## Abstract

This article describes emerging perspectives on contextualizing, complementizing, and complexifying—three processes involved when individuals ascribe meaning to mathematical objects of their thinking. The article is oriented toward a dialectic between theory and empirical research and is structured in two parts. The first part focuses on an evolving theoretical framing that acknowledges the significance of these three processes in mathematical cognition. In the second part, the evolving theoretical framing is used to analyze one student’s knowing of the limit concept of a sequence. This analysis directs one’s attention to the emergence and function of this student’s knowledge resource, which was generic in usage and complex in structure, allowing the activation of productive ideas and contextual meaning-making as needed. Through this analysis, theoretical and interpretative possibilities were generated that inform research on mathematical cognition and elucidate the emerging theoretical perspectives of contextualizing, complementizing, and complexifying.

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## Notes

- 1.
The term “knowledge resource” is used in the sense of Smith et al. (1993), designating “any feature of the learner’s present cognitive state that can serve as significant input to the process of conceptual growth” (p. 124).

- 2.
Duval (2006) revealed students’ confusion of a representation with the object that is being represented, substantiated with what he called a “cognitive paradox”: “how can they [individuals] distinguish the represented object from the semiotic representation used if they cannot get access to the mathematical object apart from the semiotic representation?” (p. 107).

- 3.
The subscript F indicates that the terms sense

_{F}, reference_{F}, thought_{F}, and idea_{F}refer to Frege (1892b). - 4.
For a discussion of generative and convergent approaches, see Clement (2000).

- 5.
The case study presented here has been subject of previous reports including Pinto and Tall (2002), Gray et al. (1999), and Tall et al. (2001). In those reports, the focus was on describing, exploring, and elaborating the strategy of giving meaning, often contrasted to extracting meaning. Further, those reports remained on a descriptive level when speaking about the student’s “mental actions with a mental object” (see Pinto & Tall, 2002): “Chris interprets the definition in terms of his old knowledge, explores the concept through thought experiment and reconstructs his understanding of the concept definition” (p. 5). The present article outstrips these reports by providing an explanatory account of the nature and function of the student’s knowledge resource as discussed here—an account that goes well beyond the descriptive approaches of previous reports.

- 6.
For a recent discussion on the ‘generic use’, see Yopp and Ely (2016).

- 7.
This is similar to Mason’s (1989) proposal that the essence of abstraction is coming to look at something differently, but differs as it is not so much a shift of attention but an expansion of attention.

- 8.
We use the notion of “conceptual unity” to stress the idea of coordinating diversity rather than looking for similarity. It shares Barnard and Tall’s (1997) view of a “cognitive unit” as “a piece of cognitive structure that can be held in the focus of attention all at one time” (p. 41), but extends their formulation as it suggests the emergence of a transcendent unity when two or more diverse ideas

_{F}are coordinated. - 9.
For a detailed account of conceptual blending, see Fauconnier and Turner (2002).

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## Acknowledgments

We are grateful to Annie Selden for her thoughtful comments and helpful suggestions given throughout the development of this paper. The first author wants to thank for support of this work both the Foundation of German Business through the Klaus Murmann Fellowship and Macquarie University through the Research Excellence Scholarship.

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Scheiner, T., Pinto, M.M. Emerging perspectives in mathematical cognition: contextualizing, complementizing, and complexifying.
*Educ Stud Math* **101, **357–372 (2019). https://doi.org/10.1007/s10649-019-9879-y

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### Keywords

- Mathematical cognition
- Meaning-making
- Theory advancement
- Giving meaning
- Limit concept