AAAS Conference on Developing Textbooks That Promote Science Literacy

February 27-March 2, 2001
American Association for the Advancement of Science
Washington, D.C.

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Developing Coherent, High Quality Curricula:
The Case of the Connected Mathematics Project

A background paper commissioned for the
AAAS Project 2061 Science Textbook Conference
Washington, D.C.
February 27-March 2, 2001

Elizabeth Difanis Phillips
Michigan State University

Glenda Lappan
Michigan State University

Susan N. Friel
University of North Carolina

James T. Fey
University of Maryland

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Table of Contents

• About the Project
  • Prior Experiences
  • Emerging Philosophy: A Problem-Centered Curriculum
• Development of Connected Mathematics
  • Collaborators
  • Setting Goals
  • Design of Student Materials
  • Providing Support for Teachers
  • Management
  • Field Testing
  • Evaluation
  • Publishing
• Sustaining Momentum
  • Professional Development
  • The Show-Me Center
  • The CMP Web Site and Getting to Know CMP
• Summary
• References

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Developing Coherent, High Quality Curricula:
The Case of the Connected Mathematics Project (CMP)

The Connected Mathematics Project is one of the 13 curriculum development projects that were funded by NSF between 1990 and 1993 to help realize the National Council of Teachers of Mathematics’ Curriculum and Evaluation Standards for School Mathematics in classrooms. Connected Mathematics, the project’s middle grades curriculum, was awarded an “Exemplary” designation by the Mathematics Panel of the U.S. Department of Education and was the most highly ranked middle grades curriculum material in an evaluation of mathematics textbooks conducted by the American Association for the Advancement of Science.

Connected Mathematics is a complete curriculum for teachers and students in grades six, seven, and eight. The curriculum is published in eight modules for each grade level accompanied by a comprehensive teacher’s guide for each module. While the modules might appear to be interchangeable within a grade level, they are carefully sequenced to provide the most powerful mathematics learning experience for students. Within a module, the mathematics is developed through a series of investigations, each focusing on key mathematical concepts, skills, processes, and ways of thinking. Investigations involve work on two to five mathematically rich situations that engage students in extended problem-solving activity, often in collaboration with their classmates.

About the Project

In the Connected Mathematics Project (CMP), the curriculum materials for students and teachers reflect a commitment to the philosophy that teaching and learning are not distinct—“what to teach” and “how to teach it” are inextricably linked. We believe that

The circumstances in which students learn affect what is learned. We have worked to produce a curriculum that helps teachers and those who work to support teachers to examine their expectations for students and analyze the extent to which classroom mathematics tasks and teaching practices are aligned with their goals and expectations. (Lappan & Phillips, 1998)

Prior Experiences

A critical factor in the development of Connected Mathematics was the collective experience of the authors. Each of us has had extensive prior experiences with research and with innovative approaches to mathematics curriculum and teaching. In the Middle Grades Mathematics Project (MGMP), Glenda Lappan, Elizabeth Phillips, and William Fitzgerald did extensive research on creative new teaching strategies and professional development of teachers. They developed five MGMP curriculum units (Mouse and Elephant, Spatial Visualization, Factors and Multiples, Probability, and Similarity) that challenge students to become active learners, both intellectually and physically. These units were built on a teaching model that employed mathematical investigations as the primary locus for mathematics learning.

The MGMP materials were used in a variety of research studies. One such study explored teacher professional development through expert coaching (Lappan et al., 1988). Several studies were conducted on student learning, in proportional reasoning (Lappan & Friedlander, 1987) and spatial visualization (Ben-Chaim et al., 1986, 1988, 1989; Lappan et al., 1985). A study of the learning and teaching of growth relationships was conducted by Fitzgerald and Shroyer in 1979. From 1975 to –1990, Fitzgerald, Lappan, and Phillips spent many hours in the classroom doing teaching experiments around drafts of units, conducting research, and working with teachers and students in a variety of settings.

In 1988, Glenda Lappan and Perry Lanier were funded through the National Center for Research on Teacher Learning to conduct a multi-year study of the development of a cadre of 24 elementary teacher education students in both mathematics and the teaching of mathematics. We used the MGMP curriculum as a core set of ideas from which to develop a sequence of university mathematics courses for these students. This research project gave us another opportunity to examine carefully the mathematical development of students and then to follow them into their classrooms as elementary teacher over the first two years of teaching. We learned a great deal about curriculum design and effective teaching that has been of use to us in our current work.

In the Used Numbers Project, Susan Friel had done similar curriculum development work focusing on making statistics ideas accessible to upper elementary grades students. She followed that curriculum work with the Teach-Stat Project that focused on the professional development of teachers. Jim Fey’s work on the Computer Intensive Algebra Project (CIA) explored strategies for teaching algebraic ideas by embedding them in authentic problem-solving activities and by employing calculators and computers as learning and problem-solving tools. Fey is also an author of Core-Plus, a high school curriculum project funded by the National Science Foundation (NSF).

Each of the research and development projects in which the authors participated prior to CMP was related to curriculum, teaching, and/or student learning. Each led to publications in refereed journals, presentations at meetings, and to the development of curriculum materials that were implemented in many schools. The CMP authors also had a deep understanding of the National Council of Teachers of Mathematics (NCTM) standards. Lappan was Grades 5–8 Chair of the NCTM Curriculum and Evaluation Standards for School Mathematics and Chair of the NCTM Professional Standards for Teaching Mathematics (1989). Friel was Chair of the Professional Development Writing Group for the NCTM Professional Standards for Teaching Mathematics (1991). Phillips was an author of Patterns and Functions for Middle Grades—the NCTM Addenda to the Standards (1991). Fey wrote a chapter, “Quantity,” for On the Shoulders of Giants (National Research Council, 1990) and was a member of the NRC committee that produced curriculum recommendations for Reconstructing School Mathematics (NRC, 1990). Lappan was also NCTM president during the development of NCTM Principles and Standards 2000.

Emerging Philosophy: A Problem-Centered Curriculum

Formal mathematics begins with undefined terms, axioms, and definitions and deduces important conclusions logically from those starting points. However, mathematics is produced and used in a much more complex combination of exploration, experience-based intuition, and reflection. The CMP team began its work with some shared assumptions about mathematics, learning, and teaching. Over time and with experience, many of these assumptions have evolved into a philosophical framework to guide our design of curriculum materials. Among these guiding philosophies are those that relate to:

Curriculum materials:

• An effective curriculum has coherence—it builds and connects from investigation to investigation, unit to unit, and grade to grade.

• The “big,” or key, mathematical ideas around which the curriculum will be built should be identified.

• Each key idea may be related to a number of smaller concepts, skills, or procedures. These need to be identified, elaborated, exemplified, and connected.

• Ideas must be explored in sufficient depth to allow students to make sense of them. Superficial treatment of an area produces shallow and short-lived understanding and does not support the making of connections among ideas.

• Mathematical tasks that students work on inside and outside the classroom are the primary vehicle for student engagement with the mathematical concepts to be learned.

• Posing mathematical tasks in context (i.e., a problem-centered curriculum) provides support both for making sense of the ideas and for processing them so that they can be recalled.

Field testing and evaluation:

• Effective curriculum materials development requires careful field-testing, evaluation, and rewriting over several trials.

• Research—including formative and summative student evaluation as well as other relevant studies—is essential in determining what seems to be working and where revision is needed.

• Research is needed on student and teacher learning from the curriculum to provide evidence for school districts that may consider adopting the materials.

Teacher support:

• Successful implementation of materials depends on teachers enacting the curriculum in a way that supports the philosophy of learning and teaching underlying the materials’ development. In order for this to be the case, teachers need effective, ongoing professional development that focuses on the materials themselves rather than generic teacher enhancement.

• Teachers need opportunity and support to collaborate with each other in study and planning for teaching the curriculum.

• Teacher support materials for the curriculum are needed to provide help with mathematics, assessment, and pedagogy.

Administrative and community support:

• Superintendents, principals, and other administrators as well as school boards and parents must have access to clear information about the project materials. The development staff needs a well-developed strategy for providing the mechanisms through which such information is made available and kept updated.

The validity and importance of these guiding principles were not completely obvious to us at the outset of our work. However, we can see from experience over the past decade that each has been significant, particularly those that relate to focusing on key ideas and providing the right context within which students can engage mathematical concepts and skills.

Key ideas. Our development team made a commitment to studying a small and select set of important ideas deeply rather than skimming a larger set of ideas in a shallow manner. This means that time is allocated to developing understanding of key ideas in contrast to “covering” the book. The concept of “spiraling” is philosophically appealing; but, too often, not enough time is spent initially with a new concept to build on it at the next stage of the spiral. Without this deeper understanding of concepts and how they are connected, students come to view mathematics as 1,001 different techniques to be memorized. They cannot apply or communicate about mathematical ideas in any way that makes sense.

This commitment to developing key ideas in depth is illustrated in the way that Connected Mathematics treats proportional reasoning, an important topic for middle school mathematics. Conventional treatments of this central topic are often limited to a brief expository presentation of the ideas of ratio and proportion, followed by training in techniques for solving proportions. In contrast, the CMP curriculum materials develop core elements of proportional reasoning in at least four major units:

1. Stretching and Shrinking introduces proportionality concepts in the context of geometric problems involving similarity. Students connect visual ideas of enlarging and reducing figures, numerical ideas of scale factors and ratios, and applications of similarity through work on problems focused around the question: “What would it mean to say two figures are similar?”

2. Comparing and Scaling connects fractions, percents, and ratios through investigation of various situations in which the central question is: “What strategies make sense in describing how much greater is one quantity than another?” Through a series of problem-based investigations, students explore the meaning of ratio comparison and develop, in a progression from intuition to articulate procedures, a variety of techniques for dealing with such questions.

3. Moving Straight Ahead is a unit on linear relationships and equations. Proportional thinking is connected and extended to the core ideas of linearity—constant rate of change and slope.

4. Data Around Us extends the theme of finding sensible ways to compare numbers by giving students tasks that call for developing and using number sense to analyze quantitative data. Differences, rates, and ratios again occur as possible strategies. The meaning and power of these mathematical tools are compared and contrasted.

Providing a context. In trying to help students make sense of mathematics we found that embedding the concepts and skills within a context or problem not only helped students to make sense of the mathematics, it also helped them to process the mathematics in a retrievable way. The following passage from Getting To Know CMP (Lappan et. al, 2001) provides a short summary of why we chose to develop a problem-centered curriculum.

Over the past three to four decades, a growing body of knowledge from the cognitive sciences has supported the notion that students develop understanding as they construct and evaluate solutions to problems. This is quite different from the previous assumption that students learn by observing a teacher as she demonstrates how to solve a problem and then practicing that method on similar problems.

Students’ perceptions about a discipline come from the tasks or problems they are asked to engage in. For example, if students in a geometry course are asked to memorize definitions, they think geometry is about memorizing definitions. If students spend a majority of their mathematics time practicing paper-and-pencil computations, they come to believe that mathematics is about calculating answers to arithmetic problems as quickly as possible. They may become faster at performing specific types of computations, but they may not be able to apply these skills to other situations or to recognize problems that call for these skills.

On the other hand, if the purpose of studying mathematics is to be able to solve a variety of problems, then students should spend most of their mathematics time solving problems. If time is spent solving problems, reflecting on solution methods, examining why the methods work, comparing methods, and relating methods to those used in previous situations, then students are likely to build more robust understandings and strategies. (pp. xx)

In our curriculum work important mathematical ideas are embedded in the context of interesting problems. As students explore a series of connected problems, they develop understanding of the embedded ideas and with the aid of the teacher, abstract powerful mathematical ideas, problem-solving strategies, and ways of thinking. (pp. xx)

To help students develop mathematical understanding and skill through work on substantial and interesting problems, a key curriculum development task was constructing problems that were engaging for students as well as carriers of important mathematical ideas. We found that problems could take a variety of forms, from analysis of mathematical games to practical tasks like interpreting product advertisements or news reports based on data, organizing and modeling data from science experiments, and optimizing design of geometric shapes. This variety is illustrated in the following sample of problems taken from several points in the CMP curriculum.

In the Roller Derby game, student teams place 12 markers on a board that looks like the following table:

1
2	X
3	X
4	X
5	XX
6	X
7	XX
8	X
9	X
10	X
11
12	X

Two dice are rolled in turns, and when the sum of faces matches a number on which students have a marker, they can remove one such marker. The goal is to remove one’s markers first. This game leads students naturally to consideration of empirical probability concepts and also to the role that finite sample spaces can play in modeling probability experiments. (p. xx)

The Comparing and Scaling unit begins with a task that asks students to think about comparisons often made in advertising:

Advertisements often refer to surveys showing that people prefer one product over another. For example, an ad for Bolda Cola starts like this:

Which Soft Drink Do You Like Better?

Bolda Cola or Cola Nola

Take the Cola Taste Test Yourself!

To complete the ad, Bolda Cola wants to report the results of a taste test. A copywriter for the advertising department has proposed four possible concluding statements.

In a taste test, people who preferred Bolda Cola out-numbered those who preferred Cola Nola by a ratio of 17,139 to 11,426.

In a taste test, 5,713 more people preferred Bolda Cola.

In a taste test, 60% of the people preferred Bolda Cola.

In a taste test, people who preferred Bolda Cola outnumbered those who preferred Cola Nola by a ratio of 3 to 2.

Problem 1.1

A. Which of the proposed statements do you think would be most effective in advertising Bolda Cola?

B. Is it possible that all four advertising claims are based on the same survey data?

C. What other concluding statements could express the same survey data?

D. Based on the survey results used to write the Bolda Cola ad, what data would you expect in a new survey of 1000 cola drinkers? (pp. xx)

In the Mathematical Modeling unit, students explore direct and inverse variation by analyzing data from classroom experiments in which they build paper bridges and compare carrying load (in pennies) with length and thickness of the bridges. They are asked to make some predictions about the relationships involved before experimenting. Then they collect data, display it graphically, and describe the patterns revealed in those displays. (p. xx)

Field trials of the materials in which these investigations were used revealed remarkable instructional power in such context-based teaching. Students with many different prior experiences and interests became engaged in the tasks. When the same underlying mathematical ideas were encountered in later work, students often connected the new problems to the settings in which those mathematical structures were first studied. However, we also learned that without careful questioning and reflection, it is quite possible for students to leave the classroom with nothing more than a pleasant and lively experience. Furthermore, we found that if a particular context for mathematics is carried on for too long, students lose interest. As a result, we have limited the use of any single problem storyline and introduced a variety of features in student and teacher materials that prompt reflection on problem-solving experiences and encourage students to articulate the underlying mathematical relationships and procedures used to solve the problems.

Each investigation includes Application/Connection/Extension exercises that push students to use their new ideas in different but related contexts and to connect new ideas to related familiar ideas. Each investigation also closes with a set of Mathematical Reflection questions that ask students to articulate (orally and in journal writing) their understanding about the key mathematical ideas of the preceding material. Finally, each unit closes with a series of overall reflection questions and additional problems that test student understanding of the key ideas.

When mathematical ideas are embedded in problem-based investigations of rich context, the teacher has a critical responsibility for ensuring that students get the mathematical messages. In a problem-centered classroom, teachers take on new roles—moving from always being the source of knowledge to one of guiding and facilitating the learner in making sense of the mathematics. Teachers become an even more integral part of the learning process. To teach a problem-centered curriculum requires a deep knowledge of mathematics; a broad and coherent view of the subject matter; and an understanding of effective ways to conduct a class based on inquiry. The teacher support materials must provide these kinds of help for the teacher. Our goal from the beginning of the project was to develop a curriculum from which both teachers and students could learn.

Development of Connected Mathematics

After the release of the National Council of Teachers of Mathematics (NCTM) Standards for School Mathematics in 1989, the National Science Foundation put forth a call for proposals to develop mathematics curricula for grades K-5, 6-8, and 9-12. It was clear from our past experiences, that curriculum materials were an essential part of improving school mathematics. Even though we had already written five mathematics modules that were widely in use, teachers wanted a more comprehensive curriculum that reflected the NCTM standards. In 1991 CMP submitted a proposal to NSF to write a complete mathematics curriculum for teachers and students in the middle grades, building on and expanding our earlier work. Our original research and development team of Fitzgerald, Lappan, and Phillips now included James Fey, who brought high school experience to the task, and Susan Friel, who provided elementary school experience. Together, we would have the expertise to build on what good elementary curricula were doing, while also preparing students for what was to come in high school. An advisory board consisting of mathematicians, scientists, mathematics educators, teachers, parents, and business representatives was created. They played a critical role in deciding what content to emphasize, in the format of materials, and in planning the research and evaluation conducted throughout the development process.

Five professional development centers (PDC) were established, each with a well-known mathematics educator at its head. These centers were established in San Diego, CA; Portland, OR; Pittsburgh, PA; Queens, NY; and in a consortium representing Michigan (Portland, Flint, Shepherd, Traverse City, Bloomfield Hills, Sturgis, and Waverly School Districts). The PDCs served as the primary sites for field-testing. Additional sites in Tallahassee, FL; Toledo, OH; Evanston, IL; and Durham, NC were added during the trial phases of the project. Collectively, the field test sites represented diversity of geographic location, academic ability, ethnicity, and socioeconomic condition.

Collaborators

A research and evaluation team was formed under the direction of Diana Lambdin at the University of Indiana, along with Judith Zawojewski at Purdue University and Sandra Wilcox at Michigan State University. Their role was also critical to the development of materials. They trained observers at each of the professional development centers; and they collected information on classroom environment, teacher needs and practices, student attitudes and achievement, and management and related school issues. In addition, during the second and third years of the field trials, they conducted a large-scale comparative study of CMP students and students in classrooms using more traditional texts. The Iowa Test of Basic Skills and a problem-solving test developed by the Balanced Assessment Project were used to assess student performance. This information was critical to the author team as we prepared the final versions of the modules for students and the supporting teacher guides.

Classroom teachers also helped to shape Connected Mathematics. A number of collaborating teachers worked with us on all phases of the development and, most important, served as our eyes in the classroom. A full-time middle school teacher headed up the development of the assessment package that was included in each module’s teacher guide. Graduate students in mathematics and mathematics education contributed fresh ideas to the project and learned a great deal from their experiences. As the next generation of curriculum development professionals, these graduate students will have a better understanding of what it takes to develop effective materials.

Setting Goals

We began with some assumptions about what students would know when they entered grade six. Then, working with our Advisory Board, our next task was to define the knowledge and skills in each content strand that students should have by the time they leave grade eight. To aid in the process and help us think more broadly about the curriculum, we wrote a set of papers that outlined the exit goals for grade eight in the four mathematical strands that we would develop—number, algebra, probability and statistics, and geometry and measurement. These papers were our touchstone for the development of each strand. Key ideas or clusters of related ideas were identified in each strand and incorporated into units. Initial sequencing of units and appropriate grade level placements were initially identified. Subsequent revisions were based on extensive field tests.

The overarching goal of Connected Mathematics is to help students and teachers develop mathematical knowledge, understanding, and skill along with an awareness and appreciation of the rich connections among mathematical strands and between mathematics and other disciplines. As the CMP materials were developed, the authors synthesized multiple mathematical goals into a single standard that has been a guide for all of our curriculum development:

All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency. (Getting to Know CMP, p. xx)

Another goal for CMP was to align teaching, learning, and assessment with each other as integral parts of the material. To accomplish that broad program goal, development of student and teacher materials was guided by the following five fundamental mathematical and instructional themes:

• CMP is organized around a selected number of important mathematical content and process goals, each of which is studied in depth.

• CMP emphasizes significant connections among various mathematical topics and between mathematics and problems in other disciplines that are meaningful to students.

• The instruction in CMP emphasizes inquiry and discovery of mathematical ideas through the investigation of rich problem situations.

• CMP helps students grow in their ability to reason effectively with information represented in graphic, numeric, symbolic, and verbal forms and to move flexibly among these representations.

• The goals and teaching of CMP approaches reflect the information-processing capabilities of calculators and computers and the fundamental changes such tools are making in the way people learn mathematics and apply their knowledge of problem-solving tasks. (Getting To Know CMP, p. xx)

As materials for students and teachers began to take shape, difficult problems and decisions surfaced at every stage of the process. The major development issues fall into three categories—student materials, teacher support, and project management.

Design of Student Materials

As we set out to write a complete connected curriculum for grades six, seven, and eight, certain basic questions about the mathematical content and format of student materials quickly surfaced:

• What are the overarching mathematical goals in each content strand?

• What size problem is feasible for the teacher and students to explore?

• What kind of sequencing or scaffolding is needed to support investigations?

• How much help is needed to move from a contextual setting to mathematical understanding that transcends specific situational factors?

• What computational skills should be developed and how?

• What combination of whole class, small group, and individual work is most effective?

• What kinds of practice or homework and reflection are needed to ensure solid understanding of key ideas and automaticity of critical skills?

• What forms of assessment will best inform teachers and students of progress and needed next steps in teaching and learning?

• How much help do teachers need in understanding and implementing the content and pedagogy of the curriculum?

It took classroom trials and observations, discussions, revisions, more trials, research, and reflections to resolve such issues into a coherent problem-centered curriculum. Although our initial analysis of overarching goals remained sound, our notions of what could be taught in one year changed. To foster students’ deep understanding of the key ideas would require more time.

Characteristics of good problems. To be effective, problems must embody critical concepts and skills and have the potential to engage students in making sense of mathematics. And, since students build understanding by reflecting and communicating, the problems need to encourage them to use these processes.

Each problem in Connected Mathematics satisfies some or all of the following criteria:

• The problem must have important, useful mathematics embedded in it.

• Students must be able to approach the problem in multiple ways, using different solution strategies.

• The problem should allow various solution strategies or lead to alternative decisions that can be taken and defended.

• The problem should engage students and encourage classroom discourse.

• Solution of the problem should require higher-level thinking and problem solving.

• Investigation of the problem should contribute to students’ conceptual development.

• The mathematical content of the problem should connect to other important mathematical ideas.

• Work on the problem should promote skillful use of mathematics and opportunities to practice important skills.

• The problem should create opportunities for the teacher to assess what students are learning and where they are experiencing difficulty. (Lappan & Phillips, 1998, pp. 87-88)

Undoubtedly the most common challenge we faced was deciding how big a problem can be. While rich, open problems encourage students to generate many interesting patterns and conjectures and to solve problems in a variety of ways, it can be difficult to decide which conjectures and problem-solving strategies to follow up on. We struggled to select and sequence problems in a way that (a) would allow both students and teachers a chance to make sense of the mathematics, (b) wouldn’t lead to activity that was irrelevant to the mathematical goals (e.g., mathematics projects that become art projects), and (c) would be manageable for teachers, both mathematically and pedagogically.

Our decisions about the grain-size of a problem were generally based on classroom experience. We considered whether the time required to develop an idea fully was time well spent; whether students could grasp the mathematical subtlety of the ideas; and whether students were reaching useful closure on important concepts, strategies, or skills.

For example, as we formulated plans for a unit on elementary data analysis and informal inference, we experimented with a student investigation comparing the length of first and last names. In its most open form, students could be given the simple task of collecting and organizing data to decide whether there seemed to be any significant patterns in such data. We found it useful to allow some initial exploration of this open question (to get a reading of students’ prior knowledge and experiences). However, this sort of open task rarely leads students to utilize useful standard techniques for data display (line plots, box plots, or scatter plots) or standard concepts for data summary (median, mean, and measures of variation). Pilot teachers almost always recommend that such materials need to provide “more structure” for students.

Another challenge was to find engaging contextual problems that would embody the mathematical concepts we wanted to target. Ideally, the context helps make concepts meaningful to students and enables them to retrieve that meaning more readily later on. We often found that when students were eventually presented with mathematical ideas symbolically—without a contextualizing storyline--they were able to go back to the original context in which they had learned those ideas to help them make sense of the symbols. It is important to note, however, that students’ ability to generalize knowledge in this way does not occur without being attended to specifically. (Lappan & Phillips 1998, p. 90).

Organization of the units. Each CMP unit is centered on an important mathematical idea or cluster of related ideas. A unit consists of four to seven investigations, each containing two to five problems for the class to explore. Each investigation ends with a set of problems called Applications, Connections, and Extensions (ACE). The ACE problems provide homework practice with applications of key ideas. We found a relatively effective mix of skills and applications through our extensive pilots and field-testing (each unit went through a three-year development and testing cycle). Finally, the Mathematical Reflections at the end of each investigation and the assessment package provide opportunities for teachers and students to assess their understanding throughout the unit (Lappan & Phillips, 1998, p. 90).

Providing Support for Teachers

We developed the CMP curriculum materials from the perspective that students develop deep understanding of mathematical concepts, skills, procedures, and processes through:

• solving problems;

• observing patterns and relationships among variables in a situation;

• conjecturing, testing, discussing, verbalizing, and generalizing these patterns and relationships;

• discovering salient mathematical features of patterns and relationships and abstracting from these underlying mathematical concepts, processes, and relationships;

• developing a language for talking about problems and for representing and communicating their ideas; and

• striving to make sense of and to connect the mathematical concepts they abstract from their experiences.

These kinds of student learning activities require a specific set of teaching practices to support them. (Lappan & Phillips, 1998, p. 85)

The teaching model. As we developed the CMP student materials and supporting teacher materials, we took into account the demands of problem-centered teaching, formulating an instructional model that provides a lesson-planning template. This model looks at instruction in three phases—launching, exploring, and summarizing.

During the first phase, the teacher launches the investigation with the whole class by setting the context for the problem. This involves making sure the students understand the setting or situation in which the problem is posed. More important, the problem must be launched in such a way that the mathematical context and challenge are clear. In planning a lesson launch, teachers need to consider questions like these:

• What mathematical questions are or can be asked in this situation?

• What are students expected to do?

• How are they expected to record and report their work?

• Will they be working individually, in pairs, or in groups?

• What tools are available that might be helpful?

This is also the time when, if necessary, the teacher introduces new ideas, clarifies definitions, reviews old concepts, and connects the problem to students’ prior knowledge. Teachers need to make sure they are launching tasks in such a way that the challenge of the task is left intact even though students are given a clear picture of what is expected. It is easy to tell too much, thus lowering the challenge of the task to something fairly routine.

After the task is launched, students explore the task as the teacher circulates, asking focusing questions when a student or group is struggling and extending questions when students have solved the problem but have not taken it as far as possible. The teacher takes stock of who is understanding and who needs help; who has a strategy, a good idea, a generalization, or an interesting way of explaining the problem solution that needs to be shared in the summary.

The final phase of the instructional model is the summary. This is the most important and, perhaps, the hardest phase to do well. Here the students and the teacher work together to make the mathematics of the problem more explicit; to generalize certain situations; to abstract useful mathematical ideas, processes, and concepts; to make connections; and to foreshadow mathematics that is yet to be studied.

As an example of how this launch/explore/summarize lesson is implemented, consider one of the tasks in a CMP algebra unit:

When a sports team wins a big game, they often celebrate by saluting each other with high fives all around. How many high fives will there be for winning teams with different numbers of members? (p.xx)

To launch this problem, students could share the sport teams on which they play, describe (briefly) how they celebrate a win, and simulate the “high five” routine with others. For the exploration phase, students could calculate the number of “high fives” for teams of different size to see if they can come up with a general rule for counting. In the summarize phase of the lesson, the teacher could ask groups to report their findings and explain their reasoning, the aim being to arrive at understanding that expressions equivalent to n(n-1)/2 will give the desired number for teams with n members. Follow-up ACE problems will extend student understanding of the underlying principle by work on tasks like finding the number of games played in an n-team round-robin tournament, where each team plays every other team exactly once. A connection problem might ask how many games are played in an n-team league where each team plays every other team twice, at home and away.

The CMP teacher materials provide help with the launch/explore/summarize teaching model. For each problem in the investigation, the teacher notes provide questions to ask during all three phases of instruction, examples of student thinking that can be expected, and extensions of the lesson content. These notes reflect the voices and experiences of our collaborating teachers and the field test teachers.

The teacher materials also provide help in understanding the mathematical goals of the curriculum. Each unit includes an overview of the development of mathematics in the unit. Interspersed throughout the unit are additional mathematical notes for the teacher. Some provide the teacher with insights into how the mathematics is connected to mathematics in later grades or how the ideas can be extended to apply to other problems and in other contexts. While we have evidence that teachers find the materials helpful, we know there is a need for further help. Inquiry-based teaching requires the teacher to be a careful listener, to make instant analysis of student thinking, and to make judgements about appropriate subsequent action.

Our experience shows that teachers need about three years of support to become comfortable with the CMP curriculum. Professional development became a central focus toward the end of the development of the curriculum materials.

Some practical issues. As we developed the CMP curriculum, we believed that asking students to write about and explain their ideas would help them to clarify their thinking and understanding. In early drafts of the text, however, the students were writing and explaining constantly! The trial teachers were overwhelmed. It became clear that there should be good mathematical reasons for writing assignments: ideas to pull together, clarify, and record for future reference. One result of our struggle with this issue is Connected Mathematics’ writing prompts, a feature that has become a real strength of the material. At the end of each collection of related problems is a set of questions designed to help students reflect on what has been learned, why it is important, when it is useful, and how it fits in with prior knowledge. The questions point to the key mathematical ideas of each investigation and how they fit together (abstracting and connecting). They point to important skills (how-tos) and to decisions about when to use specific ideas and skills (when-tos). They raise issues about how the problems of the current investigation are similar to and different from problems encountered earlier (connecting, discriminating, and elaborating). Finally, they point toward questions to ask in similar situations (questioning habits). These Mathematical Reflections are discussed by the teacher and the class together. Then students write their own answers to these questions in journals as a part of a self-assessment. Each unit contains four to seven Mathematical Reflections that tell a story about the unit (Lappan & Phillips, 1998, pp. 86-87)

Management

Problem-centered mathematics teaching raises a number of management issues. For example, a curriculum that develops student understanding through guided inquiry requires longer classroom periods—at least 45 minutes is needed for students to engage in a given problem and to carry the momentum over to the next day. Pacing is also an issue as the teacher tries to orchestrate the launch, explore, and summary phases of instruction. Unfortunately, variations in class time across the country make it difficult to set pacing guidelines for teachers. A 10-minute difference in the length of a class period is roughly equivalent to teaching one more CMP unit during the year. We also found that in many schools, other activities compete for class time. Some middle schools take off a week to go camping. Others have several days during the month for assemblies, retreats, tests, or other special events. Grading and dealing with special needs students are also problematic. Communicating to parents and administration about a curriculum that looks so different from the traditional curriculum is an added burden. Getting To Know CMP and the CMP web site provide suggestions from experienced CMP classroom teachers for dealing with some of these issues.

Field Testing

Our work began with the sixth grade units. Several were written and field-tested during the 1991-92 school year, with the first full field test of grade six during 1992-93. Meanwhile, seventh grade units were being developed with some field testing in ‘92-93 and the first full field test during 1993-94. A similar schedule was followed for the eighth grade. An outline of the testing sequence is given in the following table.

Dates	6/1991- 6/1992	6/1992- 6/1993	6/1993- 6/1994	6/1994- 6/1995	6/1995- 6/1996	6/1996- 6/1997
6th grade student & teacher materials	first draft; initial field testing	first full field test	Review & revision; 2nd fieldtest	Review & revision; 3rd field test	Final revision. sent to publisher
7th grade student & teacher materials		first draft; initial field testing	first full field test	Review & revision; 2nd field test	Review & revision; 3rd field test	Final revision. sent to publisher
8th grade student & teacher materials		first draft; initial field testing	first full field test	Review & revision; 2nd field test	Review & revision; 3rd field test	Final revision. sent to publisher

* The student and teacher materials include an assessment package for each unit.

Field-test teachers. The field-test teachers participated in the trials for a variety of reasons. Some were acquainted with our earlier work; some were disenchanted with their students’ mathematical understanding; some had been involved with other reform movements; and some were interested in the perks. Some schools decided that all of their teachers would be involved.

The role of the evaluation team. The evaluation team collected extensive data and provided information in four broad categories: curriculum and pedagogy, student assessment, implementation, and project operation. The following questions suggest the kinds of information that were gathered.

Curriculum and pedagogy:

• What assumptions can reasonably be made about what students bring to the grade that begins a curriculum?

• Given that time in the school year is finite, what are the most powerful and important mathematical ideas, concepts, skills, and ways of thinking that should be included in the curriculum?

• Is the curriculum working? Are the problems too big? Too small? Are there holes in the conceptual development? Are the contexts appealing to students?

• What help do teachers need to teach a unit?

Student assessment:

• How can teachers know what their students are understanding?

• How can we help teachers become more comfortable with unfamiliar methods of assessment?

• What is the influence of standardized tests that do not fit the goals of the project?

• How can we present new curricula in such a way that local districts can make informed decisions about he tradeoffs in student learning that these materials introduce?

Implementation:

• What kinds of concerns do teachers have about teaching a new curriculum, and what kinds of help do they need at different stages?

• How can we help schools and teachers inform their community about new curricula and elicit their support?

• How can teacher materials and teacher inservice help teachers with the management—both physical and intellectual—of teaching a new curriculum?

• What is the impact on new curricula of traditions like tracking that are embedded in community expectations?

• How can tracking-related problems be ameliorated?

• What are the special dilemmas of urban schools in dealing with new curricula? Large suburban schools? Rural schools?

Project evaluation:

• How can we determine how effective the curriculum is? What have students learned at the end of a year? Two years? Three years?

• How can we decide, describe, and evaluate how teachers use the materials?

• What factors seem to shape how the curriculum is used?

• What kinds of support do teachers need to use the curriculum as intended?

Other feedback. We struggled intensely with a concern for equity—trying to ensure that the materials were appealing and appropriately challenging to a wide range of students. Among the questions we grappled with were these: To what extent should the curriculum attempt to reflect rather than broaden students’ interests? Can one curriculum serve all students, including those from different ethnic and economic backgrounds as well as different ability levels? Does the curriculum work with heterogeneous classes? We had no definitive answers to these questions, but issues relevant to equity were always on our minds and were part of our ongoing dialogue with the directors and lead teachers from the professional development centers. As more school districts, including some very large urban districts, adopt CMP we will continue to collect evidence of how well the program is meeting our expectations.

Other kinds of valuable feedback dealt with the specifics of the curriculum itself. Are the flow and sequence of problems effective? Does the curriculum include an appropriate set of problems? Are there big jumps between problems? What mathematical ideas or strategies are missing? The teachers involved in the field tests provided us with a broad sense of whether a problem, investigation, or unit was working and where there were consistent weaknesses in students’ understanding or skills. We also relied on a smaller set of collaborating teachers who had been involved with us for a number of years to help us with these issues. Nevertheless, it was often left to us to figure out why some element of the material wasn’t working and how we might revise it.

Our advisory board, which represented a wide spectrum of the community, urged us to make changes that would help allay some of the concerns that teachers, administrators, and parents might have about this very different kind of curriculum material. Early in the development process, the board pointed out that it was difficult for a reader (parent or teacher) to get a sense of the key ideas covered in a unit without working out all of the problems and reading the teaching notes. To respond to this concern, we added a feature at the beginning of the book that highlights the mathematical goals for each unit; another feature at the end of each investigation reflects on the ideas and skills that have just been studied.

Because Connected Mathematics basic skill-building strategies are not obvious (i.e., there are not pages and pages of computation drills), the board also encouraged us to conduct an evaluation study of the development of students’ skills. Teachers would need this data, the board believed, to convince their administrators and parents to support the curriculum. Eventually, we settled on a two-part evaluation study: the Iowa Test of Basic Skills and a problem-solving/conceptual test developed by the Balanced Assessment Project (Zawojewski et. al, 1997). Again, results of this study proved to be very important for reassuring schools about the development of basic skills.

Evaluation

During the fourth and final year of development of Connected Mathematics, the Iowa Test of Basic Skills and a standards-based, problem-solving test were administered (pre- and post-instruction) in grades six, seven, and eight in CMP classes and non-CMP classes. The CMP students significantly outperformed the non-CMP students on the standards-based, problem-solving test at all three grade levels. While the CMP students held their own on the Iowa Test of Basic Skills in the sixth and seventh grade, there was a statistically significant difference in the eighth grade in favor of the CMP classes.

In addition, a study of proportional reasoning was carried out using pre- and post-instruction tests with matched samples of CMP and non-CMP students in the seventh grade. CMP students again scored significantly better than the non-CMP students. We continue to collect evidence from doctoral dissertations, other research studies, and state and local trend data. These research reports are listed on our web site as well as in a book produced by the publisher on evaluation of CMP.

CMP and some of the other NSF-funded curricula are considered by some to be experimental and are challenged to demonstrate their effectiveness. We are asked, “What is your evidence?” Our evaluation data has helped us answer the question. Increasingly, educators and the public alike are now asking for similar evidence of student achievement from developers and publishers of other more traditional textbooks.

Publishing

To have some assurance that the materials they funded would actually be used, NSF required that the author teams have a signed contract with a publisher before receiving the second half of their funding. Curriculum materials such as those created by CMP are not like typical products of the large mainstream publishers, so we decided to go with a small but innovative company. One advantage of the smaller publishing companies is their understanding of your goals and your curriculum. A major drawback is their lack of a national sales force. In our case, shortly after we signed in 1994, our publisher was bought by one of the larger mainstream publishing companies, which in turn soon merged with another, bigger company. Since the first signing, we have had three different publishers. We are now with a major publishing company and have developed a productive and collaborative relationship. However, our biggest continuing concern is keeping control of the intellectual content of the materials, including their mathematical content, pedagogy, and format.

Sustaining Momentum

CMP has been recognized nationally by both the American Association for the Advancement of Science and the U.S. Department of Education, which awarded Connected Mathematics the only “Exemplary” rating out of all the middle school curricula reviewed. The use of the material is growing rapidly—it is currently used in all 50 states. But such a curriculum needs professional development, and this is a problem. In the past, most new textbooks did not need more than a half-day introduction to teachers. Connected Mathematics, like other reform curricula, needs extensive support for its teachers, administrators, and parents.

Professional Development

Our experiences over the past 30 years show that teachers need long-term, coherent professional development support to implement the kind of standards-based instruction that is central to CMP.

In 1995, we received an NSF-funded teacher enhancement grant to train and assist 22 teams of leaders from across the country in their efforts to implement CMP into their schools. Over the course of four years, each site adapted our professional development to meet the needs of their districts. From this, a rich set of professional development experiences, including some with intensive coaching components, emerged. A subset of the participants were selected and given additional leadership training at the end of the four years. This cadre of CMP leaders is now called upon by other districts to assist them in their implementation efforts.

Based on all of our experiences, we now have a module that describes our professional development program and can be used as a guide for districts implementing CMP. (See the CMP Web site at www.math.msu.edu/cmp.) We also have a full-time professional development consultant who answers questions and advises potential CMP adopters on implementation and professional development.

The Show-Me Center

In 1997, NSF funded three centers—elementary, middle, and high school—to support the dissemination and implementation of standards-based curricula. The Show-Me Center, under the direction of Dr. Barbara Reys at the University of Missouri, is responsible for supporting the dissemination of information about the NSF middle grades curriculum projects. The Show-Me grant has provided CMP with funding to continue some of our implementation efforts. To meet one of the Show-Me Center project goals, we created a professional development module that can be used by all of the NSF-funded curriculum projects. Working with the Show-Me Center, we also conduct two “Getting to Know CMP” workshops and one CMP users’ conference during the year.

The CMP Web site and Getting To Know CMP

The Getting To Know CMP book contains information about CMP—its goals, units, format, and evaluation data—along with help especially for parents. We also depend heavily on our web site disseminating the most up to date information about CMP. For example, results from studies of student achievement with CMP are immediately posted on the Web. Over time, the types of questions and the needs of various people responsible for implementing CMP have stabilized. We can now concentrate on doing carefully focused research to meet these needs.

Summary

Development work of the sort that led to the Connected Mathematics materials requires a commitment on the part of a great number of people. The teachers who welcomed us into their classrooms, who were painfully honest with us when the materials were not working, and who were our biggest cheerleaders when the units began to take shape are the real heroes of our story. Without the trust built up over years of working with these teachers across the country, we could not have had the access to classrooms for experimentation and trials of the material in its infant stages. Teachers who worked with us on this project knew that we respected them, their knowledge, and their expertise. They often commented on how different their experiences with CMP were from their other interactions with university faculty.

Teachers know things that we do not know. If we are to improve the curriculum, teaching, and learning of mathematics and science in this country, teachers must be included in our efforts.

We have tried in this paper to provide a history of the development of CMP and the challenges we faced. The publications referenced below may be helpful to readers who want more details about aspects of our work. We hope that readers will find our ideas useful. It would be a shame if new curriculum development efforts started from scratch with no opportunity to build on what others have learned.

References

A selection of authors’ papers prior to 1991

A selection of papers about CMP or based on CMP classrooms (after 1991)

Other references