Key Features of CMP2:

• Problem Centered – Students learn mathematical concepts as they work on problems by themselves, in a group or as a whole class.
• Practice – Students practice skills and algorithms and reinforce concepts through homework and class work.
• Research Based – Each unit was tested in school. The results were used to evaluate and revise the material.
• It Works – CMP students outperform non-CMP students on tests of problem-solving, understanding, and proportional reasoning. CMP students do as well as or better than other students on basic skills tests.

CMP2 Principles
The curriculum identifies important mathematical ideas and the concepts, skills or procedures to support them. Students learn those ideas through working on problems in class and in homework. The curriculum connects these ideas from one investigation to another, one unit to another and one grade to another.

Problems are the main way that students learn mathematical concepts. Students are able to explore the ideas as deeply as they need to in order to understand them thoroughly. They are able to connect ideas and build on the concepts they have learned before.

This curriculum helps students learn to reason effectively, a critical skill in the 21st century. They learn to work with information that appears in different forms – numbers, symbols, words and graphs. They understand and use the capabilities of calculators and computers, applying them to problem-solving in effective ways. They do not use calculators as a substitute for developing their own ability to calculate.

A Problem-Centered Curriculum
Students see a subject through the way they experience it. Students who learn geometry by memorizing definitions will, of course, believe geometry is about definitions. Students who spend a lot of time working through answers to lists of problems know math to be the subject of calculation. Students in the 21st century, however, need more than definitions and calculation only: they need to learn problem-solving using skills such as planning, reasoning, thinking, evaluating and computing. They will need to understand and use definitions properly so they share a common language and can communicate how they solved a problem. All of these skills are important to problem solving.

The problem-solving approach is backed up by research from the cognitive sciences. This research supports the theory that students can learn mathematics if the concepts and skills are embedded in a context or problem. If students explore interesting situations, reflect on potential solutions, look at possible methods and those used in similar situations, then they are more likely to thoroughly understand the concepts and procedures involved. They learn through working out these carefully selected problems, as opposed to a situation in which they observe a teacher demonstrate how to solve a problem and then practice the same method on similar problems. This kind of curriculum centered on problems helps students to make sense of the mathematics. It also helps them process the mathematics in a way they can remember when faced with similar problems in the future.

CMP locates important ideas in interesting problems. Students explore a series of connected problems that build on one another. They develop an understanding of the embedded ideas. With a teacher’s help, they also learn abstract powerful mathematical ideas, problem solving strategies and ways of thinking. They learn both mathematics and how to learn mathematics.

Good Problems
To be effective, the problems used in class and in homework must embody critical mathematical concepts and skills. They also have to have the potential to engage students. Since students build understanding by reflecting, connecting, and communicating, the problems need to encourage them to use these processes.

Each problem that is used in Connected Mathematics must meet these criteria:

• Have important mathematics in it.
• Lead to investigations that contribute to the conceptual development of important ideas.
• Promote skillful use of mathematics and practice of skills.
• Create opportunities for teachers to assess what students are learning and where they are having a hard time.

Depth versus Spiraling
A “spiraling” curriculum is philosophically appealing but, too often, not enough time is spent with a new concept to create a foundation that can be built on at the next stage of the spiral. Teachers end up spending a great deal of time re-teaching the same ideas over again. Students come to view mathematics as a collection of different techniques and algorithms to be memorized because they do not develop a deeper understanding of the concepts behind them.

An in-depth curriculum, on the other hand, takes time to allow ideas to be carefully developed. That means students have a solid foundation on which to build. Students can connect new ideas to those they have previously learned.

At each grade level, a small, select set of important mathematical concepts, ideas, and related procedures are studied in depth. Practice on related skills and algorithms are provided in way that students not only practice these skills and algorithms to become competent in carrying out computations but they also learn to put their growing body of skills together to solve new problems.