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:|
|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:|
|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:|
|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:|
|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.|
|Q||Should the Core-Plus Mathematics Project curriculum be used with heterogeneous or homogeneous classroom groupings of students?|
The CPMP curriculum is a flexible curriculum that can be used in a variety of ways with different groupings of students. Thus, each school should make the decision about grouping students that best fits their student body and community. In whatever configuration the school chooses, the availability of extension activities in the student text, and of maintenance tasks in the Teacher Resource materials, as well as different styles of practice problems, allows teachers to provide appropriate challenge or review for each student.
At this time, the CPMP curriculum is being successfully used in math/science magnet schools, in high schools with heterogeneous classrooms, with accelerated 8th graders, in accelerated tracks that move more quickly through the materials starting in 9th grade, and in schools that use several different curricula. In many schools, the CPMP materials are used successfully with all the students, whether they are tracked or untracked.
|Q||Can the Core-Plus Mathematics Project curriculum be used with students having limited English proficiency (LEP), or with English language learners (ELL)?|
Bill Bokesch was a Core-Plus field-test teacher in a southern California high school. Over 70% of the students in his school did not have English as their first language. In a recent article in MathLink, Bill described the techniques he learned in a 45-hour Professional Development course required by the California legislature to help teachers teach LEP and ELL students who are assigned to regular instructional classes (as all California students are now). The course is called "Specially Designed Academic Instruction in English" (SDAIE). Bill compared the SDAIE recommendations with what he was already doing as a CMIC teacher.
Bill concludes: "By using appropriate settings, discussion, investigation, and tools, I help students in my CMIC classes learn mathematics in a way that strengthens their language skills and allows them to build on their existing mathematics knowledge." To read Bill Bokesch's entire article, download ELL-LEP.pdf (58 kb).
|Q||Are the topics recommended by the College Board as preparation for Advanced Placement Calculus covered in the Core-Plus Mathematics curriculum?|
|A||Yes, all of the topics for Algebra and Trigonometry, for Geometry, and for Coordinate Geometry are developed in the CPMP four-course curriculum. In some cases, topics appear in multiple courses. For a list of all topics recommended by the College Board, including Graphing Calculators and Other Topics, and corresponding page references in CPMP Courses 1-4, download this printable PDF file.|
|Q||When should students who are taught using the Core-Plus Mathematics Project curriculum take AP Calculus and AP Statistics?|
completing three years of the CPMP curriculum, students are very
well prepared to take AP Statistics. Some schools are finding that
one semester is enough to teach the statistics in the AP syllabus
that had not already been learned in the first three CPMP courses.
After completing four years of the CPMP curriculum, students are very well prepared to take AP Calculus. (See School Reports for AP results from some schools using the published CPMP curriculum, Contemporary Mathematics in Context.) Just as in a traditional four-year curriculum (Algebra-Geometry-Advanced Algebra-Precalculus), some form of acceleration is needed for students who want to take calculus as seniors. The second question in the section on Local Implementation questions identifies several acceleration options that schools have used successfully to enable students to complete the four years of the CPMP curriculum prior to their senior year.
|Q||Does the Core-Plus Mathematics Project curriculum prepare students for college?|
Evaluation research has shown that students using the field-test
version of the CPMP curriculum do as well as, or better than, non-CPMP
students on the SAT and ACT college
entrance exams. Also, a study at the
University of Michigan of two Michigan high schools found that
in collegiate mathematics courses at the University of Michigan,
graduates of the CPMP program performed as well, or better than,
graduates of a traditional mathematics curriculum. Finally, students
completing pilot and field-test versions of the CPMP curriculum
have been accepted at over 450 schools around the country, including
Harvard University, Stanford, Duke University, Massachusetts Institute
of Technology (MIT), Notre Dame, the University of Michigan, the
University of Chicago, the University of California at Berkeley,
Clemson University, the University of Virginia, Purdue University,
Boston College, the University of Wisconsin - Madison, Rice University,
the University of Washington, Georgetown University, the Air Force
Academy, Northwestern University, Morehouse College, the University
of Arizona, Vanderbilt University, the University of Hawaii, and
Pennsylvania State University. Students who study the final published
version of the CPMP curriculum should be even better prepared for
(Other indicators of preparedness for college based on SAT, ACT, and AP Calculus and AP Statistics from schools using the published version of Core-Plus Mathematics are reported at School Reports.)
|Q||Does the Core-Plus Mathematics Project curriculum provide the mathematics necessary to be successful on traditional college math placement exams?|
|A||Yes. In particular, there is a set of skill practice problems at the end of every lesson in Course 4. These problem sets are designed to provide practice on the specific types of problems that are often found on college math placement exams. As with any curriculum, the students' degree of preparation will depend on their own efforts.|
For more information
on evaluation evidence, see the Evaluation page
and the annotated list of Research Publications.
|Q||What do evaluation studies say about the effectiveness of the Core-Plus Mathematics Project curriculum?|
There is a large and growing body of rigorous research documenting the effectiveness of the CPMP curriculum. Based on evidence from nationally standardized tests (ITED, SAT, ACT, NAEP), course-specific tests, researcher-developed tests, interviews, and surveys, the CPMP curriculum has been shown to enhance students' mathematical achievement and attitudes toward mathematics.
and Mathematical Modeling
Mathematics in Addition to Algebra and Geometry
Assessment of Educational Progress (NAEP)
Perceptions and Attitudes
on State Assessments
on College Math Placement Tests
in College Mathematics Courses
results are drawn from several sources, including two research
papers presented at the 1998 Annual Meeting of the American Educational
two field-test progress reports:
and a paper appearing in the Journal for Research in Mathematics Education, "Effects of Standards-based Mathematics Education: A Study of the Core-Plus Mathematics Project Algebra/Functions Strand," Vol. 31, No. 3 (2000).
|Q||How well do Core-Plus students perform on standardized tests like the Iowa Tests of Educational Development?|
On the quantitative section of the Iowa Tests of Educational Development (ITED-Q), Core-Plus students significantly outperformed both the nationally representative norm group and comparison students in the same school who had a traditional mathematics curriculum.
The Ability to Do Quantitative Thinking (ITED-Q or ATDQT) is the mathematical subtest of the Iowa Test of Educational Development, a nationally standardized battery of high school tests. The ITED-Q is a 40-item multiple-choice test with the primary objective of measuring students' ability to employ appropriate mathematical reasoning in situations requiring the interpretation of numerical data and charts or graphs that represent information related to business, social and political issues, medicine, and science. The ITED-Q administered in CPMP national field test schools at the beginning of Course 1 served as the pretest for all courses, so the pretest-posttest analyses for Courses 1, 2, and 3 are for one, two, and three years of mathematics instruction, respectively. For the first and second years, there was a comparison group of ninth- or tenth-grade students in traditional mathematics courses in some field-test schools with both curricula.
Results for the following three cohort groups of CPMP students were analyzed: (1) all students who completed both the Course 1 Pretest and the Course 1 Posttest, (2) all students who completed both the Course 1 Pretest and the Course 2 Posttest, and (3) all students who completed both the Course 1 Pretest and the Course 3 Posttest. Table 1 below gives median (middle) ITED-Q percentiles of the CPMP and comparison distributions.
The results given in Table 1 are illustrated in the following graph.
Pretest to posttest growth in percentiles indicates growth by CPMP students beyond that of the national norm group. Such increases appear consistently across the CPMP distribution for each year. For example, the median CPMP Course 1 student increased the equivalent of nearly two years in just one year's time. Allowing for pretest differences, CPMP posttest means in schools with comparison groups are significantly greater than those of the comparison students.
|Q||How well do Core-Plus students perform on the SAT?|
On the SAT-I Mathematics test, students completing Core-Plus mathematics field-test courses performed at least as well as students in traditional mathematics curricula.
SAT data for 1997 from 13 CPMP schools were separated into groups according to the secondary mathematics courses the students had completed. SAT Mathematics scores of students who had completed Courses 1, 2 and 3 were compared to students who completed traditional algebra, geometry and advanced algebra. In Table 1, these groups are labeled "CPMP 3" and "Advanced Algebra," respectively. The CPMP 3 average (mean) is greater than that of the Advanced Algebra students, but the difference is not significant at the 0.05 level.
In one field-test school at the beginning of the CPMP field test (Fall 1994), all ninth-grade students who qualified for pre-algebra or algebra were randomly assigned by computer to CPMP Course 1 or to a traditional course. Many of these students completed Advanced Algebra or CPMP Course 3 in their junior year and took the SAT either in spring or summer of their junior year or in fall of their senior year. As shown in Table 2, the average Grade 8 ITBS Mathematics scores are nearly identical for the CPMP students and those in the traditional curriculum. Thus, these two groups were well-matched on mathematical achievement prior to high school. They learned mathematics in the same school and sometimes from some of the same teachers. The only apparent systematic difference between the groups is the curriculum. The average SAT Math score for the CPMP group is greater than that of the traditional group, but the difference is not statistically significant at the 0.05 level.
The results in Tables 1 and 2 are illustrated in the following graph.
|Q||How well do Core-Plus students perform on the ACT?|
On the ACT Mathematics test, students completing Core-Plus mathematics field-test courses performed as well as students in traditional mathematics curricula.
The 2,944 CPMP and 527 traditional students in the original CPMP field-test sample had nearly identical average scores on the ITED-Q pretest administered at the beginning of Grade 9. ACT scores were available from a reasonably large subset of these students, and their average ACT Mathematics and ACT Composite scores are given in Table 1. There is no significant difference (0.05 level) between the CPMP and traditional averages (means) for either the Mathematics or Composite score.
In one school district at the beginning of the CPMP field test (Fall 1994), all ninth-grade students in the two CPMP field-test schools who qualified for remedial mathematics through algebra were randomly assigned by computer to CPMP Course 1 or to the appropriate traditional course. Many of these students completed Advanced Algebra or CPMP Course 3 in their junior year and took the ACT either in spring or summer of their junior year or in fall of their senior year. The average sixth-grade CAT Mathematics percentiles for the CPMP students and those in the traditional curriculum are similar as shown in Table 2, so these two groups were well-matched on mathematical achievement prior to high school. They learned mathematics in the same schools and sometimes from some of the same teachers. The only apparent systematic difference was the curriculum. For this set of students, the average ACT Math scores for the CPMP group is almost identical to that of the traditional group. The average ACT Composite score for the CPMP group is greater than that of the traditional group, but the difference is not statistically significant at the 0.05 level.
The results in Tables 1 and 2 are illustrated in the following graph.
|Q||How well do Core-Plus students perform on mathematics placement tests at the college level?|
On a Mathematics Department Placement Test from a large Midwestern university, students completing field-test versions of Core-Plus Mathematics Courses 1-3 plus the precalculus path of Course 4 performed as well as students in traditional precalculus on basic algebra and advanced algebra subtests and better on the calculus readiness subtest.
The Mathematics Placement Test, compiled from a bank of items developed by the Mathematical Association of America, that is presently used at a major university was administered in several field-test schools in May 1999 at the end of CPMP Course 4 and traditional Precalculus courses. This test contains three subtests - Basic Algebra (15 items), Advanced Algebra (15 items) and Calculus Readiness (20 items). The first two subtests consist almost entirely of algebraic symbol manipulation, and the third subtest measures some of the important concepts that underlie calculus. A graphing calculator (with no symbol manipulation capability) is allowed on this test.
The CPMP Course 4 students included in the comparison below are all those in the 1998-99 Course 4 field test who completed the 6-unit "preparation for calculus" path as the last course in their sequence of CPMP Courses 1-4 (N = 164). The Precalculus students, also from field-test schools, completed a traditional precalculus course following a sequence of Algebra, Geometry and Advanced Algebra (N = 177). The two groups were further restricted to those students who indicated on a written survey their intention to attend a four-year college or university in the next school year. Eighth-grade mathematics standardized test scores for both groups were, on average, at about the 85th national percentile. Means by group and subtest are plotted in Figure 1. The CPMP Course 4 mean was significantly (p < 0.05) greater than the Precalculus mean on the Calculus Readiness subtest, while the group means did not differ significantly on the Basic Algebra or Advanced Algebra subtests.
The Mathematics Department at the university that provided this placement test combines the subtest scores by a formula to recommend enrollment for each student in one of four college mathematics courses - Calculus I, Precalculus, Intermediate Algebra, and Beginning Algebra. Using that formula, the percent of CPMP Course 4 and Precalculus students who would be recommended for each course is illustrated in Figure 2. A much higher percentage of CPMP Course 4 students (50.6%) than traditional Precalculus students (39.0%) would be recommended for Calculus I suggesting that the CPMP curriculum with this sequence of Course 4 units better prepares students for this examination and presumably for college calculus.
|Q||How well do Core-Plus students perform in college mathematics courses?|
CPMP Course 4 was field tested nationally during the 1998-99 school year and some preliminary evidence on how CPMP graduates perform in collegiate mathematics courses is beginning to appear. A study completed at the University of Michigan examined the performance of students from two Michigan high schools in the same district, Andover High School and Lahser High School. In 1995 and 1996, a traditional mathematics curriculum was in place at both schools, and Lahser continued to use their traditional curriculum through 1998-99. In 1997, all Andover students who had not previously been accelerated had studied the CPMP curriculum, and by 1998 all Andover students were in the CPMP program.
Computer files provided by the University of Michigan registrar were used to generate the achievement data summarized in the following table. The table includes the number of matriculants from the school under the year, the mathematics courses taken in the first year of study at the University of Michigan together with the grade point averages, and numbers of elections and the course averages in each year. The mathematics courses are 105/110 (precalculus), 115 (calculus I), 116 (calculus II), 215 (calculus III), 216 (introduction to differential equations), and honors (all honors math courses open to freshmen). The grade point averages were calculated using the University of Michigan system as follows: A+ (4.3), A (4), A- (3.7), B+ (3.3), B (3), ..., D (1), D- (0.7), E+ 0.3), and E (0).
The Andover achievement for the years 1997 and 1998 when CPMP was in place is stronger than both pre-CPMP Andover (i.e., 1995 and 1996) and 1997 and 1998 Lahser achievement. Similarly, the number of Andover matriculants at the University of Michigan for the last two years is greater than that for the previous two years. These achievement and admissions data clearly support the view that in collegiate mathematics courses at the University of Michigan, graduates of the CPMP program perform as well as, or better than, graduates of a traditional mathematics curriculum.
Graduates of the CPMP program at Andover have, themselves, commented on their preparedness for collegiate mathematics and mathematics-related fields. The following comments are from three students who studied the pilot version of CPMP Course 4. The first two students enrolled at the University of Michigan.
Comments such as the above are not unique to students at the University of Michigan. The following is a quote from an Andover graduate from the same class who enrolled at Stanford University.
|Q||What are students' perceptions and attitudes about the Core-Plus Mathematics Project curriculum?|
A written, Likert-type survey of students' perceptions and attitudes about various aspects of their mathematics course experience was administered at the end of each school year during the field test. In four field-test schools, both CPMP Course 2 students (n = 221) and traditional geometry students (n = 134) completed this survey at the end of their respective courses. (Course 2 results are presented since the newness effect of the CPMP approach is likely to have disappeared by then). Each of the following findings was consistent across levels of pretest student achievement.
|Q||What are some tips for effectively implementing the Core-Plus Mathematics Project curriculum?|
|A||Based on our experiences working with schools to implement the CPMP curriculum, Contemporary Mathematics in Context, we recommend that careful consideration be given to the form of implementation in a district and to the groundwork needed to build support for school mathematics reform. In addition, a professional development plan to support teachers is crucial to effective implementation of the curriculum. Some things to consider prior to implementation are the following:|
|Q||How can students be accelerated in the Core-Plus Mathematics Project curriculum?|
|Q||What is the role of professional development in implementing the Core-Plus Mathematics Project curriculum?|
Because much of the content in statistics, probability, and discrete mathematics is new for many teachers, and because some of the familiar material is developed more fully than in traditional mathematics, teachers need advice and support from other teachers and administrative support in order to implement the curriculum effectively. (Professional Development Opportunities)
Active involvement of students also requires a different type of planning by teachers. The Teacher Resource materials encourage teachers to be listening, observing, questioning, facilitating student work, and orchestrating class discussions in new ways. Professional development programs organized around reflecting on practice enable teachers to hone their skills in these areas.
At the very least, teachers should attend a professional development workshop led by an experienced CPMP teacher. In addition, schools should strongly consider providing the following supports:
|Q||What behaviors and characteristics of Core-Plus Mathematics teachers are associated with students' growth in mathematics achievement?|
We examined the classroom practices of 20 teachers during the field test of CPMP Course 1. Ten of these teachers comprised the top quartile of field-test teachers and the other 10 the bottom quartile with respect to their students' growth in mathematical achievement over the one-year course. Achievement was measured by a nationally standardized test called the Ability to Do Quantitative Thinking which is the mathematics subtest of the Iowa Tests of Educational Development. The primary data sources were: trained observer's holistic rating of the alignment of the instructional practice and classroom climate with CPMP's teaching for understanding model, self-perceptions of practice by the teachers, and expressed concerns of the teachers about the new curriculum.
results from this study, summarized below, are reported in a
peer-reviewed article published in the Journal for Research
in Mathematics Education:
The description of the "effective" (i.e., first-quartile) teacher that emerged from analyzing the data from these sources follows. This teacher may be of either gender, but we will use female pronouns for convenience.
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