Children's Mathematics Learning: The Struggle to Link Form and Understanding

Most children enter school with reasonably good problem-solving strategies (Carpenter, Moser, & Romberg 1982). A significant feature of these strategies is that they reflect a careful analysis of the problems to which they are applied. However, after several years many children abandon their analytic approach and solve problems by selecting a memorized algorithm based on a relatively superficial reading of the problem (National Assessment of Educational Progress [NAEP] 1983). Why is it that many children begin school with simple but sound problem-solving procedures, and then, as they receive formal mathematics instruction, replace them with shallow and sometimes meaningless procedures? The question is fundamental. The answer is not entirely clear, but re-

[1]  Robert M. Gagné,et al.  Some Issues in the Psychology of Mathematics Instruction. , 1983 .

[2]  James G. Greeno,et al.  The development of semantic categories for addition and subtraction , 1982 .

[3]  Herbert P. Ginsburg,et al.  The development of mathematical thinking , 1983 .

[4]  H. Engen,et al.  An Analysis of Meaning in Arithmetic. I , 1949 .

[5]  Margariete Montague Wheeler,et al.  The Calculator for Concept Formation: A Clinical Status Study. , 1981 .

[6]  John Seely Brown,et al.  Diagnostic Models for Procedural Bugs in Basic Mathematical Skills , 1978, Cogn. Sci..

[7]  Max S. Bell,et al.  Algebraic and arithmetic structures: A concrete approach for elementary school teachers , 1976 .

[8]  John Richards,et al.  The Acquisition and Elaboration of the Number Word Sequence , 1982 .

[9]  Klaus Hasemann.  On difficulties with fractions , 1981 .

[10]  James Hiebert,et al.  The effect of instruction on children's solutions of addition and subtraction word problems , 1983 .

[11]  Carolyn Kieran.  Concepts associated with the equality symbol , 1981 .

[12]  James Hiebert,et al.  The Position of the Unknown Set and Children's Solutions of Verbal Arithmetic Problems. , 1982 .

[13]  Lieven Verschaffel,et al.  Children's solution processes in elementary arithmetic problems: Analysis and improvement. , 1981 .

[14]  Developmental models of thinking , 1980 .

[15]  Margaret C. Donaldson.  Children's Minds , 1978 .

[16]  Douglas A. Grouws.  Using Research in Teaching: Open Sentences: Some Instructional Considerations from Research. , 1972 .

[17]  M. Behr.  How Children View the Equals Sign. , 1980 .

[18]  C. Gallistel,et al.  The Child's Understanding of Number , 1979 .

[19]  L. Brush,et al.  Preschool Children's Knowledge of Addition and Subtraction. , 1978 .

[20]  Robert E. Reys,et al.  Developing Computational Skills: 1978 Yearbook. , 1978 .

[21]  Elizabeth Fennema,et al.  Models and Mathematics. , 1972 .

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

[23]  Erik De Corte,et al.  Estimating the outcome of a task as a heuristic strategy in arithmetic problem-solving - a teaching experiment with 6th-graders , 1982 .

[24]  Guy J. Groen,et al.  Can Preschool Children Invent Addition Algorithms , 1977 .

[25]  Jean Piaget,et al.  The child's conception of geometry , 1962 .

[26]  R. Skemp.  The psychology of learning mathematics , 1979 .

[27]  James Hiebert,et al.  Development of the Fraction Concept in Two Physical Contexts: An Exploratory Investigation. , 1978 .

[28]  Leslie P. Steffe,et al.  Children’s Counting in Arithmetical Problem Solving , 1982 .

[29]  Arthur J. Baroody,et al.  The Effects of Instruction on Children's Understanding of the "Equals" Sign , 1983, The Elementary School Journal.

[30]  Daiyo Sawada,et al.  PARTITIONING: THE EMERGENCE OF RATIONAL NUMBER IDEAS IN YOUNG CHILDREN , 1983 .

[31]  C. Lindvall,et al.  Factors Associated with the Ability of Kindergarten Children to Solve Simple Arithmetic Story Problems , 1982 .

[32]  Edwin W. Schreiber.  The National Council of Teachers of Mathematics , 1936 .

[33]  J. Hiebert,et al.  Students' Conceptions of Decimal Numbers. , 1983 .

[34]  Malcolm Swan,et al.  Choice of operation in verbal problems with decimal numbers , 1981 .

[35]  John Seely Brown,et al.  Towards a Generative Theory of “Bugs” , 2020, Addition and Subtraction.

[36]  Thomas R. Post,et al.  Fractions: Results and implications from national assessment , 1981 .

[37]  Thomas P. Carpenter,et al.  Problem Structure and First-Grade Children's Initial Solution Processes for Simple Addition and Subtraction Problems , 1981 .

[38]  C. M. Lindvall,et al.  Information Processing Capabilities Used by Kindergarten Children When Solving Simple Arithmetic Story Problems. , 1981 .

[39]  Mary S. Riley,et al.  Development of Children's Problem-Solving Ability in Arithmetic. , 1984 .

[40]  Adolf af Ekenstam.  On children's quantitative understanding of numbers , 1977 .

[41]  Charles J. Brainerd,et al.  Children’s Logical and Mathematical Cognition , 1982 .

[42]  Anne S. Grossman.  Decimal Notation: An Important Research Finding. , 1983 .

[43]  Richard Lesh,et al.  Rational number concepts , 1983 .

[44]  C. Mauritz Lindvall,et al.  An Analysis of Incorrect Procedures Used by Primary Grade Pupils in Solving Open Addition and Subtraction Sentences. , 1978 .

[45]  Edward C. Rathmell,et al.  Using Thinking Strategies to Teach the Basic Facts. , 1978 .

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

[47]  James W. Hall,et al.  THE TRANSITION FROM COUNTING-ALL TO COUNTING-ON IN ADDITION , 1983 .