Abstract
Arithmetical knowledge is viewed as simply the coordinated schemes of actions and operations the child has constructed at a particular point in time. A fundamental distinction is drawn between operative and figurative schemes in the context of additive schemes. Numerical extension, an operative scheme, is distinguished from intuitive extension, a figurative scheme. The basis of the distinction resides in (1) the anticipatory nature of the schemes, (2) the adaptability of the schemes, and (3) the nature of the concepts which serve as a basis for what triggers the schemes. This fundamental distinction is crucial in the mathematical instruction of the child for the child necessarily has to construct his or her own mathematical reality even if that reality eventually becomes compatible with the reality of the social group in which the child has to operate.
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References
BoothL. R.: 1981, ‘Child methods in secondary school mathematics’, Educational Studies in Mathematics 12, 29–41.
Brownell, W. A.: 1935, ‘Psychological consideration in the learning and teaching of arithmetic’, in W. D. Reeve (ed.), Teaching of Arithmetic. The Tenth Yearbook of the National Council of Teachers of Mathematics, Bureau of Publications, Teachers College, Columbia University.
CaramuelJ.: 1680/1977, ‘Meditatio prooemialis, Mathesis biceps’, in A.Parea, P.Soriano, and P.Terzi (eds.), L'arithmetica binaria e le altre aritmetiche di Giovanni Caramuel, Vescovo di Vigevano, Vigevano, Italy: Academia Tiberina.
Easley, J. A., Jr.: 1975, ‘Thoughts on individualized instruction’, in H. Bauersfeld, M. Otte and H. C. Steiner (eds.), Schriftenreihe des IDM, Institut fur Didaktic der Mathematik. Universitat Bielefeld.
ErlwangerS. H.: 1973, ‘Benny's concept of rules and answers in IPI mathematics’, Journal of Children's Mathematical Behavior 5, 34–41.
FreudenthalH.: 1981, ‘Major problems of mathematics education’, Educational Studies in Mathematics 12, 133–150.
GelmanR. and GallistelC. R.: 1978, ‘The child's understanding of number’, Cambridge, Massachusetts: Harvard University Press.
Hatfield, L.: 1976, ‘Child-generated computational algorithms: An exploratory investigation of cognitive development and mathematics instruction with Grade II Children’, unpublished manuscript.
InhelderB. and PiagetJ.: 1969, The Early Growth of Logic in the Child: Classification and Seriation, New York: The Norton Library, W. W. Norton.
JamesG. and JamesR. C.: 1968, Mathematics Dictionary, New Jersey: Van Nostrand.
MassimoPiatelli-Palmarini.: 1980, Language and learning: The debate between Jean Piaget and Noam Chomsky, Cambridge: Harvard University Press.
McKnightC. and DavisR. B.: 1980, ‘The influence of semantic content on algorithmic behavior’, The Journal of Mathematical Behavior 3, 39–79.
OsborneA. R.: 1976, ‘Conditions for algorithmic imagination’, in M.Suydam and A. R.Osborne (eds.), Algorithmic Learning, ERIC/SMEAC Columbus, Ohio.
PiagetJ.: 1970, Genetic Epistemology, New York: Columbia University Press.
Piaget, J.: 1980, ‘The psychogenesis of knowledge and its epistemological significance’, in Massimo Piatelli-Palmarini (1980).
ResnickL. B.: 1982, ‘Syntax and semantics in learning to subtract’, in T.Carpenter, J.Moser, and T.Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Hilsdale, New Jersey: Lawrence Erlbaum Associates.
RiedesalC. A.: 1973, Guiding Discovery in Elementary School Mathematics, New York: Appleton Century-Crofts.
SematH.: 1959, Introduction to Atomic and Nuclear Physics, New York: Rinehart.
SkempR.: 1978, ‘Relational understanding and instrumental understanding’, Arithmetic Teacher 26, 9–15.
SteffeL. P., ThompsonP., and RichardsJ.: 1982, ‘Children's counting in arithmetic problem solving’, in T.Carpenter, J.Moser, and T.Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Hilsdale, New Jersey: Lawrence Erlbaum Associates.
SteffeL. P., vonGlaserfeldE., RichardsJ., and CobbP.: 1983, ‘Children's counting types: philosophy, theory, and application, New York: Praeger.
Thompson, P.: 1982, ‘A theoretical framework for understanding young children's concepts of whole number numeration’, Doctoral Dissertation, Department of Mathematics Education, University of Georgia.
vonGlasersfeldE.: 1980, ‘The concept of equilibration in a constructivist theory of knowledge’, in F.Benseler, P. M.Hejl, and W. K.Kock (eds.), Autopoiesis, Communication, and Society, New York: Campus.
vonGlasersfeldE.: 1981, ‘An attentional model for the conceptual construction of units and number’, Journal for Research in Mathematics Education 12, 83–94.
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The comments and criticisms of Paul Cobb, Ernst von Glaserfeld, Lauren Resnick, and an anonymous reviewer on an earlier version of this paper are gratefully acknowledged.
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Steffe, L.P. Children's algorithms as schemes. Educ Stud Math 14, 109–125 (1983). https://doi.org/10.1007/BF00303681
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DOI: https://doi.org/10.1007/BF00303681