Abstract
This chapter is a discussion of the design and implementation of instruction using APOS Theory. For a particular mathematical concept, this typically begins with a genetic decomposition, a description of the mental constructions an individual might make in coming to understand the concept (see Chap. 4 for more details). Implementation is usually carried out using the ACE Teaching Cycle, an instructional approach that supports development of the mental constructions called for by the genetic decomposition.
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Notes
- 1.
Preservice refers to college or university students who are preparing to become school teachers.
- 2.
The notation Line # does not actually appear on an ISETL screen. It is being used for convenience here and in other examples with longer lines of code.
- 3.
The application of Actions to physical (real world) Objects is considered in detail in Chap. 9.
- 4.
In Dubinsky et al. (2005a, b), the ability to see a Process as a Totality was considered to be a part of encapsulation. The instruction on which the study was based (Weller et al. 2009, 2011; Dubinsky et al. 2013) showed evidence of Totality as a separate stage between Process and Object. This distinction is considered later in this chapter and explored in depth in Chap. 8.
- 5.
In these discussions, decimal expansions are referred to as strings. This means finite or infinite sequences of digits that correspond to the decimal expansion of a rational number.
- 6.
ISETL recognized decimal expansions using the notation a.b(c). Here a, b, and c are nonnegative integers, where a denotes the integer part of the decimal expansion, b the decimal portion that appears before the repeating cycle, and c the repeating cycle. For repeating digits such as \( 0.{\bar{3}} \) and \( 0.{\overline{35}} \), where the cycle begins in the tenths place, the computer recognized the notation 0.3(3) and 0.3(53), respectively.
- 7.
The preloaded mystery strings were denoted m1, m2, m3, m4, m5, m6, m7, and m8.
- 8.
Because positions \( n = -3, - 4, - 5\ {\rm are}\ 0 \), they do not appear, according to convention.
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Arnon, I. et al. (2014). The Teaching of Mathematics Using APOS Theory. In: APOS Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7966-6_5
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