**Mastery Over Time–What are the Implications**?

by Michelle Chenal-Ducey

"I don't know why fraction busters gave me so much trouble when I first started out, but now I'm the one who can help others." "At first I dreaded radical and exponent problems. Though I'm still not perfect, I'm getting the hang of it better as I practice. I'm becoming confident!"

These comments from student portfolios underscore one of CPM's key phrases and major philosophical tenets: mastery takes time. What does this mean, and how much time are we talking about?

If we want students to learn something important–say the notion of a ratio–we cannot give a two-week unit on ratios and realistically expect them to understand the concept. Some students may grasp the idea quickly, but most will have no real understanding–even if they have managed to learn a pattern imitating the solution.

Of all the adjustments teachers are learning to make in using the CPM materials, mastery over time often causes the most difficulty. It is a thorny concept for most math teachers to internalize because of how they themselves were taught. Quizzes and tests assessing mastery at the end of each chapter were the norm. However, understanding mastery over time means that instead of having many mastery questions for unit six on the unit six test, teachers should be looking for mastery of most of the concepts and skills introduced in units three or four and for a beginning understanding of the material in unit six. In other words, a mastery over time assessment might have 30% devoted to new topics, with the remaining 70% covering revisited topics, including complex questions about them. Similarly, short, graded team assignments which allow for student dialogue, questioning and peer teaching, are integral parts of a program based on mastery over time.

Mastery over time also affects instruction. If we expect mastery of an important topic will occur after a long period of practice, and we aim to introduce several topics during the year, it follows that the curriculum must involve an enormous amount of spiraling. In this way, a new idea is introduced and reinforced in either a familiar or a new form, every day for several days, then every other day, and eventually about once a week, for the rest of the year. Understanding mastery over time is recognizing that material will be spiraled and that students will have additional chances to review concepts and find other applications or uses for an idea; in short, they will have time to find success in mathematics As a student put it, "While learning the new ideas in this unit, I also picked up the ideas I had missed in earlier units."

It is a source of both pleasure and surprise to teachers that the mastery over time idea actually works. Students eventually acquire understanding of an idea without further formal instruction. Understanding mastery over time means that we not re-teach nor re-test, but that we re-cycle, spiral, be patient, and wait for students' cheers and satisfaction as they finally master concepts that were stumbling blocks for them!

From the student's perspective, knowing that material will be repeated and revisited can be an opportunity to show success with previously mysterious and difficult concepts. Spiraled practice gives the teacher clues to the progress of individual students as they move through the year. Especially if combined with some form of written evaluation, often embedded in subproblems, the teacher and student can diagnose student progress.

Belief in mastery over time suggests that both teaching and assessments will allow for student growth-in-understanding before assuming that it is time for closure.

Try it! You'll like it!

Presidential Award-winner Michelle Chenal-Ducey teaches at Mt. Whitney High in Visalia, CA.

**Math 1 Second Edition Update**

**Answers for Students**

The Math 1 writers and piloters discussed at length the inclusion of answers in the student text for the second edition. There are strong arguments (and feelings!) for both adding them and leaving them out. The compromise position is that we will add a section to the teacher text that will have the answers for the homework portion of each lesson ready to photocopy and/or make an overhead transparency. This resolution of the issue lets teachers retain control over when and how to provide students answers but makes doing so quick, easy, and cost-effective.

**Solutions for Teachers**

There was also considerable discussion about whether to provide a solutions manual for teachers. We have decided to produce one that may be purchased by teachers. CPM will work on producing this resource as soon as possible. We are undecided about the format of the solution pages. We also want to provide multiple solutions to problems wherever possible. If you have written complete solutions for each lesson in the preliminary second edition of Math 1 and would like to have them considered for use in the solutions manual, please mail a few sample pages to Brian at the CPM business address. If we decide to use your solutions, we will contact you to obtain them and to arrange to compensate you for them. The deadline to submit samples is May 31st.

**Look Here! Advice for the CPM classroom teacher**

*What should I say to my students who ask for more examples?*

The developmental problems are guided investigations. By doing the problems, student study teams create the examples for future reference. Student work, coupled with teacher-provided solutions, actually give students more examples than a traditional course, but after the student has worked with the idea. To learn well, we must experience mathematics for ourselves.

In addition, life itself presents many challenges without ready-made solutions. We have to learn how to find a job, a place to live, buy a car, pay bills, advance in our careers, etc. We learn how to do all of these things by trying to do them, making errors along the way, learning from the errors we make and then correcting them. Most of us eventually succeed at doing all (or most) of these things. CPM is simply taught in the same manner as the way we successfully learn to do most of the other things we need to learn in our lives.

*Do you have any suggestions for acknowledging the various levels of class participation by the students?*

I use class participation points as part of my students' grade. What works for me is to simply use the SCANTRON attendance sheets for tallying the raw score. I allow up to three points per day of credit or debit.

During class I put tally marks on a copy of my seating chart based on a student's level of participation. I use a different color pencil for each day. I use pencil so if the student gets her/his act together during the period, I can erase deducted points.

Roughly, for every 15 minutes of productive or off-task work I see, I add a tally to the collection for that student: just use "+" or "-". I let them know what I'm recording without being overly positive or negative–more just acknowledging what I'm seeing. If behavior is off-task, I encourage getting to work.

These seating charts are a good tool to have when talking to parents to help them understand how a student's class participation, or lack thereof, is impacting the student's progress.

**Presidential Teaching Awards Finally Announced!**

**California**

We have been patiently waiting to hear the results for the winners of the Presidential Awards for Excellence in Mathematics Teaching. The California secondary finalists were Jeanne Shimizu-Yost (north), Michelle Chenal-Ducey (central) and Joan Kennelly (south). Congratulations to all three of these outstanding teachers, two of whom are our very own CPM teacher-leaders: Jeanne and Michelle! The decision must have been tough. But who is off to accept the award in Washington, DC?

The winner for Excellence in Secondary Mathematics Teaching in the State of California is Ms. Michelle Chenal-Ducey. Michelle excelled in her own education, earning two Bachelors Degrees: in Mathematics and in French. She is a 1989 fellow of the San Joaquin Valley Math Project. Michelle has been teaching Mathematics at Mount Whitney High School in Visalia, CA for the past eight years. Prior to that she taught in Southern California.

Family math nights have allowed Michelle to share her love for math; she has developed and facilitated family math nights in her district for the past five years. She truly believes that we, as colleagues, must support each other as we work to improve education for our students. This belief played into her decision to join the CPM leadership team this year as a CPM 2 teacher-leader. She credits the support of her family and colleagues with her personal successes in education, including winning this award.

As part of the 1996 Presidential Award application, teachers were required to write a "Commentary on Education." In hers Michelle states, "I now believe that making learning student-centered is a key to helping students find meaning and become good problem-solvers. I have not relinquished my role as teacher, I have broadened it broadened it to include the students. Thought-provoking!"

She ends with a plea to teachers of mathematics to be passionate about what they do: "In my classroom in 1996," she says, "the search to make math relevant and alive goes beyond the textbook, mere pencil and paper, and the walls of the classroom per se. Mathematics is not dull, rigid and narrowly defined; it is exciting, open-ended, expressive, and artistic! Try it, you'll like it!"

Again, congratulations to all of the finalists and especially to Michelle Chenal-Ducey, the 1996 Presidential Award Winner for Excellence in Secondary Mathematics Teaching in the State of California!

Note: Check out Michelle's article "Mastery over Time: What are the Implications?" on page 1 of this newsletter.

**Pennsylvania**

by Irene Eizen

Gary Plummer, who teaches 8th grade mathematics, including CPM 1, is a Pennsylvania state finalist for the 1997 Presidential Award for Excellence in Mathematics Teaching at the secondary level.

Gary's accomplishments are impressive. He is certified in both elementary education and secondary mathematics. He regularly conducts staff development workshops on mathematics for teachers in the School District of Philadelphia. He has presented numerous sessions and workshops on all aspects of mathematics at local, state and national professional conferences. He has served on many math education committees at the local and state level. His greatest accomplishment, however, is teaching kids. His ability to bond with his students and make learning mathematics exciting, meaningful and accessible to every child whose life he touches is exemplary.

Visit Gary's class and you will see students working in study teams to solve problems, using technology, manipulatives and other sources, and speaking and writing the language of mathematics through presentations to the class. Mathematics excitement and enthusiasm are the essence of Gary's teaching and his students' learning.

As part of the 1997 Presidential Award application, Gary submitted student work of CPM problems along with photographs of group work and student presentations. Students explained problem solutions enhanced with supporting charts, tables and diagrams. The pictures are worth a thousand words! Gary's commitment to his students to remain a classroom teacher, to continue his own professional growth and to share his experiences and expertise with other teachers will ultimately touch thousands of lives! Congratulations, Gary.

Irene Eizen is CPM's East Coast Regional Coordinator.

**Community Education: Don't forget the Counselors!**

We have highlighted the importance of keeping parents, administrators, and school board members informed about the nature and effectiveness of the CPM Program. Don't forget the counselors! Counselors must understand the Program since they place students and work directly with students to meet other academic needs.

Have you distributed the letter to counselors that is included in the resource section of the Teacher Version? What about giving them the Overview Packet? It is also a good idea to talk individually with each counselor to make sure s/he understands that CPM is a college prep program and to ask if s/he has any questions.

**So You Wish You Had More Problems to Assign Your Students?**

Just a reminder that the Assessment Handbook that accompanies each teacher edition of Math 1, Math 2 and Math 3 has several investigations and other resources for projects in it. For students who want more practice with problems similar to those in the text, the Math 1 Supplement and Math 2 Supplement each have over 1,000 problems in them.

Look for the Math 3 Supplement next fall. The Math 1 Second Edition Parent Guide and Supplement is set to be published as a single bound book this summer. Second Edition assessment resources will also be available this summer and included with the 1997 Teacher Editions. There will be details in the June newsletter about how those who used the Preliminary Second Edition can get the revised assessment.

**May Mastery Tips**

By May of each year, many of your students will show signs of mastery. "I get it!" may fill the room and study teams may need less of a guiding hand from you.

Look for conceptual understanding first, not for memorization of vocabulary. Students will develop, over time, a variation of formal math language for concepts. We may be erecting roadblocks by expecting precise, formal language too soon, e.g., "Write an expression using x that describes the perimeter..." could be replaced by "Find the perimeter (using x)..." The students will learn math vocabulary, but CPM places first priority on conceptual understanding.

Tip: Replace questions that focus only on vocabulary and/or technical points with questions that ask students to explain concepts, e.g., "What does 'similarity' mean?," "Write an explanation of the difference between perimeter and area."

Encourage students to use their unit summaries, portfolios, and tool kits. By using the tool kits, students gradually connect ideas with their formal terminology. Furthermore, language and memory difficulties do not conceal what they know how to do. You can also ask more challenging questions. Students who are developing good study habits are thus rewarded for their work.

Tip: Remind students on test day to take out these resources. Remind them (especially at the start of the year) to record important information in their tool kits.

Encourage student reflection and thoroughness on tests. Reflection takes time! Filling a test with too many questions in the name of comprehensiveness stymies the students' ability to both finish and work thoroughly.

Tip: Ask fewer, but harder, questions on tests, especially later in the year.

Focus assessments on how well students are progressing toward understanding the five or six main ideas each course emphasizes.

Tip: Observe study team interactions, portfolio work, explanations from group and individual tests. Be willing to ask questions many times. For example, you may have to ask students "which number is vertical and which is horizontal" in a slope ratio dozens if not hundreds of times during the year, but gradually more and more of them grow comfortable with the idea.

Trust that your students will learn the concepts–over time and in their own way.

Remember that we are teaching students the fundamentals of mathematics. As they move through the college prep sequence, automatic understanding of essential procedures and algorithms increases in importance. This emphasis, however, is more appropriate in Math Analysis and perhaps Algebra 2. It is largely premature in Algebra 1 and Geometry.

**California Math Project–High School Initiative**

In October of 1996, the California Math Project's High School Initiative (HSI) came together for the first time. The Initiative, directed by Richard Curci from San Francisco State University, gathers high school mathematics teachers from all over the state to discuss issues specifically related to secondary mathematics education. Each project sends two or three representatives; CPM is well represented by Lonnie Bellman, Dolores Dean, George DiMundo, Edna Murphy, Chris Odell and Jeanne Shimizu-Yost. During that first meeting in San Francisco, the group discussed such issues as the emerging standards and a K-12 perspective of the Scholastic Aptitude Test (SAT). They then spent much of their time in regional breakout groups.

The HSI's second meeting was held in San Diego in mid-March. Dr. Nicholas Branca, Director of the California Math Project, began by leading an activity to explore the definition of "basic facts." During the three-day meeting teachers presented many topics pertinent to secondary instruction, including the Interactive Math Project (IMP), AP Statistics, applications of graphing calculators (led by Chris), the TIMSS report (led by Lori) and one presentation by Jeanne, Dolores and Lonnie on the fourth year of the CPM curriculum. These three led the teachers through activities exploring the meaning of area under a curve. It was inspiring - high level mathematics presented in a very understandable way. In fact, one of them said that their pre-calculus students are helping the calculus students understand their work!

This Initiative is fueled by a vision of improving secondary mathematics education in California. Those of us who attend the High School Initiative meetings enthusiastically encourage each of you to seek out your local Math Project and apply for membership. You'll become part of an exciting K-14 network in your region. It's worth looking into!

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Executive Editor: Brian Hoey

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e-mail: bhamada@cpm.org

*Thanks to all!*