A new hypothesis concerning children’s fractional knowledge ☆

The basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because if a new scheme is constructed by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. There are two basic ways of understanding the reorganization of a prior scheme. The first is that the child constructs the new scheme by operating on the preceding scheme using operations that can be, but may not be, a part of the operations of that scheme. In this case, the new scheme is of the same type as the preceding scheme. But it solves problems and serves purposes that the preceding scheme did not solve or did not serve. It also solves all of the problems the preceding scheme solved, but it solves them better. It is in this sense that the new scheme supersedes the preceding, more primitive, scheme.

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

[2]  Thomas A. Romberg,et al.  Rational numbers : an integration of research , 1993 .

[3]  Ernst von Glasersfeld,et al.  An Attentional Model for the Conceptual Construction of Units and Number. , 1981 .

[4]  Leslie P. Steffe,et al.  Theories of Mathematical Learning , 1996 .

[5]  John Olive,et al.  Computer Tools for Interactive Mathematical Activity in the Elementary School , 2000, Int. J. Comput. Math. Learn..

[6]  Thomas E. Kieren,et al.  Recent Research on Number Learning , 1982 .

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

[8]  Charles Hubbard Judd.  The psychology of number. , 1926 .

[9]  Richard Lesh,et al.  Curriculum implications of research on the learning, teaching and assessing of rational number concepts: An integration of research , 1993 .

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

[11]  Herbert P. Ginsburg,et al.  Children's arithmetic: The learning process , 1977 .

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

[13]  L. Brouwer,et al.  Intuitionism and formalism , 1913 .

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

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

[16]  Barbara J. Reys.  Developing Number Sense. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. , 1991 .

[17]  Leslie P. Steffe,et al.  Cognitive Play and Mathematical Learning in Computer Microworlds. , 1994 .

[18]  I. Lakatos,et al.  Criticism and the Growth of Knowledge: Falsification and the Methodology of Scientific Research Programmes , 1970 .

[19]  D. DeTemple,et al.  Mathematical Reasoning for Elementary Teachers , 1995 .

[20]  N. Pa,et al.  Children's fractional schemes , 1987 .

[21]  Susan J. Lamon,et al.  The Development of Unitizing: Its Role in Children's Partitioning Strategies. , 1996 .

[22]  Leslie P. Steffe,et al.  Schemes of action and operation involving composite units , 1992 .

[23]  I. Lakatos.  Falsification and the Methodology of Scientific Research Programmes , 1976 .

[24]  Barry D. Biddlecomb.  Theory-Based Development of Computer Microworlds , 1994 .

[25]  Barbara Reys,et al.  Developing number sense in the middle grades , 1991 .

[26]  Noam Chomsky,et al.  Opening the debate , 1980 .

[27]  Robert P. Hunting.  ALAN: A CASE STUDY OF KNOWLEDGE OF UNITS AND PERFORMANCE WITH FRACTIONS , 1983 .

[28]  T. Broadbent,et al.  Criticism and the Growth of Knowledge , 1972 .

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

[30]  A. Sáenz-Ludlow,et al.  Michael's Fraction Schemes. , 1994 .

[31]  Leslie P. Steffe,et al.  Epistemological foundations of mathematical experience , 1991 .

[32]  Leslie P. Steffe,et al.  Teaching experiment methodology: Underlying principles and essential elements , 2000 .

[33]  F. Curcio,et al.  Understanding rational numbers and proportions , 1994 .

[34]  Lesley R. Booth.  Child-methods in secondary mathematics , 1981 .

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

[36]  Massimo Piattelli-Palmarini,et al.  Language and Learning: The Debate Between Jean Piaget and Noam Chomsky , 1980 .

[37]  J. Dewey,et al.  The Psychology of Number and Its Applications to Methods of Teaching Arithmetic , 2007 .

[38]  Paul Cobb,et al.  Construction of Arithmetical Meanings and Strategies , 1987 .

[39]  Merlyn J. Behr,et al.  Number concepts and operations in the middle grades , 1988 .

[40]  Dina Tirosh,et al.  Implicit and explicit knowledge : an educational approach , 1994 .