Paper

Core-Plus Mathematics Project (CPMP)

The article describes the Core-Plus Mathematics curriculum with a focus on how it works for all students, including heterogeneous groups in terms of race, educational background, gender, and ability. A curriculum overview and samples from a model lesson are provided, together with a brief description of achievement results.

These texts are for the first two years of a four-year integrated curriculum for high schools. Each year features strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. The content is developed through investigations of rich, applied problem situations. The curriculum emphasizes mathematical modeling and capitalizes on graphing calculator technology. The materials promote small-group collaborative learning and assessment as a means of guiding instruction.

Flowers, J. (1995). A study of teachers' indirect influence in Core-Plus Mathematics Project classes.Unpublished paper, University of Michigan.

Flowers researched verbal interactions, comparing traditional and Core-Plus classrooms. Consideration was also given to time allocated between teacher lecture and class discussions or small group work, with particular attention given to the level at which the CPMP teachers followed the instructional model set forth by the project authors.

Hart, E. W. (1997). Discrete mathematical modeling in the secondary curriculum: Rationale and examples from the Core-Plus Mathematics Project. In J. Rosenstein and F. Roberts (eds.), Discrete Mathematics in the Schools.Providence, RI: DIMACS Series in Theoretical Computer Science and Discrete Mathematics, American Mathematical Society.

The article defines discrete mathematics, describes which areas of discrete math should be incorporated into the high school curriculum, and suggests methods for integrating these topics. Included are two examples of the discrete mathematics strand from the Core-Plus Mathematics curriculum.

Hart, E. W. (in press). Algorithmic problem solving in discrete mathematics. In L. Morrow and M. J. Kenney (eds.), Teaching and Learning of Algorithms in School Mathematics,1998 Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council.

The article discusses algorithmic problem solving and how it is developed in the discrete mathematics strand of the Core-Plus Mathematics curriculum. Included are two model lessons.

Hart, E. W., & Stewart, J. (In press). Composing a curriculum: Reflections on high school reform and implications for middle schools. In L. Leutzinger (Ed.), Mathematics in the Middle Grades.Reston, VA: National Council of Teachers of Mathematics.

Using a symphony as a metaphor for the entire school curriculum, the authors describe some of the changes in content, teaching, and assessment that are common to the new reform high school programs, with examples from the Core-Plus Mathematics Project. They also consider implications for the part that middle school mathematics education plays in the development of common themes.

Hirsch, C. R., Coxford, A. F., Fey, J. T., & Schoen, H. L. (1995). Teaching sensible mathematics in sense-making ways with the CPMP. Mathematics Teacher,88(8), 694-700.

The article gives an overview of the Core-Plus Mathematics curriculum, focusing on mathematics as "sense-making." Included are the organization of the curriculum, the mathematical strands, the four-phase instructional model, the three-year "core" program accommodating all students, the curriculum-embedded assessment, and lesson samples from a Course 1 unit.

Hirsch, C. R. & Coxford, A. F. (1997). Mathematics for all: Perspectives and promising practices. School Science and Mathematics,97(5), 232-241.

This article describes CPMP perspectives on a new curriculum organization for high school mathematics, identifies implications of these perspectives for promoting access and equity for all students, and reports some of the supporting oral data from the ongoing formative evaluation of the curriculum. The focus is on diversity issues arising in ability, prior knowledge, gender, interests, and learning styles.

Hirsch, C. R. (in press). The Core-Plus Mathematics Project (CPMP). In L. S. Grinstein and S. I. Lipsey (eds.), Mathematics Education: An Encyclopedia. Washington, DC: Taylor and Francis.

An overview of the Core-Plus Mathematics curriculum highlights the alignment with the NCTM Standards in terms of a "core" of mathematics for all students, connectedness of topics, and use of various assessment alternatives.

Hirsch, C. R. & Weinhold, M.L. W. (in press). Everybody counts-Including the mathematically promising. In L. Sheffield (ed.), Developing Mathematically Promising Students.Reston, VA: National Council of Teachers of Mathematics.

The article examines curricular and instructional issues of teaching mathematically promising students, and draws from experiences of teachers using the Core-Plus Mathematics Project curriculum materials with promising students in heterogeneous classes, in accelerated classes, and in special mathematics and science centers.

Kett, J. R. (1997). A portrait of assessment in mathematics reform classrooms. Unpublished doctoral dissertation, Western Michigan University.

Kett's case studies of four CPMP teachers describe their views on assessment, how assessment was used in each of their classrooms, and what problems were encountered in implementation of their assessment plans.

Lloyd, G. M. (1996). Change in teaching about functions: Content conceptions and curriculum reform.Paper presented at the Eighteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Pp. 200-205.

A high school mathematics teacher was studied as he taught the Core-Plus curriculum for two years, and this paper focuses primarily on year two findings about the subtle but meaningful changes in his conceptions and instruction evidenced as he gained comfort with the new curriculum. In particular, the paper illustrates how the teacher revised his pedagogical content conceptions through complex interaction of his mathematical content conceptions and classroom experiences with students.

Lloyd, G. M. (submitted for publication). Re-learning to teach about mathematical functions: How pedagogical content conceptions can change in the context of reform.Paper submitted for publication, Virginia Polytechnic Institute and State University.

A high school teacher's flexible and comprehensive mathematical content conceptions supported implementation of the Core-Plus mathematics curriculum's approach to functions over a two-year period. In both years, he communicated that understanding functions means accessing covariation patterns through consideration of multiple representation. In year 2, with increased comfort and familiarity witht he materials, he domonstrated the development of a variety of powerful strategies for using ghe curriuclum materials and associated student work to more extensively enact his goals and conceptions in the Core-Plus classroom. Furthermore, in year two the teacher dispalyed evidence of having changed some of his pedagogical contenct conceptions of functions.

Lloyd, G. M. (submitted for publication). Ways of experiencing curriculum: A phenomenographic analysis of two high school teachers' attempts to promote cooperation and exploration in their mathematics classrooms.Paper submitted for publication, Virginia Polytechnic Institute and State University.

This paper describes the ways in which two high school teachers experienced the Core-Plus mathematics curriculum. In elaborating how the teachers experienced the curriculum, the focus is on the teachers' conceptions of cooperative explorations of mathematical situations. The results elaborate how the curriculum materials were experienced as a challenging vision of instructional practice for one teacher, and as a constraint of a personal vision for another teacher. Discussion of the contrasting ways the two teachers experienced the curriculum advances and illustrated the notion that it does not make sense to view the curriculum "as such" - for it is always a curriculum experienced by some person in some way.

Lloyd, G. M., & Wilson, M. R. (1997). Secondary mathematics teachers' experiences using a reform-oriented curriculum to encourage student cooperation and exploration.Paper presented at the Nineteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

This paper describes two high school teachers' interpretations of and classroom experiences with a reform-oriented mathematics curriculum (Core-Plus Mathematics Project). The study focuses on the teachers' conceptions of cooperative explorations of mathematical situations. The results elaborate how the curriculum materials presented a challenging vision of instructional practices for one teacher, and a constraint to the fulfillment of a personal vision for another teacher.

Lloyd, G. M., & Wilson, M. R. (in press). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education.

This case study investigates the content conceptions of an experienced high school mathematics teacher and links those conceptions to their role in the teacher's first implementation of reform-oriented curricular materials during a 6-week unit on functions.

Lloyd, G. M., & Wilson, M. R. (1997). The role of high school mathematics teachers' beliefs about student cooperation and exploration in their interpretations of a reform-oriented curriculum.Paper presented at the Annual Meeting of the Eastern Educational Research Association.

This paper describes the conceptions and experiences of two veteran high school teachers attempting to implement a reform-oriented mathematics curriculum (Core-Plus Mathematics Project) that explicitly supports the goals of the Standards (National Council of Teachers of Mathematics [NCTM], 1989). It focuses on the teachers' beliefs about the meaning and importance of certain mathematical activities, specifically cooperation and exploratory problem-solving. And, it reveals how teachers' beliefs about these issues relate to their interpretations of innovative curricula.

Lloyd, G. M., & Wilson, M. R. (in press). The impact of teachers' beliefs about student cooperation and exploration on their interpretations of a secondary mathematics curriculum. In J. Dossey (ed.), Proceedings of the Nineteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.Columbus, OH: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Schoen, H. L., Bean, D. L., & Ziebarth, S. W. (1996). Embedding communication throughout the curriculum. In P. C. Elliott and M. J. Kenney (eds.), Communication in Mathematics: K-12 and Beyond,1996 Yearbook of the National Council of Teachers of Mathematics. (pp. 170-179). Reston, VA: The Council.

The main theme of this article is that students learn to communicate mathematically by being in an environment where such communication is a regular, natural, and valued occurrence. Examples of curriculum-embedded communication from the Core-Plus Mathematics curriculum materials highlight communication in group investigations, written reflections, quizzes and examinations, and extended projects.

Schoen, H. L., Hirsch, C. R., & Ziebarth, S. W. (1998). An emerging profile of the mathematical achievement of students in the Core-Plus Mathematics Project.Paper presented at the Annual Meeting of the American Educational Research Association. San Diego, CA.

Achievement results are reported for the three-year Core-Plus field test (1994-97) on the standardized Ability to Do Quantitative Thinking (ATDQT) test for all schools with school means as the statistical unit. ATDQT results are also reported by school setting (urban, rural, or suburban), by make-up of classes (heterogeneous, high ability, low ability, and so on), by gender, by English or non-English first language, and for three classrooms of students with exceptionally high mathematical aptitudes. Results are also given for the various subtests of both the CPMP Posttest, an open-ended assessment instrument, and a test comprised of released items from the 1992 National Assessment of Educational Progress.

Schoen, H. L., & Pritchett, J. (1998). Students' perceptions and attitudes in a standards-based high school mathematics curriculum.Paper presented at the Annual Meeting of the American Educational Research Association. San Diego, CA.

This paper presents the results from an attitude scale called Attitude Toward Your Mathematics Course during the three-year Core-Plus field test (1994-97). This scale contains 15 likert-type items and an open-ended writing prompt. Comparisons are made between Core-Plus students and students in more traditional college preparatory mathematics curricula in the same field test schools. Students perceive the Core-Plus curriculum as difficult, at least as challenging as traditional college-prep mathematics courses, but Core-Plus students were more positive about various aspects of the curriculum and of their experience in the classroom than were students in traditional mathematics classes.

Schoen, H. L., & Ziebarth, S. W. (1997). A progress report on student achievement in the Core-Plus Mathematics Project field test.Unpublished manuscript, University of Iowa.

Achievement results based on standardized tests (ITED) and CPMP tests compare traditionally tracked students and Core-Plus Mathematics students. Results from both Course 1 and Course 2 are discussed.

Schoen, H. L., & Ziebarth, S W. 1997. A progress report on student achievement in the Core-Plus Mathematics Project field test. NCSM Journal of Mathematics Education Leadership,1(3), 15-23.

This article is a shorter version of the previous one that is re-focused for a mathematics education leadership audience.

Schoen, H. L., & Ziebarth, S. W. (1998). Assessment of students' mathematical performance: A Core-Plus Mathematics Project field test progress report.Unpublished manuscript, University of Iowa.

This report focuses on standardized achievement test results aggregated across 33 field test schools who were on a regular two-semester schedule. On the Ability to Do Quantitative Thinking, the mathematical subtest of the Iowa Tests of Educational Development, CPMP students in both Course 1 and Course 2 performed better across the distribution than comparison students in more traditional mathematics classes. CPMP students also grew more from the beginning of grade 9 to the end of each of grades 9, 10, and 11 than the nationally representative norm group for this test. At the end of Course 3, CPMP students performed particularly well on NAEP-developed measures of data analysis, probability and statistics and on measures of conceptual understanding. Their performance was somewhat lower in some other content areas and on items assessing procedural outcomes, but still considerably higher than a nationally representative sample of twelfth-grade students.

Schoen, H. L., & Ziebarth, S. W. (1998). High school mathematics curriculum reform: Rationale, research, and recent developments. In P. S. Hlebowitsh & W. G. Wraga (eds.), Annual Review of Research for School Leaders.Pp. 141-191. New York: Macmillan Publishing Company.

The present mathematics curriculum reform effort is examined in light of research on the process of changing curriculum and instruction. Extended discussions of four of the most important and perhaps the most ambitious goals are included: (1) mathematical literacy; (2) core high school mathematics curriculum; (3) technological tools in mathematics instruction; and (4) assessment alternatives. Related research as well as examples and evaluation results from the Core-Plus Mathematics curriculum are included for each goal area.

Schoen, H. L., & Ziebarth, S. W. (1998). Mathematical achievement on standardized tests: A Core-Plus Mathematics Project field test progress report.Unpublished manuscript, University of Iowa.

This report focuses on performance assessments of student achievement aggregated across 33 field test schools who were on a regular two-semester schedule. On project-developed open-ended posttests of mainly algebraic and geometric content, CPMP students in both Course 1 and Course 2 performed significantly better on conceptual, application and problem-solving tasks than comparison students in more traditional mathematics classes. At the end of Course 1, the comparison students performed somewhat better on tasks assessing algebraic procedures, but that difference had disappeared by the end of Course 2. Problem-based interviews were used to gain a better insight into the differences in the mathematical understanding of the CPMP and comparison students.

Tyson, V. (1995). An analysis of the differential performance of girls on standardized multiple-choice mathematics achievement tests compared to constructed response tests of reasoning and problem solving.Unpublished doctoral dissertation, University of Iowa.

This study involved the analysis of gender differences on performance tests administered to Core-Plus Mathematics students. Boys performed significantly better on pre-tests, but no significant gender differences were found on post-tests. Tyson notes that this result is consistent with the goals of the Core-Plus Mathematics curriculum and its compatibility with female learning styles.

Van Zoest, L. R., & Ritsema, B. E. (1998). Fulfilling the call for mathematics education reform. NCSM Journal of Mathematics Education Leadership,1(4), 5-15.

This article addresses the implementation challenges created by adoption of innovative curricula and provides information regarding a professional development effort based on the Core-Plus Mathematics curriculum.

Wilson, M. R., & Lloyd, G. (1995). Sharing mathematical authority.Paper presented at the Seventeenth Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

This article explores the experiences of three mathematics teachers as they implemented a non-traditional high school curriculum (Core-Plus Mathematics Project). During their implementation of the new curriculum, all three veteran teachers struggled with their own and their students' expectations about what constitutes appropriate mathematical activity in the classroom; in particular, re-negotiation of mathematical authority.

Wilson, M. R., & Lloyd, G. M. High school teachers' experiences in a student-centered mathematics curriculum.Paper presented at the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

This paper describes three mathematics teachers and ten of their ninth grade students as they implemented student-centered and exploration-based curriculum materials (Core-Plus Mathematics Project). The observations and interviews were conducted over a six-week period. One teacher claimed that her main challenge involved student ability to make the right connections without her explanations. Another teacher struggled with the dynamics of operating both small-group and whole-class discussions and ultimately decided not to hold whole-class discussions. A third teacher achieved a more equal balance between teacher-directed and student -centered activities. All three teachers demonstrated more difficulty than did students in changing their expectations about appropriate mathematical activity.

Wilson, M. R., & Lloyd, G. M. (1997). Ways of experiencing curriculum: A phenomenographic analysis of two high school teachers' attempts to promote cooperation and exploration in their mathematics classrooms. Manuscript submitted for publication. (Journal for Research in Mathematics Education)

In light of the impressive durability of traditional teacher-centered, procedure-oriented instruction in the mathematics classroom, the question addressed is, "How do experienced teachers interpret and implement innovative curriculum?" This represents an analysis of two teachers' attempts to promote student cooperation and exploratory problem solving.