Guiding Principles for Development
Student Learning: Rationale for a Problem Centered Curriculum
Characteristics of Good Problems
Practice with Concepts, Skills, and Algorithms
Rationale for Depth versus Spiraling
Developing Depth of Understanding and Use
CMP Instructional Model
Support for Classroom Teachers
The authors were guided by the following principles in the development of the Connected Mathematics materials. These statements reflect both research and policy stances in mathematics education about what works to support students' learning of important mathematics.
The "big" or key mathematical ideas around which the curriculum is built are identified.
The underlying concepts, skills, or procedures supporting the development of a key idea are identified and included in an appropriate development sequence.
An effective curriculum has coherence-it builds and connects from investigation to investigation, unit-to-unit, and grade-to-grade.
Classroom instruction focuses on inquiry and investigation of mathematical ideas embedded in rich problem situations.
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.
Ideas are explored through these 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.
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.
The curriculum reflects 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.
Connected Mathematics is different from many more familiar curricula in that it is problem centered. The following section elaborates what we mean by this and what the value added is for students of such a 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 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 understanding 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 practices that method on similar problems.
A problem-centered curriculum not only helps students to make sense of the mathematics, it 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 Seven 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 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.
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 problem in Connected Mathematics satisfies the following criteria:
The problem must have important, useful mathematics embedded in it.
Investigation of the problem should contribute to students' conceptual development of 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 teachers to assess what students are learning
In addition each problem satisfies some or all of the following criteria:
The problem should engage students and encourage classroom discourse.
The problem should allow various solution strategies or lead to alternative decisions that can be taken and defended.
Solution of the problem should require higher-level thinking and problem solving.
The mathematical content of the problem should connect to other important mathematical ideas.
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 should be related to the situations in which the ideas have been developed and learned.
Continued practice should use skills and procedures in situations that connect to ideas that students have already encountered.
Students need opportunities to use the ideas and skills in situations that extend beyond familiar situations. These opportunities allow students to use skills and concepts in new combinations to solve new kinds of problems.
Students need practice distributed over time to allow ideas, concepts and procedures to reach a high level of fluency of use in familiar and unfamiliar situations and to connect to other concepts and procedures.
Students need guidance in reflecting on what they are learning, how the ideas fit together, and how to make judgments about what is helpful in which kinds of situations.
Throughout the Number and Algebra Strands development, students need to learn how to make judgments about what operation or combination of operations or representations is useful in a given situation, as well as, how to become skilful at carrying out the needed computation(s). Knowing how to, but not when to, is insufficient. Skills in CMP under Mathematics Content and Algebra in CMP under Mathematics Content.
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 leads to teachers spending 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 to allow the ideas studied to be more carefully developed means that when these ideas are met in future units, 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 ideas and make substantive advances in knowledge.
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 understanding of key ideas in contrast to "covering" a book. The Teacher's Guides accompanying CMP materials were developed to support teachers in planning for and teaching a problem-centered curriculum. Practice on related skills and algorithms are 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. Field Testing
Through the field trials process we were able to develop units that result in student understanding of key ideas in depth. An example is illustrated in the way that Connected Mathematics treats proportional reasoning-a fundamentally important topic for middle school mathematics and beyond. 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 a seventh grade unit, Comparing and Scaling, with the groundwork for this unit having been developed in four prior units. Five succeeding units build on and connect to students' understanding of proportional reasoning. These units and their connections are summarized as follows:
Grade 6 Bits and Pieces I and II introduce students to fractions and their various meanings and uses. Models for making sense of fraction meanings and of operating with fractions are introduced and used. These early experiences include fractions as ratios. The extensive work with equivalent forms of fractions builds the skills needed to work with ratio and proportion problems. These ideas are developed further in the probability unit How Likely Is It? in which ratio comparisons are informally used to compare probabilities. For example, is the probability of drawing a green block from a bag the same if we have 10 green and 15 red or 20 green and 30 red?
Grade 7 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 with problems focused around the question: "What would it mean to say two figures are similar?"
The next unit in grade seven is the core proportional reasoning unit, Comparing and Scaling, which 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 one quantity is 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.
A seventh grade unit that follows, 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. Then in the probability unit What Do You Expect?, students again use ratios to make comparisons of probabilities.
Grade 8 Thinking With Mathematical Models; Looking For Pythagoras; Growing, Growing, Growing, and Frogs, Fleas, and Painted Cubes extend the understanding of proportional relationships by investigating the contrast between linear relationships and inverse, exponential, and quadratic relationships. Also in Grade Eight, Samples and Populations uses proportional reasoning in comparing data situations and in choosing samples from populations.
These unit descriptions show two things about Connected Mathematics-the in-depth development of fundamental ideas and the connected use of these important ideas throughout the rest of the units.
Problem-centered teaching opens the mathematics classroom to exploring, conjecturing, reasoning, and communicating. The Connected Mathematics teacher materials are organized around an instructional model that supports this kind of teaching. This model is very different from the "transmission" 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. The following text describes the three instructional phases and provides the general kinds of questions that are asked. Specific notes and questions for each problem are provided in the Teacher's Guides.
In the first phase, the teacher launches the problem with the whole class. This involves helping students understand the problem setting, the mathematical context, and the challenge. The following questions can help the teacher prepare for the launch:
What are students expected to do?
What do the students need to know to understand the context of the story and the challenge of the problem?
What difficulties can I foresee for students?
How can I keep from giving away too much of the problem solution?
The launch phase is also the time when the teacher introduces new ideas, clarifies definitions, reviews old concepts, and connects the problem to past experiences of the students. It is critical that, while giving students a clear picture of what is expected, the teacher leaves the potential of the task intact. He or she must be careful to not tell too much and consequently lower the challenge of the task to something routine, or to cut off the rich array of strategies that may evolve from a more open launch of the problem.
The nature of the problem suggests whether students work individually, in pairs, in small groups, or occasionally as a whole class to solve the problem during the explore phase. The Teacher's Guide suggests an appropriate grouping. As students work, they gather data, share ideas, look for patterns, make conjectures, and develop problem-solving strategies.
It is inevitable that students will exhibit variation in their progress. The teacher's role during this phase is to move about the classroom, observing individual performance and encouraging on-task behavior. The teacher helps students persevere in their work by asking appropriate questions and providing confirmation and redirection where needed. For students who are interested in and capable of deeper investigation, the teacher may provide extra questions related to the problem. These questions are called Going Further and are provided in the explore discussion in the Teacher's Guide. Suggestions for helping students who may be struggling are also provided in the Teacher's Guide. The explore part of the instruction is an appropriate place to attend to differentiated learning.
The following questions can help the teacher prepare for the explore phase:
How will I organize the students to explore this problem? (Individuals? Pairs? Groups? Whole class?)
What materials will students need?
How should students record and report their work?
What different strategies can I anticipate they might use?
What questions can I ask to encourage student conversation, thinking, and learning?
What questions can I ask to focus their thinking if they become frustrated or off-task?
What questions can I ask to challenge students if the initial question is "answered"?
As the teacher moves about the classroom during the explore, she or he should attend to the following questions:
What difficulties are students having?
How can I help without giving away the solution?
What strategies are students using? Are they correct?
How will I use these strategies during the summary?
It is during the summary that the teacher guides the students to reach the mathematical goals of the problem and to connect their new understanding to prior mathematical goals and problems in the unit. The summarize phase of instruction begins when most students have gathered sufficient data or made sufficient progress toward solving the problem. In this phase, students present and discuss their solutions as well as the strategies they used to approach the problem, organize the data, and find the solution. During the discussion, the teacher helps students enhance their conceptual understanding of the mathematics in the problem and guides them in refining their strategies into efficient, effective, generalizable problem-solving techniques or algorithms.
Although the summary discussion is led by the teacher, students play a significant role. Ideally, they should pose conjectures, question each other, offer alternatives, provide reasons, refine their strategies and conjectures, and make connections. As a result of the discussion, students should become more skillful at using the ideas and techniques that come out of the experience with the problem.
If it is appropriate, the summary can end by posing a problem or two that checks students' understanding of the mathematical goal(s) that have been developed at this point in time. Check For Understanding questions occur occasionally in the summary in the Teacher's Guide. These questions help the teacher to assess the degree to which students are developing their mathematical knowledge. The following questions can help the teacher prepare for the summary:
How can I help the students make sense of and appreciate the variety of methods that may be used?
How can I orchestrate the discussion so that students summarize their thinking about the problem?
What questions can guide the discussion?
What concepts or strategies need to be emphasized?
What ideas do not need closure at this time?
What definitions or strategies do we need to generalize?
What connections and extensions can be made?
What new questions might arise and how do I handle them?
What can I do to follow up, practice, or apply the ideas after the summary?
When mathematical ideas are embedded in problem-based investigations of 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-moving from always being the one who does the mathematics to being the one who guides, interrogates, and facilitates the learner in doing and making sense of the mathematics.
The Teacher's Guides and Assessment Resources developed for Connected Mathematics provide these kinds of help for the teacher:
The Teacher's Guide for each unit engages teachers in a conversation about what is possible in the classroom around a particular lesson. Goals for each lesson are articulated. Suggestions are made about how to engage the students in the mathematics task, how to promote student thinking and reasoning during the exploration of the problem, and how to summarize with the students the important mathematics embedded in the problem. Support for this Launch-Explore- Summarize sequence occurs for each problem in the CMP curriculum.
An overview and elaboration of the mathematics of the unit is located at the beginning of each Teacher's Guide, along with examples and a rationale for the models and procedures used. This mathematical essay helps a teacher stand above the unit and see the mathematics from a perspective that includes the particular unit, connects to earlier units, and projects to where the mathematics goes in subsequent units and years.
Actual classroom scenarios are included to help stimulate teachers' imaginations about what is possible.
Questions to ask students at all stages of the lesson are included to help teachers support student learning.
Reflection questions are provided at the end of each investigation to help teachers assess what sense students are making of the 'big" ideas and to help students abstract, generalize, and record the mathematical ideas and techniques developed in the Investigation.
Diverse kinds of assessments are included in the student units and the Assessment Resources that mirror classroom practices as well as highlight important concepts, skills, techniques, and problem solving strategies.
Multiple kinds of assessment are included to help teachers see assessment and evaluation as a way to inform students of their progress, apprise parents of students' progress, and guide the decisions a teacher makes about lesson plans and classroom interactions. Components of CMP