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Material representations in mathematical research practice

  • S.I. : MathCogEncul
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Abstract

Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process.

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Notes

  1. The aspects of the interviews not discussed in the analysis in this paper mainly relate to the selection of mathematical problems and the choice of representation connected to different communicative contexts. These aspects of the investigation are reported in Misfeldt and Johansen (2015) and Johansen and Misfeldt (2016). We refer to the analysis from these papers where appropriate. A general description of the educational implication of the study has been given in Johansen and Misfeldt (2014). In contrast to the papers previously published, the paper at hand focuses on cognitive aspects of the use of representations and seeks to understand the relationship between the cognitive and the social aspects of representation use.

  2. It should be noted that we do not interpret this to suggest that mathematics is always performed in solitude behind closed doors as an individual activity. Rather, the point is that the thinking process (alone or in groups) most often involves writing and sketching.

  3. As pointed out by one of the anonymous reviewers the role of diagrams in mathematics is very diverse, and diagrams may also in some cases be used as epistemic artefact in a way similar to symbols (see e.g. De Toffoli and Giardino 2014).

  4. As one of the anonymous reviewers kindly pointed out such dynamic processes are not confined to mathematical reasoning or reasoning with diagrams, but could also occur in connection to other representations (see e.g. Clark 1998).

  5. Note that by this we do not mean that diagrams cannot support formal thinking. Furthermore, certain diagrams can also be subject to purely formal manipulations (for in depth treatment of this aspect, see Shin 1994, Mumma 2010).

  6. In mathematics, a permutation is an operation that reorganizes the sequential ordering of the elements of a set.

  7. See also Carter (2010) where the role played by the same type of circular diagrams in finding and constructing a mathematical proof is discussed in depth.

  8. Square matrices have the same number of rows and columns and will consequently typically have a square typographical outline in the usual representational form. In contrast, in rectangular matrices, the number of rows does not coincide with that of the columns.

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Acknowledgements

This paper was developed as part of an investigation of mathematicians’ practice. The investigation and the analysis presented here springs from conversations we have had with a number of mathematicians. These mathematicians were extremely generous with their time and allowed us to learn about their concerns and strategies while conducting mathematical research. We are truly thankful for their indispensable contribution to our work.

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Authors and Affiliations

  1. Department of Science Education, University of Copenhagen, Øster Voldgade 3, 1350, Copenhagen K, Denmark

    Mikkel W. Johansen

  2. Department of Learning and Philosophy, Aalborg University, Copenhagen, Denmark

    Morten Misfeldt

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  1. Mikkel W. Johansen
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  2. Morten Misfeldt
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Correspondence to Mikkel W. Johansen.

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Johansen, M.W., Misfeldt, M. Material representations in mathematical research practice. Synthese 197, 3721–3741 (2020). https://doi.org/10.1007/s11229-018-02033-4

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