Summary
The above catalog contains fifteen headings, each of which indicates a collection of families of models for multiplication and division of whole numbers. The catalog refers to somewhat more than sixteen families of models which are easily distinguished one from the other.
Not included in the catalog thus far developed are several interpretations of multiplication and division that are also of interest. Among these are models based on the equivalency of denominations of money and various units of measurement. Other interpretations which are of historical interest are those of McLellan and Dewey [15] and Thorndike [24]. The relation between models of operations on whole numbers and models of operations defined on larger universal sets is also of interest. One aspect of this area of interest is the process of constructing models of multiplication and division of whole numbers from such models by altering the rules of the model or delimiting its universal set. For example, one can begin with one of Diénès' models of multiplication of integers [8, pp. 57–58] and make approapriate adjustments and result in a model of multiplication of whole numbers. Other interpretations developed by Diénès are of interest because they involve concretizations of whole numbers which are operators as opposed to states [8, pp. 12, 30; 9, p, 36].
These are a great many strategies available for the use of models in teaching the operations on whole numbers. In one such strategy, an educator can define either multiplication or division on some basis (most likely in terms of a model) and then the other can be defined as its inverse.
Another strategy is to define each operation in terms of a different model. For example, one might define multiplication in terms of the repeated addition model and division in terms of the repeated subtraction model.
Still another type of procedure involves a multiple embodiment strategy in which several interpretations are taught as representing each operation.
The choice of a particular strategy would depend upon a great many factors. Some of the factors would be the type of culture and students for which the program is written, the psychological assumptions adopted by the writer, and the writer's knowledge of the domain of models for the operations as well as their relation to the abstract mathematical domain which they represent. This article has contributed to a basis for intelligent decisions in this area by presenting a characterization of the domain of models for multiplication and division of whole numbers and their relation to the abstract operations.
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Vest, F.R. A catalog of models for multiplication and division of whole numbers. Educ Stud Math 3, 220–228 (1971). https://doi.org/10.1007/BF00305450
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DOI: https://doi.org/10.1007/BF00305450