The need for theories of conceptual learning and teaching of mathematics

One of the important challenges in mathematics education is the ­development of pedagogical theory that can guide mathematics instruction, instructional design, teacher education, and research in each of these areas. In this chapter, I begin with a brief review of the current state of the field with respect to ­pedagogical theory. I then offer a vision of what might constitute needed pedagogical theory. Such theory would include useful ways of describing the mechanisms of ­mathematics conceptual learning and, derived from the characterization of these mechanisms, design and instructional principles for fostering mathematics conceptual learning. Using the research program I am involved in as an example, I describe some of the methodological challenges in producing an empirical basis for such theory development.

[1]  H. Koirala,et al.  Development of a performance assessment task and rubric to measure prospective secondary school mathematics teachers’ pedagogical content knowledge and skills , 2008 .

[2]  Jere Confrey,et al.  The Development of multiplicative reasoning in the learning of mathematics , 1995 .

[3]  Terezinha Nunes,et al.  Learning and teaching mathematics : an international perspective , 1997 .

[4]  Jesse L. M. Wilkins,et al.  The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices , 2008 .

[5]  Paola Sztajn,et al.  Expanding the instructional triangle: conceptualizing mathematics teacher development , 2008 .

[6]  V. V. Davydov,et al.  The concept of theoretical generalization and problems of educational psychology , 1988 .

[7]  Paul Cobb,et al.  Constructivism in Mathematics and Science Education , 1994 .

[8]  J. Wertsch.  The Concept of Activity in Soviet Psychology , 1981 .

[9]  Jean Piaget,et al.  Studies in Reflecting Abstraction , 2001 .

[10]  Patrick W Thompson,et al.  The development of the concept of speed and its relationship to concepts of rate , 1994 .

[11]  Evan McClintock,et al.  A Developing Approach to Studying Students’ Learning through Their Mathematical Activity , 2010 .

[12]  R. Siegler,et al.  How Does Change Occur: A Microgenetic Study of Number Conservation , 1995, Cognitive Psychology.

[13]  Jeannie Oakes,et al.  Chapter 3: Opportunities, Achievement, and Choice: Women and Minority Students in Science and Mathematics , 1990 .

[14]  C. Hirsch.  Perspectives on the Design and Development of School Mathematics Curricula , 2007 .

[15]  Paul Cobb,et al.  Interaction and learning in mathematics classroom situations , 1992 .

[16]  R. Siegler.  Emerging Minds: The Process of Change in Children's Thinking , 1996 .

[17]  Despina Potari,et al.  A primary teacher’s mathematics teaching: the development of beliefs and practice in different “supportive” contexts , 2009 .

[18]  M. Windschitl.  Framing Constructivism in Practice as the Negotiation of Dilemmas: An Analysis of the Conceptual, Pedagogical, Cultural, and Political Challenges Facing Teachers , 2002 .

[19]  H. Freudenthal.  Mathematics as an Educational Task , 1972 .

[20]  D. Ball,et al.  Policy and Practice: An Overview , 1990 .

[21]  Lynn Arthur Steen,et al.  On the shoulders of giants: new approaches to numeracy , 1990 .

[22]  Charalambos Y. Charalambous,et al.  Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study , 2008 .

[23]  Jo Towers,et al.  Learning to teach mathematics through inquiry: a focus on the relationship between describing and enacting inquiry-oriented teaching , 2010 .

[24]  Margaret Kinzel,et al.  Explicating a Mechanism for Conceptual Learning: Elaborating the Construct of Reflective Abstraction , 2004 .

[25]  A. N. Leont’ev.  The Problem of Activity in Psychology , 1974 .

[26]  An Exchange: Constructivism in Mathematics and Science Education , 1994 .

[27]  Doug Clarke,et al.  A designer speaks , 2011 .

[28]  Ron Tzur,et al.  Fine grain assessment of students’ mathematical understanding: participatory and anticipatory stagesin learning a new mathematical conception , 2007 .

[29]  Martin A. Simon,et al.  Explicating the Role of Mathematical Tasks in Conceptual Learning: An Elaboration of the Hypothetical Learning Trajectory , 2004 .

[30]  Juliet A. Baxter,et al.  Social and analytic scaffolding in middle school mathematics: managing the dilemma of telling , 2010 .

[31]  J. Confrey,et al.  Splitting, covariation, and their role in the development of exponential functions , 1995 .

[32]  K.P.E. Gravemeijer.  Mediating between concrete and abstract , 1997 .

[33]  H. Freudenthal.  Didactical Phenomenology of Mathematical Structures , 1983 .

[34]  Martin A. Simon.  Reconstructing Mathematics Pedagogy from a Constructivist Perspective. , 1995 .

[35]  Colette Laborde,et al.  Design Tools in Didactical Research: Instrumenting the Epistemological and Cognitive Aspects of the Design of Teaching Sequences , 2009 .

[36]  G. Brousseau,et al.  Rationals and decimals as required in the school curriculum: Part 3. Rationals and decimals as linear functions , 2008 .

[37]  Martin A. Simon,et al.  Characterizing a Perspective Underlying the Practice of Mathematics Teachers , 2000 .

[38]  Diana Lambdin Kroll,et al.  An Overview of Research on Cooperative Learning Related to Mathematics. , 1991 .

[39]  Edward A. Silver,et al.  The Quasar Project , 1996 .

[40]  A. Treffers.  Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction ― The Wiskobas Project , 1986 .

[41]  G. Brousseau.  Theory of didactical situations in mathematics , 1997 .

[42]  T. P. Carpenter,et al.  Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics , 1996 .

[43]  D. Ball,et al.  Relations Between Policy and Practice: A Commentary , 1990 .

[44]  C. Hirsch,et al.  The Case of Core-Plus Mathematics , 2007 .

[45]  Leslie P. Steffe,et al.  Fractional commensurate, composition, and adding schemes , 2003 .