Abstract machines and grammars (Little, Brown computer by Walter J Savitch

By Walter J Savitch

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Something constructed in one’s mind) that an individual uses to make sense of mathematical situations. A source for a mental structure is a description of where that structure comes from. A mental mechanism is a means by which that structure might develop in the mind(s) of an individual or a group of individuals. (p. 98) The constructions of mathematical knowledge described in this chapter illustrate how making the most basic constructions is fundamental for an individual to construct more robust structures.

Although the most primitive of structures (and often, the only one stressed in traditional teaching), Actions are fundamental to APOS Theory. An Action conception is necessary for the development of other structures. In particular, Processes are interiorized Actions, and mental Objects arise because of the application of Actions. New Actions lead to the development of higher order structures. For instance, in the case of functions, performing operations on them spurs their encapsulation as Objects.

In a similar vein, Dubinsky points out that “[i]n describing this construction we reiterate the point that, in the context of this theory, it is never clear (nor can it be) whether we are talking about a schema that is present or a schema that is being (re-)constructed” (p. 112). It is interesting to note that this genetic decomposition reveals a cognitive step, which research has pointed out as providing a serious difficulty for students, that is not apparent when considering induction from a purely mathematical point of view.

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