– Gail Burrill, President, National Council of Teachers of Mathematics (NCTM)

"I think that CPM was designed well because it allows us to think and wants us to think. The book is designed to be tough; it is preparing us for future mathematics."

– Erin Glines, CPM student, Sacramento, CA

The College Preparatory Mathematics (CPM) curriculum integrates basic skills with higher order thinking processes. Students learn and understand both basic algebraic operations and complex problem-solving skills, as recommended by the National Council of Teachers of Mathematics' (NCTM) Curriculum and Evaluation Standards. CPM began when a group of high school math teachers and college professors joined together. The high school teachers felt there had to be a better way to teach math to their students; the college professors, spurred by the incoming freshmen's lack of math understanding, felt they had to act.

The CPM curriculum aims to actively engage high school students in learning mathematics. Recent brain research shows that long-term learning occurs best when students are engaged in puzzling through problems to create their own solutions. This knowledge became a CPM guiding principle: students should be actively involved in their learning. An Eisenhower grant of $700,000 funded the writing of this four-year math curriculum which parallels Algebra 1, Geometry, Algebra 2, and Math Analysis. A subsequent grant of $600,000 assisted in the development of an assessment component and helped CPM to assess the expanding program. Assessment results show test scores improving for students of all ability levels.

The CPM curriculum has two goals–to integrate essential mathematics in a problem-solving, student-centered environment and to teach students how to take responsibility for their own learning. CPM consciously seeks to transform pupils into:

• independent problem-solvers who can think for themselves;
• students who master basic math skills;

• students who truly understand the underlying mathematics of their studies;
• individuals who can work as team members; and
• future leaders who can communicate their knowledge to others.

When CPM began operation in 1989, 30 teachers from three districts in Sacramento, California used the curriculum with 1,500 students. As of the 1996-97 school year, use of the CPM curriculum has grown to 2,100 teachers in 700 schools with 400,000 students each year. Teachers, parents and administrators want to use CPM because they see that students of various ability levels who use CPM take more high school math classes and learn math better than their counterparts in traditional classrooms. They want that success for their students.

This booklet presents research conducted with students who are using CPM materials. Some material shows comparative results with students taught using traditional methods, some focuses on before and after pictures of schools using CPM. Both types of comparisons have had impressive results. William H. Schmidt, National Research Coordinator for the U.S. for the recent National Science Foundation and the National Center for Education Statistics funded Third International Mathematics and Science Study (TIMSS), said:

Curricula, textbooks and teaching should be part of an integrated solution united around common goals, approaches and measurements. Together, we must consider what is basic in science and math at every level and have all our children receive such instruction.

In science and mathematics, we are not where we want to be... Our children, our teachers, and our schools are working hard. We must think clearly together so that we may help them to work smart as well. (printed statement by William H. Schmidt, 11/20/96, p. 3)

College Preparatory Mathematics has been integrating curriculum, texts and better teaching methods since it began. The CPM materials have helped over 1,000,000 students to work smart and to think mathematically. Today it continues to help students bridge their way to the NCTM Standards and the kind of mathematical skills that will be needed in the 21st Century.

"Based on the number of years I have taught (35), I should probably think about retirement, but since I've been involved in math reform my batteries are fully charged and retirement is the furthest thing from my mind. I see the students struggling to understand math and with the use of manipulatives, hands-on activities, and cooperative grouping, they are now developing a love for math and are glad that we no longer use 'drill and kill' techniques."

– Lonnie Bellman, CPM Teacher now piloting Math 4, Lemoore, CA

CPM materials encourage students to find multiple solutions to a problem. This helps prepare them for tests with out-of-context problems, such as the Scholastic Aptitude Test (SAT). The SAT is not like a unit test where students can think, "We learned this theorem this week, so it must work on this problem." CPM students have not had the crutch of traditional units–each CPM homework assignment is composed of a variety of types of problems, so students must learn how to analyze and understand each problem. This practice is one of the reasons CPM students do so well on the SAT.

Results from two schools are graphed at right. Results from a dozen more schools will be available soon.
"I believe that CPM is the best thing that I could do for my students."

– Duane Frankiewicz, CPM Teacher, Spooner, WI

To investigate achievement in high-ability students, we looked for schools in California that have been using CPM in honors algebra or honors geometry classes for at least four years. Three responded with the data below.

When schools switched either their honors algebra or honors geometry class to CPM, the percentage of students who took calculus went up 59% (from 51% to 81%) in one case and 63% (from 49% to 80%) in the other.

The third school did not offer calculus before CPM had been introduced in an eighth grade honors class. Four years later, a calculus class was added and 22 of 24 CPM honors students (92%) took calculus.

Each year the California State Department of Education recognizes students for high achievement on the state's Golden State Exam (GSE). The three award levels are "school recognition," "honors," and "high honors." The school profiled at right is typical of the notable improvement in student honors that happens after the introduction of CPM.

It is not uncommon for CPM schools to have three times as many honored students as students at non-CPM schools within the same district. In fact, when CPM and another math curriculum are used side by side in the same school, the CPM class(es) often show this same high level of honors, as compared to their counterparts. We have seen these kinds of GSE results for tests 1993-96 in rural, urban, and suburban schools and from Title I to GATE schools.

"The students of El Camino High School continue to excel in mathematics. Over the past five years, students have improved their performance on the Golden State Exams in Algebra and Geometry. May, 1996 El Camino students outperformed the state in all three rankings (High Honors, Honors, School Recognition). The Math Department and I attribute the students' success to the implementation of the UC Davis integrated math curriculum (College Preparatory Mathematics) at El Camino."

– Mary Urelius, Principal of El Camino High School, Oceanside, CA

"The physical manipulation and technology (graphing labs) make algebra knowledge possible for these [special education] students even if they have serious reading problems."

– John McBride, CPM Teacher - Philadelphia, PA, as quoted in the Tribune Magazine, March 1997.

Every child can succeed at math. To tell children otherwise is to diminish their chances for success. Use of CPM benefits all students, especially those students traditionally underserved by the status quo curricula–namely students of color and girls.

"Low expectations and limited opportunity to learn have helped drive dropout rates among Blacks and Hispanics much higher (than the national average)–unacceptably high for a society committed to equality of opportunities. It is vitally important for society that all citizens benefit equally from high-quality mathematics education." from Everybody Counts: A Report to the Nation on the Future of Mathematics Education, p. 7

"Other teachers have been astounded by what I am doing with my students."

– Ronald Lyford, CPM Teacher, Rockford, IL

"Since changing to CPM geometry, the course has become meaningful to the students not only as a step to college, but as a step to life in the real world. No longer am I asked 'when am I ever gonna use this?'"

– Dan Bourassa, CPM Teacher, Oconomowoc, WI

A major problem with most traditional mathematics classes is that students can memorize how to solve problems in one chapter, take the test and then forget all about the concepts. The unit is over and students won't see that material again until they start cramming for the final. As with any skill, mathematics skills which are not used are forgotten. Conservative estimates indicate that most students forget 50% of the mathematics they learn each year.

The CPM teacher-authors addressed this problem when they designed their books. Every major topic is seen over and over again. New ideas are first foreshadowed, then introduced, developed and practiced. These new concepts are reintroduced either in a familiar form or in a new form, at first every day, then frequently as part of the homework, and then as subproblems in applications for the rest of the year or, with core concepts, such as ratios, functions, and graphs, into the next course.

For example, ratios, a central topic in mathematics and science, is the subject of four different chapters in algebra and geometry. As a result of this careful spiraling and integration of ideas, students do not have a chance to forget the big ideas and can call upon them whenever they are needed.

"My enjoyment in the classroom comes when the expression of "EUREKA!" fills the room after I have worked to develop a concept in mathematics. It is also joyous to hear students buzzing when they are confident enough about math to converse with their classmates while working to solve problems. CPM has increased this satisfaction. Helping students to be accountable for their own actions by encouraging them to stay on task and work in study teams, has been a rewarding but most challenging experience this year."

– Jacqueline Bush-Campbell, CPM Teacher, Philadelphia, PA

"The technological boom is forcing us to rethink education. The basics of tomorrow will be drastically different from the basics of today."

– Gail F. Burrill, President of NCTM

In this information age, the needs of business are changing rapidly. Business's requirements for new employees have shifted radically. Job applicants need to understand how to:

• problem solve and communicate understandings in a team setting;
• recognize the underlying mathematical principles of a problem;
• be comfortable with open-ended problems, since most real-life problems are not cleanly formatted; and
• value mathematics and its logical structure.
  

A 1992 study with open-ended test questions of the type deemed most valuable by employers, taken by 2,350 students, showed significant gains by CPM students in both algebra and geometry, as compared to traditionally taught students. This success was shared across both genders and all ethnic groups. This was not a normal classroom test designed to result in a typical bell-curve. It was a difficult test designed to challenge students to show clearly what they know and how well they know it.

"Previous to teaching mathematics I worked in various capacities for a steel fabricator. My last position there was production manager (my degree is in civil engineering, not mathematics). Working in the real world affected my approach to mathematics. I saw critical thinking and problem solving as the key factors to learning mathematics rather than memorization and solving routine problems taught in isolation. I found out just how important it is to work cooperatively. I also found that work is purposeful, which meant I needed to be able to use tools of the trade effectively and to communicate my knowledge both verbally and in writing. The CPM curriculum and philosophy parallel both my beliefs about mathematics education and my work experience."

– Karen Arth, CPM Teacher, Fresno, CA

The Third International Mathematics and Science Study (TIMSS), released in October 1996, ranked U.S. students in traditional math classes 28th out of 41 countries. This survey is the largest of its kind to date, comparing the test scores of 500,000 students, as well as analyzing teacher practices and analyzing curricula.

TIMSS studied–and the graph at left refers to–students in traditionally taught math classes. U.S. students in these traditional classes rarely learn the mathematical reasoning that underlies concepts, and even more rarely do they get to figure out solutions on their own. In contrast, CPM students, like the high-scoring Japanese students, are taught problem-solving techniques and learn how to devise solutions.

TIMSS found that Japanese teachers develop concepts to a full level and encourage students to use problem solving techniques to develop solutions. Japan, which scored among the top nations on TIMSS, follows teaching methods similar to those recommended by the National Council of Teachers of Mathematics and used in the CPM curriculum. German teachers also tend to develop concepts but here the teachers present the work rather than letting students work with it. Interestingly, German students scored very near U.S. students on the TIMSS assessment. CPM is more like the Japanese model than either of the other two models.

"All kids should have the opportunity to think and problem solve," says CPM teacher Gary Plummer. "CPM provides students the ability to do exactly that."

– Tribune Magazine, March 1997