In Connected Mathematics, teachers are an integral part of the learning process. From the beginning, the authors have viewed Connected Mathematics as a curriculum for both students and teachers. The extensive try-outs of the curriculum have produced teacher materials that are rich with examples of student work, classroom dialogues, and teacher reactions. These teacher materials are such that teachers can learn from them. Connected Mathematics provides teachers with ways to think about and enact problem- centered teaching that address the following aspects of instruction.
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?
The extensive field-testing of Connected Mathematics has helped produce teacher materials that are rich with field teachers' successful strategies, classroom dialogues and questions, and examples of student solutions and reasoning (see Skills in CMP under Mathematics Content). The Teacher's Guide for each unit contains a discussion of the mathematics underlying the Investigations, mathematical and problem-solving goals for each Investigation, connections to other units, in-depth teaching notes, and an extensive assessment package.
The teacher materials are designed as a resource to facilitate teaching Connected Mathematics. The features and organization of the Teacher's Guide are described below.
Teachers can use the material in the Unit Introduction to prepare for teaching the unit. The following features are included in this section:
Goals of the Unit
Developing Students' Mathematical Habits
The Mathematics in the Unit
Summary of the Investigations
Mathematics Background A detailed description designed to assist teachers in understanding the content
Content Connections to Other Units
A chart highlighting how the big ideas of the unit connect to ideas from previous and future units
Planning for the Unit
Pacing Suggestions and Materials
Program Resources (See Resources under Components of CMP)
Assessment Summary (See Assessment under Components of CMP)
Ongoing Informal Assessment
Correlations to Standardized Tests
Launching the Unit
Using the Unit Opener
Using the Mathematical Highlights
Introducing the Unit Project
Detailed teaching notes are included for each Investigation. These include the following:
Mathematical Goals for the Investigation For each problem:
Specific Mathematical and Problem-Solving Goals
Detailed Teaching Notes Includes problem- by-problem discussions with examples of the instructional role of the teacher during the three phases of problem instruction, as well as samples of student responses to questions.
Going Further The explore sections include occasional "Going Further" questions for students who finish early or need another challenge.
Check for Understanding The summary section, when appropriate, may end with questions for the teacher to use to check students' understanding.
Lesson At a Glance This is a two-sided one-page lesson guide for each problem. (A blank At a Glance template is included in each Teacher's Guide to facilitate a teacher's personalization of the lesson plan.) At a Glance contains:
The mathematical goal for the Problem
Materials needed for the Problem
Definitions that need to be addressed
Key questions for the Launch, Explore, and Summarize phases of the instruction
Answers to the Problem
Homework assignment guide
At a Glance. Shapes and Designs
Blackline Masters of Labsheets and other things are provided. Students use these blackline masters as they work on the problem sets.
Descriptive Glossary/Index Key concepts are summarized, often with illustrations or examples, in both English and Spanish.
In addition to the Teacher's Guide for each unit, there are several resources that are designed to assist teachers. They are:
Teaching Transparencies support problems from the student books. All the Getting Ready features are available on transparencies.
Parent materials for Connected Mathematics include a parent letter for each unit with the goals of the unit and examples of worked problems, as well as a website for parents to help with homework for each unit.
Special Needs Handbook for Teachers includes suggestions for adapting instruction, examples of modified problems and ACE exercises from the student books, and assessment items for each unit.
Assessment Resources include blackline masters for Partner Quizzes, Check-Ups, Unit Tests, multiple- choice items, Question Bank, Notebook Check and Self-Assessment for each grade level. They are also available in Spanish and on a CD-ROM.
Additional Practice and Skills Workbook for each grade level provides practice exercises for each investigation as well as additional skills practice to reinforce student learning.
Technology A Student Activities CD-ROM provides activities to enhance and support classroom learning in the Problems/Investigations of the Student books.
Teacher Express™ CD-ROM includes lesson planning software, the Teacher's Guide pages, and all the teaching resources.
Exam View® Test Generator CD-ROM includes all the items from the Assessment Resources and the Additional Practice and Skills Workbook in both English and Spanish. Items can be edited electronically and saved. Many items are dynamic, and can be used to create multiple versions of practice sheets.
See also Resources under Components of CMP and also our Professional Development area of the site.