Educational Studies in Mathematics

, Volume 91, Issue 2, pp 165–183 | Cite as

New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies



The initial assumption of this article is that there is an overemphasis on abstraction-from-actions theoretical approaches in research on knowing and learning mathematics. This article uses a critical reflection on research on students’ ways of constructing mathematical concepts to distinguish between abstraction-from-actions theoretical approaches and abstraction-from-objects theoretical approaches. Acknowledging and building on research on knowing and learning processes in mathematics, this article presents a theoretical framework that provides a new perspective on the underlying abstraction processes and a new approach for interpreting individuals’ ways of constructing concepts on the background of their strategies to make sense of a mathematical concept. The view taken here is that the abstraction-from-actions and abstraction-from-objects approaches (although different) are complementary (rather than opposing) frameworks. The article is concerned with the theoretical description of the framework rather than with its use in empirical investigations. This article addresses the need for more advanced theoretical work in research on mathematical learning and knowledge construction.


Cognition Learner types Reflective abstraction Reflectural abstraction Sense making strategies Structural abstraction Theory development 



This article is a restructured and deeply extended version of an invited presentation given at the Federal University of Rio de Janeiro (Brazil) in November 2013. I express my gratitude for the comments given by Márcia M. F. Pinto. Discussions with David O. Tall have been particularly helpful and insightful in the elaboration of several key ideas put forward in this article. Special thanks to Gabriele Kaiser and Klaus Hasemann for their encouragement and ongoing advice. I am also grateful to the anonymous reviewers for their suggestions for improvement. The views expressed in this article do not necessarily reflect those of the researchers mentioned.


  1. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.Google Scholar
  2. Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321–339.CrossRefGoogle Scholar
  3. Bikner-Ahsbahs, A., Dreyfus, T., Kidron, I., Arzarello, F., Radford, L., Artigue, M., & Sabena, C. (2010). Networking of theories in mathematics education. In M. M. F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vo.1, pp. 145–175). Belo Horizonte, Brazil: PME.Google Scholar
  4. Bikner-Ahsbahs, A., & Prediger, S. (2006). Diversity of theories in mathematics education—how can we deal with it? ZDM—The International Journal on Mathematics Education, 38(1), 52–57.Google Scholar
  5. Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education. Heidelberg: Springer.Google Scholar
  6. Bruner, J. S. (1966). Towards a theory of instruction. Cambridge, MA: Harvard University Press.Google Scholar
  7. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thornas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, l5, 167–192.CrossRefGoogle Scholar
  8. Davydov, V. V. (1972/1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (Soviet studies in mathematics education, Vol. 2) (J. Teller, Trans.). Reston, VA: NCTM.Google Scholar
  9. diSessa, A. A. (1991). If we want to get ahead, we should get some theories. In R. G. Underhill (Ed.), Proceedings of the 13 th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 220–239). Blacksburg, VA: PME-NA.Google Scholar
  10. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  11. Dubinsky, E., & Harel, G. (1992). The concept of function: Aspects of epistemology and pedagogy (Vol. 25). Washington, DC: Mathematical Association of America.Google Scholar
  12. Dubinsky, E., & Lewin, P. (1986). Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness. Journal of Mathematical Behavior, 5, 55–92.Google Scholar
  13. Duffin, J. M., & Simpson, A. P. (1993). Natural, conflicting, and alien. Journal of Mathematical Behavior, 12(4), 313–328.Google Scholar
  14. Duval, R. (1995). Sémiosis et pensée humaine. Bern: Peter Lang.Google Scholar
  15. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.CrossRefGoogle Scholar
  16. Ernest, P. (1994). Social constructivism and the psychology of mathematics education. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematics education (pp. 62–72). London: Falmer Press.Google Scholar
  17. Ernest, P. (2006). Reflection on theories of learning. ZDM—The International Journal on Mathematics Education, 38(1), 3–8.Google Scholar
  18. Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1), 97–124.CrossRefGoogle Scholar
  19. Frege, G. (1892a). Über Begriff und Gegenstand (on concept and object). Vierteljahresschrift für wissenschaftliche Philosophie, 16, 192–205.Google Scholar
  20. Frege, G. (1892b). Über Sinn und Bedeutung (on sense and reference). Zeitschrift für Philosophie und philosophische Kritik, 100, 25–50.Google Scholar
  21. Gray, E. M., Pinto, M., Pitta, D., & Tall, D. (1999). Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, 38(1–3), 111–133.CrossRefGoogle Scholar
  22. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.Google Scholar
  23. Harel, G., Selden, A., & Selden, J. (2006). Advanced mathematical thinking. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 147–172). Rotterdam, The Netherlands: Sense.Google Scholar
  24. Hershkowitz, R., Hadas, N., Dreyfus, T., & Schwarz, B. (2007). Abstracting processes, from individuals’ constructing of knowledge to a group’s ‘shared knowledge’. Mathematics Education Research Journal, 19(2), 41–68.CrossRefGoogle Scholar
  25. Hershkowitz, R., Schwarz, B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32, 195–222.CrossRefGoogle Scholar
  26. Ilyenkov, E. V. (1982). The dialectics of the abstract and the concrete in Marx’s Capital. Moscow: Progress.Google Scholar
  27. Mitchelmore, M. C., & White, P. (2007). Abstraction in mathematics learning. Mathematics Education Research Journal, 19(2), 1–9.CrossRefGoogle Scholar
  28. Pegg, J., & Tall, D. O. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. ZDM—The International Journal on Mathematics Education, 37(6), 468–475.CrossRefGoogle Scholar
  29. Piaget, J. (1961/1969). The mechanisms of perception (G. N. Seagrim, Trans.). New York: Basic Books.Google Scholar
  30. Piaget, J. (1973). Introduction à l’épistémologie génétique (2nd ed.). Paris: Presses Universitaires de France. (Original work published 1950).Google Scholar
  31. Piaget, J. [and his collaborators] (1977/2001). Studies in reflecting abstraction (Recherches sur l’abstraction réfléchissante) (R. L. Campbell, Trans.). Philadelphia: Psychology Press.Google Scholar
  32. Pinto, M. M. F. (1998). Students' understanding of real analysis (Unpublished doctoral dissertation). University of Warwick, Coventry UK.Google Scholar
  33. Pinto, M. M. F., & Tall, D. O. (1999). Student constructions of formal theory: Giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 65–73). Haifa, Israel: PME.Google Scholar
  34. Pinto, M. M. F., & Tall, D. O. (2002). Building formal mathematics on visual imagery: A case study and a theory. For the Learning of Mathematics, 22(1), 2–10.Google Scholar
  35. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM—The International Journal on Mathematics Education, 40(2), 165–178.CrossRefGoogle Scholar
  36. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189.CrossRefGoogle Scholar
  37. Scheiner, T. (2013). Mathematical concept acquisition: Reflective, structural, and reflectural learners. Paper presented at the Working Group ‘Factors that Foster or Hinder Mathematical Thinking’ of the 37th Conference of the International Group for the Psychology of Mathematics Education. Kiel, Germany.Google Scholar
  38. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.CrossRefGoogle Scholar
  39. Sfard, A. (1998). On two metaphors for learning and the danger of choosing just one. Educational Researcher, 27(2), 4–13.CrossRefGoogle Scholar
  40. Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—the case of algebra. Educational Studies in Mathematics, 26, 191–228.CrossRefGoogle Scholar
  41. Simpson, A. (1995). Developing a proving attitude. Conference Proceedings: Justifying and Proving in School Mathematics (pp. 39–46). London, England: Institute of Education.Google Scholar
  42. Skemp, R. R. (1986). The psychology of learning mathematics (2nd ed.). London: Penguin Group. (Original work published 1971)Google Scholar
  43. Tall, D. O. (2004). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.Google Scholar
  44. Tall, D. O. (2013). How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  45. Tall, D. O., Gray, E. M., Ali, M., Crowley, L. R. F., DeMarois, P., McGowen, M., et al. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics, and Technology Education, 1(1), 81–104.CrossRefGoogle Scholar
  46. Tall, D. O., Thomas, M., Davis, G., Gray, E. M., & Simpson, A. (1999). What is the object of the encapsulation of a process? Journal of Mathematical Behavior, 18(2), 223–241.CrossRefGoogle Scholar
  47. Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.CrossRefGoogle Scholar
  48. van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. New York: Academic.Google Scholar
  49. van Oers, B. (1998). From context to contextualizing. Learning and Instruction, 8(6), 473–488.CrossRefGoogle Scholar
  50. Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability. The Montana Mathematics Enthusiast, 5(2–3), 231–248.Google Scholar
  51. von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3–17). New Jersey: Lawrence Erlbaum.Google Scholar
  52. von Humboldt, W. (1975/1908). Werke (Vol. 7, part 2). Berlin: Leitmann.Google Scholar
  53. Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 192–203). Norwood, NJ: Ablex Publishing Corporation.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Faculty of EducationUniversity of HamburgHamburgGermany

Personalised recommendations