What is CPM?

CPM Educational Program, a California non-profit corporation, has provided problem-based instructional materials and professional development for teachers since its inception in 1989. “College Preparatory Mathematics (CPM)” was originally an Eisenhower-funded grant program. Its teaching strategies focus on how students best learn and retain mathematics. Teaching strategies rely on the recommendations of the National Council of Teachers of Mathematics, and are based solidly on the methodological research in teaching mathematics. The research-based principles that guide the course are:

  • Students should engage in problem-based lessons structured around a core idea.
  • Guided by a knowledgeable teacher, students should interact in groups to foster mathematical discourse.
  • Practice with concepts and procedures should be spaced over time; that is, mastery comes over time.

CPM Overview

Although CPM predates the CCSS Standards for Mathematical Practice by about 20 years, very similar practices have always been a core and integral part of CPM curriculum materials. Because of CPM’s broad experience and long history with these practices, the CCSS Mathematical Practices are deeply and seamlessly interwoven into the fabric of the daily lessons. The course balances procedural fluency (algorithms and basic skills), deep conceptual understanding, strategic competence (problem solving), and adaptive reasoning (application and extension).

On a daily basis, students using CPM Core Connections employ problem solving strategies, question, investigate, analyze critically, gather and construct evidence, and communicate rigorous arguments to justify their thinking. With the CPM instructional materials, students can tackle mathematical ideas set in everyday contexts to help them make sense of otherwise abstract principles. Students are taught how to gather and organize information about problems, break problems into smaller parts, and look for patterns that lead to solutions. Students often learn in collaboration with others, sharing information, expertise, and ideas.

Consistent with the requests we frequently hear from leaders of business and industry, CPM routinely has students solve non-routine problems. That is, students develop their skills of synthesis and analysis so that they can confidently make connections between varied mathematical concepts and deal with problems they have never seen before. Students will build problem-solving strategies that apply to most academic disciplines, the workplace, and daily life.

While students are solving complex mathematical problems, they are communicating their thinking and understanding, both formally and informally, whether they are writing or speaking out loud. Communication helps to clarify students’ thinking, prepares them for sharing their ideas in professional settings, and formal or informal feedback that allows for revision. Communication lets teachers and peers assess students’ thinking and depth of understanding. In turn, all students get the chance to improve the quality of their work.

CPM Core Connections courses are the products of classroom teachers who created lessons that work with the diverse student population of California. The teaching strategies outlined in the CPM instructional materials were initially informed by theory and scholarly research into how children learn and how teaching should occur in the ideal classroom. Care was taken to pilot and field test the lessons during the development of the first edition with thousands of students to assure the effectiveness of the lessons. But ultimately the development was informed in practice by the 4000 teachers and over 4 million students that use CPM, the specific suggestions over the last 20 years from hundreds of teachers, and even comments and suggestions by students and parents.

More than two dozen studies have examined the results of both high- and low-performing students on statewide standardized tests, and on the SAT, and ACT. All of these studies, as well as detailed investigations of individual schools, show that CPM students learn the basic mathematical skills and procedures that appear on standardized tests at least as well as students who use other programs. Most of the studies show that they do better. Studies that measure the other elements of a complete curriculum—conceptual understanding, problem solving ability, the mathematical practices—show that they do considerably better in these areas. These studies are available at www.cpm.org/learn.html, along with the research base of the CPM program.

CPM Homework Design

The Problem with Massed Practice

Traditional math courses employ “massed practice,” which are homework assignments consisting of lots of similar problems to be practiced all at once. Massed practice creates the illusion of mastery. As the CPM course developers state, “Students believe they have learned what they were supposed to learn because they can follow a pattern, and teachers believe that they have taught it because they see students getting the right answers. So everyone is happy on that day. The problem is that the effect fades away quickly.” At Bellarmine, we have often seen this problem manifest itself when a student is asked to apply a method or a concept to a new or slightly different problem. He either cannot recall a method or does not recognize that the problem before him is presenting an opportunity to use the method. This problem is particularly prevalent in our upper division courses, which rely on a student’s ability to apply and extend foundational knowledge from algebra and geometry.

Mixed, Spaced Practice

Mixed, spaced practice is more challenging than traditional massed practice, but there are clear benefits.

The Benefits of Mixed Practice

Based both on the research and on our teachers’ experience, mixed practice, or the interweaving of different types of mathematics problems (from different class lessons) in a single homework assignment, improves students' ability to pair a problem with the appropriate concept or procedure. This is a key problem-solving skill and a key success factor in our upper division courses and beyond. It’s not sufficient for a student to have a collection of mathematical tools in his toolkit. He much also know how and when to use them when problem-solving. With massed practice, a student is never asked to practice this skill—the procedure he is being asked to practice is clear. But with mixed practice, he is asked to draw on his previous knowledge of concepts and procedures and select which one to apply. Sometimes he may choose incorrectly the first time and get stuck or reach a wrong answer. This in itself presents a great learning opportunity. We tell students not to get discouraged (practice resilience), to try a different approach (practice persistence), and to use all of the resources available to them, such as the homework help web page, the textbook, the student notebook or asking a question in class or during office hours (practice resourcefulness).

As it happens, resilience, persistence, and resourcefulness are among the key success factors for college. Homework is not graded based on right or wrong precisely because it is practice. But we want it to be meaningful practice, not just the rote implementation of a procedure. We want students to be practicing the skills they will need to succeed in our more advanced math courses

The Benefits of Spaced Practice

Most people are familiar with the case of the student who “crams” for a quiz, urgently storing facts in short-term memory that are quickly forgotten once the quiz is over. Similarly, we have found in math, when a student practices a procedure or a concept only once and then moves on (massed practice), his retention is low. Spaced practice involves spreading a collection of problems pertaining to a particular concept across multiple homework assignments over time. According to educational and professional training studies, spaced practice improves long-term retention by reinforcing ideas held in memory. Simply put, spacing provides review that improves long-term retention.

CPM Homework Advice for Students

It is true that mixed, spaced practice requires more time, energy, and thought than traditional, massed practice. For this reason, we try to assign fewer problems. CPM also recommends fewer problems, citing research that shows learning is more efficient when mixed, spaced practice is used. Because this approach to math homework is new to many students, we’d like to offer a few pieces of advice, which we hope will help:

  • You can do it. Teachers will not assign any problems that depend on material you haven’t yet seen, either in your current class or your previous class. The problems that preview new material do so in a way that asks students to recall tools they already have and apply concepts they already know.
  • Don’t get discouraged if you initially don’t recognize a problem or aren’t sure where to start. There are hints to help get you started at homework.cpm.org. You can also utilize your textbook as a resource (e.g. the Math Note boxes summarize key concepts from previous lessons). The class lecture notes in your class notebook may also help.
  • If you spend more than 30 minutes on a homework assignment (focused time…no social media or other distractions) then stop there and write down what you tried and where you got stuck. Bring your questions to class and if your teacher can’t answer them there then he/she will be glad to answer them during office hours. Either way, make sure to get them answered soon, rather than waiting!
  • Additional problems are available in the parent guide for students who want more practice. The problems in the parent guide are arranged in a more traditional format so it’s clear which section of the text to which they belong. The parent guide for each course can be accessed for free using the links below (it can also be purchased in hardcopy for $20 at shop.cpm.org):
  • Algebra 1 Parent Guide: http://cpm.org/cca-parent-guide
    Geometry Parent Guide: http://cpm.org/ccg-parent-guide
    Algebra 2 Parent Guide: http://cpm.org/cca2-parent-guide

  • Homework is practice, it is graded based on effort and completion rather than right/wrong. It is also essential preparation for graded exams later on. Solving homework problems (not just looking over solutions) is an excellent way to prepare for an exam.
  • Check your answers! Answers to some homework problems are posted on the homework help page and your teacher also posts answers to all problems on Canvas. Take advantage of these resources to make sure you’re on the right track as you solve homework problems.
  • Make sure you are using all of the resources available to you! We've dedicated an entire webpage to the topic: Math Resources.
  • When preparing for an exam, we recommend doing the review problems your teacher provides. “Doing” means actually working out the problems from start to finish to prove to yourself that you understand the problem and the concept behind it. Just reading over a problem or even a solution to a problem isn’t good enough. Without actually doing the problem it’s easy to deceive yourself into believe that you know it when you don’t.

CPM Homework Advice for Parents

  • Parent Guide
  • For more information regarding the key concepts covered in your son’s class, we recommend consulting the parent guide for that class. The parent guide is also a good source of example problems, and extra practice problems, organized by chapter. The problems in the parent guide are arranged in a more traditional format so it’s clear which section of the text to which they belong. The parent guide for each course can be accessed for free using the links below (it can also be purchased in hardcopy for $20 at shop.cpm.org):

    Algebra 1 Parent Guide: http://cpm.org/cca-parent-guide
    Geometry Parent Guide: http://cpm.org/ccg-parent-guide
    Algebra 2 Parent Guide: http://cpm.org/cca2-parent-guide

  • Questions to Use When Working with your Son
  • While not a complete list, here are some questions you might use to help your son move forward with a homework assignment and/or remind him of the resources that are available to him.

    »   What have you been doing in class or during this chapter that might be related to this problem? Let’s look at your notebook, class notes or Learning Log.
    »   Can you contact your study partner or someone from your study team?
    »   Have you checked the online homework help?
    »   What have you tried? What steps did you take?
    »   What is unknown? What do you need to know to solve the problem?
    »   Which words are most important? What does this word/phrase tell you?
    »   Can you draw a diagram or sketch to help you?
    »   Have you tried making a list/table or looking for a pattern?
    »   Does your answer make sense?
    »   Have you checked your answer to make sure it’s correct? (Using the homework help page of the answers that your teacher has posted in Canvas)
    »   If you don’t understand this one, can you try another problem in the homework assignment first and come back to this one?

  • Math Resources Web Page
  • Make sure your son is using all of the resources available to him! We've dedicated an entire webpage to the topic: Math Resources.

  • Parent Tips Web Page
  • CPM publishes weekly tips for parents that explain aspects of the program and suggestions for ways in which to help students. The Parent Tips Web Page is located at http://cpm.org/tips.

Why We Teach CPM at Bellarmine

For more information about why we adopted the program and why we believe a problem-based curriculum like CPM helps Bellarmine students achieve college and career readiness, please read Why BellarmineTeaches a Problem-Based Math Curriculum - From “Hitting the Wall” to College and Career Readiness.