Specifying Defining, Generalising and Abstracting Mathematically All Seen as Subtly Different Shifts of Attention

Mason, John (2023). Specifying Defining, Generalising and Abstracting Mathematically All Seen as Subtly Different Shifts of Attention. In: Bicudo, Maria Aparecida Viggiani; Czarnocha, Bronislaw; Rosa, Maurício and Marciniak, Małgorzata eds. Ongoing Advancements in Philosophy of Mathematics Education. Cham: Springer International Publishing, pp. 103–124.

DOI: https://doi.org/10.1007/978-3-031-35209-6_5

Abstract

Specifying, defining, generalising and abstracting are common mathematical actions. The claim being made is that each of these involves a delicate but subtly different shift in attention. Furthermore, the experience for students and teachers is different. In Mason (for the learning of mathemetics 1:8–12, 1980), it was suggested that symbols (signs) may be experienced initially as abstract entities in the sense of being unconnected to other experiences, but over time can become perfectly confidently manipulable (i.e. ‘concrete’) entities. In Mason (for the learning of mathemetics 9:2–8, 1989), it was suggested that abstracting mathematically involves a ‘delicate’ shift, not so much in what is attended to, but in how it is attended to. This has implications for choices of pedagogical actions to initiate when working with learners. Here, it is proposed that the acts of specifying, defining and generalising also involve delicate shifts of attention, subtly different from each other and from abstracting. Through the use of multiple examples, readers are invited to refine the distinctions they make in the form of their own attention, so as to work more effectively with learner attention.

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