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A Quantitative Shift Towards Multiplicative Thinking

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Abstract

When two third-graders collaboratively manipulated a multi-modal, digital learning device called TouchTimes (hereafter, TT), that introduces multiplication through visual, tangible and symbolic means, their thinking about quantity shifted from being additive to being multiplicative. In this study, I examine the children’s interactions around/with TT. My goal is two-fold: (1) to demonstrate the shift between the students’ additive and multiplicative thinking; (2) to explain how their multiplicative thinking emerged around/with TT. The emergence of multiplicative thinking does not refer to the students’ correct computations of multiplicative expressions as a response to verbal or number problems. Instead, drawing on an enactivist perspective, I identify the children’s thinking as their effective bodily reactions to a given unitizing task using TT—and I distinguish their multiplicative and additive thinking based on various researchers’ conceptions of multiplicative thinking. Data was created by video-recording the children’s interaction around/with TT. A retrospective analysis of the data reveals that the children’s effective action to solve the unitizing task developed through a history of recurrent interactions in this environment.

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taken from Maturana & Varela, 1987/1992, p. 180)

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Notes

  1. The term “level of abstraction” was used by Clark and Kamii (1996), but I interpret it here as the number of units to be operated on.

  2. These images are based, respectively, on Schwartz (1988, p. 45); Vergnaud (1988, p. 146); Confrey (1994, p. 302); Maffia and Mariotti (2018, p. 31); Izsák and Beckmann (2019, p. 92); and Polotskaia and Savard (2021).

  3. Multiplicative thinking does not necessarily indicate any reasoning. Based on Clark and Kamii’s (1996) definition, multiplicative thinking involves identification of two units of multiplication and the two levels of inclusive relationships among them, not an expression of reasoning behind these structures, even though it might implicitly involve some reasoning. Therefore, I do not use the expressions “multiplicative reasoning” and “multiplicative thinking” interchangeably. Rather, I employ the specific terminology used by the authors cited.

  4. This is based on Fig. 3 in Kaput & Pattison-Gordon (1987, p.13).

  5. A “three-pod” refers to a pod that includes three pips. In Fig. 4c, the fingers were first placed on the left side of the screen, but if they had been placed on the right side of the screen, then the pods would appear on the left side of the screen.

  6. In this paper, the term “gesture” refers to the notion of the “tangible gesture” as discussed in Sinclair and de Freitas (2014).

  7. Maturana and Varela define structure as, “the components and the relations that actually constitute a particular.

    unity” (1987/1992, p. 43).

  8. This particular video-recording was also analyzed, albeit differently, by Bakos and Pimm (2020).

  9. The author of this article was not present at this recorded event.

  10. For more discussion on certain aspects of gestures more generally, see Bakos and Pimm (2020).

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Funding

This research was funded by Social Sciences and Humanities Research Council, grant number 435‐2018‐0433.

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  1. Faculty of Education, Simon Fraser University, Burnaby, BC, Canada

    Canan Güneş

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Appendices

Appendix 1

Table 4 The records of children’s gestures from the 21st to the 30th second
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Appendix 2

Table 5 The records of children’s gestures from the 41st to the 47th second
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Güneş, C. A Quantitative Shift Towards Multiplicative Thinking. Digit Exp Math Educ 7, 361–394 (2021). https://doi.org/10.1007/s40751-021-00094-8

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