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

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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.

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Notes

  1. Whereas, there is a major focus within mathematics education on understanding, researchers, who employ non-cognitive theories, do not necessarily take understanding as a goal of mathematics instruction (c.f., Dowling, 2013).

  2. Because of the synthetic nature of this article, I do not do a literature review in advance. Rather, I bring in foundational and related literature as it fits into the development of the constructs.

  3. I use “mathematics educators” as a general term to refer to all who are involved in mathematics education (researchers, teachers, curriculum developers, university mathematicians). Key, as discussed, is their role as observer.

  4. Second-order models can be considered as a subset of Marton’s (1981) “second-order perspectives.”

  5. Ulrich, Tillema, Hackenberg, and Norton (2014) recently exemplified second-order models.

  6. Balacheff and Gaudin (2010) have taken a different approach to modeling a conception within the framework of French Didactical Theory. Rather than focusing specifically on the students’ thinking, Balacheff and Gaudin focused on the state of the “subject/milieu system.”

  7. The usefulness of this distinction will become clearer in the Section 7, “The usefulness of the constructs mathematical concept and mathematical conception.”

  8. I started with the assumption that as 4th graders, 9–10 years old, they already had conservation of area.

  9. One of the reasons for this formulation is that evolution in the learners’ goals is key to our explanation of learners’ progression through the sub-stages and stages of developing a concept. See Simon et al. (2016) for this progression.

  10. Boyce (2014) seemed to hold a similar idea, “Only an interiorized scheme can become part of another cognitive scheme.”

  11. Some researchers are working to specify goals in terms of student understanding by specifying students’ schemes and operations (c.f., Hackenberg, 2010; Thompson & Saldanha, 2004).

  12. The fact that I gave names to the mathematical conception (fraction as arrangement) and the mathematical concept (fraction as quantity) is neither typical nor important.

  13. This maxim is consistent with Steffe’s longstanding push for researchers to focus on the rationality of students’ mathematics (e.g., Steffe & Thompson, 2000).

  14. Note that logical necessity and mathematical proof are not the same. In this case, the student can reason about the unique paths of the rays, relative to the line segment, and the single point of their intersection. Unlike mathematical proof, logical necessity does not require the formal building of an axiomatic system.

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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.

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  1. Department of Teaching & Learning, New York University, 239 Greene St, #420, New York, NY, 10003, USA

    Martin A. Simon

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Simon, M.A. Explicating mathematical concept and mathematicalconception as theoretical constructs for mathematics education research. Educ Stud Math 94, 117–137 (2017). https://doi.org/10.1007/s10649-016-9728-1

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