By Ilana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller

Even if APOS thought has been used largely in several scholarly courses, within the layout of textbooks, and in educating perform, there's no unmarried references that comprises all of the proper information regarding its parts, and gives information approximately its program. The objective of this e-book is to offer the most parts of APOS concept. it's going to be important for researchers who paintings with, or wish to research extra approximately, this theoretical procedure, people who find themselves drawn to the best way which mathematical conceptions are built in accordance with this idea, arithmetic schooling researchers, graduate scholars in arithmetic schooling, and arithmetic teachers.

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**Sample text**

Specifically, to combine the disjunction P _ Q with a declarative statement R to form the statement ðP _ QÞ ^ R, the disjunction Process must be encapsulated to form a new Object ðP _ QÞ to which the statement R can be conjoined. Thus, in general, the Actions of conjunction, disjunction, implication, and negation that are applied to declarative statements must be interiorized and then encapsulated in order to construct more complex first-order propositions as Objects [note how the use of parentheses in mathematical notation corresponds here to encapsulation (Dubinsky and Lewin 1986)].

118) Another example that Dubinsky (1991) presents about the generation of a new process by the mechanism of reversal is related to integration: A calculus student may have interiorized the action of taking the derivative of a function and may be able to do this successfully with a large number of examples, using various techniques that are often taught and occasionally learned in calculus courses. If the process is interiorized, the student might be able to reverse it to solve problems in which a function is given and it is desired to find a function whose derivative is the original function.

Implementation of the instruction provides an opportunity for gathering data, usually in the form of written instruments and/or in-depth interviews. In analysis of the data, two questions are asked: (1) Did the students make the mental constructions called for by the genetic decomposition? (2) How well did the subjects learn the mathematical content? Answers to these questions may lead to revision of the genetic decomposition and/or the instruction. At this point, the genetic decomposition is no longer regarded as preliminary.