**Your CPM Classroom Check List**

Students explore and develop the ideas in each unit by completing the guided investigations (recommended in-class problems), much like lab experiments in a science class.

The majority (but not all) of class time is spent working the guided problems with a partner or as part of a study team of four students.

Solutions to the previous day's problem set are provided to the study teams so that the students may check their work and have examples available.

Teachers circulate among the study teams, listening to student conversations and observing student work so that they know when and how to intervene.

Direct teaching or whole class discussion is used to summarize, help connect ideas, clarify confusion or ambiguities in a question, and refocus or redirect students when a substantial portion of the class needs such assistance.

Students are responsible to be active participants in their learning by keeping an organized notebook, reviewing homework, writing unit summaries and creating personal course took kits.

**Listening And Responding To The Study Teams**

During the time when students are working in their study teams, the teacher should be moving around the room, observing student interaction and written work. Probably the most important teacher behavior is **LISTENING**. Especially early in the period, the teacher needs to listen to the kinds of questions students ask each other and the responses they give. What you hear, coupled with spot reading of student work, will tell you when to intervene. Brief remarks will encourage students to keep working: "Looks good." "Nice graphs." "You're moving along nicely." "You might want to read the directions again." "Maria, you and Sam have different graphs for part (b). Compare your work before you move on."

When the entire team has a question, your responsibility is to help them move forward. Some questions, of course, can be handled by having the students reread the problem or read it aloud. Often a simple question such as, "How do you undo multiplying x by 3?" "What useful information does an equation in y-form give you?" or "What did you learn about graphing in yesterday's lesson (maybe cite a problem number) that you could apply here?" will get the team moving again. The foregoing questions are examples of helping students answer their own questions. By doing so they will increase their self-confidence and self-reliance.

Sometimes you may have to engage in a series of questions and a brief dialogue with the students if their question has deeper mathematical implications or indicates a significant difficulty with a skill or concept. Sometimes you will find that a quick question gets three students moving, but you identify the need to work with one of the students individually for a few minutes because you notice confusion on her/his part beyond the question the group asked. In this context you may need to answer a few questions with the student (and at times the group) to get to the question you really want to ask the student(s) to work on next.

Each of the situations described above requires active listening by the teacher. S/he needs to recognize the individual needs of teams and students and then make instant decisions about what to do. Often the correct decision is to do nothing except listen to the students work through an issue. Remember that a perfectly good response to a request for help is to ask a good question or make a general suggestion, then break from the group by saying, "Try that. I'll come back in three or four minutes and see how you are doing." Then be sure to come back and see what has happened. In most cases students will report that they are making progress and do not need further assistance.

**Reflections From Ruth Tsu**

One of the important philosophical underpinnings of CPM is the recognition that teachers learn in the same way that students do. Thus the guiding principles apply to us as much as they do to our students. In this article I would like to focus on the important role of discourse in effective learning AND TEACHING.

Some of us who have been teaching for several years have a difficult time allowing others to come into our classrooms and share our struggles. But there is so much we can learn together about teaching and learning. In a videotape called "Communication," one of the teachers says, "I don't think we as educators realized this rich situation that we were going to have by having kids talk to each other. That certainly is important when they start talking about real-life kinds of problems, more intricate situations and using technology. If they don't spend time communicating about those issues...a lot of the interesting stuff gets lost."

The same can be said about teachers talking together. We have this rich resource which we have too often overlooked in the past, mistakenly thinking we need to figure out how to solve every problem by ourselves. I vividly recall an experience that happened several years ago when, at the end of a long day of classes and meetings, one of my colleagues at San Lorenzo High asked if I was okay. I briefly mentioned one of the problems which had come up in one of my classes that day and I ended by saying, "I need to go home now and figure out to make tomorrow different." My colleague responded by saying, "Let's go to dinner together and brainstorm some options." Another colleague was also there at the time and the three of us went to dinner together, came up with many possible alternatives and chose from among them. My colleagues checked back with me to learn the outcome of "our" plan.

My experience during the past several years has convinced me that the slogan **"No one is as smart as all of us"** applies to teacher teams as much as it does to student study teams. Ruth Cossey calls it "working smarter." It means we use our time differently, but it does not need to take more time. It means we realize we have more alternatives than each of us could have come up with individually. And, very often, as we talk we discover some "interesting stuff" which individually might have been lost.

* "Communication" is one of the videotapes in the series "TEACHING MATH: A Video library, 9–12,"produced by WGBH Boston and funded by the Annenberg CPB Math and Science Collection. Look for this series to be block broadcast on instructional TV this fall.

**Mathematics Program Advisory:Adopting A Balanced Mathematics Program**

A summary of the Program Advisory from the State Board of Education

NOTE: This summary is based on a draft document posted to the State Department of Education web site. An article appeared in the L.A. Times in mid-September that left the impression that the Advisory had been accepted and released by the State Board of Education. We now understand that final approval is still pending.

**WHAT IS BALANCED MATHEMATICS?**- A challenging, rigorous mathematics program for all students
- Includes basic skills, conceptual understanding, and problem solving
- Provides instructional guidance and support

**CHALLENGING, RIGOROUS, MATHEMATICS**- Includes: number, measurement, geometry, patterns and functions, statistics and probability, logic, algebra, and discrete mathematics
- Mathematical tasks increase in depth and complexity from year to year.
- ALL students have access to all courses and achieve at high levels
- Students spend at least five class hours per week learning mathematics

**A BALANCED MATHEMATICS PROGRAM**- Teaches basic skills
- Develops conceptual understanding
- Promotes effective problem solving

**BASIC SKILLS**- Skills students use routinely and automatically with understanding
- Continually practiced
- Deepened over time
- Used frequently and in new settings
- Used to solve problems

**CONCEPTUAL UNDERSTANDING**- The ability to make sense of mathematics
- Knowing when and how to apply skills
- Understanding mathematical relationships
- Understanding the structure and logic of mathematics
- Using mathematics flexibly and appropriately
- Understanding and applying mathematical patterns
- Applying old learning to new situations

**PROBLEM SOLVING**- The ability to apply skills, understanding, and experiences to resolve new or perplexing situations.
- The application of conceptual understanding
- Exercising basic skills
- Finding solutions to problems outside of the classroom
- Extensive, varied, experiences which develop persistence
- Day long, week long ,or month long problems
- The ability to determine if the process and the solution fit the problem

**INSTRUCTIONAL GUIDANCE AND SUPPORT**- Variety of Instructional Strategies: direct instruction, hands on learning, classroom discussion, discovery, individual work, and group work
- High standards established and maintained
- Ongoing professional development and support
- Parental involvement at all levels

**A BALANCED MATHEMATICS PROGRAM MUST:**- Intellectually engage and challenge ALL students with rigorous mathematical content
- Develop basic skills, conceptual understanding, and problem solving
- Provide ongoing teacher support
- Include parents as educational partners

Summary prepared by Denise Hoyt, CPM Teacher-Consultant, Yucaipa High School

**CPM Commentary**

Taking the advisory as a whole, CPM meshes well with it. Many of the statements describe necessary elements of any math program. The CPM Program builds the basics of college prep math and strong conceptual development, all in a problem solving environment. CPM has plenty of practice spread over time; indeed, if you collected the problems and placed them by type each all together, you would have the equivalent or more of practice sets in standard texts. The program has a clear scope and sequence that teaches more mathematics by the end of three years (and lots more when we add Math 4) than traditional texts. CPM expects all students to learn math. It is rigorous in the way that topics parallel the conceptual maturity of high school students, increasing in complexity and intellectual challenges through four years of math.

The one aspect of the advisory that we think is incomplete is the discussion about problem solving. Problem solving is about open-ended, universal strategies for doing mathematics as well as using specific algorithms. Part of a complete curriculum is the conscious, systematic instruction of ways to approach situations that do not lend themselves to a handy algorithm for a solution. Problem solving strategies equip students to deal with the ambiguity of realistic problems in creative ways. Problem solving provides a vehicle for grappling with the mathematical structure that underlies conceptual understanding. In that sense it leads to the development of basic skills in addition to providing a place to apply them.

You may have read various newspaper reports about the advisory. Each report presents a point of view depending on the experience of the reporter and the position held by the paper's editorial board. One headline could be, "Math Advisory Encourages Use of Best Elements of Traditional and Reform Math." Another could say, "Math Advisory Recommends Strong Teacher Training Program to Improve Scores." Yet another could say, "Basics Emphasized in Math Advisory." Because the advisory succeeds in providing some balance, those who support an extreme view will weigh the headlines in that direction.

**Community Education Resources**

Last year we wrote often of our collective obligation to explain to local communities–students, parents, teachers, administrators, and the general public–why math education is changing and how CPM addresses those changes with a balanced program. We sent you resources to do this in every newsletter. We sent copies of most of these materials to every teacher new to CPM who submitted the teacher "permission to use CPM" agreement during the past five months. We hope that everyone has these materials in a folder to give to those who inquire about CPM or math reform as well as for use at parent information nights. They are listed in the next section of the newsletter.

**RESOURCE MATERIALS: WHAT'S AVAILABLE & HOW TO OBTAIN THEM**

Over the years we have developed resources for the courses to assist students, teachers and parents with the course materials. Teacher Versions come with an introduction and overview (philosophy), suggestions for using study teams (groups), resources for community education (letters to parents, students and counselors, back to school night plans, summaries of CPM research, and question and answer pages about frequently asked questions about CPM.) Math 1 and Math 2 have parent guides. Math 1 also has a plan for using the course in sheltered classrooms, while Math 2 has suggestions for using the course with Key Curriculum's Geometer's Sketchpad and Patty Paper Geometry.

Last year the September Newsletter contained a blue reply card that offered new assessment materials for each course. YOU HAD TO RETURN THE CARD TO RECEIVE THE MATERIALS. Since last September each Teacher Version has included the course assessment resources on a disk formatted for Macintosh using Microsoft Word. Each course has two assessment resources:

**Assessment Resource Folder:**

This document has dozens of problems for constructing your own unit quizzes and tests and final examinations. There are also suggestions for constructing tests. We do not provide photo ready tests because we believe that you should construct tests that are appropriate for your students. In particular, the lengths of class periods are so variable that there is no such thing as a "one size fits all" test.

**Assessment Handbook:**

This document has several sections, including: rubrics and grading, projects and larger investigations, ideas for how to do alternative forms of assessment, including portfolios, student presentations, and management issues (homework, block schedules, tool kits, etc.).

The assessment materials are sent on disk so that you can customize them for your classes. You must have the application program "Word" installed on your hard drive to use the disk. It is not compatible with ClarisWorks or other word processing programs. If you have a PC (IBM platform), follow the instructions that came with the disk (i.e., return the Mac disk) and we will send you a PC version. Sending the electronic copy saved us production and mailing costs, saved you the time and expense of ordering the disk separately, and made the materials more convenient for you to use. But we suggest that you print out the entire Handbook. Many teachers have told us that they did not realize what a valuable resource it could be until they saw the entire paper version.

**Newsletter Enclosures: 1995-96**

- CPM Overview Packet: 14 pages with program description, research, sample problems
- School board presentation of the CPM program
- Research results that suggest the need for change
- One page CPM background information
- Models for asking classroom questions
- Questions and answers about CPM
- Letter to introduce CPM to your principal
- CPM teacher and school summary by county
- Excerpt from California Mathematics Task Force
- Pro-active ideas to help successfully implement CPM
- University professors comment on math education
- Content outlines of Math 1, Math 2 and Math 3
- Comparative data: international and national test results
- Position papers from CPM
- Curriculum choices for districts
- Classroom methodologies
- The role of "guess and check" in the CPM classroom

- Resources for community education
- Suggestions of actions to take in your community
- Editorials from major papers
- Quotations from prominent business people
- Excerpts from media commentaries

**How to Obtain Resource Materials**

If you need anything on the list, or if you need other CPM resources, MAIL a request listing what you need, along with a self-addressed stick-on mailing label. Put "Educational Resources" on the outside of the envelope. Include a phone number with your list so that we can contact you if we need to clarify your request. We will send you the materials by return mail within two weeks. Sorry, we cannot accept phone or email requests. We have allocated staff time to process materials by mail. Send requests to:

Educational Resources

CPM Educational Program

1233 Noonan Drive

Sacramento, CA 95822

**Block Schedules**

In late August we mailed block schedule adaptations of Math 1, 2 and 3 to everyone who requested this information on the green postcards. These resources have day to day assignments (including which problems to omit) for block schedules that teach an entire year's course in one semester. They have been used with classes in Santa Barbara. If you need a copy, use the directions later in this newsletter that explain how to order support materials.

**Update: The Second Edition Of MATH 1**

We are pleased to report that the preliminary 2nd edition of Math 1 was completed on time. Almost 20,000 copies are now in print! We mailed everyone a copy of the table of contents in the July Newsletter. The revision was more extensive than we described in the May and June newsletters. However, the writing team felt that in order to meet some of the recommendations from the teacher dinner meetings, the topic sequence needed more restructuring than we originally expected to do. The course is still faithful to CPM methodology and concentrates on the same main threads. Most consensus recommendations have been incorporated into the scope, sequence, and design of the text. If you have made the switch from the original to the new version, please send Leslie a note about what you like as well as what you think still needs work.

During the 1996-97 school year we will be working on the final version of the 2nd edition. A team of a dozen teachers is using the text and submitting suggestions for revisions to the managing editor, Leslie Dietiker. We will also revise the support material (assessment, parent guide, supplement, etc). We expect to have everything finished by June and the final version of the 2nd edition available for the 1997-98 school year. If you are using the new preliminary version and have suggestions for making a problem work better, send your suggestion to:

Leslie Dietiker

10 Montana Street

San Francisco, CA 94112

Or email her at dietiker@cpm.org.

**How To Order Math 1 And Math 2 Supplements**

We have a good inventory of Math 1 Supplements for the original edition (©1994) and will have Math 2 (Geometry) Supplements in stock by the end of October. We strongly encourage you to have a supply of supplements at school and sell them to parents when they ask for them. This will save everyone time, money and hassles. However, if you cannot handle sales through the school (yourself, Math Club, or PTA), parents may order copies directly from CPM. The information is contained on the photo-ready form below.

The completion of the Math 2 Supplement was delayed due to the intensive effort needed to revise Math 1 and write Math 4 last summer. It will go to press this week and be available by the end of October. It is approximately 140 pages long and will be bound like the textbooks. As soon as it is complete we will send copies to everyone who indicated they are teaching Math 2 on last June's green postcard (about 1,100 teachers).

**Course Supplement Order Form**

CPM Educational Program has developed supplements for Math 1 (Algebra 1, 1st ed.) and Math 2 (Geometry). These resources are designed to assist parents and students who want more practice with the topics in the courses or who wish to see examples and explanations of ideas in the student text in an alternative format. To order a supplement, complete this form and mail it along with your check to the CPM business office. You will have your order in seven to ten days.

Ordering information: specify how many of each book you want. Add local sales tax (CA only) and shipping, make checks payable to "CPM," and mail your order to:

CPM Educational Program: Supplements

1233 Noonan Drive

Sacramento, CA 95822

Shipping: 5% of order plus:

- 1 book, $3.00
- 2-4 books, $4.00
- 5 or more books, $5.00

Number of each book you wish to order:

- Math 1 Supplement (1st ed.): $5.00 each
- Math 2 Supplement: $8.00 each

And the following information:

- Name
- Address
- City
- State
- Zip
- Phone
- Order summary
- Cost of books
- Sales tax
- Shipping
- TOTAL

**CONGRATULATIONS**

We would like to congratulate King and Willard Junior High Schools in Berkeley and Sanger High School in the San Joaquin Valley for earning California Distinguished School Awards. King and Willard use Math 1 for algebra. Sanger uses CPM texts in most of its math courses.

If your school has earned recognition–general or specific to math–let us know. We would like to acknowledge your success as well as add it to the growing documentation of CPM's effectiveness.

To be added to the News You Can Use mailing list, send a request by fax (916-444-5263) or e-mail bradley@cpm.org

Executive Editor: Brian Hoey

We welcome your comments and suggestions:

Production Manager: Brian Hamada

e-mail: bhamada@cpm.org

*Thanks to all!*