Design and Development Process

The CMP philosophy grew out of work in middle grades classrooms. With the aid of funds from several National Science Foundation (NSF) professional development grants, Glenda Lappan, Elizabeth Phillips, and William Fitzgerald, with their colleagues at Michigan State University worked with experienced teachers to find new ways to engage students with mathematics.

With teacher support and funding from NSF, they created five units that were published in 1985 for use by teachers to help make a transition from how they typically taught, ’show and practice,’ to classrooms that required engagement in mathematical thinking, reasoning, solving, and proving. These units are known as the Middle Grades Mathematics Project (MGMP) materials. Each was focused on an important area of mathematical thinking, and each was designed to engage students in important problem situations that were engaging, challenging, and educative. During this time, their focus was on developing experienced teachers’ knowledge of mathematics and their skill in engaging students in making sense of mathematics.

For the next ten years, the MGMP units were used to provide professional development and leadership training for teachers and administrators in our Michigan State University summer institutes. Teachers and leaders who attended the  summer institutes spread the units across the country through their local institutes.

It was due to these MGMP materials and their success that our group at Michigan State University was asked to make substantial contributions to the development of the Curriculum and Evaluation Standards for School, published by the National Council of Teachers of Mathematics (NCTM, 1989). Lappan chaired the middle school writing group and reviewed the draft standards with teachers in our summer institutes. This document launched a new era of curriculum development in the US and our group was ready to act on our conviction that curriculum materials could make a difference” (Lappan & Phillips, 2009, p. 1). These NSF-funded activities created a rich school leadership base to support the subsequent development and use of the Connected Mathematics Project curriculum in school districts.

The NSF funded Connected Mathematics 1 (CMP1) in 1991. At this time,Maryland and Susan Friel from the University of North Carolina joined the author team at Michigan State University for the development of CMP1. Both Fey and Friel had extensive curriculum and professional development experiences—Fey at the secondary level and Friel at the elementary level. This collaboration continues today. In 2000, NSF funded a revision of the materials, Connected Mathematics 2 (CMP2), to take advantage of what was learned in the six years that the first edition of CMP had been used in schools. In 2010 the Common Core State Standards for Mathematics (CCSSM) were released. At about the time a revision of CMP3 was being considered. With funding from the University of Maryland and Michigan State University’s royalties, Connected Mathematics 3 (CMP3) was developed and the ten years later, CMP4. All four editions build on the unique strengths of the CMP philosophy. The following table illustrates the design challenges over the four revision cycles.

Curriculum Design and Teacher Enactment Tensions

Edson, A.J., Wald, S., & Phillips, E. D. (2025). The design, field-testing, and evaluation of a contextual, problem-based curriculum: Feedback analysis from mathematics teachers on the fourth edition of CMP. Education Sciences, 15(5), 628 1-29. https://doi.org/10.3390/educsci15050628.

The development of Connected Mathematics four editions was built on the extensive knowledge gained from the development, research and evaluation of the authors over the past forty years. The diagram shows the design and development cycle used for CMP 1, CMP 2, CMP3, and CMP4. Each revision went through at least three cycles of design, field trials–data feedback–revision. This process of (1) commissioning reviews from experts, (2) using the field trials with four feedback loops for each draft, (3) conducting key classroom observations by the CMP staff, and (4) monitoring of student performance on state and local tests by trial schools was done for CMP1, CMP2, CMP3, and CMP4.

The CMP philosophy fosters a focus on isolating important mathematical ideas and embedding these ideas in carefully sequenced sets of contextual problems. These problems are developed and trialed in classrooms in different states over several years. Each revision of CMP has been extensively field-tested in its development phases. We solicited iterative and in-depth input and reviews from teachers, parents, administrators, mathematics educators, mathematicians, cognitive scientists, and experts in reading, special education, and English language learners. Our materials are created to support teachers in helping their students develop deeper mathematical understanding and reasoning. This stance is the foundation of the success of CMP, which has withstood the pressures of various political changes in the Nation over time.

“Getting to know something is an adventure in how to account for a great many things that you encounter in as simple and elegant a way as possible. And there are lots of ways of getting to that point, lots of different ways. And you don’t really ever get there unless you do it, as a learner, on your own terms. ... All you can do for a learner enroute to their forming a view of their own view is to aid and abet them on their own voyage. ... in effect, a curriculum is like an animated conversation on a topic that can never be fully defined, although one can set limits upon it. I call it an “animated” conversation not only because one uses animation in the broader sense—props, pictures, texts, films, and even “demonstrations.” Conversation plus show-and-tell plus brooding on it all on one’s own.” 

Bruner, Jerome. (1992). Science education and teachers: A Karplus lecture. Journal of Science Education and Technology; 1(1), 5-12.

Developing a curriculum with a complex set of interrelated goals takes time and input from many people. As authors, our work was based on a set of deep commitments we had to creating a more powerful way to engage students in making sense of mathematics. Our Advisory Boards took an active role in reading and critiquing Units in their various iterations. In order to enact our development principles, we found that at least three full years of field trials in schools for each development phase were essential.

The testing and feedbackfrom teachers and students across the country is the key element in the success of the CMP materials. The final materials comprised the ideas that stood the test of time in classrooms across the country. Thousands of teachers around the the United States and several foreign countries (and their thousands of students) are a significant part of the team of professionals that made these materials happen. The interactions between teacher and students with the materials became the most compelling parts of the teacher support.

Without these teachers and their willingness to use materials that were never perfect in their first-versions, CMP would have been a set of ideas that lived in the brains and imaginations of the author team. Instead, they are materials with classroom heart because our trial teachers and students made them so. We believe that such materials have the potential to dramatically change what students know and can do in mathematical situations. The emphasis on thinking and reasoning, on making sense of ideas, on making students responsible for both having and explaining their ideas, and on developing sound mathematical habits provides opportunities for students to learn in ways that can change how they think of themselves as learners of mathematics.

From the authors’ perspectives, our hope has always been to develop materials that play out deeply held beliefs and firmly grounded theories about what mathematics is important for students to learn and how they should learn it. We hope that we have been a part of helping to challenge and change the curriculum landscape of our country. Our students are worth the effort.

For more information on the history and development of Connected Mathematics, see A Designer Speaks.

The CMP1 authors began by working with an outstanding advisory board to articulate the goals for what a student exiting from CMP in grade eight would know and be able to do. The understanding needed for each major strand, number, algebra, geometry, measurement, probability and statistics and in the interactions among these strands were addressed in several essays. These essays that elaborated our goals became our touchstones for the development of the materials for three grades—6, 7, and 8 for CMP1, CMP2, and CMP3. (For copies of these essays see A Designers Speaks.)

 

Connected Mathematics1	Connected Mathematics2

CMP1: NSF/ESI-93-55542, NSF/ESI-93-55542, and NSF/ESI-97-14999

CMP2: NSF/ESI-9986372 and NSF/CG004954-3

Central to the success of CMP1 was the cadre of hard-working teachers who cared deeply about their students learning of mathematics. As we set out to write a complete connected curriculum for grades, 6, 7, and 8, the following issues quickly surfaced and needed resolution:

• Identifying the important ideas and their related concepts and procedures;
• Designing a sequence of tasks to develop understanding of the idea;
• Organizing the sequences into coherent, connected curriculum;
• Balancing open and closed tasks;
• Making effective transitions among representations and generalizations;
• Addressing student difficulties and ill-formed conceptions;
• Deciding when to go for closure of an idea or algorithm;
• Staying with an idea long enough for long-term retention;
• Balancing skill and concept development;
• Determining the kinds of practice and reflection needed to ensure a desired degree of automaticity with algorithms and reasoning;
• Writing for both students and teachers; and
• Meeting the needs of all fifty states and diverse learners.

With the help of the field teachers, advisory board and consultants, these issues were resolved over the six years of development, field-testing, and evaluation.

Curriculum Design and Teacher Enactment Tensions

Edson et al. (under review). Adapted from Edson et al. (2025).

 

Before starting the design phase for the CMP2 materials, we commissioned individual reviews of CMP material from 84 individuals in 17 states and comprehensive reviews from more than 20 schools in 14 states. Individual reviews focused on particular strands over all three grades (such as number, algebra, or statistics) on particular subpopulations (such as students with special needs or those who are commonly underserved), or on topical concerns (such as language use and readability). Comprehensive reviews were conducted in groups that included teachers, administrators, curriculum supervisors, mathematicians, experts in special education, language, and reading-level analyses, English language learners, issues of equity, and others. Each group reviewed an entire grade level of the curriculum. All responses were coded and entered into a database that allowed reports to be printed for any issue or combination of issues that would be helpful to an author or staff person in designing a Unit.

In addition, we made a call to schools to serve as pilot schools for the development of CMP2. We received 50 applications from districts for piloting. From these applications, we chose 15 that included 49 school sites in 12 states and the District of Columbia. We received evaluation feedback from these sites over the five-year cycle of development.

Based on the reviews, what the authors had learned from CMP pilot schools over a six-year period, and input from our Advisory Board, the authors started with grades 6 and 7 and systematically revised and restructured the Units and their sequence for each grade level to create a first draft of the revision. These were sent to our pilot schools to be taught during the second year of the project. These initial grade-level Unit drafts were the basis for substantial feedback from our trial teachers.

Here are examples of the kinds of questions we asked classroom teachers following each revision of a Unit or grade level.

"Big Picture" Unit Feedback

• Is the mathematics important for students at this grade level? Explain. Are the mathematical goals clear to you? Overall, what are the strengths and weaknesses in this Unit?
• Please comment on your students’ achievement of mathematics understanding at the end of this Unit. What concepts/skills did they “nail”? Which concepts/skills are still developing? Which concepts/skills need a great deal more reinforcement?
• Is there a flow to the sequencing of the Investigations? Does the mathematics develop smoothly throughout the Unit? Are there any big leaps where another Problem is needed to help students understand a big idea in an Investigation? What adjustments did you make in these rough spots?

Problem-by-Problem Feedback

• Are the mathematical goals of each Problem/Investigation clear to you?
• Is the language and wording of each Problem understandable to students?
• Are there any grammatical or mathematical errors in the Problems?
• Are there any Problems that you think can be deleted?
• Are there any Problems that needed serious revision?

Applications-Connections-Extensions

• Does the format of the ACE exercises work for you and your students? Why or why not?
• Which ACE exercises work well, which would you change, and why?
• What needs to be added to or deleted from the ACE exercises? Is there enough practice for students? How do you supplement and why?
• Are there sufficient ACE exercises that challenge your more interested and capable students? If not, what needs to be added and why?
• Are there sufficient ACE exercises that are accessible to and helpful to students that need more scaffolding for the mathematical ideas?

Mathematical Reflections and Looking Back/Ahead

Are these reflections useful to you and your students in identifying and making more explicit the “big” mathematical ideas in the Unit? If not, how could they be improved?

Assessment Material Feedback

• Are the check-ups, quizzes, tests, and projects useful to you? If not, how can they be improved? What should be deleted and what should be added?
• How do you use the assessment materials? Do you supplement the materials? If so, how and why?

Teacher Content Feedback

• Is the teacher support useful to you? If not, what changes do you suggest and why?
• Which parts of the teacher support help you and which do you ignore or seldom use?
• What would be helpful to add or expand in the Teacher support?

Year-End Grade-Level Feedback

• Are the mathematical concepts, skills and processes appropriate for the grade level?
• Is the grade-level placement of Units optimal for your school district? Why or why not?
• Does the mathematics flow smoothly for the students over the year?
• Once an idea is learned, is there sufficient reinforcement and use in succeeding Units?
• Are connections made between Units within each grade level?
• Does the grade-level sequence of Units seem appropriate? If not, what changes would you make and why?
• Overall, what are the strengths and weaknesses in the Units for the year?

Final Big Question

What three to five things would you have us seriously improve, change, or drop at each grade level?

The development of CMP3 was built on the knowledge we had gained over the past 20 years of working with teachers and students who used CMP1 and CMP2. In addition, for 17 years we solicited information from the field through our website and CMP mailing list and through our annual CMP week-long workshops and two-day conferences. The experiences with development processes for CMP1 and CMP2 and the ongoing gathering of information from teachers resulted in a smaller but more focused development process for CMP3.

The process of revision for CMP3 was similar to the preceding iterations except on a smaller scale. A group of field-test teachers from CMP2 trialed the versions of the Units for CMP3 that had substantive changes from CMP2. They also contributed to the development of the assessment items and suggested many of the new features in the student and teacher materials. They were influential in designing many new features such as the “focus questions” for each problem, a more streamlined set of mathematical goals, and Mathematical Reflections. Their feedback was invaluable in making sure that our adjustment for CCSSM resulted in materials from which students and teachers could learn. CMP3 is fully aligned with the Common Core State Standards for Mathematics (CCSSM) and Mathematical Practices and reflects the thoughtful concern and care of the authors and CMP3 trial teachers. This process guided by the following design and enactment tensions has produced a mathematical experience that is highly educative for students and teachers in the middle grades.

Curriculum Design and Teacher Enactment Tensions

Edson, A.J., Wald, S., & Phillips, E. D. (2025). The design, field-testing, and evaluation of a contextual, problem-based curriculum: Feedback analysis from mathematics teachers on the fourth edition of CMP. Education Sciences, 15(5), 628 1-29. https://doi.org/10.3390/educsci15050628

For the Development of CMP4 see Connected Mathematics® 4.

The fundamental features of the CMP program reflect the authors’ core beliefs about essential features of effective teaching and learning of mathematics. These features include a focus on big ideas and the connections among them, teaching through student-centered exploration of mathematically rich problems, and using continuous assessment to inform instruction.

Through more than thirty years of CMP work, our personal knowledge and beliefs about mathematics and its teaching have been richly enhanced by advice from teachers and students who used field-test versions of the materials. We have gained valuable insights from mathematicians, teacher educators, and other curriculum developers. Our work has also been influenced in significant ways by insights from theoretical and empirical research in mathematics education, cognitive science, educational research about teaching and teacher development, and education policy implementation.

From the broad array of pertinent research findings seven major themes continue to influence development and implementation of the CMP curriculum and teaching resources. These themes are explored below.

Collaborative Learning and Classroom Discourse

We are in general agreement with constructivist explanations of the ways that knowledge is developed, especially the social constructivist ideas about the influence of discourse on learning. There is a consistent and growing body of research indicating that when students engage in collaborative work on challenging problem-solving tasks, their mathematical and social learning will be enhanced. Recent research has also shown that the discourse of a classroom—the ways of representing, thinking, talking, agreeing and disagreeing—is central to what students learn about mathematics. Effective teachers ask open-ended questions to elicit student thinking and ask students to explain their thinking and comment on one another’s work.

This complex of related research findings is reflected in our decision to write materials that support student-centered investigation of mathematical problems and in our attempt to design problem content and formats that encourage student- student and student-teacher dialogue about the work.

Teaching Through Problem Solving

Mathematics education has always sought to develop student problem solving skills. But teaching mathematics through problem solving presents different challenges for both teachers and students.

Over the past several decades, researchers have engaged in extensive classroom studies to assess the effectiveness of problem-based learning and the practices that make it effective. A summary of that research by Stein, Boaler, and Silver identified five critical teacher actions: “(a) (appropriate) scaffolding of students’ thinking; (b) a sustained press for students’ explanations; (c) thoughtful probing of students’ strategies and solutions; (d) helping students accept responsibility for, and gain facility with, learning in a more open way; and (e) attending to issues of equity in the classroom.” The CMP instructional materials and teacher resources have been designed with the explicit aim of engaging and sustaining diverse groups of students in high-level thinking.

Equity for Learning

Connected Mathematics is grounded in the latest research on equity in education, ensuring that instructional strategies, assessments, and materials are inclusive and responsive to diverse student needs. The problem based provides access to rigorous mathematical concepts, while addressing systemic barriers and promoting a classroom environment where every learner feels valued and empowered to succeed. 

We have also done some research of our own to explore aspects of mathematics and teaching that are most effective for engaging student attention and interest. We have also attended to research that explores the response of diverse students to teaching through problem solving.

<see Attending to Individual Needs Framework in the What’s New in CMP4 section.>

Conceptual and Procedural Knowledge

We have been influenced by theory and research indicating that mathematical understanding is fundamentally a web of logical and psychological connections among ideas. Knowledge that is rich in connections will be retained well and retrieved effectively for subsequent problem solving and reasoning tasks. Furthermore, the  research on the interplay of conceptual and procedural knowledge suggests that sound conceptual understanding is an important foundation for procedural skill, not an incidental and delayed consequence of repeated rote procedural practice.

These findings are reflected in the CMP instructional materials that are rich in connections between topics in the four major content strands of the curriculum— number and operations, geometry and measurement, data analysis and probability, and algebra—and in our careful attention to laying conceptual foundations for learning of mathematical procedures.

Formative Assessment

Extensive recent research has demonstrated convincingly that student learning improves significantly when teachers provide frequent feedback on their progress and when teachers use that assessment as a core input to their planning for instruction. The CMP instructional resources provide a variety of classroom-tested tools for such helpful formative assessment. The CMP authors have continued to study the role of assessment as an integral part of the teaching and learning cycle.

Information about CMP's Formative Assessment Research Project.

Mathematical Knowledge for Teaching

Research over the past several decades has shown quite convincingly that effective mathematics teachers have a special kind of understanding of the subjects they teach—knowledge that includes ways of representing concepts and procedures in different forms and ways that students are likely to have difficulty in learning or to form misconceptions. This special kind of understanding of

mathematics is often referred to as mathematical knowledge for teaching. Some of the keys to this valuable teacher knowledge are generic—applying across all content strands of the subject—while others are particularly salient in specific topics.

In design and development of the CMP instructional materials, we have paid close attention to both general and specific kinds of mathematical knowledge for teaching. We have included tasks for students that will help them succeed when confronted by common obstacles to learning. We have included advice to teachers about those likely stumbling blocks and strategies for addressing or avoiding them. In particular, the research findings in the following content areas of mathematical reasoning have been incorporated into the student and teacher materials.

Multiple Representations

An important indication of students’ connected mathematical knowledge is their ability to represent ideas in a variety of ways. This suggests that instructional materials should frequently provide and ask for knowledge representation using graphs, number patterns, written explanations, visual images, and symbolic expressions. Students also should be challenged to interpret information provided in one representational style through use of another representation.

Rational Numbers/Proportional Reasoning

The extensive psychological literature on the development of rational numbers and proportional reasoning has guided our development of curriculum materials to address this important middle school topic. Furthermore, the implementation of CMP materials in real classrooms has allowed us to contribute to that literature with research publications that show the effects of new teaching approaches to traditionally difficult topics.

Probability and Statistical Reasoning

The interesting research literature related to the development of and the cognitive obstacles to student learning of statistical concepts, such as mean and graphic displays, and probability concepts, such as the law of large numbers, has been used as we developed the statistics and probability Units for CMP3.CMP4 is based on the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report Endorsed by the American Statistical Association. (August 2005)

Algebraic Reasoning

The different conceptualizations of algebra described and studied in the research literature contributed to the treatment of algebra in CMP4. Various scholars describe algebra as a study of modeling, functions, generalized arithmetic, and/or as a problem-solving tool. CMP3 develops each of these foci for algebra but attends more directly to functions and the effects of rates of change on representations.

The research literature illuminates some of the cognitive complexities inherent in algebraic reasoning and offers suggestions for helping students overcome difficulties. We have drawn on that research in the design of our approach to concepts of equivalence, functions, the equal sign, algebraic variables, the use of graphs and other representations, and the role of technology.

Geometric/Measurement Reasoning

Results from national assessments show that achievement in geometry and measurement is weak among many American students. The theoretical ideas of the van Hieles and other specific studies about student understanding of shape and form, and learning of geometric/ measurement concepts, such as angle, area, perimeter, volume, congruence, transformations, and processes such as visualization, contributed to the development of geometry/measurement Units in CMP4 materials.

Teacher Development and School Change

Education is a social institution with traditional practices that are deeply resistant to change. Thus, implementation of strikingly different curricular and instructional practices is a challenge for teachers and students and for the broader school community.

In the process of helping teachers through professional development, we have paid close attention to what is known about effective teacher professional development and the school strategies that are most effective. The sections of this guide reflect our reading of the research on teacher and school change and our experience over 35 years of work with teachers and schools introducing the CMP curriculum.

While each of the twelve themes described above indicates influence of theory and research on design and development of the CMP curriculum, teacher, and assessment materials, it would be misleading to suggest that the influence is direct and controlling in all decisions. As the authors have read the research literature reporting empirical and theoretical work, research findings and new ideas have been absorbed and factored into the creative, deliberative, and experimental process that leads to a comprehensive mathematics program for schools.

Additional references to consult for more insight into what research says in these areas and are:

 Kilpatrick, J., Martin, W. G., & Schifter, D. (Eds.) (2003) A Research Companion to Principles and Standards for School Mathematics, Reston, VA: National Council of Teachers of Mathematics.

Cai, J. (Ed.) (2017) A Compendium for Research in Mathematics Education, Reston, VA: National Council of Teachers of Mathematics.

Hatti, J. (2016)  Visible Learning for Mathematics, Grades K-12:What Works Best to Optimize Student Learning. Corwin Mathematics.

National Council of Teachers of Mathematics (2014).  Principles to Actions: Ensuring Mathematical Success for All. Reston, Va.: NCTM.

Newton, J. & Grant, Y., (2017) “The Right Stuff”: Curriculum to Support the Vision of NCTM and CCSSM.” In Enhancing Classroom Practice with Research behind Principles to Actions, edited by Denise A. Spangler and Jeff Wanko. Reston, VA: National Council of Teachers of Mathematics.