Frequently Asked Questions
About the Core-Plus Mathematics Project

Last Updated: 28 January 2005

 

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Evaluation Evidence

Evaluation Evidence

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?
A

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.

Quantitative Thinking
CPMP students outperform comparison students on the mathematics subtest of the nationally standardized Iowa Tests of Educational Development ITED-Q.

Conceptual Understanding
CPMP students demonstrate better conceptual understanding than students in more traditional curricula.

Problem Solving Ability
CPMP students demonstrate better problem solving ability than comparison students.

Applications and Mathematical Modeling
CPMP students are better able to apply mathematics than students in more traditional curricula.

Algebraic Reasoning
CPMP students perform better on tasks of algebraic reasoning than comparison students.

Algebraic Procedural Skills
This is the one area for which field-test research indicates mixed results. On some evaluation tests, CPMP students do as well or better, on others they do less well than comparison students. As part of the curriculum development process, revisions have been made to strengthen students' algebraic skills. The final and published version of the Core-Plus Mathematics curriculum maintains the well-documented effectiveness of the curriculum, while strengthening students' algebraic procedural skills.

Important Mathematics in Addition to Algebra and Geometry
CPMP students perform well on mathematical tasks involving probability, statistics, and discrete mathematics.

National Assessment of Educational Progress (NAEP)
CPMP students scored well above national norms on a test comprised of released items from the National Assessment of Educational Progress.

Student Perceptions and Attitudes
CPMP students have better attitudes and perceptions about mathematics than students in more traditional curricula.

Performance on State Assessments
The pass rate on the 2004-05 Tenth-Grade Washington Assessment of Student Learning Mathematics test for 22 state of Washington high schools that were in at least their second year using the Core-Plus Mathematics curriculum was significantly higher than that of a sample of 22 schools carefully matched on prior mathematics achievement, percent of students from low-income families, percent of underrepresented minorities, and student enrollment.

College Entrance Exams - SAT and ACT
CPMP students do as well as, or better than, comparable students in more traditional curricula on the SAT and ACT college entrance exams.

Performance on College Math Placement Tests
On a mathematics department placement test used at a major midwestern university, CPMP students performed as well as students in traditional precalculus courses on basic algebra and advanced algebra subtests, and they performed better on the calculus readiness subtest.

Performance in College Mathematics Courses
CPMP students completing the four-year curriculum perform as well as, or better than, comparable students in a more traditional curriculum in college mathematics courses at the calculus level and above.

The above results are drawn from several sources, including two research papers presented at the 1998 Annual Meeting of the American Educational Research Association:

  • An Emerging Profile of the Mathematical Achievement of Students in the Core-Plus Mathematics Project
  • Students' Perceptions and Attitudes in a Standards-based High School Mathematics Curriculum,

two field-test progress reports:

  • Mathematical Achievement on Standardized Tests
  • Assessments of Students' Mathematical Performance,

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?
A

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.

Table 1

CPMP Students Comparison Students
N Median N Median
Course 1 Pretest
Course 1 Posttest
2,944
2,944
54
62
527
527
54
55
Course 1 Pretest
Course 2 Posttest
2,270
2,270
59
66
201
201
59
62
Course 1 Pretest
Course 3 Posttest
1,457
1,457
65
71

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?
A

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.

Table 1: Means and Standard Deviations of 1997 SAT Mathematics Scores

CPMP3 Advanced Algebra
Number of Students Average Number of Students Average
371 552.0 190 543.4

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.

Table 2: ITBS Math and SAT Math Means and Standard Deviations for CPMP and Well-Matched Traditional Students in One High School

Grade 8 ITBS Math Percentile SAT Math
Group Number of Students Average Average
CPMP 54 57.1 484.6
Traditional 44 57.5 467.0

The results in Tables 1 and 2 are illustrated in the following graph.



Q How well do Core-Plus students perform on the ACT?
A

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.

Table 1

ACT Mathematics ACT Composite
Group Number of Students Average Average
CPMP 531 19.2 20.4
Traditional 111 19.8 20.3

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.

Table 2

CAT Math Percentile ACT Mathematics ACT Composite
Group Number of Students Average Average Average
CPMP 71 66.3 18.3 20.3
Traditional 42 68.5 18.4 19.1

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?
A

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.

Figure 1:

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.

Figure 2:



Q How well do Core-Plus students perform in college mathematics courses?
A

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).

Table 1: Mean Grade Point Averages (Number of Students) by School, Course, and Year

Andover High School Lahser High School
College Class 1995
(50)
1996
(74)
1997
(87)
1998
(72)
1995
(34)
1996
(57)
1997
(45)
1998
(35)
105 3.18(4) 2.29(6) 2.74(13) 2.98(6) 1.46(7) 3.00(4) 2.60(5) 2.97(3)
115 2.86(14) 2.60(19) 3.08(32) 2.89(25) 2.33(7) 2.82(13) 2.58(15) 2.87(7)
116 2.67(14) 3.33(12) 3.17(19) 3.49(12) 2.45(6) 3.21(18) 2.63(8) 2.29(8)
215 2.66(5) 3.10(4) 2.95(6) 2.99(8) 2.50(2) 3.17(11) 3.34(6) 2.34(5)
216 2.15(2) 4.00(1) 4.00(2) 3.30(2) --- 3.67(3) 3.65(2) ---
Honors --- 3.28(5) --- --- 3.30(1) 3.77(3) 4.23(4) ---
All Courses 2.76(39) 2.89(47) 3.06(72) 3.07(53) 2.15(23) 3.15(52) 2.92(40) 2.57(23)

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.

Student 1:
In high school, I looked forward to math as one of my favorite subjects. The way I was taught and the instructors who taught it, made Core-Plus math extremely interesting to me. My sophomore year of high school is when I developed such a love for math and science that I decided to go into engineering. In my senior year of high school, I took Calculus BC and placed into Calculus 116 [second semester calculus] here at Michigan. The Core-Plus mathematics system and the calculus class I took [in high school] gave me such a strong base in mathematics that I received an A+ in Calculus 116.

The real-life examples of Core-Plus Mathematics gave me an excellent background for demanding engineering courses. Because of my Core-Plus background, I feel I am two steps ahead of students who did not take Core-Plus math in high school. ... I am able to problem solve much faster than students who do not have a Core-Plus mathematics background.

Student 2:
The first political science class that I took at U of M was Comparative Politics. I was lucky because in Core-Plus Mathematics I learned many different kinds of charts, many different data tables, and many different methods for analysis of data. While many of my peers at college were left wondering what a Pearson's r correlation was, I was asking the professor questions like, "Did you, and by what method, screen out any outliers in the data sets?" I think the biggest advantage of Core-Plus math is that the diversity of topics allows me to feel comfortable in any math setting, whether it is politics, economics, or any subject.

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.

Student 3:
It is my firm belief that my Core-Plus education in fact better prepared me for the mathematics I encountered in college, as well as for preceding Advanced Placement Examinations, than would have a traditional mathematics program. For any student who intends to study math at the level of single-variable calculus or beyond, I believe that the conceptual-based style of education stressed in the Core-Plus program will prove far more beneficial than the memorization of what would otherwise be meaningless formulas and algorithms.


Q What are students' perceptions and attitudes about the Core-Plus Mathematics Project curriculum?
A

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.

  • Students perceive the CPMP curriculum to be quite difficult, at least as challenging as traditional college-prep mathematics courses. A common perception of students is that CPMP is challenging and makes them think, but they say that with effort they are able to understand the mathematical ideas and their applications.

  • Over three-fourths of CPMP and geometry students agreed that cooperative-group work was enjoyable and helped them learn mathematics. The advantages of learning in groups most often cited by students were seeing how other people attack problems and the support of group members during problem-solving efforts.

  • A significantly higher percent of CPMP students than of geometry students agreed that their mathematics course made them feel more confident that they could solve mathematical problems (71.1% compared to 55.6%), that they learned to reason mathematically (68.8% to 53.0%), and that the course helped them see that mathematical ideas make sense (64.7% to 51.1%).

  • A significantly higher percent of CPMP students than of geometry students agreed that their mathematics course contained realistic problems (76.5% to 47.8%), made the mathematical ideas interesting (70.1% to 41.4%), and increased their ability to talk about (68.2% to 42.9%) and to write about mathematics (66.5% to 40.6%).

  • CPMP and geometry students (over 85% of each) agreed that they enjoyed using the calculator in mathematics class. About 70% of both groups also agreed that they learned more mathematics by using the calculator.

  • CPMP students were much more likely than geometry students to want to take a mathematics course taught in the same way the next year (75.0% compared to 43.0% agreement), and 27% of CPMP students at the end of Course 3 agreed that it was mainly because of CPMP that they took a third year of mathematics. These findings coupled with substantial increases in enrollments in junior and senior mathematics courses in many field-test schools provide strong evidence that the CPMP curriculum is a factor in keeping more students in mathematics courses longer.

 

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