# A cognitive gap between arithmetic and algebra

- 1.1k Downloads
- 93 Citations

## Abstract

Serious attempts are being made to improve the students' preparation for algebra. However, without a clear-cut demarcation between arithmetic and algebra, most of these undertakings merely provide either an earlier introduction of the topic or simply spread it out over a longer period of instruction. The present study investigates the upper limits of the students' informal processes in the solution of first degree equations in one unknown prior to any instruction. The results indicate the existence of a*cognitive gap* between arithmetic and algebra, a cognitive gap that can be characterized as*the students' inability to operate spontaneously with or on the unknown*. Furthermore, the study reveals other difficulties of a pre-algebraic nature such as a tendency to detach a numeral from the preceding minus sign in the grouping of numerical terms and problems in the acceptance of the equal symbol to denote a decomposition into a difference as in 23=37−*n* which leads some students to read such equations from right to left.

## Keywords

Minus Sign Early Introduction Informal Process Numerical Term Degree Equation## Preview

Unable to display preview. Download preview PDF.

## References

- Behr, M., Erlwanger, S., and Nichols, E.: 1976,
*How Children View Equality Sentences*, Project for the Mathematical Development of Children, Technical Report No. 3, Florida State University, Tallahassee.Google Scholar - Booth, L.: 1989, ‘Grade 8 students' understanding of structural properties in mathematics’, in G. Vergnaud, J. Rogalski, and M. Artigue (eds.),
*Proceedings of PME-XIII*, Paris, France, 141–148.Google Scholar - Boyer, C. B.: 1991,
*A History of Mathematics*, 2nd edition, revised by U. C. Merzbach, John Wiley and Sons, New York.Google Scholar - 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, Virgina.Google Scholar - Chalouh, L. and Herscovics, N.: 1984, ‘From letter representing a hidden quantity to letter representing an unknown quantity’, in J. M. Moser (ed.),
*Proceedings of PME-NA-VI*, Madison, Wisconsin, 71–76.Google Scholar - Chalouh, L. and Herscovics, N.: 1983, ‘The problem of concatenation in early algebra’, in J. C. Bergeron and N. Herscovics (eds.),
*Proceedings of PME-NA-V*, Montreal, Canada, 153–160.Google Scholar - Collis, K. F.: 1975,
*The Development of Formal Reasoning*, Report of a Social Science Research Council sponsored project (HR 2434/1) carried out at the University of Nottingham, University of Newcastle, NSW, Australia.Google Scholar - Collis, K. F.: 1974,
*Cognitive Development and Mathematics Learning*, paper prepared for the Psychology of Mathematics Education Workshop, published at the Shell Mathematics Unit Centre for Science Education, Chelsea College, University of London, U.K.Google Scholar - Cortes, A., Vergnaud, G., and Kavafian, N.: 1990, ‘From arithmetic to algebra: negotiating a jump in the learning process’, G. Booker and T. De Mendicutti (eds.),
*Proceedings of PME XIV, Mexico*2, 27–34.Google Scholar - Davis, R. B.: 1975, ‘Cognitive processes involved in solving simple algebraic equations’,
*Journal of Children's Mathematical Behavior*1(3), 7–35.Google Scholar - Ebos, F., Robinson, B., and Tuck, B.: 1984,
*Math is/2*, second edition, Nelson Canada, Scarborough, Ontario.Google Scholar - Educational Testing Service and the College Entrance Examination Board: 1990:
*Algebridge: Concept Based Instructional Assessment*, Janson Publications, Providence, Rhode Island.Google Scholar - Edwards, Jr., E. L. (ed.): 1990,
*Algebra for Everyone*, NCTM, Reston, Virginia.Google Scholar - Filloy, E.: 1987, ‘Modelling and the teaching of algebra’, in J. C. Bergeron, N. Herscovics, and C. Kieran (eds.),
*Proceedings of PME-XI*, Montreal, Canada, Vol. 1, 295–300.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 - Filloy, E. and Rojano, T.: 1985a, ‘ Obstructions to the acquisition of elemental algebraic concepts and teaching strategies’, in L. Streefland (ed.),
*Proceedings of PME-IX*, OW & OC, State University of Utrecht, The Netherlands, 154–158.Google Scholar - Filloy, E. and Rojano, T.: 1985b, ‘Operating the unknown and models of teaching’, in S. Damarin and M. Shelton (eds.),
*Proceedings of PME-NA VII*, Columbus, Ohio, 75–79.Google Scholar - Filloy, E. and Rojano, T.: 1984, ‘From an arithmetical thought to an algebraic thought’, in J. Moser (ed.),
*Proceedings of PME-NA VI*, Madison, Wisconsin, 51–56.Google Scholar - Gallardo, A. and Rojano, T.: 1987, ‘Common difficulties in the learning of algebra among children displaying low and medium pre-algebraic profiency levels’, in J. C. Bergeron, N. Herscovics, and C. Kieran (eds.),
*Proceedings of PME-XI*, Montreal, Canada, Vol. 1, 301–307.Google Scholar - Herscovics, N.: 1989, ‘Cognitive obstacles encountered in the learning of algebra’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Reston, Virginia: NCTM, and Hillsdale, N.J.: Erlbaum, 60–68.Google Scholar - Herscovics, N.: 1980, ‘Constructing meaning for linear equations: a problem of representation’,
*Recherches en Didactiques des Mathématiques*, vol. 1, no. 3, Grenoble, France, 351–385.Google Scholar - Herscovics, N.: 1979, ‘A learning model for some algebraic concepts’, in W. Geeslin and K. Fuson (eds.),
*Explorations in the Modelling of Learning Mathematics*, ERIC-SMEAC, Columbus, Ohio, 98–116.Google Scholar - Herscovics, N. and Chalouh, L.: 1985, ‘ Conflicting frames of reference in the learning of algebra’, in S. Damarin and M. Shelton (eds.),
*Proceedings of PME-NA VII*, Ohio University, Columbus, Ohio, 123–131.Google Scholar - Herscovics, N. and Chalouh, L.: 1984, ‘Using literal symbols to represent hidden quantities’, in J. M. Moser (ed.),
*Proceedings of PME-NA-VI*, Madison, Wisconsin, 64–70.Google Scholar - Herscovics, N. and Kieran, C.: 1980, ‘ Constructing meaning for the concept of equation’,
*The Mathematics Teacher*73(8), 572–580.Google Scholar - Kieran, C.: 1989, ‘The early learning of algebra: a structural perspective’, in S. Wagner and C. Kieran (eds.),
*Research Issues in the Learning and Teaching of Algebra*, Reston, Virginia: NCTM, and Hillsdale, N.J.: Erlbaum, 33–56.Google Scholar - Kieran, C: 1984, ‘Cognitive mechanisms underlying the equation-solving errors of algebra novices’, Southwell et al. (eds.),
*Proceedings of PME-VIII*, Sydney, Australia, 70–77.Google Scholar - Kieran, C.: 1981a, ‘Concepts associated with the equality symbol’,
*Educational Studies in Mathematics*12, 317–326.Google Scholar - Kieran, C: 1981b, ‘Pre-algebraic thinking among 12- and 13-year olds’, Equipe de Recherche Pédagogique (eds.),
*Proceedings of PME-V*, Grenoble, France, 158–164.Google Scholar - Kieran, C.: 1979, ‘Children's operational thinking within the context of bracketing and the order of operations’, in D. Tall (ed.),
*Proceedings of the Third International Conference for the Psychology of Mathematics Education*, Coventry, England: Mathematics Education Research Centre, Warwick University, 128–133.Google Scholar - Kuchemann, D.: 1981, ‘Algebra’, in K. Hart (ed.),
*Children's Understanding of Mathematics: 11–16*, London: John Murray, 102–119.Google Scholar - Kuchemann, D.: 1978, ‘Children's understanding of numerical variables’,
*Mathematics in School*7(4), 23–26.Google Scholar - Lodholz, R.: 1990, ‘The transition from arithmetic to algebra’, in E. L. Edwards Jr. (ed.),
*Algebra for Everyone*, NCTM: Reston, Virginia, 24–33.Google Scholar - MacGregor, M. and Stacey, K.: 1993, ‘ Cognitive models underlying students' formulation of simple linear equations’,
*Journal for Research in Mathematics Education*24(3), 217–232.Google Scholar - Sfard, A. and Linchevski, L.: 1994, ‘The gains and pitfalls of reification: the case of algebra’, to appear in a special issue of
*Educational Studies in Mathematics*.Google Scholar - Swafford, J. O. and Brown, C. A.: 1989, ‘ Variables and relations’, in M. Montgomery Lindquist (ed.),
*Results from the Fourth Mathematics Assessment of the National Assessment of Educational Progress*, NCTM: Reston, Virginia, 55–63.Google Scholar - Travers, K. J. (ed.): 1985,
*Second Study of Mathematics: Summary Report-United States*. Champaign, University of Illinois.Google Scholar