Abstract
APOS Theory has been successful in describing and predicting the types of mental structures students need to construct in order to learn abstract concepts. As new research is carried out and complex research projects are undertaken, it has become necessary to widen the scope of the theory. This has been achieved by expanding the researchers’ understanding of various theoretical constructs. Although there has been less research using these constructs, they already form part of the theory or are being tested in current research. One of these constructs is Schema; another is the mechanism of thematization and another, to be discussed in Chap. 8, is a possible new stage, Totality, between Process and Object.
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Notes
- 1.
Problems for the excerpts of students’ responses shown in this section appear in the Appendix at the end of this chapter.
- 2.
The term totality was used as part of the encapsulation of a Process into an Object before it was proposed as a new stage in Dubinsky et al. (2013). The former is the meaning of its use here. In Chap. 8, this term is used differently as a possible new stage in APOS Theory between Process and Object.
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Appendix: Problems for the Interview in the Chain Rule Study (Cottrill 1999)
Appendix: Problems for the Interview in the Chain Rule Study (Cottrill 1999)
Compute the derivative of each of the following functions. Show all your work.
11. \( f(x) = 11{x^5} - 6{x^3} + 8 \) | 12. \( g(x) = 3/{x^2} \) |
13. \( h(x) = ({x^2} - 3) \) | 14. \( y = 3{e^{x\ }} - 4\tan (x) \) |
15. \( y = {x^2}\sin (x) \) | 16. \( F(x) = {{(1 - 4{x^3})}^2} \) |
17. \( G(x) = 2{{\left( {5{x^2} + 1} \right)}^4} - 4x{{\left( {5{x^2} + 1} \right)}^4} \) | 18. \( H(x) = \sin(5{x^4}) \) |
19. \( y = {\cos^3}(t) \) | 20. \( y = {e^{{-{t^2}}}} \) |
Additional question for interview:
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Arnon, I. et al. (2014). Schemas, Their Development and Interaction. In: APOS Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7966-6_7
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