Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach


This article is concerned with cognitive aspects of students’ struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors, and roots in the domain of reals and in the domain of complex numbers. The article introduces a polysemous approach for structuring students’ concept images in these situations. Post-exchanges from an online forum and excerpts from an interview were analyzed for illustrating the potential of the approach for indicating possible sources of students’ misconceptions and meta-ways of thinking that might make students aware of their mistakes.

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  1. 1.

    The standard definition that was provided in the classroom stated the following: Let p(x) and q(x) be polynomials in [x]. If there is a polynomial r(x) such that  p(x)=q(x)·r(x), then p(x) is said to be divisible by q(x) and we denote q(x)∣p(x).

  2. 2.

    The students were exposed to the formal definition of addition of vector spaces in the classroom. However, similar to the case of Johnny, there was no evidence of that in their post exchange.


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I am grateful to Arthur Bakker and to anonymous reviewers for their thorough criticism and insightful suggestions. I wish to thank Sze Looi Chin for proofreading.

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Correspondence to Igor’ Kontorovich.

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The title of this article was inspired by Dreyfus’s (1999) work entitled Why Johnny can’t prove (which is an adaptation from the famous Kline, 1973). In his work, Dreyfus identified some of the reasons for students struggling with proofs and clarified why this struggle cannot be avoided. In this article, I focus on another struggle that many students go through when familiar mathematical concepts are reconsidered in a new (mathematical) domain. My approach and the answer to the question in the title resonate with Dreyfus’s.

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Kontorovich, I. Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach. Educ Stud Math 97, 5–20 (2018). https://doi.org/10.1007/s10649-017-9778-z

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  • Concept image
  • Conceptual change
  • Cross-curricular concepts
  • Epistemological obstacles
  • Polysemy