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

- 1.5k Downloads
- 10 Citations

## Abstract

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.

## Keywords

Mathematical concept Mathematical conception Mathematical understanding Mathematics learning Instructional goals## Notes

### Acknowledgments

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.

## References

- Anderson, D. (1986). The evolution of Peirce’s concept of abduction.
*Transactions of the Charles S. Peirce Society, 22*, 145–164.Google Scholar - Balacheff, N., & Gaudin, N. (2010). Modeling students’ conceptions: The case of function.
*CBMS Issues in Mathematics Education, 16*, 207–234.CrossRefGoogle Scholar - Boyce, S. J. (2014).
*Modeling students’ units coordinating activity*. Unpublished doctoral dissertation. Virginia Polytechnic Institute and State University, Virginia, USA.Google Scholar - 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 - 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 - 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 - Dowling, P. (2013). Social activity method (SAM): A fractal language for mathematics.
*Mathematics Education Research Journal, 25*, 317–340.CrossRefGoogle Scholar - Doyle, A. C. (1930).
*The Complete Sherlock Holmes*(Vol. 1). New York, NY: Random House.Google Scholar - 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 - 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 - 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 - Hackenberg, A. J. (2010). Students’ reversible multiplicative reasoning with fractions.
*Cognition and Instruction, 28*, 383–432.CrossRefGoogle Scholar - 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 - 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 - Lave, J., & Wenger, E. (1991).
*Situated learning: Legitimate peripheral participation*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Llewellyn, A. (2012). Unpacking understanding: the (re) search for the Holy Grail of mathematics education.
*Educational Studies in Mathematics, 81*, 385–399.CrossRefGoogle Scholar - Marton, F. (1981). Phenomenography—describing conceptions of the world around us.
*Instructional Science, 10*, 177–200.CrossRefGoogle Scholar - Mathematics Learning Study Committee. (2001).
*Adding it up: helping children learn mathematics*. Washington, DC: National Academies Press.Google Scholar - Montangero, J., & Maurice-Naville, D. (1997).
*Piaget, or, the advance of knowledge*. New York: Psychology Press.Google Scholar - National Council of Teachers of Mathematics. (1989).
*Curriculum and evaluation standards for school mathematics*. Reston, VA: Author.Google Scholar - 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 - Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of complex ideas.
*International Journal of Educational Research, 27*, 37–48.CrossRefGoogle Scholar - Piaget, J. (1952).
*The origins of intelligence in children*. New York: International Universities Press.CrossRefGoogle Scholar - Piaget, J. (2001).
*Studies in reflecting abstraction*. Sussex, England: Psychology Press.Google Scholar - 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 - Ricco, R. (1993). Revising the logic of operations as a relevance logic: From hypothesis testing to explanation.
*Human Development, 36*, 125–146.CrossRefGoogle Scholar - 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 - Siegler, R. S. (1995). How does change occur: A microgenetic study of number conservation.
*Cognitive Psychology, 28*, 225–273.CrossRefGoogle Scholar - Sierpinska, A. (1994).
*Understanding in mathematics*. London: Falmer.Google Scholar - Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
*Journal for Research in Mathematics Education, 26*, 114–145.CrossRefGoogle Scholar - Simon, M. A. (2004). Raising issues of quality in mathematics education research.
*Journal for Research in Mathematics Education, 35*, 157–163.CrossRefGoogle Scholar - 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 - Simon, M. A. (2009). Amidst multiple theories of learning in mathematics education.
*Journal for Research in Mathematics Education, 40*, 477–490.Google Scholar - 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 - Skemp, R. (1986).
*The psychology of learning mathematics*(2nd ed.). Harmondsworth, UK: Penguin.Google Scholar - Steffe, L. P. (1983). Children’s algorithms as schemes.
*Educational Studies in Mathematics, 14*, 109–125.CrossRefGoogle Scholar - Steffe, L. P. (1992). Schemes of action and operation involving composite units.
*Learning and Individual Differences, 4*, 259–309.CrossRefGoogle Scholar - 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 - 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 - Steffe, L. P., & Wiegel, H. G. (1994). Cognitive play and mathematical learning in computer microworlds.
*Educational Studies in Mathematics, 26*, 11–134.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - von Glasersfeld, E. (1995).
*Radical constructivism: A way of knowing and learning*. London, United Kingdom: Falmer Press.CrossRefGoogle Scholar - Vygotsky, L. S. (1962).
*Thought and language*. (E. Hanfmann & G. Vakar, Trans.). Cambridge, MA: MIT.Google Scholar - 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