Educational Studies in Mathematics

, Volume 94, Issue 1, pp 97–116 | Cite as

Rupture or continuity: The arithmetico-algebraic thinking as an alternative in a modelling process in a paper and pencil and technology environment

  • Fernando Hitt
  • Mireille Saboya
  • Carlos Cortés Zavala


Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico–algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.


Early Algebra Arithmetico-algebraic thinking Institutional representation Spontaneous representation Technology 


  1. Artigue, M. (2012). Enseignement et apprentissage de l’algèbre. Accessed 24 may, 2014.
  2. Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approches to algebra. Perspectives for research and teaching (pp. 115–136). Dordrech: Kluwer Academic Publishers.CrossRefGoogle Scholar
  3. Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor: NFER-Nelson.Google Scholar
  4. Booth, L. R. (1988). Children’s difficulties in beginning algebra. In: The ideas of algebra, K-I2, 1988 NCTM Yearbook (pp. 20–32). Reston, VA.: NCTM.Google Scholar
  5. Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 137–160). New York: Springer.CrossRefGoogle Scholar
  6. Brownell, W.-A. (1942). Problem solving. In N. B. Henry (Ed.), The psychology of learning (41st Yearbook of the National Society for the Study of Education. Part 2). Chicago: University of Chicago Press.Google Scholar
  7. Brownell, W. A. (1947). The place and meaning in the teaching of arithmetic. The Elementary School Journal, 4, 256–265.CrossRefGoogle Scholar
  8. Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. New York: Springer.Google Scholar
  9. Carpenter, T., Ansell, E., Franke, M., Fennema, E., & Weisbeck, L. (1993). Models of problem-solving: A study of kindergarden children’s problem-solving process. Journal for Research in Mathematics Education, 24, 429–441.CrossRefGoogle Scholar
  10. Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school. Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra: Proceedings of the 12 th ICMI Study Conference (pp. 155–162). Melbourne: The University of Melbourne.Google Scholar
  11. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.Google Scholar
  12. Carraher, D., Schliemann A., & Brizuela B. M. (2000). Early algebra, early arithmetic: Treating operations as functions. Annex to the PME-NA XXII proceedings (pp. 1–24). Tucson.Google Scholar
  13. Cooper, T., & Warren, E. (2011). Students’ ability to generalise: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187–214). New York: Springer.CrossRefGoogle Scholar
  14. Davis, R. B., Jokusch, E., & McKnight, C. (1978). Cognitive process in learning algebra. Journal of Children’s Mathematical Behavior, 2(1), 10–320.Google Scholar
  15. Davydov, V.V., & J. Kilpatrick (Eds.). (1990). Soviet Studies in Mathematics Education. Vol. 2. Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (J. Teller, Trans.). Reston: NCTM (Original work published 1972).Google Scholar
  16. Eco, U. (1988). Le signe. Bruxelles: Labor.Google Scholar
  17. Eco, U. (1992). La production des signes. Paris: Livre de Poche.Google Scholar
  18. Engeström, Y. (1999). Activity theory and individual and social transformation. In Y. Engeström, R. Miettinen, & R.-L. Punamäki (Eds.), Perspectives on activity theory (pp. 19–38). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  19. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–26.Google Scholar
  20. Healy, L., & Sutherland, R. (1990). The use of spreadsheets within the mathematics classroom. International Journal of Mathematics Education in Science and Technology, 21(6), 847–862.CrossRefGoogle Scholar
  21. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.CrossRefGoogle Scholar
  22. Hitt, F. (1994). Visualization, anchorage, availability and natural image: Polygonal numbers in computer environments. International Journal of Mathematics Education in Science and Technology, 25(3), 447–455.CrossRefGoogle Scholar
  23. Hitt, F. (2007). Utilisation de calculatrices symboliques dans le cadre d’une méthode d’apprentissage collaboratif, de débat scientifique et d’auto-réflexion. In M. Baron, D. Guinet, & L. Trouche (Eds.), Environnements informatisés et ressources numériques pour l’apprentissage. Conception et usages, regards croisés (pp. 65–88). Paris: Hermès.Google Scholar
  24. Hitt, F. (2013). Théorie de l’activité, interactionnisme et socioconstructivisme. Quel cadre théorique autour des représentations dans la construction des connaissances mathématiques ? Annales de Didactique et de Sciences Cognitives. Strasbourg, 18, 9–27.Google Scholar
  25. Hitt, F., & González-Martín, A. (2015). Covariation between variables in a modelling process: The ACODESA (Collaborative learning, Scientific debate and Self-reflexion) method. Educational Studies in Mathematics, 88(2), 201–219.CrossRefGoogle Scholar
  26. 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.CrossRefGoogle Scholar
  27. Kaput, J. (1995). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. Paper presented at the Annual Meeting of the NCTM, Boston, MA.Google Scholar
  28. Kaput, J., (2000). Transforming Algebra from an Engine of Inequity to an Engine ofMathematical Power By “Algebrafying” the K-12 Curriculum. Paper from National Center for Improving Student Learning and Achievement in Mathematics and Science, Dartmouth, MA. (ERIC Document Reproduction Service No. ED 441 664).Google Scholar
  29. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Greenwich: Information Age Publishing.Google Scholar
  30. Lee, L. (1996). An initiation into algebraic culture through generalisation activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approches to algebra. Perspectives for research and teaching (pp. 87–106). Dordrech: Kluwer Academic Publishers.CrossRefGoogle Scholar
  31. Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20, 41–54.CrossRefGoogle Scholar
  32. Lins, R., & Kaput, J. (2012). The early development of algebraic reasoning: The courrent state of the field. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra (pp. 45–70). Massachusetts: Kluwer Academic Publishers.Google Scholar
  33. Prusak, N., Hershkowits, R., & Schwarz, B. (2013). Conceptual learning in a principled design problem solving environment. Research in Mathematics Education, 15(3), 266–285.CrossRefGoogle Scholar
  34. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.CrossRefGoogle Scholar
  35. Radford, L. (2011). Grade 2 students’ non – symbolic algebraic thinking. In J. Cai & E. Knuth (Eds.), Early algebrization, advances in mathematics education (pp. 303–322). Dordrecht: Kluwer Academic Publishers.Google Scholar
  36. Saboya, M., Bednarz, N., & Hitt, F. (2015). Le contrôle en algèbre: Analyse de ses manifestations chez les élèves, éclairage sur sa conceptualisation. Partie 1: La résolution de problèmes. Annales de Didactique et de Sciences Cognitives, 20, 61–100.Google Scholar
  37. Schliemann, A., Carraher, D., & Brizuela, B. (2012). Algebra in elementary school. In L. Coulange & J.-P. Drouchard (Eds.), Enseignement de l’algèbre élémentaire (pp. 107–122). Paris: Éditions La Pensée Sauvage.Google Scholar
  38. Thompson, P., & Carlson, M. (2016). Variation, covariation and functions: Foundational ways of mathematical thinking. In J. Cai (Ed.), Third Handbook of Research in Mathematics Education. Reston: NCTM.Google Scholar
  39. Vergnaud, G. (1988). Long terme et court terme dans l’apprentissage de l’algèbre. In C. Laborde (Ed.), Actes du Premier Colloque Franco-Allemand de Didactique des Mathématiques et de l’informatique (pp. 189–199). Grenoble: La Pensée Sauvage.Google Scholar
  40. Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, 10(23), 133–170.Google Scholar
  41. Verschaffel, L., & De Corte, E. (1996). Number and arithmetic. In A. J. Bishop et al. (Eds.), International handbook of mathematical education (pp. 99–137). Dordrecht: Kluwer Academic Publishers.Google Scholar
  42. Voloshinov, V. N. (1973). In L. Matejka & I. R. Titunik (Eds.), Marxism and the phylosophy of langage. Cambridge: Harvard University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Fernando Hitt
    • 1
    • 2
  • Mireille Saboya
    • 1
    • 2
  • Carlos Cortés Zavala
    • 1
    • 2
  1. 1.Département de MathématiquesUniversité du Québec à MontréalMontréalCanada
  2. 2.Universidad Michoacana de San Nicolás de HidalgoMoreliaMéxico

Personalised recommendations