Abstract
According to commognitive conceptualization, development of mathematical thinking, whether historical or ontogenetic, requires periodic transitions to mathematical discourse incommensurable with the one that has been practiced so far. In this new discourse, some familiar mathematical words will be used in a new way. Historically, such discursive transformations were usually greeted with resistance and led to heated, centuries-long debates. Also for today’s learners, a discourse incommensurable with the one currently in use constitutes a source of much trouble. This is particularly true in those cases in which the change in the use of words occurs tacitly, leading to apparent paradoxes. In this paper, I argue that discourses of finite and infinite sets are mutually incommensurable, and thus the case of students grappling with Sierpiński triangle (ST) may lead to insights about ways in which learners act in the face of incommensurability. Here, possible sources of the confusion reported by the participants are identified with the help of specially designed discourse-analytic tools. It is shown that the students, imperceptibly to themselves, oscillate between the discourses of area-as-a-segment-of-a-plane and of area-as-a-number. The analysis is followed with discussion on theoretical, methodological and practical implications of this study.
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Notes
Most current uses of the term story or narrative imply chronological interconnection of the different parts, and thus this definition leads to a wider understanding of this term than some readers may be used to.
To avoid confusion, let me remark that in the work of some other authors (e.g. Radford, 2013), the word objectification is understood differently, and one should be wary of mistaking one for the other.
Both italicized words, graph and tree, are used here in the sense in which they appear in the mathematical graph theory.
In his later work, Kuhn himself tried to solve the problem of conceptual ambiguity by proposing languages as the objects that can be in the relation of incommensurability. If this proposal did not seem to work, it might be for reasons similar to those I presented above while explaining my preference for defining mathematics as discourse rather than language.
This position has been presented in many different ways by the increasingly vocal and influential opponents of the postmodernist movement. It may be best reflected in the famous ‘Sokal hoax’ (Hilgartner, 1997).
More precisely, perimeter and area are quantities, which are expressible in numbers of units, such as centimeter and square centimeter, respectively. By saying that for us area and perimeter are numbers, the intention is to stress the form in which the answer to the question ‘What is the area/perimeter?’ is usually given.
To summarize, in this classroom the word area was sometimes used as standing for a concrete shape, and sometimes as a metonymy for such shape (the use is called metonymical if a word that signifies a property of an object is employed to signify the object as such).
Note that the prefixes CS, N in the names of the discourses indicate the discourse-specific ontology of the objects denoted by the words area and perimeter. It is important to keep in mind that the three discourses include also objects other than those denoted by those prefixes, with those among them that are signified by the geometric names such as triangle (note that also the ontology of these latter, geometric objects is different from one of the discourses to another).
In the full transcript of the session we did find a single use of the noun infinity: in SGW6, Joy said ‘Well, in one sense it's infinity, because you keep adding a little bit more’. However, even here, the hedge ‘in a sense’ shows her insecurity with regard to the use of this word.
Because of the scarcity of instructor’s talk, we can only speculate on the extent to which this claim held for him as well. As argued by one of the reviewers of this paper, instructor’s questions such as "Can you assign an area to this shape?" "Can you assign a perimeter to this shape?" posed in the worksheet may signal his awareness to at least the incommensurability between N and AS discoruses.
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Acknowledgements
This special issue has been conceived at a workshop supported by the Israel Science Foundation under grant number 2821/19. The data described above and used in the papers in this special issue have been collected in the framework of research funded by the Israel Science Foundation under grant number 843/15. The author wishes to thank the Editors of this special issue for giving her the opportunity to engage with their work and have her own analytic take on their data.
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Highlights
• Learning mathematics requires occasional transitions to a new mathematical discourse.
• The transition, while invisible to the learner, may produce narratives incommensurable with those the student endorsed so far.
• In the analyzed conversation, the students oscillated between two discourses of area strongly incommensurable with those of experts.
• To make the transition to the discourse of experts the students need first to reify infinity and limits.
• A conversation about incommensurability of the new discourse and about its history may help students cope with the transition.
Sfard, A. Taming fantastic beasts of mathematics: Struggling with incommensurability International Journal for Research in Undergraduate Mathematics Education.
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Sfard, A. Taming Fantastic Beasts of Mathematics: Struggling with Incommensurability. Int. J. Res. Undergrad. Math. Ed. 9, 572–604 (2023). https://doi.org/10.1007/s40753-021-00156-7
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DOI: https://doi.org/10.1007/s40753-021-00156-7