Summary
The above material contains twelve headings that indicate a collection of families of models. Many of these collections consist of several families. For example, the description of the collection of number-line models refers to at least seven families. There are listed here more than 20 families of models for addition and subtraction on the whole numbers which are easily distinguished from one another. Research has revealed that there are also more than 20 well-known models for the other operations of multiplication and division of whole numbers.
It is interesting to note the variety of model representations of whole numbers (referents of whole number symbols) occurring in these many models. They have been cardinal numbers, ordinal numbers, measurements, scale numbers, states, operators, operands, relative length, color, set, and pattern. Several models involve more than one type of number representation.
A list such as this presents in an inductive manner the view of the operations as multivalent abstract mathematical models that reflect a large number of more primitive and concrete models. It is this property of the operations that makes them so useful and powerful in the mental equipment of the elementary school child. This interpretation of the operations suggests the need for continued investigation into the design of multiple embodiment strategies and the possible learning outcomes resulting from them [8]. It also suggests that some of the current debate as to which models are best [6, 18], might be replaced with concern that the child understand the multivalent nature of the operations in terms of several models as well as other ‘true mathemathical’ properties of the operations as they are embedded in number systems.
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Vest, F.R. A catalog of models for the operations of addition and subtraction of whole numbers. Educ Stud Math 2, 59–68 (1969). https://doi.org/10.1007/BF00368985
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DOI: https://doi.org/10.1007/BF00368985