Learning Mathematics pp 87-124 | Cite as

# The Gains and the Pitfalls of Reification — The Case of Algebra

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## Abstract

Algebraic symbols do not speak for themselves. What one actually sees in them depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice. It is this last statement which becomes the leading theme of this article. The main focus is on the versatility and adaptability of student’s algebraic knowledge

The analysis is carried out within the framework of the theory of reification according to which there is an inherent process-object duality in the majority of mathematical concepts. It is the basic tenet of our theory that the operational (process-oriented) conception emerges first and that the mathematical objects (structural conceptions) develop afterward through reification of the processes. There is much evidence showing that reification is difficult to achieve.

The nature and the growth of algebraic thinking is first analyzed from an epistemological perspective supported by historical observations. Eventually, its development is presented as a sequence of ever more advanced transitions from operational to structural outlook. This model is subsequently applied to the individual learning. The focus is on two crucial transitions: from the purely operational algebra to the structural algebra ‘of a fixed value’ (of an unknown) and then from here to the functional algebra (of a variable). The special difficulties experienced by the learner at both these junctions are illustrated with much empirical data coming from a broad range of sources.

## Keywords

Functional Approach Algebraic Expression Equality Sign Propositional Formula Algebraic Thinking## Preview

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## References

- Behr, M., Erlwanger, S. and Nichols, E.: 1976,
*How Children View Equality Sentences*,(PMDC Technical Report No. 3), Florida State University (ERIC Document Reproduction Service No. ED 144802).Google Scholar - Bell, A.: 1992, ‘School algebra — what it is and what it might be’, paper presented at the meeting of Algebra Working Group at ICME 7, Quebec, Canada.Google Scholar
- Beth, E. W. and Piaget, J.: 1966,
*Mathematical Epistemology and Psychology*, D. Reidel Publishing Company, Dordrecht, The Netherlands.Google Scholar - Booth, L.: 1988, ‘Children’s difficulties in beginning algebra’, in A. F. Coxford (ed.),
*The Ideas of Algebra*,*K-12*( 1988 Yearbook), NCTM, Reston, VA, pp. 20–32.Google Scholar - Boyer, C. B.: 1985, A
*History of Mathematics*, Princeton University Press, Princeton, NJ (originally published in 1968 ).Google Scholar - Breidenbach, D., Dubinsky, E., Hawks, J. and Nichols, D.: 1992, ‘Development of the process conception of function’,
*Educational Studies in Mathematics***23**, 247–285.CrossRefGoogle Scholar - Chalouh, L. and Herscovics, N.: 1988, ‘Teaching algebraic expressions in a meaningful way’, in A. F. Coxford (ed.),
*The Ideas of Algebra*( 1988 Yearbook), NCTM, Reston, VA, pp. 33–42.Google Scholar - Clement, J., Lochhead, J. and Soloway, E.: 1979,
*Translation Between Symbol Systems: Isolating a Common Difficulty in Solving Algebra Word Problems.*COINS technical report No. 79–19Google Scholar - Department of Computer and Information Sciences, University of Massachusetts, Amherst. Cobb, P.: 1988, ‘The tension between theories of learning and instruction in mathematics education’,
*Educational Psychologist***23**, 87–104.Google Scholar - Collis, K. F.: 1974, ‘Cognitive development and mathematics learning’, paper presented at Psychology of Mathematics Education Workshop, Center for Science Education, Chelsea College, London.Google Scholar
- Crowe, M.: 1988, ‘Ten misconceptions about mathematics and its history’, in W. Asprey and P. Kitcher (eds.),
*History and Philosophy of Modern Mathematics*, Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, Minneapolis.Google Scholar - Davis, P.: 1975, ‘Cognitive processes involved in solving simple arithmetic equations’,
*Journal of Children’s Mathematical Behavior***1**(3), 7–35.Google Scholar - Davis, R.: 1989, ‘Research studies in how humans think about algebra’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Lawrence Erlbaum, NCTM, Hillsdale, NJ, pp. 266–274.Google Scholar - Davis, R.: 1988, ‘The interplay of algebra, geometry, and logic’,
*Journal of Mathematical Behavior***7**, 9–28.Google Scholar - Douady, R.: 1985, ‘The interplay between different settings: Tool-object dialectic in the extension of mathematical ability–Examples from elementary school teaching’, in L. Streefland (ed.),
*Proceedings of the Ninth International Conference for the Psychology of Mathematics Education*, Vol. 2, State University of Utrecht, Subfaculty of Mathematics, OWandOC, Utrecht, The Netherlands, pp. 33–52.Google Scholar - Dreyfus, T. and Halevi, T.: 1988, ‘Quadfun - A case study of pupil-computer interaction’, paper presented to the theme group on Microcomputers and the Teaching of Mathematics at ICME 6, Budapest, Hungary.Google Scholar
- Dubinsky, E.: 1991, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.),
*Advanced Mathematical Thinking*, Kluwer Academic Publishers, Dordrecht, pp. 95–123.Google Scholar - Dubinsky, E. and Hard, G. (eds.): 1992,
*The Concept of Function: Aspects of Epistemology and Pedagogy*,MAA Notes, Vol. 25, Mathematical Association of America.Google Scholar - Even, R.: 1988, ‘Pre-service teachers’ conceptions of the relationship between functions and equations’, in Borbas (ed.),
*Proceedings of the Twelfth International Congress of the PME*, Vol. I, Vesprem, Hungary, pp. 304–311.Google Scholar - Fauvel, J. and Gray, J.: 1987,
*The History of Mathematics -A Reader*, Macmillan Education, London, The Open University, Milton Keynes.Google Scholar - Filloy, E. and Rojano, T.: 1985, ‘Operating the unknown and models of teaching’, in S. K. Damarin and M. Shelton (eds.),
*Proceedings of the Seventh Annual Meeting of PME-NA*, Ohio State University, Columbus, pp. 75–79.Google Scholar - 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 - Frege, G.: 1970, ‘What is function?’, in P. Geach and M. Black (eds.),
*Translations from the Philosophical Writings of Gottlob Frege*, Blackwell, Oxford (German original published in 1904 ).Google Scholar - Freudenthal, H.: 1978,
*Weeding and Sowing*, D. Reidel Publishing Company, Dordrecht, The Netherlands.Google Scholar - Garcia, R. and Piaget, J.: 1989,
*Psychogenesis and the History of Science*, Columbia University Press, New York.Google Scholar - Greeno, G. J.: 1983, ‘Conceptual entities’, in D. Genter and A. L. Stevens (eds.),
*Mental Models*, pp. 227–252.Google Scholar - Gregory, D. F.: 1840,
*On the Nature of Symbolic Algebra*,Trans. Roy. Soc., Vol. 14, Edinburgh, pp. 208–216.Google Scholar - Gray, E. and Tall, D. O.: 1991, ‘Duality, ambiguity and flexibility in successful mathematical thinking’, in F. Furinghetti (ed.),
*Proceedings of the Fifteenth Conference for the Psychology of Mathematics Education*, Vol. 2, Assisi, Italy, pp. 72–79.Google Scholar - Hadamard, J. S.: 1949,
*The Psychology of Invention in the Mathematics Field*, Princeton University Press, NJ.Google Scholar - Harel, G. and Kaput, J.: 1991, The role of conceptual entities in building advanced mathematical concepts and their symbols’, in D. Tall (ed.),
*Advanced Mathematical Thinking*, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 82–94.Google Scholar - Harper, E.: 1987, ‘Ghost of Diophantus’,
*Educational Studies in Mathematics***18**, 75–90.CrossRefGoogle Scholar - Herscovics, N. and Linchevski, L.: 1991, ‘Pre-algebraic thinking: range of equations and informal solutions used by seventh graders prior to any instruction’, in F. Furinghetti (ed.),
*Proceedings of the Fifteenth Conference for the Psychology of Mathematics Education*, Vol. 2, Assisi, Italy, pp. 173–180.Google Scholar - Herscovics, N. and Linchevski, L.: 1993, ‘The cognitive gap between arithmetics and algebra’, forthcoming in
*Educational Studies in Mathematics*.Google Scholar - Kaput, J. J.: 1989, ‘Linking representations in the symbol system of algebra’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Lawrence Erlbaum, Hillsdale, NJ, pp. 167–194.Google Scholar - Kieran, C.: 1981, ‘Concepts associated with equality symbol’,
*Educational Studies in Mathematics***12**(3), 317–326.CrossRefGoogle Scholar - Kieran, C.: 1988, ‘Two different approaches among algebra learners’, in A. F. Coxford (ed.),
*The Ideas of Algebra*( 1988 Yearbook), NCTM, Reston, VA, pp. 91–96.Google Scholar - Kieran, C.: 1991, ‘A procedural-structural perspective on algebra research’, in F. Furinghetti (ed.),
*Proceedings of the Fifteenth International Conference for the Psychology of Mathematics Education*, Vol. 2, Assisi, Italy, pp. 245–254.Google Scholar - Kieran, C.: 1992, ‘The learning and teaching of school algebra’, in D. A. Grouws (ed.),
*The Handbook of Research on Mathematics Teaching and Learning*, Macmillan, New York, pp. 390–419.Google Scholar - Kleiner, I.: 1989, ‘Evolution of the function-concept: A brief survey’,
*College Mathematics Journal***20**(4), 882–300.CrossRefGoogle Scholar - Lakoff, G. and Johnson, M.: 1980,
*The Metaphors We Live By*, University of Chicago Press, Chicago.Google Scholar - Linchevski, L. and Sfard, A.: 1991, ‘Rules without reasons as processes without objects–The case of equations and inequalities’, in F. Furinghetti (ed.),
*Proceedings of the Fifteenth Conference for the Psychology of Mathematics Education*, Vol. 2, Assisi, Italy, pp. 317–324.Google Scholar - Mason, J. H.: 1989, ‘Mathematical subtraction as the results of a delicate shift of attention’,
*For the Learning of Mathematics***9**(2), 2–8.Google Scholar - Moschkovich, J., Schoenfeld, A. and Arcavi, A.: 1992, ‘What does it mean to understand a domain: A case study that examines equations and graphs of linear functions’, paper presented at the 1992 Annual Meeting of the American Educational Research Association, San Francisco, CA.Google Scholar
- Novy, L.: 1973,
*Origins of Modern Algebra*, Noordhoff International Publishing, Leiden, The Netherlands.Google Scholar - Schoenfeld, A. H., Smith, J. P. and Arcavi, A.: 1993, ‘Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain’, forthcoming in R. Galser (ed.),
*Advances in Instructional Psychology*, Vol. 4, Lawrence Erlbaum, Hillsdale, NJ.Google Scholar - Schwartz, B., Dreyfus, T. and Bruckheimer, M.: 1990, ‘A model of function concept in a three-fold representation’,
*Computers and Education***14**, 249–262.CrossRefGoogle Scholar - Sfard, A.: 1987, ‘Two conceptions of mathematical notions: operational and structural’, in
**J**. C. Bergeron, N. Hershcovics and C. Kieran (eds.),*Proceedings of the Eleventh International Conference for Psychology of Mathematics Education*, Vol. III, Université de Montréal, Montréal, Canada, pp. 162–169.Google Scholar - Sfard, A.: 1989, ‘Transition from operational to structural conception: The notion of function revisited’, in G. Vergnaud, J. Rogalski and M. Artigue (eds.),
*Proceedings of the Thirteenth International Conference for the Psychology of Mathematics Education*, Vol. 3, Laboratoire PSYDEE, Paris, pp. 151–158.Google Scholar - 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.CrossRefGoogle Scholar - Sfard, A.: 1992, ‘Operational origins of mathematical notions and the quandary of reification–the case of function’, in E. Dubinsky and G. Hazel (eds.),
*The Concept of Function: Aspects of Epistemology and Pedagogy*, MAA Notes, Vol. 25, Mathematical Association of America, pp. 59–84.Google Scholar - Sfard, A. and Linchevski, L.: 1993, ‘Processes without objects -the case of equations and inequalities’, forthcoming in a special issue of
*Rendiconti del Seminario Matematico dell’Universita e del Politecnico di Torino*.Google Scholar - Soloway, E., Lochhead, J. and Clement, J.: 1982, ‘Does computer programming enhance problem solving ability? Some positive evidence on algebra word problems’, in R. J. Seidel and R. E. Anderson (eds.),
*Computer Literacy*, Academic Press, New York.Google Scholar - Struik, D. J.: 1986, A
*Source Book in Mathematics*,*1200–1800*, Princeton University Press, Princeton, NJ (originally published in 1969 ).Google Scholar - Vergnaud, G., Benhdj, J. and Dussouet, A.: 1979,
*La coordination de l’enseignement des mathématiques entre les cours moyen 2e année et la classe de sixième*, Institut National de Recherche Pédagogique, Paris.Google Scholar - Wagner, S.: 1981, ‘Conservation of equation and function under transformations of variable’,
*Journal for Research in Mathematics Education***12**, 107–118.CrossRefGoogle Scholar - Wagner, S. and Kieran, C.: 1989, ‘An agenda for research on the learning and teaching of algebra’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Lawrence Erlbaum, Hillsdale, NJ, pp. 220–237.Google Scholar - Waits, B. K. and Demana, F.: 1988, ‘New models for teaching and learning mathematics through technology’, paper presented to the theme group on Microcomputers and the Teaching of Mathematics at ICME 6, Budapest, Hungary.Google Scholar
- Wheeler, D.: 1989, ‘Context for the research on teaching and learning algebra’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Lawrence Erlbaum, Hillsdale, NJ, pp. 278–287.Google Scholar