Distinguishing Two Stages of Mathematics Conceptual Learning

In this theoretical article, we distinguish two stages of learning a new mathematical concept – participatory and anticipatory. We use a recently developed mechanism for explaining mathematical conceptual learning – reflection on activity-effect relationship – as well as von Glasersfeld’s tripartite model of a scheme, to explain qualitative distinctions between the two stages. We use this distinction to explain why instructional interventions (including inquiry-based approaches) may not bring about the intended instructional goals.

[1]  J. Dewey.  How we think : a restatement of the relation of reflective thinking to the educative process , 1934 .

[2]  Leslie P. Steffe,et al.  A new hypothesis concerning children’s fractional knowledge ☆ , 2001 .

[3]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

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

[5]  Ed Dubinsky,et al.  Reflective Abstraction in Advanced Mathematical Thinking , 2002 .

[6]  B. Schwarz,et al.  Abstraction in Context: Epistemic Actions , 2001 .

[7]  Paul Cobb,et al.  Modeling, Symbolizing, and Tool Use in Statistical Data Analysis , 2002 .

[8]  Juan Pascual-Leone,et al.  C. A view of cognition from a formalist's perspective , 1976 .

[9]  A. Sfard,et al.  The gains and the pitfalls of reification — The case of algebra , 1994 .

[10]  Leen Streefland,et al.  Fractions in Realistic Mathematics Education , 1991 .

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

[12]  Ron Tzur.  An Integrated Study of Children's Construction of Improper Fractions and the Teacher's Role in Promoting That Learning , 1999 .

[13]  Ron Tzur,et al.  An Integrated Research on Children's Construction of Meaningful, Symbolic, Partitioning-related Conceptions and the Teacher's Role in Fostering That Learning , 1999 .

[14]  J. Piaget.  Adaptation and Intelligence: Organic Selection and Phenocopy , 1980 .

[15]  R. Tzur.  Teacher and Students' Joint Production of a Reversible Fraction Conception. , 2003 .

[16]  Jacques Bergeron,et al.  Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (5th, Montreal, Quebec, Canada, September 29-October 1, 1983). Volumes 1 and 2. , 1983 .

[17]  Richard Lesh,et al.  ORDER AND EQUIVALENCE OF RATIONAL NUMBERS: A CLINICAL TEACHING EXPERIMENT , 1984 .

[18]  Patrick W Thompson,et al.  Didactic Objects and Didactic Models in Radical Constructivism , 2002 .

[19]  Koeno Gravemeijer,et al.  Emergent Models as an Instructional Design Heuristic , 2002 .

[20]  John Olive,et al.  From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis. , 1999 .

[21]  Ed Dubinsky,et al.  Development of the process conception of function , 1992 .

[22]  J. Piaget,et al.  The equilibration of cognitive structures : the central problem of intellectual development , 1985 .

[23]  Ernst von Glasersfeld.  Abstraction, Re-Presentation, and Reflection, an Interpretation of Experience and Piaget's Approach. , 1991 .

[24]  Anna Sfard,et al.  The development of algebra: Confronting historical and psychological perspectives , 1995 .

[25]  C. Bereiter.  Toward a Solution of the Learning Paradox , 1985 .

[26]  E. Glasersfeld.  Radical Constructivism: A Way of Knowing and Learning. Studies in Mathematics Education Series: 6. , 1995 .

[27]  Susan E. B. Pirie,et al.  Creating constructivist environments and constructing creative mathematics , 1992 .

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

[29]  J. Piaget,et al.  Child's Conception Of Geometry , 1960 .

[30]  Understanding for Teaching for Understanding. , 1990 .

[31]  K.P.E. Gravemeijer,et al.  Developing realistic mathematics education , 1994 .

[32]  Leen Streefland,et al.  Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research , 1991 .