APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research

  • Ed Dubinsky
  • Michael A. Mcdonald
Part of the New ICMI Study Series book series (NISS, volume 7)


In this paper, we have mentioned six ways in which a theory can contribute to research and we suggest that this list can be used as criteria for evaluating a theory. We have described how one such perspective, APOS Theory, is being used in an organized way by members of RUMEC and others to conduct research and develop curriculum. We have shown how observing students’ success in making or not making mental constructions proposed by the theory and using such observations to analyze data can organize our thinking about learning mathematical concepts, provide explanations of student difficulties and predict success or failure in understanding a mathematical concept. There is a wide range of mathematical concepts to which APOS Theory can and has been applied and this theory is used as a language for communication of ideas about learning. We have also seen how the theory is grounded in data, and has been used as a vehicle for building a community of researchers. Yet its use is not restricted to members of that community. Finally, we point to an annotated bibliography (McDonald, 2000), which presents further details about this theory and its use in research in undergraduate mathematics education.


Mathematics Education Mathematical Concept Mathematical Thinking Constructivist Theory Concept Image 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D. and Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1–32.Google Scholar
  2. Asiala, M., Dubinsky, E., Mathews, D., Morics, S. and Oktac, A. (1997). Student understanding of cosets, normality and quotient groups. Journal of Mathematical Behavior, 16(3), 241–309.CrossRefGoogle Scholar
  3. Baker, B., Cooley, L. and Trigueros, M. (2000). A Calculus graphing schema. Journal for Research in Mathematics Education, 31(5), 557–578.Google Scholar
  4. Clark, J., Cordero, P., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D., Tolias G. and Vidakovic, D. (1997). Constructing a schema: The case of the chain rule. Journal of Mathematical Behavior, 16(4), 345–364.CrossRefGoogle Scholar
  5. Cottrill, J. (1999). Students’ understanding of the concept of chain rule in first year calculus and the relation to their understanding of composition of functions. Doctoral dissertation, Purdue University.Google Scholar
  6. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking, pp. 95–126. Dordrecht: Kluwer Academic Publishers.Google Scholar
  7. McDonald, M. A., Mathews, D. and Strobel, K. (2000). Understanding sequences: A tale of two objects. Research in Collegiate Mathematics Education IV, CBMS Issues in Mathematics Education, 8, 77–102.Google Scholar
  8. Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.Google Scholar
  9. Piaget, J. and Garcia, R. (1989). Psychogenesis and the History of Science, New York: Columbia University Press.Google Scholar
  10. Tall, D. and Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Ed Dubinsky
  • Michael A. Mcdonald
    • 1
  1. 1.Occidental CollegeUSA

Personalised recommendations