Educational Studies in Mathematics

, Volume 30, Issue 1, pp 39–65 | Cite as

Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations

  • Liora Linchevski
  • Nicolas Herscovics
Article

Abstract

The objective of the teaching experiment reported in this article was to overcome the “cognitive gap”, that is, students' inability to spontaneously operate with or on the unknown. Following an analysis of the cognitive obstacles involved, this paper reports the results of an alternative approach. We designed an individualized teaching experiment which was tested in six case studies. In the first part the students' natural tendency to group singletons in the unknown within the equations was expanded to a process of grouping like terms. In the second part we introduced a reverse process to grouping like terms, that of decomposition of a term into a sum. This process, combined with the cancellation of identical terms, provides a procedure for the solution of first degree equations with the unknown on both sides of the equality sign. The last part of the teaching experiment involved the decomposition of an additive term into a difference. The first two parts proved very successful and the students developed procedures on their own that were more efficient than the initial ones. The results of the third part, however, revealed the limits of this approach. The students experienced difficulties in choosing the required decomposition. It seems that some of these obstacles are rather robust and perhaps should not be dealt with incidentally but should be addressed as part of a pre-algebra course.

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References

  1. Bher M. J., Lesh R., Post T. R. and Silver E. A.: 1983, ‘Rational number concepts’, in Lesh R. and Landau M. (eds.), Acquisition of Mathematical Concepts and Processes, Academic Press, New York, pp. 92–128.Google Scholar
  2. Bell, A.: 1988, ‘Algebra-choices in curriculum design’, in Borbas, A. (ed.), Proceedings of the 12th International Conference for the Psychology of Mathematics Education, 1, 147–153.Google Scholar
  3. Bell A., Malone J. A. and Taylor P. C.: 1987, Algebra—an Exploratory Teaching Experiment, Nottingham, England: Shell Center for Mathematics Education.Google Scholar
  4. Booth L. R.: 1988, ‘Children's difficulties in beginning algebra’, in The Ideas of Algebra, K-12, 1988 NCTM Yearbook, National Council of Teachers of Mathematics: Reston, Virginia, pp. 20–32.Google Scholar
  5. Boyer C. B./ Merzbach V. C.: 1991, A History of Mathematics, 2nd edition, revised by Merzbach, John Wiley & Sons, New York, pp. 228–9.Google Scholar
  6. Bruner J.: 1967, The Process of Education, New York: Random House.Google Scholar
  7. Carpenter T. P., Corbin M. K., Kepner H. S., Montgomery Lindquist M. and Reys R. E.: 1981, Results from the Second Mathematics Assessment of the National Assessment of Educational Progress, NCTM: Reston, Virginia.Google Scholar
  8. Cauzinille, E., Mathieu, J. and Resnick, B.: 1984, ‘Children's Understanding of Algebraic and Arithmetic Expressions’, paper presented at the annual meeting of the American Educational Research Association, New Orleans, La.Google Scholar
  9. Chaiklin, S. and Lesgold, S.: 1984, ‘Pre-Algebra Students’ Knowledge of Algebraic Tasks with Arithmetic Expressions’, paper presented at the annual meeting of the American Educational Research Association, New Orleans, La.Google Scholar
  10. Chalouh L. and Herscovics N.: 1988, ‘Teaching algebraic expressions in a meaningful way’, in The Ideas of Algebra, K-12, 1988 NCTM Yearbook, National Council of Teachers of Mathematics: Reston, Virginia, pp. 33–42.Google Scholar
  11. Chalouh, L. and Herscovics, N.: 1984, ‘From letter representing a hidden quantity to letter representing an unknown quantity’, in Moser, J. M. (ed.), Proceedings PME-NA VI, Madison, Wisconsin, pp. 71–76.Google Scholar
  12. Collis K. F.: 1975, The Development of Formal Reasoning, Report of a Social Science Research Council sponsored project (HR2434/1) carried out at the University of Nottingham, University of Newcastle, NSW, Australia.Google Scholar
  13. Collis K.F.: 1974, ‘Cognitive Development & Mathematics Learning’, paper prepared for the Psychology of Mathematics Education Workshop, published at the Shell Mathematics Unit Center for Science Education, Chelsea College, University of London, UK.Google Scholar
  14. Cooney T. J.: 1985, ‘A beginning teacher's view of problem solving’, Journal for Research in Mathematics Education 16, 324–336.Google Scholar
  15. Davis R. B.: 1985, ‘Algebraic thinking in the early grades’, Journal of Mathematical Behavior 2, 310–320.Google Scholar
  16. Davis R. B.: 1975, ‘Cognitive processes involved in solving simple algebraic equations’, Journal of Children's Mathematical Behavior 1, (3), 7–35.Google Scholar
  17. Demana F. and Leitzel J.: 1988, in The Ideas of Algebra, K-12, 1988 NCTM Yearbook, National Council of Teachers of Mathematics: Reston, Virginia, pp. 61–69.Google Scholar
  18. Filloy E. and Rojano T.: 1989, ‘Solving equations: the transition from arithmetic to algebra’, For the Learning of Mathematics 9, (2), 19–25.Google Scholar
  19. Filloy E. and Rojano T.: 1985a, ‘Obstructions to the acquisition of elemental algebraic concepts and teaching strategies’, in Streefland L. (ed.), Proceedings of PME-9, OW & OC, State University of Utrecht: The Netherlands, pp. 154–158.Google Scholar
  20. Filloy, E. and Rojano, T.: 1985b, ‘Operating the unknown and models of teaching’, in Damarin, S. and Shelton, M. (eds.), Proceedings of PME-NA-7, Columbus, Ohio, pp. 75–79.Google Scholar
  21. Filloy, E. and Rojano, T.: 1984, ‘From an arithmetical to an algebraic thought’, in Moser, J. M. (ed.), Proceedings of PME-NA-6, Madison, Wisconsin, pp. 51–56.Google Scholar
  22. Greeno, J.: 1982, ‘A Cognitive Learning Analysis of Algebra’, paper presented at the annual meeting of the American Educational Research Association, Boston, MA.Google Scholar
  23. Harper E.: 1987, ‘Ghosts of Diophantus’, Educational Studies in Mathematics 18, 75–90.Google Scholar
  24. Herscovics N.: 1989, ‘Cognitive obstacles encountered in the learning of algebra’, in Wagner S. and Kieran C. (eds.), Research Issues in the Learning and Teaching of Algebra, Reston, Virginia: NCTM, and Hillsdale, N.J.: Erlbaum, pp. 60–68.Google Scholar
  25. Herscovics N. and Kieran C.: 1980, ‘Constructing meaning for the concept of equation’, The Mathematics Teacher 73, (8), 572–580.Google Scholar
  26. Herscovics N. and Linchevski L.: 1994, ‘The cognitive gap between arithmetic and algebra’, Educational Studies in Mathematics 27, (1), 59–78.Google Scholar
  27. Herscovics, N. and Linchevski, L.: 1992, ‘Cancellation within-the-equation as a solution procedure’, in Geeslin, W. and Graham, K. (eds.), Proceedings of the XVI PME Conference, University of New Hampshire, Durham, 1, pp. 265–272.Google Scholar
  28. Kieran C.: 1992, ‘The learning and teaching of school algebra’, in Grouws D. A. (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan Publishing Company, New York, pp. 390–419.Google Scholar
  29. Kuchemann D.: 1981, Children's Understanding of Mathematics: 11–16, London: John Murray, pp. 102–119.Google Scholar
  30. Kuchemann D.: 1978, ‘Children's understanding of numerical variables’, Mathematics in School 7, (4), 23–26.Google Scholar
  31. Linchevski, L. and Sfard, A.: 1991, ‘Rules without reasons as processes without objects-the case of equations and inequalities’, in Furinghetti, F. (ed.), Proceedings of PME XV, Assisi, Italy, pp. 317–325.Google Scholar
  32. Lodholz R.: 1990, ‘The transition from arithmetic to algebra’, in Edwards E. L.Jr. (ed.), Algebra for Everyone, NCTM: Reston, Virginia, pp. 24–33.Google Scholar
  33. Menchinskaya N. A.: 1969, ‘The psychology of mastering concepts: Fundamental problems and methods of research’, in Kilpatrick J. and Wirszup I. (eds.), Soviet Studies in the Psychology of Learning and Teaching Mathematics, V1, SMSG, Stanford, pp. 75–92.Google Scholar
  34. Peck D. M. and Jencks S. M.: 1988, ‘Reality, arithmetic, algebra’, Journal of Mathematical Behavior 7, 85–91.Google Scholar
  35. Sfard A.: 1991, ‘On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin’, Educational Studies in Mathematics 22, 1–36.Google Scholar
  36. Sfard A. and Linchevski L.: 1994, ‘The gains and pitfalls of reification: The case of algebra’, Educational Studies in Mathematics 26, 191–228.Google Scholar
  37. Steinberg R. M., Sleeman D. H. and Ktorza D.: 1991, ‘Algebra students’ knowledge of equivalence of equations’, Journal for Research in Mathematics Education 22, (2), 112–121.Google Scholar
  38. Streeter J. and Hutchison D.: 1989, Intermediate Algebra, McGraw-Hill, New York, p. 75.Google Scholar
  39. Usiskin Z.: 1988, ‘Children's difficulties in beginning algebra’, in The Ideas of Algebra, K-12, 1988 NCTM Yearbook, National Council of Teachers of Mathematics: Reston, Virginia, pp. 8–20.Google Scholar
  40. Yerusalmy M.: 1988, Effects of Graphic Feedback on the Ability to Transform Algebraic Expressions when Using Computers, The University of Haifa, Haifa, Israel.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Liora Linchevski
    • 1
    • 2
  • Nicolas Herscovics
    • 1
    • 2
  1. 1.School of EducationHebrew UniversityJerusalemIsrael
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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