By Ilana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller
In spite of the truth that APOS concept has been used largely in different scholarly guides, within the layout of textbooks, and in educating perform, there is not any unmarried references that comprises all of the appropriate information regarding its elements, and offers counsel approximately its software. The objective of this ebook is to provide the most parts of APOS concept. it may be necessary for researchers who paintings with, or wish to examine extra approximately, this theoretical process, those people who are drawn to the way in which which mathematical conceptions are developed in response to this idea, arithmetic schooling researchers, graduate scholars in arithmetic schooling, and arithmetic instructors.
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Additional info for APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education
1 Actions According to Piaget and adopted by APOS Theory, a concept is first conceived as an Action, that is, as an externally directed transformation of a previously conceived Object, or Objects. An Action is external in the sense that each step of the transformation needs to be performed explicitly and guided by external instructions; additionally, each step prompts the next, that is, the steps of the Action cannot yet be imagined and none can be skipped. For example, in the case of the function concept, “an individual who requires an explicit expression in order to think about the concept of function and can do little more than substitute for the variable in the expression and manipulate it is considered to have an action understanding of functions” (Dubinsky et al.
Sfard 1991, p. 30) As will be seen in Chap. 5, APOS-based instruction has had considerable success in dealing with this difficulty. 4 De-encapsulation, Coordination, and Reversal of Processes Once a Process has been encapsulated into a mental Object, it can be de-encapsulated, when the need arises, back to its underlying Process. In other words, by applying the mechanism of de-encapsulation, an individual can go back to the Process that gave rise to the Object. 2 Description of Mental Structures and Mechanisms 23 The mechanism of coordination is indispensable in the construction of some Objects.
The logic Schema needs to include a Process called modus ponens. This Process is the interiorization of an Action applied to implications (assuming, as above, that they have been encapsulated into Objects). The Action begins with the hypothesis, determining that it is true, followed by asserting the truth of the conclusion. These constructions make it possible for a student to coordinate the function Schema, as it applies to an implication-valued function Q (obtained from a proposition-valued function P), and the logic Schema, as it applies to the Process modus ponens.