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The relationality in/of teacher–student communication

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Abstract

In mathematics education, student–teacher communication is recognized to constitute an important dimension in/of mathematical learning. Significant effort has been made in recent decades to depart from a focus on the individual in which teachers and student simply use communication to express, to and for others, their private knowledge or thinking. In this paper, we continue this departure taking as a starting point the observation that (mathematical) communication is possible only when there is a relation with others: Communication is the relation with others. That is, we present a way of thinking about student–teacher communication in which geometrical being-in-the-know is conversationally produced. Using fragments of elementary classroom conversations involving three-dimensional geometry as a tool to flesh out this theoretical study, we illustrate (a) how being-in-the-know-with can be recognized in asking and responding to questions involving mathematical concepts and (b) how conversations are then the fine-tuning of being-in-the-know relations in which mathematical ideas can come forth even in those instances where not-being-in-the-know is asserted.

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Notes

  1. With citations, we spell Vygotsky’s name as it appears on the book cover in romanized form; within the text, we use the normal English spelling.

  2. All translations are ours.

  3. We use the term signification—which some translators incorrectly and inconsistently render as “meaning”—which better reflects K. Marx’s German Bedeutung and F. de Saussure’s French signification, which Vološinov (Bakhtin) and Vygotsky have taken up in Russian as značenie.

  4. Transaction is a category that does not reduce a relation to the collaboration or interaction of individuals but rather constitutes a unit that encompasses all actors and their world (Dewey and Bentley 1999), which makes it a better category for understanding situated actions and situated cognition (Roth and Jornet 2013).

  5. The verb “to sublate” translates Hegel’s German term aufheben, which has the contradictory senses of “to do away with” and “to keep.”

  6. Precise mechanisms for the emergence of language from physical movements have been provided in the study of Bonobos (Hutchins and Johnson 2009) and for mathematical activity (Roth 2012c, in press).

  7. The transcription of the sound makes use of the language-independent conventions of the International Phonetics Association.

  8. It has been shown that even where a sound-word appears for the very first time, it is inherently shared (Roth 2013b). A word, even when it has never occurred before, is a reality for two (Vygotskij 2005).

  9. We subscribe here to a socio-phenomenological understanding of embodiment, which draws an important distinction between the material body and the flesh (e.g., Roth 2011), stressing that there would be no embodiment without the experiences of other material bodies, whereas the flesh is what is within those experiences.

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Acknowledgments

This study was made possible by several grants from the Social Sciences and Humanities Research Council of Canada. We express our appreciation to J. Thom, the teachers, students, and the research assistants who contributed to the experimental unit on three-dimensional geometry.

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  1. Département de Mathématiques, Université du Québec à Montréal, Pavillon President-Kennedy, C.P. 8888, Succursale Centre-ville, Montréal, QC, Canada, H3C 3P8

    Jean-Francois Maheux

  2. University of Victoria, MacLaurin Building A548, Victoria, BC, Canada, V8W 3N4

    Wolff-Michael Roth

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  1. Jean-Francois Maheux
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Correspondence to Jean-Francois Maheux.

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Maheux, JF., Roth, WM. The relationality in/of teacher–student communication. Math Ed Res J 26, 503–529 (2014). https://doi.org/10.1007/s13394-013-0096-1

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