The history of mathematics education provides ample evidence of the dichotomous distinction that has been made over the years between concepts and procedures, between concepts and skills, and between “knowing that” and “knowing how.” In no field of school mathematics learning has this dichotomy been so damaging as in algebra. While reform efforts of the past decade have attempted to imbue algebra learning with meaning by focusing on “real-life” problems and their various representations, these efforts have missed the main point with respect to the literal-symbolic: that is, that conceptual aspects of algebra abound within the literal-symbolic and that these are integral to most of the so-called procedures of algebra. Both theoretical and empirical arguments will be used to make the point for adopting a different vision of the literal-symbolic domain, in which the procedural is so permeated with the conceptual as to render obsolete a primarily procedure-based view of algebra in school mathematics.
- Mathematics Education
- Conceptual Understanding
- Conceptual Knowledge
- Procedural Knowledge
- Computer Algebra System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The research presented in this article was made possible by a grant from the Social Sciences and Humanities Research Council of Canada (Grant # 410-2007-1485). I also express my appreciation to research colleagues, as well as to the students and their mathematics teacher who participated in the research.
This is a preview of subscription content, access via your institution.
Tax calculation will be finalised at checkout
Purchases are for personal use onlyLearn about institutional subscriptions
Lagrange, J.-B. (2000, p. 16).
Lagrange, J.-B. (2003, p. 271).
Donald, M. (2001, pp. 52–57).
Please note that, henceforth in this chapter, the terms procedure and technique will be used synonymously, except in those cases where the context clearly indicates otherwise.
Our research team has included over various periods of this program of research André Boileau, Caroline Damboise, Paul Drijvers, José Guzmán, Fernando Hitt, Ana Isabel Sacristán, Luis Saldanha, Armando Solares, and Denis Tanguay—as well as our project consultant, Michèle Artigue.
Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274. doi:10.1023/A:1022103903080.
Artigue, M., Defouad, B., Duperier, M., Juge, G., & Lagrange, J.-B. (1998). Intégration de calculatrices complexes dans l’enseignement des mathématiques au lycée [Integration of complex calculators in the teaching of high school mathematics]. Paris, France: Université Denis Diderot Paris 7, Équipe DIDIREM.
Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115–131. doi:10.2307/30034952.
Brownell, W. A. (1935). Psychological considerations in the learning and teaching of arithmetic. In W. D. Reeve (Ed.), The teaching of arithmetic: Tenth yearbook of the National Council of Teachers of Mathematics. New York, NY: Columbia University, Teachers College.
Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge: Implications from research on the initial learning of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113–132). Hillsdale, NJ: Lawrence Erlbaum.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique [The analysis of teaching practice within the anthropological theory of didactics]. Recherches en Didactique des Mathématiques, 19, 221–266.
Davis, E. J. (1978). Suggestions for teaching the basic facts of arithmetic. In M. N. Suydam (Ed.), Developing computational skills: Yearbook of the National Council of Teachers of Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Donald, M. (2001). A mind so rare: The evolution of human consciousness. London, England: W. W. Norton & Company.
Fehr, H. (Ed.). (1953). The learning of mathematics: Its theory and practice. Washington, DC: National Council of Teachers of Mathematics.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as tool. Journal for Research in Mathematics Education, 19, 3–25. doi:10.2307/749108.
Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum.
Hitt, F., & Kieran, C. (2009). Constructing knowledge via a peer interaction in a CAS environment with tasks designed from a Task-Technique-Theory perspective. International Journal of Computers for Mathematical Learning, 14, 121–152. doi:10.1007/s10758-009-9151-0.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York, NY: Macmillan.
Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11, 205–263. doi:10.1007/s10758-006-0006-7.
Kilpatrick, J. (1992). A history of research in mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 3–38). New York, NY: Macmillan.
Lagrange, J.-B. (1996). Analyzing actual use of a computer algebra system in the teaching and learning of mathematics. International DERIVE Journal, 3, 91–108.
Lagrange, J.-B. (2000). L’intégration d’instruments informatiques dans l’enseignement: Une approche par les techniques [Integrating instruments of informatics in teaching: A technique-oriented approach]. Educational Studies in Mathematics, 43, 1–30.
Lagrange, J.-B. (2002). Étudier les mathématiques avec les calculatrices symboliques. Quelle place pour les techniques? [Studying mathematics with symbolic calculators. What role for techniques?]. In D. Guin & L. Trouche (Eds.), Calculatrices symboliques. Transformer un outil en un instrument du travail mathématique: Un problème didactique (pp. 151–185). Grenoble, France: La Pensée Sauvage.
Lagrange, J.-B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. T. Fey (Ed.), Computer Algebra Systems in secondary school mathematics education (pp. 269–283). Reston, VA: National Council of Teachers of Mathematics.
Mounier, G., & Aldon, G. (1996). A problem story: Factorisations of x n-1. International DERIVE Journal, 3, 51–61.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101, 529–544. doi:10.1037/a0014224.
Ruthven, K. (2002). Instrumenting mathematical activity: Reflections on key studies of the educational use of Computer Algebra Systems. International Journal of Computers for Mathematical Learning, 7, 275–291. doi:10.1023/A:1022108003988.
Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual procedural dichotomy. European Journal of Psychology in Education, XIX, 19–43.
Shumway, R. J. (Ed.). (1980). Research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.
Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181–198). Hillsdale, NJ: Lawrence Erlbaum.
Sowder, L. K. (1980). Concept and principle learning. In R. J. Shumway (Ed.), Research in mathematics education (pp. 244–285). Reston, VA: National Council of Teachers of Mathematics.
Star, J. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411. doi:10.2307/30034943.
Suydam, M. N., & Dessart, D. J. (1980). Skill learning. In R. J. Shumway (Ed.), Research in mathematics education (pp. 207–243). Reston, VA: National Council of Teachers of Mathematics.
Thorndike, E. L. (1921). The psychology of drill in arithmetic: The amount of practice. Journal of Educational Psychology, 12, 183–194.
Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3), 1–7. Retrieved from http://www.aft.org/newspubs/periodicals/ae/.
Editors and Affiliations
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kieran, C. (2013). The False Dichotomy in Mathematics Education Between Conceptual Understanding and Procedural Skills: An Example from Algebra. In: Leatham, K. (eds) Vital Directions for Mathematics Education Research. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6977-3_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6976-6
Online ISBN: 978-1-4614-6977-3