Abstract
The focus of this chapter is a discussion of the emergence of a possible new stage or structure and the use of levels in APOS Theory. The potential new stage, Totality, would lie between Process and Object. At this point, the status of Totality and the use of levels described in this chapter are no more than tentative because evidence for a separate stage and/or the need for levels arose out of just two studies: fractions (Arnon 1998) and an extended study of the infinite repeating decimal \( 0.\bar{9} \) and its relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. in press). It remains for future research to determine if Totality exists as a separate stage, if levels are really needed in these contexts, and to explore what the mental mechanism(s) for constructing them might be. Research is also needed to determine the role of Totality and levels for other contexts, both those involving infinite processes and those involving finite processes. It seems clear that explicit pedagogical strategies are needed to help most students construct each of the stages in APOS Theory and that levels which describe the progressions from one stage to another may point to such strategies. Moreover, observation of levels may serve to help evaluate students’ progress in making those constructions.
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- 1.
\( 0.\bar{0}1 \) refers to repeating 0s, with 1 at the end.
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Arnon, I. et al. (2014). Totality as a Possible New Stage and Levels in APOS Theory. In: APOS Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7966-6_8
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