# Philosophy

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.

- Guiding Principles
- A Problem Centered Curriculum
- Criteria for Mathematical Tasks
- Mathematical Fluency
- Depth vs Spiraling
- Instructional Model

## Guiding Principles

The authors were guided by the following principles in the development of the CMP materials. These statements reflect both research and policy stances in mathematics education about what works to support students' learning of important mathematics.

**CMP is problem-centered.**Mathematical tasks for students in class and in homework are the primary vehicle for student engagement with the mathematical concepts to be learned. The key mathematical goals are elaborated, exemplified, and connected through the Problems in an Investigation.**CMP identifies big ideas and goes for depth.**Ideas are explored through these mathematical tasks in the depth necessary to allow students to make sense of them. Superficial treatment of an idea produces shallow and short-lived understanding and does not support making connections among ideas.**CMP has coherence.**The underlying concepts, skills, or procedures supporting the development of a key idea are identified and included in an appropriate development sequence. The curriculum builds and connects from Problem to Problem, Investigation to Investigation, Unit to Unit and grade to grade.**CMP intertwines conceptual and procedural knowledge.**The curriculum 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 to produce fluency in both conceptual and procedural knowledge.**CMP develops skills and concepts as needed.**Concepts and skills are developed, as appropriate, to solve interesting and challenging problems.**CMP promotes inquiry-based instruction.**Classroom instruction focuses on inquiry and investigation of mathematical ideas embedded in rich problem situations through rich classroom discourse and collaborations.**CMP promotes effective use of technology.**The curriculum reflects the information-processing and delivery 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 to new tasks.**CMP has high expectations of all students.**All students are asked sophisticated mathematical questions and are expected to persevere in their explorations to these questions, looking for patterns, generalizing, validating, and sharing and critiquing each others work.

## A Problem Centered Curriculum

### Rationale for a Problem-Centered Curriculum

Students' perceptions about a discipline come from the tasks or problems with which they are asked to engage. 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. 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. If the purpose of studying mathematics is to be able to solve a variety of problems, then students need to spend significant portions of their mathematics time solving problems that require thinking, planning, reasoning, computing, and evaluating.

A growing body of evidence from the cognitive sciences supports the theory that students can make more sense of mathematics if the concepts and skills are embedded within a context or problem. If time is spent exploring interesting mathematics situations, 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 of mathematical concepts and related procedures. This method is quite different from the assumption that students learn by observing a teacher as he or she demonstrates how to solve a problem and then practicing that method on similar problems. A problem-centered curriculum not only helps students to make sense of the mathematics, but also helps them to process the mathematics in a retrievable way.

Teachers of CMP report that students in succeeding grades remember and refer to a concept, technique, or problem-solving strategy by the name of the Problem in which they encountered the ideas. For example, the Basketball Problem from What Do You Expect? in grade 7 becomes a trigger for remembering the processes of finding compound probabilities and expected values.

Results from the cognitive sciences also suggest that learning is enhanced if it is connected to prior knowledge and is more likely to be retained and applied to future learning. Critically examining, refining, and extending conjectures and strategies are also important aspects of becoming reflective learners.

In CMP, important mathematical ideas are embedded in the context of interesting problems. As students explore a series of connected problems, they develop an understanding of the embedded ideas and, with the aid of the teacher, abstract powerful mathematical ideas, problem-solving strategies, and ways of thinking. They learn mathematics and learn how to learn mathematics.

## Criteria for Mathematical Tasks

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, connecting, and communicating, the problems need to encourage them to use these processes.

### Each CMP Problem has some or all of the following characteristics:

- Embeds important, useful mathematics
- Promotes conceptual and procedural knowledge
- Builds on and connects to other important mathematical ideas
- Requires higher-level thinking, reasoning, and problem solving
- Provides multiple access points for students
- Engages students and promotes classroom discourse
- Allows for various solution strategies
- Creates an opportunity for teacher to assess student learning

## Mathematical Fluency

Students need to practice mathematical concepts, ideas, and procedures to reach a level of fluency that allows them to “think” with the ideas in new situations. To accomplish this, we were guided by the following principles related to skills practice.

- Immediate practice is related to the situations in which the ideas have been developed and learned.
- Continued practice uses reasoning, skills, and procedures in situations that connect to ideas that students have already encountered as well as to new situations.
- Practice is distributed over time to allow ideas, concepts and procedures to reach a level of fluency of use in familiar and unfamiliar situations and to allow connections to be made to other concepts and procedures.
- Students learn how to make judgments about what operation or combination of operations or representations is useful in a given situation, as well as become skillful in carrying out the needed computation(s);
*knowing how to, but not when to, is insufficient.*

## Depth vs Spiraling

Example of Depth and Connectedness

The concept of a “spiraling” curriculum 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. This causes teachers to spend a great deal of time re-teaching the same ideas over and over again. Without a deeper understanding of concepts and how they are connected, students come to view mathematics as a collection of different techniques and algorithms to be memorized.

Problem solving based on such learning becomes a search for the correct algorithm rather than seeking to make sense of the situation, considering the nature and size of a solution, putting together a solution path that makes sense, and examining the solution in light of the original question. Taking time for the ideas studied to be more carefully developed means that when these ideas are met in the future, students have a solid foundation on which to build. Rather than being caught in a cycle of relearning the same ideas at a superficial level, which are quickly forgotten, students are able to connect new ideas to previously learned knowledge and make substantive advances in learning.

With any important mathematical concept, there are many related ideas, procedures, and skills. At each grade level, a small, select set of important mathematical concepts, ideas, and related procedures are studied in depth rather than skimming through a larger set of ideas in a shallow manner. This means that time is allocated to develop an understanding of key ideas in contrast to “covering” a book. The teacher support materials accompanying CMP were developed to support teachers to plan for and teach a problem-centered curriculum. Practice on related skills and algorithms is provided in a distributed fashion so that students not only practice these skills and algorithms to reach facility in carrying out computations, but they also learn to put their growing body of skills together to solve new problems.

## Instructional Model

### An Inquiry-Based Classroom

Problem-centered teaching opens the mathematics classroom to exploring, conjecturing, reasoning, and communicating. The CMP teacher materials are organized around an instructional model that supports this kind of “inquiry-based” teaching. This model is very different from the “transmission” or “direct instruction” model, in which teachers tell students facts and demonstrate procedures and then students memorize the facts and practice the procedures. The CMP model looks at instruction in three phases: launching, exploring, and summarizing.

### Teacher Roles

When mathematical ideas are embedded in problem-based investigations with rich context, the teacher has a critical responsibility for ensuring that students abstract and generalize the important mathematical concepts and procedures from their experiences with the problems. In a problem-centered classroom, teachers take on new roles. They move from always being the one who does the mathematics to being the one who guides, interrogates, and facilitates the learner to do and make sense of the mathematics. As teachers grow into their new roles, satisfaction comes from their increasing ability to recognize and use student thinking more effectively in developing a deeper understanding, reasoning, and skills. Witnessing students collaborating, challenging each other, thinking in new ways, and communicating these ideas is worth the effort. Extensive help is provided in the teacher support. See Components and The CMP Classroom for more information about the teacher support.

Teacher Materials

Classroom

Teacher Support

### Student Roles

The problems in which students engage form the perceptions they have about mathematics. For student learning of mathematics, the “what to teach” and the “how to teach it” are inextricably linked. To develop a deep understanding of mathematical concepts, skills, procedures, and processes, students take responsibility for their learning by taking on the following roles during the instruction.

During the Launch, students make sense of the context and the nature of the question.

In the Explore, students continue to make sense of the challenge as they work with their classmates in small groups, pairs, or individually to:

- Observe patterns and relationships in a situation;
- Recognize and employ prior understandings to conjecture, test, discuss, verbalize, and generalize patterns;
- Justify their reasoning, including the expectation to make sense of the reasoning of their classmates;
- Discover salient mathematical features of patterns and relationships and abstract the underlying mathematical concepts, processes, and relationships; and,
- Discover salient mathematical features of patterns and relationships and abstract the underlying mathematical concepts, processes, and relationships; and,

During the Summarize the teacher and students work as a class to:

- Clarify their understanding and strategies by sharing it with others and critiquing the work of other students who have followed different paths to the result;
- Make visible the mathematics embedded in the problem;
- Develop a mathematical language for representing and communicating ideas; and
- Make sense of and connect mathematics abstracted from their experiences to prior learnings.

At the end of an Investigation, the students have a chance to reflect on the essential understandings and strategies of the Investigation in the *Mathematical Reflections*. Then at the end of the unit, students distil their learnings into one large reflection in the *Looking Back*.

Even though CMP was developed long before the publication of the National Academy of Science, Adding It Up (NRC, 2001)[1], the stances in this National Research Council publication’s five interrelated strands that comprise student proficiency are reflected in CMP. The first four NRC strands are conceptual understanding, procedural fluency, strategic competence (the ability to formulate, represent, and solve mathematical problems), and adaptive reasoning (the capacity to think logically and to informally and formally justify one’s reasoning). The fifth strand, productive disposition, is “the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics” (Lappan & Phillips 2008)[2] More recently, the Standards for Mathematical Practice^{3} support the roles for engaging students in CMP classrooms.

[1] National Research Council. (2001). *Adding it up: Helping children learn mathematics.* J. Kilpatrick, J. Swafford, & B. Findell (Eds.) Washington, DC: National Academy Press.

[2] Lappan, G., Phillips, E. (2009). A designer speaks: Glenda Lappan and Elizabeth Phillips. *Educational Designer, Journal of the International Society for Design and Development in Education, 1*(3), 1-19. Retrieved from: http://www.educationaldesigner.org/ed/volume1/issue3/article11

3 National Governors Association Center for Best Practices & Council of Chief State School Officers. *Common Core State Standards for Mathematics.* Washington, DC: Authors, 2010.