilensky, 1997). In the case of univariate data sets, for example, this enabled us to treat measures of center, spreadoutness, skewness, and relative frequency as characteristics of the way the data are distributed. In addition, it allowed us to view various conventional graphs such as histograms and box-and-whiskers plots as different ways of structuring distributions. Our instructional goal was therefore to support students’ gradual development of a single, multi-faceted notion, that of distribution, rather than a collection of topics to be taught as separate components of a curriculum unit. Multiplicative reasoning A distinction that we made while preparing for the teaching experiments which later proved to be important is that between reasoning additively and reasoning multiplicatively about data (cf. Harel and Confrey, 1994; Thompson, 1994; Thompson and Saldanha, 2000). As an illustration, in reasoning additively, students might focus on the absolute frequency of the data in particular intervals (e.g., 15 cars are going faster than the speed limit and 30 are going slower). In contrast, in reasoning multiplicatively, students might focus on the relative frequency of the data (e.g., one-third of the cars are going faster than the speed limit). Multiplicative reasoning is inherent in the proficient use of a number of conventional inscriptions such as histograms and box-and-whiskers plots. In addition, it cuts across a number of content strands and constitutes the overarching goal for American mathematics instruction at the middle-school level. An explicit focus on multiplicative reasoning therefore opens up the possibility of coordinating statistics instruction with other areas of the mathematics curriculum. We explicitly focused on supporting students’ ability to reason multiplicatively in the last eight sessions of the seventh-grade teaching experiment. Integration of computer-based tools Statistics as a discipline has changed profoundly in the last 20 years, fueled in large part by the accessibility of computers (Biehler, 1993; Biehler, 1994; Biehler et al., 1988; Hancock, in press). The computational methods of statistical inference have been complemented by exploratory techniques that involve the creation and manipulation of graphical inscriptions. We therefore developed two computer-based tools for students to use as integral aspects of the instructional sequence. In doing so, we had to address an issue that is the focus of current debates about the role of technologies in supporting students’ mathematical learning. This debate is often cast in


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terms of the contrast between expressive and exploratory computer models (cf. Doerr, 1995). In one of these approaches, the expressive, students are expected to recreate conventional graphs with only occasional nudging from the teacher. In the other approach, the exploratory, students work with computer software that presents a range of conventional graphs with the expectation that the students will come to interpret them in relatively sophisticated ways as they use them. The approach that we took when designing computer-based tools offers a middle ground between the two approaches. It introduces particular tools and ways of structuring data that are designed to fit with students’ current ways of understanding, while simultaneously building toward conventional graphs (cf. Gravemeijer, et al., 2000). The computer tools were therefore designed as a primary means of supporting students’ learning while simultaneously providing them with tools for data analysis. This approach reflects the view that the development of increasingly sophisticated ways of reasoning about data are inextricably bound up with the development and use of increasingly sophisticated ways of inscribing data (Biehler, 1993; de Lange et al., 1993; Lehrer and Romberg, 1996; Roth and McGinn, 1998). More generally, the approach we took is broadly consistent with theoretical perspectives which treat tools and symbols as reorganizers rather than mere amplifiers of activity (Dörfler, 1993; Pea, 1993; Kaput, 1994; Meira, 1998). Nature of the instructional activities In developing the instructional sequences, we attempted to ensure that students’ activity was imbued with an investigative spirit that is characteristic of genuine data analyses from the outset (cf. Biehler and Steinbring, 1991). This implied that the instructional activities should involve analyzing realistic data sets for purposes that the students considered reasonable. As a consequence, the activities we developed involved analyzing either 1) a single data set in order to understand a phenomenon, or 2) comparing two data sets in order to make a decision or judgment. In completing these activities, students would have to structure the data in ways that enabled them to develop and substantiate arguments about a course of action that should be followed. The classroom activity structure A further design decision that we made when preparing for the teaching experiments stemmed from our concern that students should be involved in genuine data analysis which entailed their acting on data. Interviews conducted with the students prior to the teaching experiment revealed students’ conceptions of school math and reflected their prior instructional

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experiences of doing statistics in school. As an example, most students calculated the mean of every data set, but rarely reasoned about the mean in relation to the context or the situation at hand. Based on analysis of pre-interviews, we would argue that the mean held little meaning for these students other than the result of a calculation (cf. Cai and Moyer, 1995; Mokros and Russell, 1995; Pollatsek et al., 1981; Strauss and Bichler, 1988). As a result, an initial goal was to engage the students in actually analyzing data. We viewed this goal as transcending the dichotomy between process and content in that we envisioned the students engaged in genuine data analysis while focused on significant mathematical content. To this end, we developed an approach in which the teacher talked through the data generation process with the students. These conversations often involved protracted discussions during which the teacher and students together framed the particular phenomenon under investigation (e.g., AIDS), clarified its significance (e.g., the importance of developing more effective treatments), delineated relevant aspects of the situation that should be measured (e.g., T-cell counts), and considered how they might be measured (e.g., taking blood samples). The teacher then introduced the data the students were to analyze as being produced by this process. The resulting structure of classroom activities, which often spanned two or more class sessions, was therefore 1) a whole-class discussion of the data creation process, 2) individual or small-group activity in which the students worked at computers to analyze data, and 3) a whole-class discussion of the students’ analyses.

T HE SEVENTH - GRADE TEACHING EXPERIMENT The analysis provided in this section is intended to document the means of supporting the students’ mathematical development in the course of the seventh-grade teaching experiment. In particular, in this episode we analyze how 1) the instructional activities and accompanying computer tools 2) and the data creation process supported the emergence of the sociomathematical norm of what counts as an acceptable mathematical argument.


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Figure 2. Braking distance data on the Coupe and Sedan.

Braking distances instructional activity Intent of the instructional activity The instructional activity that is the focus of analysis was posed to the students during the fifth class session of the seventh-grade teaching experiment and was the first instructional activity posed using the first of the two computer-based tools. The instructional activity involved students analyzing data on the braking distances on ten of each of two makes of cars in order to determine which make of car had the better braking distance. A primary goal was that the measures as inscribed in the computer tool would be acted upon as data and that students would make reasoned decisions about how best to analyze this data. The first computer tool The first computer tool was designed to facilitate students’ initial explorations of univariate data sets and provide them with a means to manipulate, order, partition, and otherwise organize small sets of data. Part of our rationale in designing this tool was to orient the students to analyze data as measures of an aspect of a situation rather than simply to ‘do something with numbers’ (cf. McGatha, Cobb and McClain, 1999). When data were entered into the tool, each individual data value was shown as a bar, the length of which signified the numerical value of that single data point (see Figure 2). The bars could be either pink or green, allowing for two data sets to be easily compared. Our choice of this relatively elementary way of inscribing

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individual data values reflected our goal of ensuring that students were actually analyzing data. To this end, the initial data sets the students analyzed were also selected so that the measurements made when generating the data had a sense of linearity and thus lent themselves to being inscribed as horizontal bars such as the braking distance of a car or the longevity of batteries or light bulbs. A data set was therefore shown as a set of parallel bars of varying lengths that were aligned with an axis as shown in Figure 2. The tool also contained a value bar that could be dragged along the axis to partition data sets. In addition, there was a range tool that could be used to determine the number of data points within an interval. The use of the computer tool in the classroom made it possible for the teacher to support shifts in the students’ activity such that they came to act on data in a relatively direct way. Data creation process At the beginning of the seventh-grade teaching experiment, the teacher as well as the students was figuring out in action what might be involved in talking through the data creation process. As a result, many of the early discussions involved disjoint turn-taking during which students recounted personal experiences that often fit only loosely with the intent of the instructional activity. After these discussions, the teacher presented the data the students were to analyze and attempted to support their understanding of the process by which it was generated by offering an explanation. In these situations, the students did not participate in talking through the data creation process and as a result their analyses often were not grounded in the context of the question at hand. As an example, consider the braking distance instructional activity posed to the students during the fifth session of the seventh-grade teaching experiment. Students were asked to analyze data on the braking distances of ten each of two different makes of car, a Sedan and a Coupe3 . The teacher began by engaging the students in a conversation about what it feels like to be in a car when the brakes are applied very quickly. Students recounted stories from their personal experiences and talked about the importance of seatbelts and airbags. The following exchange then occurred.

McClain: What would be important to know about a car if you were driving and a small child or a student stepped out in front of the car? Bill: What the brake ratio is. McClain: What do you mean by brake ratio? Bill: How fast it takes the car to stop.


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McClain: Okay, I’m going to call that braking distance (writes ‘Braking distance’ on the board).

Several other students shared stories of being in accidents and about braking distances on cars. The teacher then posed the task by explaining that she was considering purchasing either a Sedan or a Coupe and stated that safety considerations were very important to her. She asked the students to help her by analyzing data on the braking distances of the two types of cars and making a recommendation about which make of car they thought she should buy based on this information. She then showed the students data on the braking distances of ten cars of each type and explained that this data had been gathered by conducting tests at a test track. The students raised numerous questions about how this could be done fairly. They referred to the road conditions, the driver, and the condition of the cars’ brakes. The teacher addressed each of these issues by further clarifying the testing procedures. After she and the students had resolved these issues, students were asked to analyze the data in order to make a recommendation about the safety of the two types of cars. It is important to note that although the teacher engaged the students in a conversation related to braking distances and the importance of being able to stop quickly, they did not discuss either what data would be needed to make a good decision or how that data could be generated. In particular, the students were not asked to consider how one might test the effectiveness of a car’s brakes or how that data could be gathered. As a result, their framing of the instructional activity was superficially grounded in discussions of personal experiences related to riding in cars rather than in a process of data creation that would generate measures to be used in answering the question at hand. This in turn contributed to some of the students’ subsequent inability to situate their analysis in the context of the investigation. Data analysis The students’ initial introduction to the data involved the teacher handing the students hard copies of the data as it would appear inscribed on the computer tool. After they understood the inscription, they were introduced to the data on the computer tool. They then worked in groups of two or three at computers and used the tool to develop and refine their arguments. As the teacher and other members of the research team monitored the students’ activity, we noted that many groups used the value bar on the computer tool to determine the numerical values of the bars and subsequently calculate the mean. The first student to present her analysis in the whole-class discussion stated that she had found the value of each bar and

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then calculated the average of each set to find which average was smaller. A shift occurred in the form of argumentation when a second student, Maria, explained that she had approached the task in a similar way. Maria: The longer lines make you think that this one (points to bars showing the braking distance of the ten Coupes) is going to have more distance. McClain: What do you mean by more distance? Maria: A farther distance when you put the brake on. But, then really it has shorter lines than the other one has. McClain: Questions? Kent? Kent: Okay, I’m just going to restate what I think Maria said. McClain: Okay, Maria, Kent is going to say what he thinks you said. Your job is to listen and see if you agree. Everybody else’s job is to try and understand Kent. Kent? Kent: I did it pretty much the same thing except I averaged too, but what she is saying is that most people, the first thing they are looking for is the biggest, okay? But then they are not looking that in the pink [Sedan] the shortest ones are not as short as the green [Coupe]. You see because those are long they add up (points to the Sedan data projected on while board). The short ones add up.

In this exchange, Kent and Maria argued that although two of the ten Coupe cars that were tested had the longest braking distances, several of the remaining eight Coupes had the shortest braking distances. These shorter distances ‘make up’ for the longer ones and result in a better average braking distance for the Coupe. In contrast to the first student who had also calculated the mean, Maria and Kent were dealing with braking distances as measures and reasoned about the means in qualitative terms, the implicit metaphor being that of equalizing. In addition, although Kent was talking in terms of pinks and greens (the colors of the bars on the computer display), there is some indication that he was reasoning in terms of quantity and that the bars were attributes of individual cases that had been measured. This was the first time that the mean had been discussed in this manner in the course of a whole-class discussion and represented a shift in the nature of the classroom discourse in which the students were participating. In particular, Maria was referring to the graph of data when she pointed to the ‘longer lines.’ In this way she was speaking of the bars as data which formed the basis of her argument. This stands in contrast to merely reciting a procedure used to yield a number (e.g. the mean). However, the issue of whether the mean might be a useful way to assess the relative safety of the two types of cars did not arise. The exchange focused on the processes of estimating the means of the two data sets rather than on justifying the relevance of the mean with respect to the decision to be


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made. We would argue that this aspect of their activity was grounded in their instructional history (cf. Cobb, 1999). At this point in the discussion, a second shift occurred in the conversation. Wes demonstrated a different way to think about the instructional activity that involved focusing on the consistency of each data set. He used the range tool to ‘capture’ the data values and talk about consistency within the sets. Wes: I got something else. See, the [Sedan] is more consistent on stopping. See, they got like right here a range (points to endpoints of range of Sedan data), it’s more consistent of stopping. McClain: That’s a very good point. Wes: But the [Coupe], it stops on different ones. McClain: So, you are saying that the [Sedan] is more consistent. Wes: And then, but with the [Coupe] if you bought the car and it happened and you’re going and this happened to you, you couldn’t rely on it to stop by a certain point but with the [Sedan] it’s pretty much within a range where it will stop. Sharon: I don’t get it. Wes: Okay, if you bought the [Sedan] car and had to stop all of a sudden, you could rely on it to stop between about 53 feet and 72 feet and that would be better. But with the [Coupe], see, it has all these different places where it stopped so you don’t know if it’s gonna stop short or you’re gonna crash into whatever it is, so you wouldn’t rely on the distance it takes to stop. Dan: What is consistent? Rob: It does the same thing every time or close to it. McClain: That’s a really nice argument that Wes is making. He thinks it’s kinda important if you own a car to know that if you have to stop it’s gonna stop within this (points to range of Sedan) . . . that the car it’s gonna stop within this range but this car (points to data on the Coupe) could stop anywhere.

Several students then volunteered that they had not previously considered Wes’ point but now agreed with his argument. It appeared that in doing so, they were beginning to take a stance to data analysis that placed a high premium on organizing data in ways that were relevant to the question at hand. Wes’s argument was accepted as valid because his focus on the spreadoutness of the data was judged to be more pertinent than a comparison of the means when deciding which type of car was safer. As a result of his explanation, Wes initiated a renegotiation in the sociomathematical norm of what constitutes a mathematical argument. In retrospect, this shift in the students’ understanding of the purpose of data analysis proved to be highly significant in that the sociomathematical norm gradually became established that students had to explicitly justify the ways in which they

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Figure 3. Data displayed in the second computer tool.

structured data with respect to the purpose or question at hand (cf. Cobb, 1999). During the following seven weeks, students continued to engage in data analysis activities using the first computer tool. The characteristics of data sets that emerged as significant during whole-class discussions included the range, maximum and minimum values, the median, and the number of data points above or below a certain value (cf. Cobb, 1999; McClain, Cobb and Gravemeijer, 2000). The accompanying mathematical arguments focused on the afore mentioned features of the data sets and were grounded in the problem or question at hand. However, although the sociomathematical norm of what counts as a mathematical argument was continually being renegotiated in the course of discussions, we would argue that at this point in the teaching experiment, data sets were constituted in classroom discourse as collections of data rather than distributions (cf. Cobb, 1999). Description of second computer tool The second computer tool was introduced during the eighth week of the seventh-grade teaching experiment. It can be viewed as an immediate successor of the first in that the endpoints of the bars that each signified a


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single data point in the first tool have, in effect, been collapsed down onto the axis so that a data set was now shown inscribed as an axis plot as shown in Figure 3. The tool offered a range of ways to structure data. Two of the options can be viewed as precursors to standard ways of structuring and inscribing data. These involve organizing the data into four equal groups so that each group contains one-fourth of the data (precursor to the box-and-whiskers plot) and organizing data into groups of a fixed interval width along the axis (precursor to the histogram). However, the three other options available to students do not correspond to graphs typically taught in school. These involve structuring the data by (1) making your own groups, (2) partitioning the data into groups of a fixed size, and (3) partitioning the data into two equal groups. The first and least sophisticated of these options simply involves dragging one or more bars to chosen locations on the axis in order to partition the data set into groups of points. The number of points in each partition is shown on the screen and adjusted automatically as the bars are dragged along the axis. The activities the students completed when the second computer-based tool was first introduced involved analyzing data sets with an equal number of data points. The way of reasoning with the second computer tool that emerged involved using the various options to partition the data sets (cf. Cobb, 1999; McClain, Cobb and Gravemeijer, 2000). For example, in the first instructional activity in which this tool was used, the students were given data on the speeds of sixty drivers on a particularly busy highway in the local area both before and after a police speed trap had been in place (see Figure 3). The task was to determine if the speed trap was effective in slowing the speed of the traffic and thus reducing accidents. A variety of solutions were discussed including 1) partitioning the data at the speed limit by using the create your own groups feature on the computer tool and comparing the number of drivers going above the speed limit before and after the speed trap was introduced, 2) using the fixed interval feature and comparing numbers of drivers in corresponding intervals (e.g. the number of drivers going between 50 and 55 mph both before and after the speed trap), and 3) envisioning the data as ‘hills’ and reasoning about the shift in the hill. We viewed this last solution as an important first step in seeing the data set in global, qualitative terms by referring to its shape in terms of qualitative relative frequency (cf. Cobb, 1999). The last series of activities in the instructional sequence also involved the second computer tool, but the data sets had different numbers of data points. Having students investigate these activities provided means for further supporting shifts in the students’ reasoning toward multiplicative

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ways of structuring the data since direct additive comparisons were insufficient in these situations (cf. Cobb, 1999; McClain, Cobb and Gravemeijer, 2000). By the end of the seventh-grade teaching experiment, all but four of the twenty-nine students developed ways of structuring data to make their arguments that involved multiplicative reasoning about distributions of data. These students were able to both structure and then subsequently reason effectively about comparisons across data sets with unequal numbers of data points. In addition, these students understood that the data in a particular interval was a qualitative proportion of the entire data set rather than a mere additive part of it (cf. Cobb, 1999; McClain, Cobb and Gravemeijer, 2000; Cobb, McClain and Gravemeijer, in press). It is important to note that we attribute the students’ developing ways of reasoning not only to the support offered by the instructional activities and the computer-based tools, but also to the increased attention given to the data creation process (cf. Tzou, 2000). Students routinely engaged in analyses in which their activity was grounded in the context of the investigation. In particular, in the course of the seventh-grade teaching experiment, there was a hand-over of responsibility such that the students viewed it their responsibility to clarify not only the goal of the analysis, but also the method of data generation. In doing so, they also clarified which attributes of the phenomenon would be relevant, and how to measure them. As a result, their activity took on the characteristics of genuine data analysis where they investigated the data in order to identify relevant trends and patterns. This, in turn, contributed to the development of more sophisticated forms of mathematical argumentation.

A NALYSIS OF PERFORMANCE ASSESSMENT INSTRUCTIONAL ACTIVITY

Ambulance response time instructional activity The ambulance response time instructional activity was the second instructional activity posed to the students during the eighth-grade teaching experiment. Our primary goal was to pose an instructional activity involving two univariate distributions with unequal number of data points and use the students’ analysis to determine if there had been any regression since the end of the seventh-grade teaching experiment. We wanted to determine if students would still reason multiplicatively in task situations that involved unequal data sets before introducing bivariate distributions. The specific instructional activity that is the focus of analysis involved analyzing the response times of two ambulance companies in order to make a


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Figure 4. Ambulance response time data on Acme and Lifeline.

recommendation to the school board about which service to use. The data on the response times of the ambulance companies was the first occasion in which the students worked with what we came to call ‘stacked data’ rather than with data that was continuous along the axis. In this case, the stacking of data was a consequence of the data collection process in which response times were rounded to the nearest minute (see Figure 4). Data creation process The teacher introduced the instructional activity by telling the students that the school board planned to arrange a contract with one of two ambulance companies. In the ensuing discussion, the students raised numerous questions about the need for ambulance service in a school system. They discussed accidents that occur in schools and the schools’ responsibility for students’ safety. They went on to question the service provided by an ambulance company and their expectations. In the next part of the conversation, the students generated a list of items on which they would want to collect data if they were making the decision. These included the response time, location, the qualifications of the staff, and the quality of the equipment for each company. Against this background, the teacher told them that the board had already considered the location and qualifications in narrowing the choice to two companies and was now faced with analyz-

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ing the data on response times. The discussion then focused on how one might collect this data and the importance of making sure that the data on both companies were collected in a similar manner. In this process, the students raised a number of concerns that included the length of the runs made by each company, the traffic conditions, and the locations of the ambulance stations. It was only after these issues were resolved to the students’ satisfaction that they began to analyze the data. Consistent with conversations that occurred at the end of the seventhgrade teaching experiment, the students actively engaged in discussion of the data creation process with little prompting from the teacher. In addition, many students delineated aspects of the task situation for which data should be collected. These students were also able to anticipate questions that might be asked of the data and could speculate about how to account for variability in the collection of data (cf. Tzou, 2000). In doing so, they were able to define aspects of the data collection process that should be considered in order for the data to be justifiable measures of the ambulance response times. This in turn suggests that these students could anticipate the implications of the data collection process for the types of conclusions that they would be able to draw when they analyzed the data. Data analysis Following this discussion, the teacher showed the students data on the response times of the two ambulance companies inscribed in the second tool as shown in Figure 4. As the students worked in pairs at the computers to analyze the data, the teacher and other members of the research team monitored their activity in order to make decisions about the most effective way of structuring the subsequent whole-class discussion. Both these initial observations and a detailed retrospective analysis indicate there was no regression in the students’ ways of structuring and organizing data (cf. Cobb, McClain, and Gravemeijer, in press). The majority of the students were able to reason multiplicatively about the data sets. In addition, most of the students hid the individual data points when conducting their analyses (a feature on the tool) and inferred trends and patterns in the data from the resulting graphs. We find this significant in that we infer that many of the students were able to imagine the shape of the data behind the inscription. As an example, consider the second pair of students who shared their analysis during the whole-class discussion. They explained that they had partitioned the data into two equal groups by using one of the features on the tool as shown in Figure 5. As they talked, the teacher projected their way of organizing the data on the white board and wrote ‘1 /2 ’ in each of the intervals to clarify their approach. The two students then explained that the


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Figure 5. Ambulance data partitioned into two equal groups.

lower 50 percent of the response time data on Acme Ambulance Company fell in the range of six to ten minutes whereas the lower 50 percent of the response time data on Lifeline Ambulance Company fell in the range of six to twelve minutes. This, in their judgment, was justification for choosing Acme Ambulance Company. The next pair of students who presented their argument built on this analysis by explaining that they had reasoned about the data in a similar manner but had partitioned the data into four equal groups, again using a feature on the tool (see Figure 6). Kent then summarized their reasoning as follows:

Kent: Seventy-five percent of Acme ambulances do their job in 13 minutes. Whereas a little over 50% of Lifeline ambulances took 13 minutes to complete their task or 75% in 16 minutes.

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Figure 6. Ambulance data partitioned into four equal groups.

They went on to explain that when they had viewed the data partitioned in four equal groups they had noticed that if the data sets are partitioned at thirteen minutes, only 50% of the Lifeline ambulances were below this value compared to almost 75% of the Acme ambulances. As the discussion continued, Kent made the following observation: Kent: Look how concentrated it is here (points to lower 50% of the Acme ambulance data). That’s 50%. In the lower half, that’s where you want the ambulances, you want it right in there as soon as it can. So all these dots are concentrated in here. But when you come down here [to the Lifeline data] there’s only one [25% interval], it’s only one so it’s all spread out. They (points to lower 50% of Acme data) are all concentrated between 6 and 8 and 8 and 10 but down here (points to the first 25% of the Lifeline data) it’s 6 and 10.

In giving this explanation, Kent based his argument in favor of Acme on a concern for how the data were distributed with a focus on the concentration or relative density and thus on the relative frequency. Although this type of reasoning was by no means universal, mathematical arguments that focused on the proportion of the data in various intervals did appear to be


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both understood and accepted as valid by all members of the classroom community. The reader will recall that one of the mathematical issues that emerged for us as we developed the instructional sequence for the seventh-grade classroom teaching experiment was the importance of students coming to reason multiplicatively about data sets. In this regard, Konold et al. (in press) argue that a focus on the rate of occurrence or proportion of data within a range of values is at the heart of what they term a statistical perspective. As discussions of solutions to the ambulance data instructional activity involved a concern for the proportion of data within various ranges of values, most of the students appeared to be developing this statistical perspective. In particular, in this episode we see evidence of some students’ ability to reason about data structured in multiplicative terms. In addition, those students were able to construct and subsequently justify arguments based on data structured in this manner. It is worth noting that throughout the discussion the students continued to raise questions about the validity of their analyses by making reference to the process that generated the data. We viewed these questions and issues as strong indicators that most of the students were developing a relatively deep understanding of data creation and its implications for the conclusions that could legitimately be drawn from the subsequent analysis. This is indicated by the manner in which they were able to step back from their analyses and critique them in light of the appropriateness of the data sampling and collection techniques, and then question the validity of predictions and inferences made from such analyses. Further, the students routinely critiqued the conclusions drawn from data by pointing to possible inadequacies in the data creation process (cf. Tzou, 2000). In doing so, they appeared to realize that the validity of conclusions depended crucially on the adequacy of the process by which the data had been generated. C ONCLUSIONS In reflecting back on the episodes presented in this paper, we have documented 1) the emergence of the sociomathematical norm of what counts as a mathematical argument in the context of data analysis, and 2) the importance of the data creation process. In doing so, it is important to note that the first episode illustrated a significant shift in orientation to data analysis. At the beginning of the seventh-grade teaching experiment, many of the students typically compared univariate data sets by calculating or estimating the means without considering whether this statistic was relevant to the question at hand. The sociomathematical norm that began

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to emerge in the first session where the computer tool was used was that of justifying the appropriateness of the ways of structuring and organizing data with respect to the question under investigation. This was made possible by the renegotiation of the sociomathematical norm of what counts as an acceptable mathematical argument. Students’ discussions of their data analysis activity provided opportunities for shifts to occur in ways of reasoning about the question under investigation. We saw evidence of this in the braking distance instructional activity in the solutions offered by both Maria and Wes. The shift from a calculational procedure to an argument grounded in an investigation of the data served to re-orient the purpose of the instructional activity. As the seventh-grade teaching experiment progressed, the data analysis activity often resulted in all students agreeing on a particular course of action or judgment. However, students would still engage in lengthy discussions focused on the adequacy of the different mathematical arguments. In doing so, they would tease out aspects of an argument and accompanying inscription that were either problematic or valid and work to establish what they would characterize as the ‘best’ way to present the data. We see evidence of this in the second episode presented in this paper when Kent and his partner built on the solution offered by the first pair of students. They were able to clarify the differences by noting what could be observed about the data when it is partitioned into four equal groups. Although both groups reached the same conclusion – ACME Ambulance Company has the better response times – the students were still engaged in teasing out the aspects of the data that would best substantiate this conclusion. In other words, they continued to refine the mathematical argument by pointing to significant features of the distributions as captured by the inscriptions. Concurrent with the development of the sociomathematical norm of what counts as a mathematical argument was the development of an understanding of the importance of the data creation process. One of the initial goals of the teaching experiment was to ensure that all students were acting on data. To support this development, the teacher and students engaged in the process of talking through the data creation process. As we have indicated in the two analyses presented in this paper, this was an emergent process that developed over the course of the seventh-grade teaching experiment. As the students were inducted into a data analyst perspective, they and the teacher began to realize the importance of the data creation process. As we have noted, these conversations initially took the form of ‘story telling’ during which different students recounted personal events sometimes only vaguely related to the issue at hand. As the sociomathematical norm of what counts as a mathematical argument was renegotiated,


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there was a concurrent need to clarify the data creation process in order to give validity to the arguments. In this way, the students and teacher came to appreciate the importance of the process with respect to the arguments. As a result, as the teaching experiment progressed, students began to assume a greater responsibility for clarifying issues that could impact decisions based on the data. In our initial design efforts, we viewed the instructional sequence and the accompanying computer tools as critical means of support. In doing so, we did not give agency to the sequence or the computer tools, but viewed them as a resource that could support the research teams’ goals. In particular, we would argue that the design of the first computer tool facilitated most of the students coming to act on data. The design decisions that the research team made about the first computer tool were, in fact, focused specifically on 1) supporting students’ ability to act on data and 2) the emergence of data as located in a space of values. The use of horizontal bars to represent the values was intended to provide the students with an inscriptional form of the data that built on the actual generation of the data values. The teacher capitalized on this in action by pointing to the bars as she described the process of testing the braking distances in the first instructional activity. She also asked the students to explain what was happening in the car at certain points along the bar (e.g. at the very beginning of the bar, in the middle, or at the end of the bar). These discussions helped build imagery of the situation through the graphs that supported the students’ subsequent data analysis activity. In addition, the value bar and range tool offered support for analyses based on partitioning data to make an argument. This type of mathematical activity was further supported by the features on the second tool which allowed the students to continue to partition data sets, but also made it possible for the introduction of larger data sets of unequal numbers of data points. This provided an opportunity to make the distinction between additive and multiplicative reasoning explicit. Multiplicative reasoning initially emerged in qualitative forms as students focused on perceptual shapes in the data (e.g. hills). As the students continued to refine their arguments, we observed a shift toward multiplicative ways of reasoning that involved graphs to describe how the data were distributed. It is important to clarify that the purpose of the instructional sequence developed for the seventh-grade teaching experiment was not that the students might come to create specified graphs in particular situations or calculate measures of central tendency correctly (most could already do the latter, although with little understanding). Instead, it was that they might develop understandings of important statistical ideas as they used

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the computer tools to structure data and create databased arguments. It was for this reason that whole-class discussions throughout the classroom teaching experiment focused on the ways in which students organized data in order to develop arguments. In addition, students seemed to reconceptualize their understanding of what it means to know and do statistics as they compared and contrasted solutions. The crucial norm that became established from the beginning class sessions of the seventh-grade teaching experiment was that of explaining and justifying solutions in the context of the problem being explored. This is a radically different approach to statistics than is typically introduced in middle schools. It highlights the importance of middle-school curricula that provide opportunities for students to engage in genuine problem solving that supports the development of central mathematical concepts (cf. Moore, 1996).

N OTES 1. Members of the research team were Paul Cobb, Kay McClain, Koeno Gravemeijer, Maggie McGatha, Jose Cortina, Lynn Hodge, Carrie Tzou, Carla Richards, Nora ShuartFarris. 2. The way that the research team functioned while conducting the classroom teaching experiments involved McClain taking primary responsibility for the teaching. However, on numerous occasions Cobb played a critical role in contributing to whole-class conversations. In addition, additional members of the research team circulate around the room to monitor students’ activity as they work individually or in small groups. The team then discusses which solutions to highlight in the subsequent discussion. McClain is the teacher quoted in the classroom episodes. 3. The actual car names have been removed and generic pseudonyms used.

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Vanderbilt University