ZDM

, Volume 48, Issue 6, pp 775–791 | Cite as

An arithmetic-algebraic work space for the promotion of arithmetic and algebraic thinking: triangular numbers

  • Fernando Hitt
  • Mireille Saboya
  • Carlos Cortés Zavala
Original Article

Abstract

This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils’ spontaneous production using a triangular numbers activity. Based on a specific collaborative learning methodology, this study explores the possibility of constructing an Arithmetic-Algebraic Work Space around the process of constructing signs as framed by both activity theory and a technological approach, showing the spontaneous representations produced by seventh grade pupils and their evolution in a socio-cultural environment.

Keywords

Mathematical work space Mathematical thinking Algebraic thinking Arithmetic-algebraic thinking Technology Spontaneous representations Institutional representations Sociocultural milieu 

References

  1. Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18(2), 147–176.CrossRefGoogle Scholar
  2. Bednarz, N., & Janvier, B. (1996). A problem solving perspective on the introduction of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and teaching (pp. 115–136). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  3. Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to algebra: perspectives for research and teaching. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  4. Blanton, M-L. & Kaput, J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: a global dialogue from multiple perspectives (pp. 5–23). Springer.Google Scholar
  5. Bourdieu, P. (1980). Le sens pratique. Paris: Éditions de Minuit.Google Scholar
  6. Britt, M., & Irwin, J. (2011). Algebraic thinking 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
  7. Brousseau, G. (1997). Theory of didactical situations in mathematics. 1970–1990, In Balacheff, N., Cooper, M., Sutherland, R. & Warfield, V. (Eds. and Trans.) Dordrecht: Kluwer.Google Scholar
  8. 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
  9. Brownell, W. A. (1947). The place and meaning in the teaching of arithmetic. The Elementary School Journal, 4, 256–265.CrossRefGoogle Scholar
  10. Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: a global dialogue from multiple perspectives. New York: Springer.Google Scholar
  11. 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
  12. 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 12th ICMI Study Conference, pp. 155–162). Melbourne: The University of Melbourne.Google Scholar
  13. 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
  14. CIEAEM. (1987). Compte rendu de la 39 e rencontre internationale de la CIEAEM, Sherbrooke.Google Scholar
  15. 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
  16. Cortés C. & Hitt F. (2012). Poly. Applet pour la construction des nombres polygonaux. UMSNH.Google Scholar
  17. Cortés J-C., Hitt F. & Saboya M. (2014). De la aritmética al álgebra: Números Triangulares, Tecnología y ACODESA. REDIMAT, 3(3), 220–252. doi:10.4471/redimat.2014.52.
  18. Duval, R. (2003). Voir en mathématiques. In F. Filloy, F. Hitt, C. Imaz, A. Rivera, & S. Ursini (Eds.), Matemática Educativa: Aspectos de la investigación actual (pp. 19–50). México: Fondo de Cultura Económica.Google Scholar
  19. Eco, U. (1992). [1975] La production des signes. Paris: Livre de Poche.Google Scholar
  20. Filloy, E & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2).Google Scholar
  21. Goupille C. & Thérien L. (1987). The role errors play in the learning and teaching of mathematics. Proceedings CIEAEM39, Sherbrooke.Google Scholar
  22. 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
  23. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59–78.CrossRefGoogle Scholar
  24. 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
  25. Hitt, F. (2006). Students’ functional representations and conceptions in the construction of mathematical concepts. An example: the concept of limit. Annales de Didactique et de Sciences Cognitives, 11, 253–268. (Strasbourg).Google Scholar
  26. 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, 18, 9–27. (Strasbourg).Google Scholar
  27. 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
  28. 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
  29. Houdement, C., & Kuzniak, A. (2006). Paradigmes géométriques et enseignement de la géométrie. Annales de Didactique et de Sciences Cognitives, 11, 175–193.Google Scholar
  30. 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 National Council of Teachers of Mathematics, Boston.Google Scholar
  31. Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by ‘algebrafying’ the K-12 curriculum. The nature and role of algebra in the K-14 curriculum (pp. 25–26). Washington: National Council of Teachers of Mathematics and the Mathematical Sciences Education Board, National Research Council.Google Scholar
  32. Kaput, J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. National Center for Improving Student Learning and Achievement in Mathematics and Science. Dartmouth. (ERIC Service No. ED 441 664).Google Scholar
  33. Karsenty, R. (2003). What adults remember from their high school mathematics? The case of linear functions. Educational Studies in Mathematics., 51, 117–144.CrossRefGoogle Scholar
  34. 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
  35. Kuzniak, A. (2011). L’espace de travail mathématique et ses genèses. Annales de Didactique et de Sciences Cognitives, 16, 9–24.Google Scholar
  36. Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and teaching (pp. 87–106). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  37. Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20, 41–54.CrossRefGoogle Scholar
  38. Lins, R., & Kaput, J. (2004). 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
  39. Malle, G. (1993). Didaktische Probleme der Elementaren Algebra. Braunschweig/Wiesbaden: Vieweg.CrossRefGoogle Scholar
  40. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and teaching (pp. 65–86). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  41. 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
  42. Radford, L. (1996). Some reflexions on teaching algebra through generalization. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and teaching (pp. 107–111). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  43. 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
  44. 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.Google Scholar
  45. Saboya, M. (2010). Élaboration et analyse d’une intervention didactique co-construite entre chercheur et enseignant, visant le développement d’un contrôle sur l’activité mathématique chez les élèves du secondaire. Thèse de doctorat non publiée, Université du Québec à Montréal.Google Scholar
  46. Saboya, M., Bernarz, N., & Hitt, F. (2015). Le contrôle exercé 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
  47. 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
  48. Sfard, A. (2008). Thinking as communicating: human development, the growth of discourse, and mathematizing. New York: Cambridge University Press.CrossRefGoogle Scholar
  49. Thompson, P. (2002). Some remarks on conventions and representations. In F. Hitt (Ed.), Mathematics Visualisation and Representations (pp. 199–206). Psychology of Mathematics Education North American Chapter and Cinvestav-IPN. Mexico.Google Scholar
  50. 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). La Pensée Sauvage: Grenoble.Google Scholar
  51. Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, 10(23), 133–170.Google Scholar
  52. 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
  53. Voloshinov, V.N. (1973). Marxism and the philosophy of language. Translated by Matejka L. & Titunik I. R. Cambridge: Harvard University Press.Google Scholar
  54. Wille, A. (2008). Aspects of the concept of a variable in imaginary dialogues written by pupils. In O. Figueras, J.-L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings PME32 and PME-NA30 (Vol. 4, pp. 417–424). México: Cinvestav-UMSNH.Google Scholar

Copyright information

© FIZ Karlsruhe 2015

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 HidalgoMoreliaMexico

Personalised recommendations