The theory we present begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental actions, processes, and objects and organizing them in schemas to make sense of the situations and solve the problems. In reference to these mental constructions we call it APOS Theory. The ideas arise from our attempts to extend to the level of collegiate mathematics learning the work of J. Piaget on reflective abstraction in children’s learning. APOS Theory is discussed in detail in Asiala, et. al. (1996). We will argue that this theoretical perspective possesses, at least to some extent, the characteristics listed above and, moreover, has been very useful in attempting to understand students’ learning of a broad range of topics in calculus, abstract algebra, statistics, discrete mathematics, and other areas of undergraduate mathematics. Here is a brief summary of the essential components of the theory.
An action is a transformation of objects perceived by the individual as essentially external and as requiring, either explicitly or from memory, step-by-step instructions on how to perform the operation. For example, an individual with an action conception of left coset would be restricted to
working with a concrete group such as Z20 and he or she could construct subgroups, such as H={0,4,8,12,16} by forming the multiples of 4. Then the individual could write the left coset of 5 as the set 5+H={1,5,9,13,17} consisting of the elements of Z20 which have remainders of 1 when divided by 4.
When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action, but no longer with the need of external stimuli. An individual can think of performing a process without actually doing it, and therefore can think about reversing it and composing it with other processes. An individual cannot use the action conception of left coset described above very effectively for groups such as S4, the group of permutations of four objects and the subgroup H corresponding to the 8 rigid motions of a square, and not at all for groups Sn for large values of n.
In such cases, the individual must think of the left coset of a permutation p as the set of all products ph, where h is an element of H. Thinking about forming this set is a process conception of cosets. An object is constructed from a process when the individual becomes aware of the process as a totality and realizes that transformations can act on it. For example, an individual understands cosets as objects when he or she can think about the number of cosets of a particular subgroup, can imagine comparing two cosets for equality or for their cardinalities, or can apply a binary operation to the set of all cosets of a subgroup.
Finally, a schema for a certain mathematical concept is an individual’s collection of actions, processes, objects, and other schemas which are linked by some general principles to form a framework in the individual’s mind that may be brought to bear upon a problem situation involving
that concept. This framework must be coherent in the sense that it gives, explicitly or implicitly, means of determining which phenomena are in the scope of the schema and which are not. Because this theory considers that all mathematical entities can be represented in terms of actions, processes, objects, and schemas, the idea of schema is very similar to the concept image which Tall and Vinner introduce in “Concept image and concept definition in mathematics with particular reference to limits and continuity,” Educational Studies in Mathematics, 12, 151-169 (1981). Our requirement of coherence, however, distinguishes the two notions.
The four components, action, process, object, and schema have been presented here in a hierarchical, ordered list. This is a useful way of talking about these constructions and, in some sense, each conception in the list must be constructed before the next step is possible. In reality, however, when an individual is developing her or his understanding of a concept, the constructions are not actually made in such a linear manner. With an action conception of function, for example, an individual may be limited to thinking about formulas involving letters which can be manipulated or replaced by numbers and with which calculations can be done. We think of this notion as preceding a process conception, in which a function is thought of as an input-output machine. What actually happens, however, is that an individual will begin by being restricted to certain specific kinds of formulas, reflect on calculations and start thinking about a process, go back to an action interpretation, perhaps with more sophisticated formulas, further develop a process conception and so on. In other words, the construction of these various conceptions of a particular mathematical idea is more of a dialectic than a linear sequence.
APOS Theory can be used directly in the analysis of data by a researcher. In very fine grained analyses, the researcher can compare the success or failure of students on a mathematical task with the specific mental constructions they may or may not have made. If there appear two students who agree in their performance up to a very specific mathematical point and then one student can take a further step while the other cannot, the researcher tries to explain the difference by pointing to mental constructions of actions, processes, objects and/or schemas that the former student appears to have made but the other has not. The theory then makes testable predictions that if a particular collection of actions, processes, objects and schemas are constructed in a certain manner by a student, then this individual will likely be successful using certain mathematical concepts and in certain problem situations. Detailed descriptions, referred to as genetic decompositions, of schemas in terms of these mental constructions are a way of organizing hypotheses about how learning mathematical concepts can take place. These descriptions also provide a language for talking about such hypotheses.
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