Educational Studies in Mathematics

, Volume 94, Issue 2, pp 117–137 | Cite as

Explicating mathematical concept and mathematicalconception as theoretical constructs for mathematics education research



Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by “mathematical understanding” is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms mathematical concept and mathematical conception so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.


Mathematical concept Mathematical conception Mathematical understanding Mathematics learning Instructional goals 



This work is supported by the National Science Foundation under Grant No. DRL-1020154. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.


  1. Anderson, D. (1986). The evolution of Peirce’s concept of abduction. Transactions of the Charles S. Peirce Society, 22, 145–164.Google Scholar
  2. Balacheff, N., & Gaudin, N. (2010). Modeling students’ conceptions: The case of function. CBMS Issues in Mathematics Education, 16, 207–234.CrossRefGoogle Scholar
  3. Boyce, S. J. (2014). Modeling students’ units coordinating activity. Unpublished doctoral dissertation. Virginia Polytechnic Institute and State University, Virginia, USA.Google Scholar
  4. Brown, M. (1998). The paradigm of modeling by iterative conceptualization in mathematics education research. In Sierpinska & Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (Vol. 2, pp. 263–276). Dordrecht, Netherlands: Kluwer.Google Scholar
  5. Campbell, R. L. (2001). Reflecting abstraction in context. In R. L. Campbell (Ed. & translator), J. Piaget, J. (2001) Studies in reflecting abstraction (pp. 1–27). Sussex, England: Psychology Press.Google Scholar
  6. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–590). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Dowling, P. (2013). Social activity method (SAM): A fractal language for mathematics. Mathematics Education Research Journal, 25, 317–340.CrossRefGoogle Scholar
  8. Doyle, A. C. (1930). The Complete Sherlock Holmes (Vol. 1). New York, NY: Random House.Google Scholar
  9. 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
  10. Duval, R. (2000). Basic issues for research in mathematics education. In Proceedings of the twenty-fourth annual meeting of International Group for the Psychology of Mathematics Education (pp. 55–69). Hiroshima, Japan: PME.Google Scholar
  11. Godino, J. D. (1996). Mathematical concepts, their meanings and understanding. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 417–424). Valencia, Spain: PME.Google Scholar
  12. Hackenberg, A. J. (2010). Students’ reversible multiplicative reasoning with fractions. Cognition and Instruction, 28, 383–432.CrossRefGoogle Scholar
  13. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  14. Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1, 75–86.CrossRefGoogle Scholar
  15. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Llewellyn, A. (2012). Unpacking understanding: the (re) search for the Holy Grail of mathematics education. Educational Studies in Mathematics, 81, 385–399.CrossRefGoogle Scholar
  17. Marton, F. (1981). Phenomenography—describing conceptions of the world around us. Instructional Science, 10, 177–200.CrossRefGoogle Scholar
  18. Mathematics Learning Study Committee. (2001). Adding it up: helping children learn mathematics. Washington, DC: National Academies Press.Google Scholar
  19. Montangero, J., & Maurice-Naville, D. (1997). Piaget, or, the advance of knowledge. New York: Psychology Press.Google Scholar
  20. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.Google Scholar
  21. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors.Google Scholar
  22. Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of complex ideas. International Journal of Educational Research, 27, 37–48.CrossRefGoogle Scholar
  23. Piaget, J. (1952). The origins of intelligence in children. New York: International Universities Press.CrossRefGoogle Scholar
  24. Piaget, J. (2001). Studies in reflecting abstraction. Sussex, England: Psychology Press.Google Scholar
  25. Radford, L. (2013). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. REDIMAT-Journal of Research in Mathematics Education, 2, 7–44.Google Scholar
  26. Ricco, R. (1993). Revising the logic of operations as a relevance logic: From hypothesis testing to explanation. Human Development, 36, 125–146.CrossRefGoogle Scholar
  27. 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, 1–36.CrossRefGoogle Scholar
  28. Siegler, R. S. (1995). How does change occur: A microgenetic study of number conservation. Cognitive Psychology, 28, 225–273.CrossRefGoogle Scholar
  29. Sierpinska, A. (1994). Understanding in mathematics. London: Falmer.Google Scholar
  30. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.CrossRefGoogle Scholar
  31. Simon, M. A. (2004). Raising issues of quality in mathematics education research. Journal for Research in Mathematics Education, 35, 157–163.CrossRefGoogle Scholar
  32. Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8, 359–371.CrossRefGoogle Scholar
  33. Simon, M. A. (2009). Amidst multiple theories of learning in mathematics education. Journal for Research in Mathematics Education, 40, 477–490.Google Scholar
  34. Simon, M. A., Placa, N., & Avitzur, A. (2016). Two stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47, 63–93.CrossRefGoogle Scholar
  35. Skemp, R. (1986). The psychology of learning mathematics (2nd ed.). Harmondsworth, UK: Penguin.Google Scholar
  36. Steffe, L. P. (1983). Children’s algorithms as schemes. Educational Studies in Mathematics, 14, 109–125.CrossRefGoogle Scholar
  37. Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259–309.CrossRefGoogle Scholar
  38. Steffe, L. P. (1995). Alternative epistemologies: An educator’s perspective. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 489–523). Hillsdale, NJ: Erlbaum.Google Scholar
  39. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Erlbaum.Google Scholar
  40. Steffe, L. P., & Wiegel, H. G. (1994). Cognitive play and mathematical learning in computer microworlds. Educational Studies in Mathematics, 26, 11–134.CrossRefGoogle Scholar
  41. Tall, D., & 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
  42. Thompson, P., & Saldanha, L. (2004). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  43. Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30, 390–416.CrossRefGoogle Scholar
  44. Tzur, R., Johnson, H. L., McClintock, E., Kenney, R. H., Xin, Y. P., Si, L., et al. (2013). Distinguishing schemes and tasks in children’s development of multiplicative reasoning. PNA, 7, 85–101.Google Scholar
  45. Ulrich, C., Tillema, E. S., Hackenberg, A. J., & Norton, A. (2014). Constructivist model building: Empirical examples from mathematics education. Constructivist Foundations, 9, 328–339.Google Scholar
  46. Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 5–28). East Sussex, UK: Psychology Press.Google Scholar
  47. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London, United Kingdom: Falmer Press.CrossRefGoogle Scholar
  48. Vygotsky, L. S. (1962). Thought and language. (E. Hanfmann & G. Vakar, Trans.). Cambridge, MA: MIT.Google Scholar
  49. White, P., & Mitchelmore, M. (2002). Teaching and learning mathematics by abstraction. In D. Tall & M. Thomas (Eds.), Intelligence, learning, and understanding in mathematics: A tribute to Richard Skemp (pp. 235–256). Flaxton, Australia: Post Pressed.Google Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Teaching & LearningNew York UniversityNew YorkUSA

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