Abstract
Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We discuss uses of the typology and how reversibility can be understood as the result of reflective abstraction. Finally, we describe two strategies for promoting reversibility that have distinct uses in terms of the types of reverse concepts they engender. We share empirical results which led to our postulation of these strategies and discuss their theoretical basis.
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Notes
For simplicity, we continue to talk about a coordination of actions, while keeping in mind that each action is fused with a goal as part of an existing concept.
Whereas the reverse of an action may be necessary for the reverse of a concept, it is not sufficient. A reversible concept involves a new level of abstraction.
We acknowledge that our elaboration of the construct of reflective abstraction may not be consistent with Steffe’s interpretation of the construct.
A reversible concept could also be developed beginning with division and using it to build a concept of multiplication.
We have eschewed the functional notation, P (I) = R, because we wanted a semi-iconic representation in which left to right indicates a temporal relationship. Thus, one starts with the input, which is the symbol on the left.
To help the reader keep track of the three quantities involved, we will keep the same three numbers (1/5, 1/3, and 1/15) as we exemplify the different types of tasks.
We do not assume an automatic correspondence between task and concept used to think about the task. Students may think about an assessment item in unanticipated ways. However, the task types in the typology are meant to characterize (for the researchers) particular related concepts.
Note that types 4, 5, and 6 can be considered more advanced than 1, 2, and 3. The former represent reverses of 1, 2, and 3, because of the incorporation of the reverse process.
We based Table 1 on the table that appeared in Carpenter, Fennema, Peterson, and Carey (1988, p. 388). We transposed the rows and columns to facilitate comparison of their table with our typology. The two additional rows of their table were not relevant to our discussion. The authors explained, “The join and separate problems… involve two distinct types of action, whereas the combine and compare problems in the third and fourth rows describe static relationships” (p. 388). The “combine” (third) row was similar to the “join” row but without the asymmetry of I and P. The compare row was parallel to the separate row. However, it was not directly relevant to our discussion, because comparing is not a reverse of joining.
In an attempt to preempt confusion, we point out that whereas R in our typology always represents the result of the type 1 task, the result in Carpenter and Moser’s result unknown is relative to whether the process or reverse process is involved (likewise with “start unknown”). Our use of the I-P-R notation in Table 1 offers a common system for our typology with the Carpenter-Moser problem types.
Note that this was not a mindless or rote application of this relationship, as she can justify it upon request.
Steffe and Olive (2010) refer to this latter knowledge as a “splitting operation.”
Whereas the side lengths do not indicate a multiplicand and a multiplier, students may use an asymmetric conception of multiplication to think about an area task (e.g., number of boxes in a column × the number of columns).
Thanks to John Mason for initiating the idea of specific versus general processes in a conversation during PME 39 in Hobart.
Thanks to Tommy Dreyfus for raising this issue.
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Acknowledgments
This work was supported by the National Science Foundation under Grant No. DRL-1020154. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Simon, M.A., Kara, M., Placa, N. et al. Categorizing and promoting reversibility of mathematical concepts. Educ Stud Math 93, 137–153 (2016). https://doi.org/10.1007/s10649-016-9697-4
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DOI: https://doi.org/10.1007/s10649-016-9697-4