Design and Development Process
The development of CMP 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 and CMP 2. Each revision went through at least three cycles of design, field trials–data feedback–revision. Building on these experiences, CMP3 underwent a similar development process but on a smaller scale. This process of (1) commissioning reviews from experts, (2) using the field trials with feedback loops for the materials, (3) conducting key classroom observations by the CMP staff, and (4) monitoring student performance on state and local tests by trial schools comprises research-based development of curriculum.
The feedback from 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. Over 425 teachers and thousands of their students in 54 school district trial sites 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 the trial teachers and students made them so. These materials have the potential to dramatically change what students know and are able to 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.
Research, Field Tests, and Evaluation
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.
The Development of CMP1 and CMP2
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 in each of the strands of mathematics under consideration—number, algebra, geometry, measurement, probability and statistics and in the interactions among these strands 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 papers, see A Designers Speaks.)
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.
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?
- 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?
- 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
The development of CMP3 was built on the knowledge we gained over the?past 20 years of working with teachers and students who used CMP1 and CMP2. In addition, for the past 17 years we have solicited information from the field through our web site 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 have 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 CCSSM and Mathematical Practices and reflects the thoughtful concern and care of the authors and CMP3 trial teachers. This process has produced a mathematical experience that is highly educative for students and teachers in the middle grades.
Co-Development with Teachers and Students
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 three full years of field trials in schools for each development phase were essential.
This feedback from 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. Nearly 200 teachers in 15 trial sites around the country (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 are able to 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.