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When the Problem Seems Answerable yet the Solution is Unavailable: Affective Reactions Around an Impasse in Mathematical Discourse

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Abstract

Situations where students encounter mathematical "impasses" – instances where their current discourse is incommensurable with the discourse demanded to solve a task – have the potential to stir emotional responses of different types. They can engender feelings of confusion and bafflement, or even embarrassment, or alternatively open students' curiosity to the ways by which more advanced mathematics can solve a problem. In this study, we closely examine the subjectifications (affective communication) of a group of graduate students encountering such an impasse in the form of a request to account for the area and perimeter of the Sierpiński triangle. We analyze these subjectifications in relation to an a priori mathematical analysis of the impasse inherent in the task. We show two major types of subjectifications, found to be communicated by distinct members of the classroom: "I'm baffled", and "this is not a problem". We show how the students subjectifying "this is not a problem" were those who avoided engagement with the impasse by attending only to the infinite process that defines the fractal, while those students who subjectified "I'm baffled" were those who engaged with the process as well as its outcome. Moreover, despite the lack of substantial contribution to the exploration of the impasse by those students who subjectified "this is not a problem", they were positioned as the "explainers", in a position of power relative to those students who expressed bafflement. We conclude by discussing the dialectical relationship between identity and engagement with mathematical impasses.

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Notes

  1. Commognition (Sfard, 2008) is the welding of communication and cognition, conveying the message that thinking is the intra-personal version of inter-personal communication.

  2. For this we must assume that “finite” binary representations are replaced with equivalent “infinite” representations (e.g. ½ is represented as \(0.0\overline{1 }\) rather than \(0.1\)).

  3. In fact, this infinite process defines the binary digits of the point’s barycentric coordinates, as is demonstrated in the “chaos game”, see for example Devaney (1995).

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Authors and Affiliations

  1. Technion – Israel Institute of Technology, Haifa, Israel

    Einat Heyd-Metzuyanim

  2. Weizmann Institute of Science, Rehovot, Israel

    Jason Cooper

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  1. Einat Heyd-Metzuyanim
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Correspondence to Einat Heyd-Metzuyanim.

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Heyd-Metzuyanim, E., Cooper, J. When the Problem Seems Answerable yet the Solution is Unavailable: Affective Reactions Around an Impasse in Mathematical Discourse. Int. J. Res. Undergrad. Math. Ed. 9, 605–631 (2023). https://doi.org/10.1007/s40753-022-00172-1

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