Frequently Asked Questions
About the Core-Plus Mathematics Project

Last Updated: 28 January 2005

 

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Characteristics of the CPMP Curriculum

Characteristics of the CPMP Curriculum
Q How is the Core-Plus Mathematics Project curriculum different from traditional US mathematics curricula?
A One way to characterize the CPMP curriculum, in contrast to traditional curricula, is to see it as an effort to achieve a better balance among skills, conceptual understanding, and problem solving. This can be summarized as follows:
 
Traditional Approach CPMP Approach
Instructional Focus facts and procedures, with some applications and problem solving conceptual understanding, problem solving, mathematics done in context, with requisite work on procedures and facts
Method direct instruction, memorize and practice, with some projects and occasional group activities active student engagement in inquiry and investigation, guided by probing teacher questions, with teacher-led introductions and summaries, and some direct instruction
Supplementary Material problem solving, applications, thinking skills additional skill practice
  The prevalence and disappointing consequences of the traditional approach in the US are verified by international comparisons such as the Third International Mathematics and Science Study.


Q How is the final version of the Core-Plus Mathematics Project curriculum different from the pilot-test version?
A There are several differences between the pilot-test version and the final version of the CPMP curriculum. These differences came about based on the four-year research, development, and evaluation cycle used for each course:
 
Year 1 initial development, in consultation with teachers, content and instructional specialists, and an international advisory board, including some small-scale local tryouts
Year 2 pilot testing in 19 Michigan schools, followed by revision based on feedback from the pilot-test teachers, consultants, advisory board, and data collected on the pilot test
Year 3 field testing in 36 schools around the country, followed by revision based on feedback from the field-test teachers, consultants, advisory board, and data collected on the field test
Year 4 final revisions and publication
  Based on feedback and data from this development process, the basic approach of the curriculum was maintained, but many changes were made from the pilot version to the final published version. These changes varied from minor rewording of questions to major reorganization of units. More practice with algebra skills was added as were structured reviews (see the next two questions for specifics).


Q Is the attention to algebraic skills in the Core-Plus Mathematics Project curriculum as thorough as that in a traditional algebra program?
A Yes and no. Yes, the CPMP curriculum carefully and thoroughly develops algebraic skills. However, there are many important skills and concepts that students need to learn, and a limited amount of instructional time. Thus, choices had to be made about what to include, what to emphasize, and what to delete. Some particularly complex and little-used algebraic skills, which may be included in some traditional algebra programs, like simplifying complicated rational and radical expressions, are not included in the CPMP curriculum. On the other hand, vital algebraic skills, like factoring, finding equivalent expressions, and solving equations, are covered thoroughly. CPMP developers devoted considerable time and effort designing the curriculum so that these skills are developed meaningfully and completely, based on solid conceptual understanding and appropriate practice. The developers learned from the pilot and field testing, and from the field-test evaluation research, that progress was being made towards achieving these goals. However, it was also found that more practice with algebraic skills was needed. So the following modifications and additions were made:
 
  • Short skill maintenance sheets now are provided for each unit (chapter), beginning with Unit 4 of Course 1, as part of the Teacher Resource materials. Most teachers assign these as homework and then discuss them in class.

  • New lessons and exercises involving symbolic reasoning strategies and algebra practice were added to the student materials for Courses 1-3, particularly in Course 3.

  • In Course 4, the primary algebra focus is on symbolic reasoning and manipulation strategies needed for calculus. The heavy emphasis on symbolic reasoning and manipulation skills in Course 4 is consistent with the original development plan. Throughout the entire curriculum, algebraic concepts and methods are developed using verbal, graphic, numeric, and symbolic representations. In Course 1, there is more emphasis on reasoning with verbal, graphic, and numeric representations, and less emphasis on symbolic manipulation. By-hand symbolic manipulation becomes more prominent as the courses progress, so that by Course 4 symbolic reasoning and manipulation is the primary focus. The reasons for using this gradual approach to teaching and learning symbolic reasoning and manipulation skills are: (a) it allows the symbolic skills to be learned based on solid conceptual understanding and with well-understood links to graphs and tables, and (b) it delays the heavy algebraic skill work to a point closer to when some students will actually need this facility, i.e., college mathematics.

  • A set of algebraic skill practice problems has been added at the end of each lesson in Course 4, specifically to ensure a smooth transition to college mathematics and college math placement exams.
  CPMP developers continue to work with teachers using the curriculum to identify other adjustments and/or supplements that might be needed. For example, a Reference and Practice book has been developed for each course. These books provide an "executive summary" of the main ideas of the preceding course, and also sets of mixed review and skill-practice exercises.


Q How are summaries and review provided in the Core-Plus Mathematics Project curriculum?
A It is crucial that students summarize and review what they have learned. Students summarize and review in the CPMP curriculum in the following ways:
  • There is a set of boxed-off review questions at the end of every investigation (1-3 days). The teacher leads a whole-class discussion of these "Checkpoint" questions, making sure all students understand the main ideas.

  • Students create summaries of important ideas, definitions, and methods in a structured section of their notes called a "math toolkit." They make entries into this toolkit based on specific prompts from their teacher suggested in the teacher notes. The math toolkits become, in essence, student-created glossaries.

  • There is a summary and review lesson at the end of every unit (3-6 weeks). These are called Looking Back lessons. A Looking Back lesson consists of tasks that summarize and review all the main ideas of the unit.

  • The short skill maintenance sheets which are part of the Teacher Resource materials for each unit, beginning with Unit 4 of Course 1, provide periodic review of material from previous units and courses.

  • Also, at the end of each Course 4 unit, students are given an outline of a unit summary. Students complete this as a record of the concepts, facts, and procedures that they have learned. As well as being an excellent summary and review, these concise summaries can be kept for easy reference later on.

  • At the end of each course, there is a 1-2 week synthesizing unit, called a Capstone. The Capstone unit is designed to pull together and review what students have learned during the year. Each Capstone involves investigating problem situations related to one context or one theme. For example, the Course 1 Capstone unit is about putting on a school carnival, including games, finances, and planning, and the Course 3 Capstone is centered around the theme of optimization.

  • Of course, homework, quizzes, projects, and tests also contribute to review and summary.
  The math toolkits and the maintenance sheets were incorporated into the final version of the curriculum as part of the development process (they were not part of the pilot-test version). These modifications were made to enable students to better organize, summarize, review, and retrieve the important mathematical concepts, principles, and methods that they have studied.


Q Are there worked-out examples in the Core-Plus Mathematics Project curriculum?
A Yes, there are many worked-out examples in the CPMP curriculum. And it is the students who work them out! Through carefully orchestrated investigations, the students produce a large number of worked examples. Students learn better when they do the work, instead of (possibly) reading examples worked out by textbook authors.

 

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