Abstract
Most students, even those who desire to succeed in school, are intellectually aimless in mathematics classes because often they do not realize an intellectual need for what we intend to teach them. The notion of intellectual need is inextricably linked to the notion of epistemological justification: the learners’ discernment of how and why a particular piece of knowledge came to be. This chapter addresses historical and philosophical aspects of these two notions, as well as ways teachers can be aware of students’ intellectual need and address it directly in the mathematics classroom.
I wish to acknowledge the helpful comments on this chapter from Evan Fuller, Evgenia Harel, and Patrick Thompson. Also, I wish to thank the anonymous reviewer of this volume for his or her excellent suggestions.
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Notes
- 1.
- 2.
See also Lakatos (1976, Footnote 3, pp. 19–20, Footnote 2, pp. 22–23, and Appendix 2, pp. 151–152) for an interesting discussion on a similar resistance “monstrous” conceptualization of function.
- 3.
Here and elsewhere in this chapter it is essential to understand the phrase “learn mathematics” in the sense described earlier, that is, in accordance with the Knowledge of Mathematics Premise and the definition of learning presented earlier.
- 4.
Since the discussion of pedagogical considerations follow the discussion of historical phenomena, it is important to state our belief that the intellectual necessity for a learner need not—and in most cases cannot—be the one that occurred in the history of mathematics.
- 5.
Pseudonyms.
- 6.
Other solutions were offered by the class. For example, one solution examined all the possible cases for the size of the quilt, and another solution simply calculated the number of white squares by subtracting the number of black squares from the total number of squares.
- 7.
This transition involved interesting cognitive disequilibria, which are not discussed in this paper.
- 8.
The use of the term “definition” here should not imply that the students’ intention was to define—in the mathematical sense of the term—the concept “\( \text{e}\) to the power of a matrix” (see the discussion on definitional reasoning).
- 9.
More precisely, intellectual need cannot be determined independently of what hypothetically satisfies it. The added qualification (“hypothetically”) is needed, for otherwise this claim would mean that the experience of disequilibrium over famous unsolved problems such as the Riemann hypothesis would not be describable.
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Harel, G. (2013). Intellectual Need. In: Leatham, K. (eds) Vital Directions for Mathematics Education Research. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6977-3_6
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