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Tools for Studying the Use of Curriculum to Enhance Teacher Knowledge and Practice,

and Develop Leadership Capacity

A Model for Professional Development

Sandra K. Wilcox Jill Newton

Michigan State University1 Introduction School-university partnerships are a key component of the Center for the Study of Mathematics Curriculum (CSMC). They provide a unique opportunity for groups of teaching professionals – practicing teachers, university faculty, and doctoral students – to work and learn together. Since early 2005, Michigan State University and the Grand Ledge Public Schools have collaborated to learn more about how the investigation of curriculum can serve as a vehicle for professional development, teacher learning, and school-based curriculum leadership development. This partnership has been truly collaborative: it builds on work that meets the needs of the district, draws on the expertise of the university faculty, and fits within the mission of the CSMC grant. This report details the scope of the work undertaken, the tools developed to enable curriculum analysis and to document teachers’ learning, and the role that high-quality mathematical tasks have played in advancing our agenda. It presents a model of professional development that has guided our work – using the study of the district adopted curriculum to: increase teachers’ subject matter knowledge; enhance their instructional practices to engage students in problem solving and reasoning, and communicating their sense-making; and make adaptations to existing curriculum programs to better reflect a trajectory of learning designed to develop students deeper conceptual knowledge of important ideas. The Grand Ledge participants include more than two dozen teachers in grades K-9, including resource teachers who work with special needs students, the head of the district’s high school mathematics department, and the Assistant Superintendent for Academic Services. The university participants include three mathematics educators and a teacher consultant with expertise in curriculum and assessment design and development, as well as a long history of collaboration with practicing teachers, and doctoral students in mathematics education. In looking back over the past two years, the work has evolved through several phases. Phase I was a period of “getting to know each other.” Phase II moved more deliberately to curriculum analysis, combining an examination of the National Council of Teachers of Mathematics’ most

1 The authors wish to acknowledge all the participants who have made this collaboration such an exciting experience. The teachers who have served on the Steering Committee, Jill Hoort, Teri Mulder, Karen Spitzley, and Mary Jo Stein, make significant contributions in shaping the work for the Learning Community. Superintendent, Marsha Wells, and Assistant Superintendent, Kathleen Peasley have given the project their whole-hearted support. The teachers have approached the work with enthusiasm and seriousness and are a model of commitment to professionalism: Joy Marshall, Erinn Barnes, Erika Rothwell, Jennifer Poplar, Sheila Bell, Erica Sampson, Julie Abbott, Kelly Smith, Chris Fizzell, Dave Schuchaskie, Beth Johnston, Lori Bucholz, Mark Buckland, Jeff Palacios, Heather Spitzley. Denise Green, Mary Ann Schmedlen, Andrea Dionise, Julie Boruta, Sharon Schneider, Katherine Palmiter, Ben Lorson, Tonya Rice. In addition to the two authors, the MSU team included CSMC PIs Glenda Lappan and Betty Phillips, Elizabeth Jones, Michigan State University and Lansing Public Schools, and doctoral student Greg Larnell.

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recent standards documents, Principles and Standards for School Mathematics (2000), with the Michigan Curriculum Standards and the Michigan Grade Level Content Expectations, and the overlay of these standards documents with the mathematics curriculum programs in use in the district. Phase III involved the selection of a specific content strand – fractions – as a site for developing teachers’ conceptions of the big ideas within the strand, and creating a tool to enable a deep analysis of the treatment of fractions across the district’s several adopted curriculum programs. In this paper, we elaborate on each of these phases, paying particular attention to the tools that advanced our work. These tools include: a Materials Use Log that documents teachers’ use of and alterations to the curriculum; Rich Tasks that provide opportunities to grapple with multiple issues of content knowledge, curriculum, classroom practice and student learning; Concept Maps that capture evidence of teacher learning; a Curriculum Analysis Framework for examining curriculum; and Reflective Writing.

Phase I: Getting to Know Each Other In January 2005, MSU approached the Grand Ledge district about a possible collaboration.2 The Assistant Superintendent invited a group of teachers representing each of the schools to attend an information meeting to elicit their interest in participating in CSMC. Initially, 17 teachers signed on and we constituted ourselves as a Steering Committee (SC). Over the next 5 months, the SC held four half-day meetings designed to help the partners better understand each other’s contexts and how collaboration might organically grow. As a first activity, the district enthusiastically agreed to participate in the CSMC Cross-Site Study, and in spring, 2005, nearly 90% of the teachers of mathematics completed the survey on beliefs, preparedness, and practices. In addition, 30 elementary and middle school teachers volunteered to complete a 10-lesson Materials Use Log. The log asked teachers to keep a record for ten consecutive teaching days of their use of instructional materials (see Appendix A). One significant finding was that nearly half the teachers at grades K-5 judged the district adopted curriculum materials to be “very poor to fair” and used them less than 75% of the time. The Materials Use Log indicated that much of the supplemental work involved additional practice on routine exercises. So that MSU might get a feel for classroom life, several teachers at the elementary and middle grades volunteered to have the graduate students observe in their mathematics classes. A major interest for the district was to learn more about the transitions between the three different mathematics curriculum programs for grades K-2, 3-5, 6-8.3 Teachers suspected that there were gaps and overlaps at the transition grades but had never undertaken an analysis of the different programs to see whether or to what extent the programs were compatible. To explore this concern, MSU doctoral students analyzed the three curriculum programs of the elementary and middle grades, with a focus on algebra, patterns and functions. They used the PSSM standards, the Michigan Curriculum Standards, and the Michigan Grade Level Content 2 Grand Ledge is a large school district (125 square miles) in mid-Michigan with a population of 31,000. It consists of six elementary schools, two middle schools, and 2 high schools with a combined enrollment of 5300 students. More than 90% of the students continue with post-secondary education and there is strong family and community involvement in the district. 3 The district uses Bridges in grades K-2, Trailblazers in grades 3-5, and MaThematics in grades 6-8. In addition, some students in grade 8 take algebra, and some students in grade 9 are in Transition Math.

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Expectations (GLCEs) as their framework and overlaid these standards documents with the mathematics curriculum programs in use in the district. The analysis provided an interesting picture of the offerings in each grade and made rather apparent some disjuncture at transitions. The analysis also raised some issues about the quality of the tasks that students encountered in their textbooks. One of the SC sessions was devoted to working through a carefully crafted task, the Pool Problem (se below), to raise issues about developing algebraic reasoning across the grades. It marked the beginning of a strategy of using rich mathematical tasks as a tool to raise issues about curriculum, teaching and learning (both of teachers and their students). The Role of High Quality Mathematics Tasks Exploring the development of big ideas across the grades. Early on in our work, high quality mathematics tasks became a staple for exploring issues of content – student and teacher understanding of the core mathematics embedded within a task, and other mathematical ideas that various approaches to solving a task might open up; and instruction – how to use high-quality tasks (a) without reducing the cognitive demand, (b) to get at students’ reasoning, including faulty reasoning that leads to incorrect responses, and (c) to support further learning, particularly how to facilitate students’ talk about the mathematics of the task and their engagement with it. Teachers were presented with the following problem context and then invited to solve problems appropriate to the grade level they taught. In addition, teachers were asked to consider a series of questions about what mathematics the task was going for in each version.

The Problem Context: Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the pool. Around each pool is a border of white square tiles. He uses blue tiles to represent the water and white tiles to represent the border.4

Directions to Teachers: As you work on the tasks at your grade level, ask yourself the following questions: • What mathematical understandings can develop from this situation? • How are the mathematical understandings in this problem building on prior mathematical

understanding? • How can these understandings be used to build new understandings in later grades? • What other questions can you pose around the task? Grades K-2 • Build the first three pools with borders using white tiles for the border and blue for the

water. What is the shape of each figure? • Sort the tiles by color. Count how many are in each pile. Describe your methods for counting

the different piles. • Are there more blue or white tiles? • Build the next-biggest pool with its borders. How many tiles are needed to build this figure? • Can you build a pool with 25 tiles? With 30 tiles? Explain why.

4 Adapted from Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with Algebraic Thinking in the Elementary Grades. Teaching Children Mathematics, 3(6), 282-288.

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Grades 3-5 • Build the first three pools with borders. Record your data in a table. Continue with your table

for the next three pools and their borders. • Describe how the border tiles and pool tiles are growing. • If there are 32 border tiles, how many tiles would it take to build the pool? Explain your

reasoning. • Can you make a square with 49 blue tiles? 12 tiles? Explain your reasoning. • In the first three pools with borders, what fraction of the total area (pool with borders) is

blue for water and what fraction is white for the border? What patterns do you see?

Grades 5-6 A pool is a square swimming pool surrounded by a border of tiles. • Make a table showing the numbers of blue tiles for water and white tiles for the border for

the first six pools. • What are the variables in this problem? How are they related? How can you describe the relationship between the number of border tiles and the number of

pool tiles? • Make a graph that shows the number of blue tiles in each pool. Make a graph that shows the

number of white tiles in each spool. • As the number of the pool increases, how does the number of white tiles change? How does

the number of blue tiles change? How do these relationships show up in a table? The graph? • Use your graph to find the number of blue tiles in the 7th square. • Can there be a border for a square pool with exactly 25 white tiles? Explain why or why not. • Find the number of blue tiles in the 10th pool. The 25th pool. The 100th pool. • If there are 144 blue tiles, how many white tiles are needed for the border? Grades 7-9 A square swimming pool is surrounded by a border of square tiles. • Write an equation for the number of border tiles, N needed to surround a square pool with

side length s. • Find more than one way to represent the relationship between the number of border tiles and

side length of the pool. • How can you convince your classmates that the expressions for the number of border tiles

are equivalent? • As the side length of the pool increases, how does the number of border tiles change? • How is this change represented in a table, graph, and equation?

At the end of the session, teachers were asked to try a grade level appropriate task in their own classroom, bring interesting pieces of work to the subsequent Steering Committee meeting, and be prepared to discuss insights and issues raised by using the task. This initiated what would become another staple of our work: teachers working with tasks in our SC meetings, then trying them with their own students, and using student work and the experience of trying a task to commence our work at the next session. Using assessment to further student learning. Another major interest of the teachers was to work on what they termed “more authentic assessments.” In June, 2005, eight teachers from the SC attended the 4-day Balanced Assessment (BA) Workshop sponsored by the Mathematics

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Assessment Resource Service at Michigan State. Much of the work centered on the use of high-quality tasks from the BA collection.

These tasks: • provide an occasion for professional discussions about what makes a good assessment

task, what a particular task has the potential to reveal about multiple aspects of students’ mathematical understanding, what teachers think students might do with the task, what kinds of learning opportunities students would need if they were to productively engage with the task;

• provide the opportunity to push at the boundaries of teachers’ mathematical knowledge so they can build their own mathematical power;

• exemplify a range of task types so that students can demonstrate mathematical performance;

• can be used by teachers for classroom-based assessments to monitor student progress toward learning goals as expressed in their state and district standards.

These tasks and students’ written responses to them: • become a site for deep analysis of students’ knowledge, problem solving and reasoning,

and communication, thereby moving teachers beyond simply looking for right and wrong answers and focusing their attention on a broader range of effort and engagement with the task;

• can lead to questions of where a teacher might go next in instruction; • can raise curricular and instructional issues: If students are to develop the mathematical

knowledge that will give them power over these kinds of tasks, what are the implications for curriculum and instruction? What resources are available to provide the kinds of learning opportunities that will prepare students for these kinds of assessments? How might these kinds of tasks also serve instructional purposes?

The Grand Ledge teachers decided to match the tasks in the BA collection with appropriate places in their curriculum for use as in-class assessment and instructional tasks.

Phase II: Examining District-Adopted Curriculum Programs and Using Tasks to Get at Students’ Thinking and Reasoning

In August, 2005, a 2-day Retreat with 25 teacher participants was held to begin planning the work for the upcoming school year.5 We reconstituted ourselves as a Mathematics Professional Learning Community and 3 teachers volunteered to serve on a Steering Committee to plan upcoming sessions with MSU folks. The specific activities of the retreat focused on: • issues of content – what are some high quality tasks from the BA collection that we might

incorporate with existing curriculum to get a better understanding of what students are learning and at what levels.

• issues of instruction – how can we use these tasks in our classrooms, to better understand students’ thinking and reasoning, and to monitor and adjust our own teaching. A central feature of this work was how to engage students in conversations around the mathematics embedded in the task so that students learn from them, and assessment and learning coexist in a supportive relationship.

5 In the four months that the SC had been meeting, word spread about the value participants were finding in our collaboration and a number of additional teachers asked to join in the 2005,06 academic year.

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In keeping with our strategy of using a high-quality task as a site for exploring multiple issues, we used Fractions of a Square from the BA collection (see Appendix B).6 This particular task was chosen for several reasons. Fractions is a topic that many teachers find challenging to teach and kids find challenging to learn. It is also a topic that seems to be getting pushed to lower grades – as evidenced by the Michigan GLCEs – and would likely have broad appeal across the grades. We considered ways that the task can be used in the classroom. We examined some samples of student work to get insights into students’ reasoning, both robust and faulty. And we explored how a teacher can use information from informal assessments to shape her/his next instructional moves to support students’ further learning. Finally, we considered where this task might appropriately be used, given the Grand Ledge curriculum programs and the GLCEs. Additional activities at the retreat drew on a range of tasks across the grades from the BA collection, with a growing emphasis on how to get at children’s thinking and reasoning and how to engage them in conversations about their thinking. At the conclusion of the retreat, the group reconstituted itself as a Mathematics Professional Learning Community, and three teachers formed a new Steering Committee to plan with MSU the upcoming sessions. Examining Number and Operations Strand: The Need for an Analytic Framework. At the first fall meeting, using the model that the doctoral students developed to analyze the Algebra, Patterns and Functions strand, the teachers attempted to examine the treatment of the Number and Operations strand in the three K-8 programs. This activity pointed out several things. First, as teachers suspected, there were considerable gaps at the transition grades between programs and there were considerable redundancies, both within and across the three programs. As one teacher commented, “Our textbook assumes previous understanding of these concepts and procedures and doesn’t do much development of it and now I see why they don’t know it based on what comes before.” And she was not alone in her assessment. Second, the close look at number led to considerable discussion about fractions. The group thought that it would be really interesting to trace the treatment of fractions, decimals, and percents across the grades. But their efforts also revealed a somewhat superficial examination of the textbooks. It was apparent that we weren’t all looking for/at the same things and so our analyses were uneven. It became clear that some kind of analytic framework would be needed to support a deeper and more substantive analysis of fractions. Using Tasks to Get at Students’ Thinking. Teachers enthusiastically brought samples of their students’ written work on BA and other tasks to the fall sessions. In addition, they brought along vignettes of students’ engagement with the tasks and discussions with kids following their work on solving the problems. Based on our work at the summer workshop on “what reasoning might have led a student to respond incorrectly to a problem,” teachers seemed quite interested in trying to make sense of their own students’ faulty reasoning and fragile knowledge. As a prompt for one of the sessions, a middle grades teacher volunteered to have her class video-taped while her students engaged with a rich task. We used the video as a site to explore student’s thinking and approaches to solving a pattern problem in a geometric context. We considered how the task was launched and what effect it might have had on how students approached the task. We attended carefully to the whole class discussion, in particular the kinds of questions that the teacher posed and the responses they elicited, what the responses revealed about their thinking, and the extent to which the task was engaging for them. By the end of 2005, teachers were ready to tackle an analysis of the treatment of fractions in the district adopted curriculum programs.

6 This task appears in Balanced Assessment, Middle Grades Package 1, published by Dale Seymour, 1999.

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Phase III: Analyzing Curriculum Treatment of Fractions and Using Tasks to Get at Big Ideas in the Domain

Our earlier work in looking at curriculum through Number and Operations led the Steering Committee to recommend that before the Learning Community undertook the analysis of another content strand, it seemed reasonable that we should have some agreements about what we would be looking for/at. To get to these agreements, we needed to better understand the content strand we were going to look at – in all its complexity. And Fractions are indeed complex. Therefore, some professional development activity to deepen our own understanding of multiple aspects of the domain was likely to be very useful. Further, we wanted to document how we developed as mathematics educators through curriculum analysis – toward one of the aims of the CSMC research agenda: To learn to what extent and under what conditions mathematics curriculum materials can promote teacher learning and effective teaching. The next 6 all-day sessions, between February and December 2006, were devoted to fractions. To begin our work, teachers were asked to individually construct a concept map on what they thought was important in the domain of fractions. The teachers who had attended the BA workshop the previous June had made concept maps on assessment. A concept map is a useful tool in several respects. First, it is a personal representation of how an individual conceives the domain. There is no one right answer or one way to make a map. Second, concept maps provide a rich context for having discussions about individual and collective ways of thinking about the domain. Participants can look across maps for similarities and differences in the ways they represent the domain and what they include or don’t include. Following a discussion about the individual maps, small groups of teachers made a collaborative concept map poster. This afforded an opportunity for small groups to build on each other’s thinking, to decide what the group considered important to include in a map, and how to represent those ideas. Groups posted collaborative maps for all to examine. As groups circulated, they considered whether they wanted to change something on their map and if so, what, how and why? Groups then made alterations to their maps with post-its so we could keep track of what was original and what was an amendment. The maps provided some baseline data about what each teacher thought was important for students to know about and be able to do. (Teachers would make another concept map at the conclusion of our work on fractions.)7 Using Tasks to Concretize Big Ideas in a Domain. A key aspect of our work together had been using rich mathematical tasks to get at issues of curriculum and learning. Because it seemed to be serving us well, the SC thought it made sense to continue with this kind of activity to help make concrete the big ideas represented (or not represented) on the concept maps. The MSU folks selected about a dozen fraction tasks for small groups of teachers to solve and to describe the core fraction idea(s) that each task entailed (see Appendix C)8. The aim here was to analyze tasks beyond surface features and really grapple with what was at the heart of each task mathematically. Tasks were chosen that could help reveal some of the complexity, even to those who may have thought they had a solid grasp on the domain. Tasks were chosen as instantiations of the different meanings of fractions, different representations or models,

7 A report on the use of concept maps in this project is being prepared by doctoral candidate Jill Newton and Grand Ledge teachers Teri Mulder and Karen Spitzley. The report will include an analysis of what the maps reveal about teacher learning, and the affordances and limitations of this tool to document teacher learning. 8 Many of these tasks were adapted from Middle-grade teachers' mathematical knowledge and its relationship to instruction: A research monograph (Sowder et al.,1998).

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partitioning, equivalence, benchmarks/estimating. The intent of this kind of activity was to construct some categories, some language, some meanings that could begin to shape a framework for the curriculum analysis. Designing a Framework for Curriculum Analysis. Following the work with fractions tasks, MSU folks proposed a Framework for analyzing the treatment of fractions at each grade across the three curriculum programs. Small cross-grade groups were constituted to use the framework to search through the text for any place where fractions appeared. The template (see Appendix D) specified documenting the following information:

• Unit and page number • Topic • Meaning of Fraction • Model/representation used • Introduced/developed/maintained • Example of problem

Part of two sessions was devoted to the grade level analyses. At the conclusion, teachers made posters for each grade indicating what their text expected students to enter with and what they could be expected to leave the grade having had an opportunity to learn. Again, the analyses highlighted serious gaps, particularly at the transition from grade 2 to 3, and inappropriate redundancies across the upper elementary grades. In addition, representations and models seemed either inadequate (little use of the number line) or inappropriate (use of pattern blocks up through grade 6). It was clear to all that some serious attention needed to be given to a reasonable learning trajectory for the teaching of fractions and that would entail modifications to the existing curriculum programs. Monitoring Progress with Reflective Writing. It had become a common practice at the end of the sessions to allow some time for teachers to do some reflective writing. The prompts were typically open-ended. After two full-day sessions working with fraction tasks and the curriculum analyses, teachers were asked to respond to the following; What new insights do you have about fractions that were prompted by this work? The responses seemed to fall into several categories:

• A need to spend more time on developing concepts before rushing to algorithms; “Wow! There are so many ways fractions are used it is no wonder kids are confused! I think our work

on this topic will be time well spent. • A need to rethink what ideas are taught at each grade and at what level’ “So much of the fraction curriculum is at end of 3rd grade book so it is glossed over quickly. Maybe

the order should be changed.” “Major concepts that are stressed in 7th/8th are already a major part of 5th grade curriculum – so

makes me think we need to really rethink – what is taught at each grade level. Some vocabulary that is used in 5th grade is obscure – never used beyond need to clean up vocabulary across the grades.”

• A need to move beyond curriculum analysis to tackling issues of teaching; “How can we teach kids in a way that will help them develop their understanding of concepts? How

can teachers gain confidence in using sound instructional methods?”

Taking Up Instructional Issues. While there was still unfinished work related to the curriculum analyses, especially how to think about a learning trajectory across the grades, teachers were anxious to move on to issues of teaching fractions. Again, rich tasks became the site for our exploration. This work was introduced by asking teachers to think of themselves as curriculum developers when they plan for instruction.

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• Setting goals and analyzing tasks: how a task will push the mathematical agenda, and what skill development, algorithm development, higher order thinking development the task will support

• Taking stock: what tools and resources students need to tackle the task, and what skills, processes, ways of thinking would contribute to students’ success with the task.

• Reaching all students: what changes in the context of the task would make it more engaging for my students? How will I reach all students – what are my core goals; what are my extension questions; what scaffolding questions will help students who are struggling, but that will not reduce the cognitive demand of the task?

• Pulling out the mathematics: what summary questions can I ask to help students make the mathematics used and invented more explicit; what reflection questions can I ask to help students make connections between the mathematics of the task and the growing mathematical skills, understandings, and ways of thinking of the student?

• Assessing and evaluating students: How will I know what sense my students have made of the task so that I can make instructional decisions, report on students’ progress, evaluate the classroom environment, celebrate student accomplishment, and hold students accountable, setting expectations?

• Assessing and evaluating curriculum: How coherent, connected, and powerful is the sequence of tasks I have used to promote understanding of the core ideas that are my goals; Do I have the “right stuff” for my students to chew on; Is the sequence of tasks powerful?

To make these ideas concrete, teachers tackled the following problem.9

Compare these four mixes for apple juice

Mix W Mix X 5 cups concentrate 3 cups concentrate 8 cups water 6 cups water Mix Y Mix Z 6 cups concentrate 3 cups concentrate 9 cups water 5 cups water

a. Which mix would make the most “appley” juice? b. Which mix would make the least “appley” juice? c. Suppose you make a single batch of each mix. What fraction of each

batch is concentrate? d. Rewrite your answ4ers to part © as percents. e. Suppose you make only 1 cup of Mix W. How much water and how much

concentrate do you need? Teachers were asked to think carefully about why these questions might be posed, how students might respond, how some questions might lead to insights into the different ways students might reason about the problem, and raise some of the typical kinds of errors and good ways of thinking that might be elicited in a classroom. Over the next two sessions, teachers explored tasks with the intent to draw out some design principles that could be used to adapt tasks in the district adopted textbooks.

9 A version of the problem appears in the Connected Mathematics unit Comparing and Scaling.

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In addition, teachers revisited their textbook analyses and began to work on scope and sequence for fractions through the grades. At the writing of this report, this aspect of the work is ongoing, outside the time of the Learning Community sessions. The plan is to develop a learning trajectory and present it as a proposal to the entire mathematics teaching staff so that decisions can be made for the 2007-08 academic year. Yvonne Grant, a teacher writer and professional development facilitator of Connected Mathematics, facilitated the final session on fractions. Yvonne is a user of CM at the middle grades, and an elementary teacher who has taught fractions in lower grades with the same text used in Grand Ledge. She brought her own lived experiences of trying to teach fractions with understanding to children in grades 3-8. She focused on how she used rich tasks drawn from the Connected Mathematics units Bits and Pieces I, II, and how a series of coherent, connected tasks could promote understanding of the core mathematics embedded in them.

Phase IV: Tackling Additional Content Domains

In early 2007, the Learning Community members formed two groups, one for grades K-5, and one for grades 6-9. The elementary group is focusing its attention on whole number sense and operations. The sessions are facilitated by Liz Jones, a teacher, and an assessment and professional development designer with Balanced Assessment and the Mathematics Assessment Resource Service. Teachers are using a curriculum from Developing Mathematical Ideas.10 The middle grades group is taking up algebra and is being led by Betty Phillips. The district will soon be selecting new algebra textbooks and the work in this group aims to equip teachers with an analytic framework for examining textbooks. This work began with an exploration of linear functions, using one of the units from Connected Mathematics. Unfortunately, despite our best efforts to assure teachers otherwise, we wondered whether some of the teachers thought we were really making a sales pitch for CM. So at the subsequent meeting, we moved toward a learning trajectory that was not dependent on any particular curriculum, starting with two very broad ideas: patterns of change (function and their representations) and equivalence (representations, symbolic expressions and equations). Although the tasks used to explore these ideas were drawn from CM, teachers’ attention was drawn away from a text and focused on a sequence of coherent, connected tasks. In addition, the teachers are reading an article that contrasts two types of algebra curricula.11 This should provide teachers with an additional lens by which to examine textbooks for the upcoming adoption. Summary. In this paper, we have focused on the tools that helped to move our work forward. Some tools were developed by others (Cross-site Survey, Materials Use Log), some we developed as a need arose (Curriculum Analysis Framework). What we have not taken up in this paper is any discussion of whether, in what ways, and to what extent this work contributed to teachers’ learning, enhanced their instructional practices, or developed local leadership capacity; this is the subject of another report. But we do wish to provide a hint of the impact. From a recent series of classroom observations conducted by the district superintendent and assistant superintendent, they report that they can tell whether or not the teacher has been a participant in this project. They note the quality of the tasks, students’ engagement with mathematics, and the level of conversation orchestrated by the teacher through questions. In 2006, 3 teachers and 2 MSU folks put together a presentation for NCSM and for the Michigan Council of Teachers of Mathematic (MCTM). In 2007, 7 teachers volunteered to present at NCSM, 5 actually presented. And they again will present at MCTM. Their presentations were first-class. Another group

10 A full description of these materials is available at http://www2.edc.org/CDT/dmi/dmicur.html 11 From Equation-based to Function-based Algebra Curricula. Texas Mathematics Teacher, Fall 2005.

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presented a BA-type workshop for colleagues, and several have spoken about the project at district Board meetings. At an upcoming retreat, the Learning Community will consider how to extend the reach of our project beyond the 25 or so who have participated. We are particularly interested in creating opportunities for taking this work back to the local schools, with the work being led by LC teachers.

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References

Balanced Assessment for the Mathematics Curriculum. (1999). Balanced assessment, middle grades package 1. White Plains, NY: Dale Seymour Publications.

Billstein, R., & Williamston, J. (1999). Math thematics. Evanston, IL: McDougal Littell.

Burk, D., & Snider, A. (2000). Bridges in mathematics. Salem, OR: The Math Learning Center. Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with algebraic thinking in the

elementary grades. Teaching Children Mathematics, 3(6), 282-288. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Difanis Phillips, E. (2006). Connected

mathematics project. Glenview, IL: Prentice Hall. Li, X. (2005). From equation-based to function-based algebra curricula. Texas Mathematics

Teacher (Fall 2005), 22-27. National Council of Teachers of Mathematics. (2000). Principles and standards for school

mathematics. Reston, VA: National Council of Teachers of Mathematics. Schifter, D., Bastable, V., & Russell, S. J. (2000). Developing mathematical ideas. White Plains,

NY: Dale Seymour Publications. Sowder, J. T., Philipp, R. A., Armstrong, B. E., & Schappelle, B. P. (Eds.). (1998). Middle-grade

teachers' mathematical knowledge and its relationship to instruction: A research monograph. Albany, NY: State University of New York Press.

TIMS Elementary Mathematics Curriculum Project. (1998). Math trailblazers. Dubuque, IL:

Kendall/Hunt Publishing Company.

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Appendix A

Materials Use Log

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Appendix B

Fraction of a Square

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Appendix C

Fraction Problems

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Appendix D

Curriculum Analysis Framework Sheet

Unit and Pages Topic Meaning of

Fraction Representations Introduced,

Developed, or Maintained

Sample Task