Skip to main content
SpringerLink
Log in

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

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  • Artigue, M. (2012). Enseignement et apprentissage de l’algèbre. http://educmath.ens-lyon.fr/Educmath/dossier-manifestations/conference-nationale/contributions/. Accessed 24 may, 2014.

  • 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.

    Chapter  Google Scholar 

  • Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor: NFER-Nelson.

    Google Scholar 

  • 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.

  • 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.

    Chapter  Google Scholar 

  • 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 

  • Brownell, W. A. (1947). The place and meaning in the teaching of arithmetic. The Elementary School Journal, 4, 256–265.

    Article  Google Scholar 

  • Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. New York: Springer.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

  • 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.

    Chapter  Google Scholar 

  • 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 

  • 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).

  • Eco, U. (1988). Le signe. Bruxelles: Labor.

    Google Scholar 

  • Eco, U. (1992). La production des signes. Paris: Livre de Poche.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–26.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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).

  • 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 

  • 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.

    Chapter  Google Scholar 

  • Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20, 41–54.

    Article  Google Scholar 

  • 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 

  • Prusak, N., Hershkowits, R., & Schwarz, B. (2013). Conceptual learning in a principled design problem solving environment. Research in Mathematics Education, 15(3), 266–285.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, 10(23), 133–170.

    Google Scholar 

  • 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 

  • Voloshinov, V. N. (1973). In L. Matejka & I. R. Titunik (Eds.), Marxism and the phylosophy of langage. Cambridge: Harvard University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, 201 Président Kennedy, Montréal, QC, H2X 3Y7, Canada

    Fernando Hitt, Mireille Saboya & Carlos Cortés Zavala

  2. Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México

    Fernando Hitt, Mireille Saboya & Carlos Cortés Zavala

Authors
  1. Fernando Hitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mireille Saboya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Cortés Zavala
    View author publications

    You can also search for this author in PubMed Google Scholar

Consortia

GRUTEAM

Corresponding author

Correspondence to Fernando Hitt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hitt, F., Saboya, M., Zavala, C.C. et al. Rupture or continuity: The arithmetico-algebraic thinking as an alternative in a modelling process in a paper and pencil and technology environment. Educ Stud Math 94, 97–116 (2017). https://doi.org/10.1007/s10649-016-9717-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-016-9717-4

Keywords

Search

Navigation