Positives About Negatives: A Case Study of an Intermediate Model for Signed Numbers

In this article, we present a study using an intermediate abstraction as a model for the acquisition of the concept of negative numbers. The intermediate abstraction is a computerized environment based on a detailed epistemological analysis of negative numbers. Four children participated in activities with the intermediate abstraction during eleven 30-min training sessions. This article outlines the development of the children's representations of negative numbers during the experiment. We analyzed how students used their representations as problem models in transfer tasks with several different referents. The results obtained in the experiment support the use of certain environments for the acquisition of higher level mathematical concepts that cannot be learned informally.

[1]  Kenneth D. Forbus.  Qualitative Process Theory , 1984, Artif. Intell..

[2]  Mitchell J. Nathan,et al.  A theory of algebra-word-problem comprehension and its implications for the design of learning environments. , 1992 .

[3]  P. Johnson-Laird,et al.  Mental Models: Towards a Cognitive Science of Language, Inference, and Consciousness , 1985 .

[4]  M. Kline.  Mathematical Thought from Ancient to Modern Times , 1972 .

[5]  K. Holyoak,et al.  Interdomain Transfer Between Isomorphic Topics in Algebra and Physics , 1989 .

[6]  J. Gibson.  The Senses Considered As Perceptual Systems , 1967 .

[7]  A. G. Howson,et al.  A Review of Research in Mathematical Education: Part A, Research on Learning and Teaching , 1983 .

[8]  H. Webb.  Addition and Subtraction Facts and Processes , 1949 .

[9]  L. Vygotsky.  Mind in Society: The Development of Higher Psychological Processes: Harvard University Press , 1978 .

[10]  R S Siegler.  Individual differences in strategy choices: good students, not-so-good students, and perfectionists. , 1988, Child development.

[11]  M. Bunge.  Sense and reference , 1974 .

[12]  Stellan Ohlsson.  Sense and Reference in the Design of Interactive Illustrations for Rational Numbers. , 1986 .

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

[14]  Robert B. Davis,et al.  Explorations in mathematics : a text for teachers , 1967 .

[15]  Tamar Katriel,et al.  A semantic analysis of addition and subtraction word problems in arithmetic , 1977 .

[16]  Thomas A. Romberg,et al.  Addition and subtraction : a cognitive perspective , 2020 .

[17]  William Berkson.  The Formal and the Informal , 1978, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[18]  W. Kintsch.  Learning From Text , 1986, Knowing, Learning, and Instruction.

[19]  James G. Greeno,et al.  Number sense as situated knowing in a conceptual domain , 1990 .

[20]  Patrick W Thompson,et al.  Integers as transformations , 1988 .

[21]  Lauren B. Resnick,et al.  From Protoquantities to Operators: building mathematical competence on a Foundation of Everyday Knowledge , 1991 .

[22]  Stephanie D. Teasley,et al.  The Construction of Shared Knowledge in Collaborative Problem Solving , 1995 .

[23]  John H. Holland,et al.  Induction: Processes of Inference, Learning, and Discovery , 1987, IEEE Expert.

[24]  M. Cole,et al.  The psychology of literacy , 1983 .

[25]  Frank E. X. Dance,et al.  What Do You Mean , 1987 .

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

[27]  Zoltan Dienes,et al.  An Experimental Study of Mathematics Learning , 1964 .

[28]  L. S. Vygotksy.  Mind in society: the development of higher psychological processes , 1978 .