Designing and Developing a Digital Mathematics Curriculum Dr. Judith Olson Curriculum Research & Development Group University of Hawai‘i at Mānoa United States
[email protected] Dr. Melfried Olson Curriculum Research & Development Group University of Hawai‘i at Mānoa United States
[email protected]
ABSTRACT: This paper is a report on the work to create a digital curriculum for struggling students. The Curriculum Research & Development Group developed A Modeling Approach to Algebra, a curriculum designed and developed to support ninth-grade students taking Algebra I. Technology that provides the opportunity to interact with dynamic representations of concepts for classroom instruction is integrated throughout the curriculum both for teacher and student use. In addition to the inclusion of technology within the curriculum, the curriculum package is available digitally. Specifically, PublishView™ created by Texas Instruments ™, was used as the vehicle to provide access to student pages, annotated student notes, teacher notes, and links to technology for explorations outside of the classroom.
INTRODUCTION To successfully complete the mathematics requirements created by adopting the Common Core State Standards in Mathematics (CCSSM) (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) raises the bar for students as well as for their teachers. This is of particular concern for Algebra I, the traditional entry point in the high school mathematics curriculum. Such is the case in Hawai’i where pre-algebra is no longer a high school course even though there are high school students who most likely will encounter difficulties when enrolled in Algebra I. Partially as a result of these policies, approximately one in three students do not succeed in high school Algebra I (Gottlieb, personal communication, Spring 2011). To address the Algebra I failure rate, a course Modeling our World I (MOW I) focusing on modeling and providing opportunities for students to learn mathematics in a more investigative manner was established. Although designed for struggling learners, it is intended that MOW I be taken concurrently with Algebra I. As such, MOW I is not viewed as a remedial course but as one in which students learn content that complements that of Algebra I. The curriculum and digital materials are briefly described below followed by a short summary.
CURRICULUM MATERIALS The Curriculum Research & Development Group (CRDG) at the University of Hawai’i was contracted by the Hawai‘i Department of Education to design and develop the curriculum materials, A Modeling Approach to Algebra (AMAA) (Olson, J., Olson, M., Slovin, Venenciano, & Zenigami, 2013), for the MOW I course. The CCSSM together with materials from CRDG’s previous curriculum research and development projects, e.g. Algebra I: A Process Approach (Rachlin, Matsumoto, Wada, & Dougherty, 2001), Reshaping Mathematics for Understanding (Slovin, Venenciano, Ishihara, & Beppu, 2003), provided a research base from which to begin the development for AMAA. The modeling standards embedded in the CCSSM were established as the framework around which to build the curriculum materials.
-176-
E-Learn 2013 The AMAA curriculum materials are designed around the premise that there is more to learning algebra than memorizing formulas and following rules to determine answers to questions. The lessons, built around problems as compared to exercises, emphasize the use of models, promote the investigation of open-ended problem solving tasks, and provide multiple opportunities for students to develop concepts, generalizations, and skills. In addition, there is heavy emphasis on the CCSSM eight Standards for Mathematical Practice throughout the investigations to develop positive student dispositions for learning mathematics, encourage student perseverance, promote classroom discourse, and emphasize communication of mathematical ideas. Students are asked to model, represent, graph, write about, and discuss their strategies for investigating and solving problems as they begin to internalize algebraic ideas and develop an understanding of algebraic techniques. The Teacher Notes that accompany each lesson give guidance for instructional strategies for enhancing the efforts of the students, helping students learn the content and communicate their thinking. The development of the teacher materials is reflective of extensive prior work in professional development projects conducted by CRDG (Olson, M., Zenigami, Slovin, & Olson, J., 2011) and feedback from teachers who piloted the AMAA materials during the 2012 – 2013 school year. The AMAA content is organized according to the five critical areas identified in the CCSSM for the Traditional Pathway: High School Algebra I. These areas are organized into five units, Unit 1 Relationships Between Quantities and Reasoning with Equations, Unit 2 Linear and Exponential Relationships, Unit 3 Descriptive Statistics, Unit 4 Expressions and Equations, and Unit 5 Quadratic Functions and Modeling. Because it was thought that the modeling approach together with the emphasis on problem solving would be new for students, a preliminary unit was also created. Unit 0 Getting Started introduces students to the problem solving investigations and processes used throughout the course. Because students for whom this course was intended often do not have experience conducting mathematical investigations, Unit 0 starts with problems to highlight features of modeling and introduces students to the specific modeling cycle suggested by the CCSSM (2010, p. 72–73). This cycle is shown in Figure 1. Teachers are encouraged to have students use the modeling cycle to write solutions to open-ended problems.
Figure 1 Unit 0 lessons also provide opportunities for the class to establish norms for an environment critical for productive classroom discourse. These norms may include how to conduct classroom discourse, working together to solve problems, listening to the ideas of others, and developing a willingness to share thinking. It also helps students develop important dispositions necessary for success in a mathematics classroom. Unit 0 further initiates the focus on standards for mathematical practice (CCSSM, 2010) that empathize actions expected of mathematically proficient students. It is expected that students will engage in and embrace these practices with greater proficiency as they progress through the materials.
DIGITAL MATERIALS In the development of the curriculum materials and associated Teacher Notes, technology was considered important in two ways. First, as much as possible, the curriculum is built around ideas that could be dynamically explored with spreadsheets, graphing, or links to internet explorations. Second, the curriculum materials are available digitally for the teachers and students. TI-Nspire™ technology has been used to accomplish this. The TINspire™ Teacher Edition PublishView™ feature has been used to digitally format the interactive curriculum materials. Teachers can display Student Pages to use with lessons or digitally access Teacher Notes and Annotated Student Pages that can be accessed for planning and instruction. The software has also been used to create TINspire™ .tns documents that consist of interactive spreadsheets and graphs teachers can use in a demonstration with the whole class or send to TI-Nspire™ handhelds via networked classrooms using TI-Nspire Navigator™. Links to
-177-
E-Learn 2013 these .tns documents are provided within the Teacher Notes and Annotated Student Pages so they can be easily accessed at appropriate times during instruction. Using TI-Nspire Navigator™ features, the screens of the students’ handhelds can be captured and individual students can be selected as class presenters as they explain their work. The teacher can also use TI-Nspire Navigator™ for real-time formative assessment through quick polls and for collecting students’ graphs, equations, and spreadsheet lists. The features of TI-Nspire technology are discussed in more detail in the sections that follow. TI-NSPIRE PUBLISHVIEW™ TI-Nspire PublishView™ provides layout and editing features for presenting mathematics concepts in a document where TI-Nspire™ applications can be interactively and dynamically linked with supporting media. Documents can be created that include formatted text, TI-Nspire™ applications (Calculator, Graphs, Lists and Spreadsheets, Data and Statistics, Geometry, Notes, Vernier DataQuest™, and a Question App), images, hyperlinks, links to videos, and embedded videos in a format that is suitable for printing on a standard piece of paper, publishing to a website or blog, or for use as an interactive worksheet. Multiple dynamically linked pages can be embedded in the same document. This allows for all Student Pages, Teacher Notes, and Annotated Student Pages in the AMAA curriculum materials to be formatted in TI-Nspire PublishView™ documents. These are intended for teachers’ instructional purposes including both planning and conducting lessons. These documents are linked so teachers can use the TI-Nspire Teacher Edition™ software to present a problem to students, link to Teacher Notes or Annotated Students Pages for assistance during instruction, link to an interactive TI-Nspire™ .tns document for whole class discussion or, if available, sent to students’ TI-Nspire™ handhelds. See Figures 2 – 7 for example of the Table of Contents in Figure 2, list of lessons in Unit 2 in Figure 3, Teacher Notes page showing links to the student page for the Painted Cube lesson on the top left and the student annotated page on the top right in Figure 4, Annotated Student Pages in Figures 5 and 6, and an example of a link to a TI-Nspire™ .tns document in Figure 7. Occasionally, links to websites are provided for introducing a problem or for background information on the topic being explored. Teacher Notes were designed and developed as educative materials (Davis & Krajcik, 2005). As educative materials, the Teacher Notes are written to expand upon the mathematical ideas within the lesson so teachers have a deeper knowledge of the mathematical ideas within the lesson. Where appropriate these explanations include a look at mathematics beyond what the students are expected to pursue. Occasionally, external links for further explanations of mathematical ideas are included. Teacher Notes are intended to help with lesson planning by providing a summary of the content and objectives for the investigation, highlighting opportunities to model with mathematics, and anticipating student thinking and possible responses, including common misunderstandings. The Teacher Notes also list materials needed and provide guidance for the use of technology by describing ways the use of technology can enhance student learning or provide an alternative approach to understanding the relationships within the lesson. The Annotated Student Pages expand the material from the Student Pages and provide teachers with notes for managing the investigation and suggest questions teachers can use to prompt classroom discussion. Questions are intended to indicate topics and ideas important to the investigation. As students become familiar with the instructional approach, they are expected to raise these issues themselves or pose the questions spontaneously to extend a problem or probe its mathematical content.
-178-
E-Learn 2013
Figure 2
Figure 5
Figure 3
Figure 6
Figure 4
Figure 7
While the materials are considered complete as related to AMAA, the PublishView™ feature allows the teacher to build upon the materials for future use. A valuable PublishView™ editing feature is the “add space tool” that teachers can use to digitally record notes about how they used a lesson. For example, a teacher can make notes about what was successful by inserting a text box and use that information later. A teacher may also insert an appropriate hyperlink or create a TI-Nspire™ spreadsheet document that has proven successful in teaching. Teachers can also add pages or problems to create lesson plans from existing lessons or TI-Nspire™ documents, link to related lesson plans or documents, build or interact with other TI-Nspire™ applications directly from the lesson plan or embed explanatory text, images, and video. If students have access to TI-Nspire™ Student software they can use the PublishView™ features to create reports or projects containing data playback, curve fits, pictures, and videos. They can also print and turn in assignments. Students essentially have access to one tool to create a document that contains problems, text, images, hyperlinks, or videos, interactive TI-Nspire™ applications, screen shots, and layout options needed to print a document. TI-NSPIRE™ Technology that enables the opportunity to interact with dynamic representations of concepts for classroom instruction is integrated throughout the curriculum. Prepared documents in TI-Nspire™ Teacher Edition software provides opportunities to enhances and extends algebraic concepts throughout the curriculum. Most lessons include a specific TI-Nspire™ .tns documents that primarily consists of spreadsheets and graphs along with suggestions in the teacher materials for ways to use these during instruction. See Figures 8 – 10 for samples from the Painted Cube lesson. This use of technology focuses on using graphical representations for data, encourages conjecturing and validation, and emphasizes relationships between quantities. The .tns documents can be used as classroom demonstrations that can be led by the teacher or by students. If students have TI-Nspire™ handheld devices, the .tns document can be sent to the students who can work on the document individually.
-179-
E-Learn 2013
Figure 8
Figure 9
Figure 10
TI-Nspire Navigator™ The TI-Nspire Navigator™ system is a classroom network of TI-Nspire™ handhelds for instant assessment and instruction consisting of features such as Question Capability, Document Transfer, Quick Poll, Screen Capture. Live Presenter, Portfolio and Auto-grading. While the AMAA curriculum does not require the use of TI-Nspire Navigator™ the digitally formatted materials in PublishView and the TI-Nspire™ .tns documents are ready to use in the TI-Nspire Navigator Teacher Software™. With the software all the AMAA content can be accessed, additional lessons created, and formative assessment student data can be collected with teacher generated questions. When teachers send students the TI-Nspire™ .tns documents included in the AMAA PublishView™ they can use screen capture to make sharing results in real-time displays possible and students can share their work using the class presenter feature. This feature affords opportunities to build discussions, to collaborate, and to examine others’ work on a public screen while performing one’s own activity. The public screen is a dynamic space of mutually exchanged information that becomes a source from which teachers can generate questions. Students receive immediate feedback and learn by comparing, sharing, and discussing. TI-Nspire Navigator™ also allows teachers to obtain immediate formative assessment data from all students on their understanding of mathematics concepts through actions such as, collecting lists, placing coordinates on a graph, specifying graph and equation representations for functions, providing explanations for open response questions, as well as responding to more traditional uses such as multiple choice, true/false, and Likertscale questions and explanations.
SUMMARY This paper describes how the development of the AMAA curriculum materials and ideas were enhanced by the use of digital technology. As such, in the minds of the curriculum development team members was a guiding principle that these materials and ideas would be made available digitally and the format of the digital materials would be useful for both teachers and students. The curriculum materials including Student Pages, Teacher Notes, and Annotated Student Pages, have been successfully piloted and are distributed statewide. Over 40 teachers and 1800 students throughout the state are using AMAA materials during the 2013 – 2014 school year and it is anticipated this will continue for the near future.
REFERENCES Davis, E. A. and Krajcik, J. (2005) Designing educative curriculum materials to promote teacher learning, Educational Researcher 34 (2005) (3), pp. 3–14. Gottlieb, D. (Spring 2011). Personal Communication to Curriculum Research & Development Group researchers when describing the need for the course Modeling Our World. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author.
-180-
E-Learn 2013 Olson, J., Olson, M., Slovin, H., Venenciano, L, & Zenigami, F. (2013). A Modeling Approach to Algebra. Honolulu, HI: Curriculum Research & Development Group. Olson, M., Zenigami, F., Slovin, H., & Olson, J. (2011). Educative curriculum materials in professional development. In Wiest, L. R., & Lamberg, T. (Eds.). Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Reno, NV: University of Nevada, Reno. Rachlin, S. L., Matsumoto, A. N., Wada, L. T., & Dougherty, B. J. (2001). Algebra 1, A process approach. Honolulu, HI: University of Hawai‘i, Curriculum Research & Development Group. Slovin, H., Venenciano, L., Ishihara, M., & Beppu, C. (2003). Reshaping mathematics for understanding: Getting started. Honolulu, HI: University of Hawai‘i, Curriculum Research & Development Group.
-181-
