Influence of Theory amd Research
Research From the Cognitive Sciences
Research From Mathematics Education
Research From Education Policy and Organization
Influence of Theory and Research on CMP2 Curriculum
The curriculum, teacher support, and assessment materials that comprise the Connected Mathematics program reflect influence from a variety of sources:
- knowledge of theory and research;
- authors' imaginations and personal teaching and learning experiences;
- advice from teachers, mathematicians, teacher educators, curriculum developers, and mathematics education researchers;
- advice from teachers and students who used pilot and field-test versions of the materials.
The fundamental features of the CMP program-focus on big ideas of middle grades mathematics, teaching through student-centered exploration of mathematically rich problems, and continual assessment to inform instruction- reflect the distillation of advice and experience from these varied sources.
Our work was influenced in significant ways by what we knew of existing theory and research in mathematics education. Here we mention and explain briefly the key themes in the theory/research basis for our work.
Research From the Cognitive Sciences
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Social Constructivism
We are in general agreement with constructivist explanations of the ways that knowledge is developed, especially the social constructivist ideas about influence of discourse on learning. This position is reflected in the authors' decision to write materials that support student- centered investigation of mathematical problems and in our attempt to design problem content and formats that encourage student-student and student- teacher dialogue about the work. -
Conceptual and Procedural Knowledge
We have been influenced by theory and research indicating that mathematical understanding is fundamentally a web of logical and psychological connections among ideas. Furthermore, we have interpreted research on the interplay of conceptual and procedural knowledge to say that sound conceptual understanding is an important foundation for procedural skill, not an incidental and delayed consequence of repeated rote procedural practice. -
Multiple Representations
An important indication of students' connected mathematical knowledge is their ability to represent ideas in a variety of ways. We have interpreted this theory to imply that curriculum materials should frequently provide and ask for knowledge representation using graphs, number patterns, written explanations, and symbolic expressions. -
Cooperative Learning
There is a consistent and growing body of research indicating that when students engage in cooperative work on appropriate problem-solving tasks, their mathematical and social learning will be enhanced. We have interpreted this line of theory and research to imply that we should design student and teacher materials that are suitable for use in cooperative learning instructional formats as well as individual learning formats-the mathematical tasks dictate the format.
Research From Mathematics Education
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Rational Numbers/Proportional Reasoning
The extensive psychological literature on the development of rational numbers and proportional reasoning has guided our development of curriculum materials to address this important middle school topic. Furthermore, the implementation of CMP materials in real classrooms has allowed us to contribute to that literature with research publications that show the effects of new teaching approaches to traditionally difficult topics. -
Probability and Statistical Reasoning
The interesting research literature related to the development of and the cognitive obstacles to student learning of statistical concepts, such as mean and graphic displays, and probability concepts, such as the law of large numbers, has been used as we developed the statistics and probability units for CMP. -
Algebraic Reasoning
The different conceptualizations of algebra described and researched in the literature contributed to the treatment of algebra in CMP. Various scholars describe algebra as a study of modeling, functions, generalized arithmetic, and/or as a problem-solving tool. CMP has aspects of each of these descriptions of algebra, but focuses more directly on functions and on the effects of rates of change on representations. The research literature illuminates some of the cognitive complexities inherent in algebraic reasoning and offers suggestions for helping students overcome difficulties. Research concerning concepts, such as equivalence, functions, the equal sign, algebraic variables, graphical representations, multiple representations, and the role of technology, were used to develop the algebra units in CMP. -
Geometric/Measurement Reasoning
Results from national assessments and research findings show that student achievement in geometry and measurement is weak. Research about student understanding of shape and form, and learning of geometric/measurement concepts, such as angle, area, perimeter, volume, and processes such as visualization, contributed to the development of geometry/measurement units in CMP materials.
Research From Education Policy and Organization
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Motivation
One of the fundamental challenges in mathematics teaching is convincing students that serious effort in study of the subject will be rewarding and that learning of mathematics can also be an enjoyable experience. We have paid careful attention to literature on extrinsic and intrinsic motivation, and we have done some informal developmental research of our own to explore aspects of mathematics and teaching that are most effective for engaging student attention and interest. -
Teacher and School Change
The most attractive school mathematics curriculum materials will be of little long-term value or effect if they are not put into use in schools. In the process of helping teachers through professional development, we have paid close attention to what is known about effective teacher professional development and the school strategies that seem to be most effective.
A good reference book to read for more insight into what research says in these areas is Kilpatrick, J., Martin, W.G., & Schifter, D. (Eds.) (2003) A Research Companion to Principles and Standards for School Mathematics, Reston, VA:NCTM. (ISBN 0-87353-537-5).
While each of these ten points indicates influence of theory and research on design and development of the CMP curriculum, teacher, and assessment materials, it would be misleading to suggest that the influence is direct and controlling in all decisions. As the authors have read the research literature reporting empirical and theoretical work, research findings and new ideas have been absorbed and factored into the creative, deliberative, and experimental process that leads to a comprehensive mathematics program for schools.
