Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers

The relationship between mental computation and number sense is complex: mental computation can facilitate number sense when students are encouraged to be flexible, but flexibility and number sense is neither sufficient nor necessary for accuracy in mental computation. It is possible for familiarity with a strategy to compensate for a lack of number sense and inefficient processes. This study reports on six case studies exploring Year 3 students’ procedures for and understanding of mental addition and subtraction, and understanding of number sense and other cognitive, metacognitive, and affective factors associated with mental computation. The case studies indicate that the mental computation process is composed of four stages in which cognitive, metacognitive and affective factors operate differently for flexible and inflexible computers. The authors propose a model in which the differences between computer types are seen in terms of the application of different knowledges in number facts, numeration, effect of operation on number, and beliefs and metacognition on strategy choice and strategy implementation.

[1]  N. Bednarz,et al.  The understanding of numeration in primary school , 1982 .

[2]  Effie Maclellan.  Mental Calculation: its place in the development of numeracy , 2001 .

[3]  Robert E. Reys,et al.  Computational Estimation Performance and Strategies Used by Fifth- and Eighth-Grade Japanese Students. , 1991 .

[4]  Alistair McIntosh,et al.  A Proposed Framework for Examining Basic Number Sense. , 1992 .

[5]  Judith T. Sowder.  Mental Computation and Number Sense , 1990 .

[6]  Robert L. Russell,et al.  The Development of Mental Addition as a Function of Schooling and Culture , 1981 .

[7]  Thomas J. Cooper,et al.  Flexibility and inflexibility in accurate mental addition and subtraction: two case studies , 2002 .

[8]  A. W. Blöte,et al.  Mental computation and conceptual understanding , 2000 .

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

[10]  C. Kamii,et al.  Young Children Continue to Reinvent Arithmetic--3rd Grade: Implications of Piaget's Theory , 1994 .

[11]  Aj McIntosh,et al.  Mental computation, number sense and general mathematics ability-are they linked? , 2000 .

[12]  Ann M. Heirdsfield,et al.  Childrens' Mental Strategies for Addition and Subtraction Word Problems , 1996 .

[13]  R. Rubenstein,et al.  Computational Estimation and Related Mathematical Skills. , 1985 .

[14]  Robert E. Reys,et al.  Mental Computation: A Snapshot of Second, Fifth and Seventh Grade Student Performance. , 1993 .

[15]  Gaea Leinhardt,et al.  Analysis of Arithmetic for Mathematics Teaching , 1992 .

[16]  Lorna J. Morrow,et al.  The Teaching and Learning of Algorithms in School Mathematics. 1998 Yearbook. , 1998 .

[17]  Computational Estimation Procedures of School Children. , 1984 .

[18]  Jack A. Hope,et al.  Characteristics of Unskilled and Skilled Mental Calculators. , 1987 .

[19]  Robert E. Reys,et al.  PROCESSES USED BY GOOD COMPUTATIONAL ESTIMATORS , 1982 .

[20]  Van Luit,et al.  Research on Learning and Instruction of Mathematics in Kindergarten and Primary School. , 1994 .

[21]  S. Paris,et al.  How metacognition can promote academic learning and instruction. , 1990 .

[22]  Erik De Corte,et al.  Learning and Instruction: European Research in an International Context: Volume 1 , 1987 .

[23]  R. Connell,et al.  Making The Difference , 1982 .

[24]  Lieven Verschaffel,et al.  Do nonsemantic factors also influence the solution process of addition and subtraction word problems , 1990 .

[25]  J. Bana,et al.  Mathematics Education Beyond 2000 , 2000 .

[26]  Stuart Plunkett,et al.  Decomposition and All That Rot. , 1979 .

[27]  Margariete Montague Wheeler,et al.  The Development of Concepts and Strategies Used in Computational Estimation. , 1989 .

[28]  R. Lesh,et al.  Acquisition of mathematics concepts and processes , 1983 .

[29]  Richard S. Prawat,et al.  Promoting Access to Knowledge, Strategy, and Disposition in Students: A Research Synthesis , 1989 .

[30]  C. Kamii.  Reform in Primary Mathematics Education: A Constructivist View. , 1991 .

[31]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[32]  Jo-Anne LeFevre,et al.  The Development of Procedural and Conceptual Knowledge in Computational Estimation. , 1993 .

[33]  David W. Carraher,et al.  Written and Oral Mathematics. , 1987 .

[34]  Douglas B. McLeod,et al.  Research on affect in mathematics education: A reconceptualization. , 1992 .

[35]  Robert E. Reys,et al.  Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8. , 1995 .

[36]  Meindert Beishuizen,et al.  Mental Strategies and Materials or Models for Addition and Subtraction Up to 100 in Dutch Second Grades. , 1993 .

[37]  R. Glaser.  Advances in Instructional Psychology , 1978 .

[38]  Lauren B. Resnick,et al.  A Developmental Theory of Number Understanding. , 1984 .

[39]  Beau Jones,et al.  Dimensions of thinking and cognitive instruction , 1990 .

[40]  Ann M. Heirdsfield.  Mental computation, computational estimation, and number fact knowledge for addition and substration in year four children , 1996 .

[41]  Judith T. Sowder,et al.  The Development of Computational Estimation: A Neo-Piagetian Analysis , 1990 .

[42]  Anne Buzeika.  INVENTED ALGORITHMS: TEACHERS FACE THE CHALLENGE , 1999 .

[43]  Lorna J. Morrow,et al.  The teaching and learning of algorithms in school mathematics , 1998 .

[44]  James M. Moser,et al.  The Acquisition of Addition and Subtraction Concepts in Grades One through Three. , 1984 .

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

[46]  Judith T. Sowder.  Making Sense of Numbers in School Mathematics , 1992 .

[47]  Unravelling the mysteries of expert mental calculation , 1985 .