La transferencia de aprendizaje algorítmico y el origen de los errores en la sustración

  1. Sánchez García, Ana Belén
  2. López Fernández, Ricardo
Zeitschrift:
Revista de educación

ISSN: 0034-8082

Datum der Publikation: 2011

Titel der Ausgabe: La formación práctica de estudiantes universitarios: repensando el Practicum

Nummer: 354

Seiten: 429-445

Art: Artikel

DOI: 10.4438/1988-592X-RE-2011-354-006 DIALNET GOOGLE SCHOLAR lock_openOpen Access editor

Andere Publikationen in: Revista de educación

Zusammenfassung

The results of the study presented in this paper demonstrate that the influence of formal, intuitive and procedural knowledge in the process of algorithmic learning, the pedagogical context of the classroom and the process of transferring mathematical knowledge are decisive in error generation through mechanisms of analogical transfer. The paper reports this fact in the subtraction algorithm. The answers given by nine children of ages between seven and ten years in a test made up of 20 subtraction problems were analysed. Altogether, the processes executed by the nine children were recorded in 180 subtraction operations. The volume of data obtained via talk-aloud protocols was analysed with the help of the Nud*ist 4.0 statistical program. The study falls within the context of a larger study where the authors analysed a database of 7,140 subtraction operations performed by 357 children between the ages of seven and thirteen years, with the aim of determining whether systematic errors occurred and typing any systematic errors found. The article shows how the errors most commonly found cluster around certain factors of the task. These factors are related to the understanding of essential concepts for the meaningful learning of the ability. In the authors' opinion, it is essential to help teachers plan how to teach the subtraction algorithm and provide teachers with a specific methodological practice to apply in the classroom; these things would help make the nature of algorithmic processes clearer and reduce error generation in subtraction. Lastly, the paper demonstrates the importance of contextual factors, such as the language used in the teaching process. When children begin learning, they build interpretations of the procedure, based on a series of concepts or specific vocabulary organised within the conceptual field of subtraction.

Bibliographische Referenzen

  • Baroody, A., Herbet, P. G. & Barbara W. (1983). Children's use of Mathematical Structures. Journal for Research in Mathematics Education, 14, 156-68.
  • Brown, J. S., Burton, R. B. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192.
  • Brown, D. & Clement, J. (1989). Overcoming misconceptions via analogical reasoning: Abstract transfer versus explanatory model construction. Instructional Science, 18, 237-261.
  • Brown, J. S. & Vanlehn, K. (1982).Towards a generative theory of «bugs». In T. P. Carpenter, J. M. Moser & T. A. Romberg (Comps.), Addition and subtraction: A cognitive perspective (pp. 117-135). Hillsdale, NJ: Erlbaum.
  • Carpenter, T. P. & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser & J. M. Romberg (Comps.), Addition and subtraction: A cognitive perspective (pp. 9-24). Hillsdale, NJ: Erlbaum.
  • Carpenter, T. P. & Moser, J. M.(1983). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Comps.), Acquisition of Mathematics: Concepts and Processes. Nueva York: Academic Press.
  • Clement, J. (1993). Using bridging analogies and anchoring intuitions to deal with students ́ preconceptions in physics. Journal of Research in Science Teaching, 30, 1241-1257.
  • Duit, R. (1991). On the role of analogies and metaphors in learning science. Science Education, 75, 649-672.
  • Fischbein, E. (1987). Intuition in science and mathematics: An Educational Approach. Dordrecht: D. Reidel.
  • Fischbein, E. (1994). The interaction between the formal and the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Straser & B. Winkelmann (Comps.), Didactics of mathematics as a scientific discipline (pp. 328-375). Dordrecht: Kluwer Academic.
  • Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38, 11-50.
  • Fischbein, E., Deri, M., Nello, M. S. & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16 (1), 3-17.
  • Fuson, K. (1986). Role of representation and verbalization in the teaching of multi-digit additions and subtractions. European Journal of Psychology of education, 35-36.
  • Fuson, K. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R. Putnam & R. A. Hattrup (Comps.), Analysis of arithmetic for mathematics teaching (pp. 53-187). Hillsdale, NJ: Erlbaum.
  • Fuson, K. (1992b). Research on Whole Number Addition and Subtraction. Handbook of Research on Mathematics Teaching and Learning: a project of the National Council of Teachers of Mathematics (pp. 243-275). New York: Maxwell Macmillan International.
  • Fuson, K. & Briars, D. J. (1990). Using base-ten blocks learning/teaching approach for first and second grace place value and multidigit additions and subtraction. Journal for Research in Mathematics Education, 21, 180-206.
  • Gentner, D., Brem, S., Ferguson, R., Markman, A., Levidow, B., Wolff, P. & Forbus, K. (1997). Analogical Reasoning and Conceptual Change: A Case Study of Johannes Kepler. The Journal of the Learning Sciences, 6 (1), 3-40.
  • Gentner, D., Loewenstein, J. & Thompson, L. (2003). Learning and Transfer: A General Role for Analogical Encoding. Journal of Educational Psychology, 2, 393-408.
  • Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Comp.), Conceptual and procedural knowledge: the case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates.
  • Hiebert, J. & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 251-284.
  • López, R. y Sánchez, A. B. (2006). Adquisición del error en la sustracción en Educación Primaria. Proceedings of Internacional Symposium on Early Mathematics. Publisher by Departament of Psychology. University of Cadiz. Research Group HUM-634, Cádiz, 249-26.
  • López, R. (2007). Los componentes generadores de errores algorítmicos. Caso particular de la sustracción. Revista de Educación, 344, 377-402.
  • Mayer, R. E. (1992). Teaching for transfer of problem-solving skills to computer programming. In E. De Corte, M. C. Linn & L. Verschaffel (Comps.), Computer-based learning environments and problem solving (pp. 193-206). Berlin: Springer-Verlag.
  • Nesher, P. (1976). The three determinants of difficulty in verbal arithmetic problems. Educational Studies in Mathematics, 7, 369-388.
  • Nesher, P. & Katriel, T. (1977). A semantic analysis of addition and subtraction word problems in arithmetic. Educational Studies in Mathematics (8), 251-269.
  • Nesher, P., Greeno, J. G. & Riley, M. S. (1982). The development o semantic categories for addition and subtraction. Educational Studies in Mathematics, 13 (4), 373-394.
  • Nesher, P. y Hershkovitz, S. (1994). The role of schemes in two-step problem: Analysis and Research findings. Educational Studies in mathematics, 26, 1-23.
  • Ohlsson, S. & Langley, P. (1988). Psychological evaluation of path hypotheses in cognitive diagnosis. In H. Mandl & A. Lesgold (Comps.), Learning issues for intelligent tutoring system (pp. 42-62). New York: Springer-Verlag.
  • Olhlson, S. & Rees, E. (1991). The function of conceptual understanding in the learning of arithmetic procedure. Cognition and Instruction, 103-180.
  • Orgill, M. & Bodner, G. (2003).What research tells us about using analogies to teach chemistry. Chemistry Education: Research and Practice, 5, 1, 15-32.
  • Orrantia, J. (2003). El rol del conocimiento conceptual en la resolución de problemas aritméticos con estructura aditiva. Infancia y Aprendizaje, 26, 451-468.
  • Resnick, L. (1982). Syntax and semantics in learning to subtract. In T. Carpenter, J. Moser & T. Romberg (Comps.), Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Lawrence Erlbaum Assoc.
  • Resnick, L. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Comp.), The development of mathematical thinking (pp. 109-151). New York: Academic Press.
  • Resnick, L. & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Comp.), Advances in instructional psychology. Hillsdale, NJ: Lawrence Erlbaum Assoc.
  • Sander, E. (2001). Solving arithmetic operations: a semantic approach. In proceedings of the 23rd Annual Conference of the Cognitive Science Society. Edinburgh, 915-920.
  • Thagard, P. (1992). Analogy, explanation, and education. Journal of Research in Science Teaching, 29, 537-544.
  • Thorton, S. (1998). La resolución infantil de problemas. Madrid: Morata.
  • Van De Walle, J. (1990). Concepts of Number. In J. N. Payne (Comp.), Mathematics for the young child (pp. 63-89). National Council of Teachers of Mathematics, Inc. Virginia.
  • VanLehn, K. (1982). Bugs are not enough: Empirical studies of bugs, impasses and repairs in procedural skills. Journal of Mathematical Behaviour, 3, 3-71.
  • VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Comp.), Conceptual and procedural knowledge: The case of mathematics (pp. 133-179). Hillsdale, NJ: Lawrence Erlbaum.
  • VanLehn, K. (1990). Mind bugs: origins of procedural misconceptions. Cambridge, Mass.: MIT Press.
  • VanLehn, K. (1983). On the Representation of Procedures in Repair Theory. In H. Ginsburg (Ed.), The Development of Mathematical Thinking. New York: Academic Press.
  • VanLehn, K. (1987). Learning one subprocedure per lesson. Artificial Intelligence, 31, 1-40.
  • VanLehn, K. & Brown, J. S. (1980). Planning nets: A representation for formalizing analogies and semantic models of procedural skills. In R. E. Snow, P. A. Federico & W. E. Montague (Comps.), Aptitude, learning, and instruction (2, 95-137). Hillsdale, NJ: Lawrence Erlbaum.
  • Young, R. M. & O'Shea, T. (1981). Errors in children's subtraction. Cognitive Science, 5, 153-177.
  • Zook, K. B. (1991). Effects of analogical processes on learning and misrepresentation. Educational Psychology Review, 3, 41-72.
  • Zook, K. B. & DiVesta, F. J. (1991). Instructional analogies and conceptual misrepresentations. Journal of Educational Psychology, 83, 246-252.
Fuente de los datos: Dialnet