# Research

CMP1 was guided by our experiences and vision of a curriculum that would engage both teachers and students around significant mathematics. Through more than twenty five years our personal knowledge about mathematics, how it is learned, how it is taught has been richly enhanced by CMP teachers and students.

We have gained valuable insights from mathematicians, teacher educators, and other curriculum developers. Our work has also been influenced in significant ways by insights from theoretical and empirical research in mathematics education, cognitive science, educational research about teaching, teacher development, and education policy implementation.

From the broad array of pertinent research findings, seven major themes continue to influence development and implementation of the CMP curriculum and teaching resources. These themes are explored below.

### Cooperative Learning and Classroom Discourse

We are in general agreement with constructivist explanations of the ways that knowledge is developed, especially the social constructivist ideas about the influence of discourse on learning.[1] There is a consistent and growing body of research indicating that when students engage in cooperative work on challenging problem solving tasks, their mathematical and social learning will be enhanced.[2] Recent research has also shown that the discourse of a classroom—the ways of representing, thinking, talking, agreeing and disagreeing—is central to what students learn about mathematics.[3] Effective teachers ask open-ended questions to elicit student thinking and ask students to explain their thinking and comment on one another's work.

This complex set of related research findings is reflected in our decision to write materials that support student-centered investigation of mathematical problems. It also informed our efforts to design problem content and formats that encourage student-student and student-teacher dialogue about the work.

### Teaching Through Problem Solving

Mathematics education has always sought to develop student problem solving skills. But teaching mathematics through problem solving presents different challenges for both teachers and students.

Over the past several decades, researchers have engaged in extensive classroom studies to assess the effectiveness of problem-based learning[4] and the practices that make it effective. A summary of that research by Stein, Boaler, and Silver[5] identified five critical teacher actions: “(a) (appropriate) scaffolding of students’ thinking; (b) a sustained press for students’ explanations; (c) thoughtful probing of students’ strategies and solutions; (d) helping students accept responsibility for, and gain facility with, learning in a more open way; and (e) attending to issues of equity in the classroom.” The CMP instructional materials and teacher resources have been designed with the explicit aim of engaging and sustaining diverse groups of students in high-level thinking.

### Equity and Motivation for Learning

In American schools, one of the greatest challenges for teachers is to provide learning tasks and support that engage and sustain the interest and effort of students with very diverse background experiences, values, interests, and abilities. This challenge is especially acute for teachers who want their students to take principal responsibility for their own learning and to do so in collaboration with other students.

In designing the CMP instructional materials, 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. We have also attended to research that explores the response of diverse students to teaching through problem solving.[6]

### 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. Knowledge that is rich in connections will be retained well and retrieved effectively for subsequent problem solving and reasoning tasks.[7] 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.

These findings are reflected in the CMP instructional materials that are rich in connections between topics in the four major content strands of the curriculum— number and operations, geometry and measurement, data analysis and probability, and algebra—and in our careful attention to laying conceptual foundations for learning of mathematical procedures.

### Formative Assessment

Extensive recent research has demonstrated convincingly that student learning improves significantly when teachers provide frequent feedback on their progress and when teachers use that assessment as a core input to their planning for instruction, teaching and reflecting.[8] The CMP instructional resources provide a variety of classroom-tested tools for such helpful formative assessment.

### Mathematical Knowledge for Teaching

Research over the past several decades has shown quite convincingly that effective mathematics teachers have a special kind of understanding of the subjects they teach—knowledge that includes ways of representing concepts and procedures in different forms and ways that students are likely to have difficulty in learning or to form misconceptions. This special kind of understanding of mathematics is often referred to as mathematical knowledge for teaching.[9] Some of the keys to this valuable teacher knowledge are generic-applying across all content strands of the subject—while others are particularly salient in specific topics.

In design and development of the CMP instructional materials, we have paid close attention to both general and specific kinds of mathematical knowledge for teaching. We have included tasks for students that will help them succeed when confronted by common obstacles to learning. We have included advice to teachers about those likely stumbling blocks and strategies for addressing or avoiding them. In particular, the research findings in the following content areas of mathematical reasoning have been incorporated into the student and teacher materials.

#### 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 instructional materials should frequently provide and ask for knowledge representation using graphs, number patterns, written explanations, and symbolic expressions. Students also should be challenged to interpret information provided in one representational style through use of another representation.

#### 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 CMP3.

#### Algebraic Reasoning

The different conceptualizations of algebra described and studied in the research literature contributed to the treatment of algebra in CMP3. Various scholars describe algebra as a study of modeling, functions, generalized arithmetic, and/or as a problem-solving tool. CMP3 develops each of these foci for algebra, but attends more directly to functions and 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. We have drawn on that research in the design of our approach to concepts of equivalence, functions, the equal sign, algebraic variables, the use of graphs and other representations, and the role of technology.

#### Geometric/Measurement Reasoning

Results from national assessments show that achievement in geometry and measurement is weak among many American students. The theoretical ideas of the van Hieles and other specific studies 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 CMP3 materials.

### Teacher Development and School Change

Education is a social institution with traditional practices that are deeply resistant to change. Thus, implementation of strikingly different curricular and instructional practices is a challenge for teachers and students and for the broader school community.

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.[10] The sections of this guide reflect our reading of the research on teacher and school change and our experience over 20 years of work with teachers and schools introducing the CMP curriculum.

### Conclusion

While each of the seven themes described above 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.

A good reference book to consult 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: National Council of Teachers of Mathematics.

Since its publication in 1996, a substantial number of research studies have been carried out in CMP classrooms. These studies have helped the field understand the conditions under which a problem-centered curriculum promotes students’ and teachers’ conceptual and procedural skills as well as their problem-solving and reasoning abilities.

### References

*[1] Yackel, E. and Cobb, P. (1996) Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. Journal for Research in Mathematics Education (27), 458–477.*

*[2] Cohen, E. G. (1994). Restructuring the Classroom: Conditions for Productive Small Groups. Review of Educational Research, 64(1), pp. 1–35.*

*[3] Smith, M. S. and Stein, M. K. (2011) 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics.*

*[4] Wirkala, C., & Kuhn, D. (2011). Problem-based Learning in K-12 Education: Is it Effective and How Does it Achieve its Effects? American Educational Research Journal, 48(5), pp. 1157–1186.*

*[5] Stein, M. K., Boaler, J., and Silver, E. A. (2003) Teaching Mathematics Through Problem Solving: Research Perspectives (p. 253). In H. L. Schoen and R. I. Charles (Eds.) Teaching Mathematics through Problem Solving Grades 6–12. Reston, VA: National Council of Teachers of Mathematics. pp. 245–256.*

*[6] Lubienski, S. T. and Stilwell, J. (2003). Teaching Low-SES Students Mathematics Through Problem Solving: Tough Issues, Promising Strategies, and Lingering Dilemmas. In H. L. Schoen and R. I. Charles, op cit.*

*[7] Hiebert, J. (Ed.) Conceptual and Procedural Knowledge: The Case of Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates (1986).*

*[8] Wiliam, D. (2007) Keeping learning on track: classroom assessment and the regulation of learning. In F. K. Lester Jr (Ed.), Second handbook of mathematics teaching and learning, Greenwich, CT: Information Age Publishing.*

*[9] Hill, H.C., Rowan, B., & Ball, D.L. (2005) Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal 42, 371–406.*

*[10] Borko, H. (2004) Professional Development and Teacher Learning: Mapping the Terrain. Educational Researcher (33) 3, pp. 3–15.*