The Interplay Between Mathematicians’ Conceptual and Ideational Mathematics about Continuity of Complex-Valued Functions

Abstract

Adopting Schiralli and Sinclair's notions of conceptual mathematics (CM) and ideational mathematics (IM), we investigated mathematicians' reasoning about continuity of complex-valued functions. While CM centers on formal mathematics as a discipline, IM focuses on how an individual perceives formal mathematics. There were four IM notions that the mathematicians used to convey the idea of continuity for complex-valued functions: control, topological features, preservation of closeness, and paths. The mathematicians' IM tended to be grounded in their embodied experiences and espoused for pedagogical reasons, in preparation for other actions, or to assist their own reasoning. Some of the mathematicians' IM metaphors conveyed a domain-first quality, which accounted for the domain of the function before mentioning any objects from the codomain. Given such metaphors did not capture the full structure of the epsilon-delta definition of continuity, the mathematicians transitioned to CM language in an effort to make their IM statements more rigorous. Our research suggests that while IM metaphors stemming from embodied experiences can serve as helpful tools for reasoning about continuity of complex-valued functions, one must be cognizant of ways in which the informal IM must be altered or extended to fully capture the CM. Given the pedagogical intent of many of the participants' domain-first IM examples, Int.

What enabled the production of mathematical knowledge in complex analysis?

“…Additionally, in the discipline there are studies such as those by Hanke (2020) and Soto-Johnson et al (2016) that aim to understand how professional mathematicians perceive and communicate different mathematical tools and concepts of CA to their peers. This kind of research has revealed that attributing geometric meanings to purely algebraic expressions is not a trivial task.…”

“…Studies in Mathematics Education have been carried out to address this type of problems. For example, some research has incorporated digital technologies to attend to different mathematical objects inscribed in the theory of complex functions (D'Azevedo & Dos Santos, 2021;Dittman et al, 2016;Ponce, 2019), while some other studies investigate how undergraduate mathematics students and professional mathematicians work geometrically with them (Hanke, 2020;Oehrtman et al, 2019;Soto & Oehrtman, 2022;Soto-Johnson et al, 2016).…”

Effectiveness of Digital Pedagogy in Higher Mathematics Education

“…For example, using examples (Durand-Guerrier, 2016;Lockwood et al, 2016), using graphical images (Zhen et al, 2016), using analogy (Dawkins & Roh, 2016), and using metaphor (Durand-Guerrier, 2016), researchers have tried to materialize mathematical abstraction. Some researchers have tried conceptual and ideational reasoning (Soto-Johnson, Hancock, & Oehrtman, 2016), metalinguistic and mathematical reasoning (Dawkins & Roh, 2016), procedural and conceptual reasoning (Bagley & Rabin, 2016), syntactic and semantic reasoning, cognitive and metacognition reasoning (Mejía-Ramos, Weber, & Fuller, 2015) to materialize proof oriented mathematics meaningfully.…”

Student difficulties with complex numbers

“…Previous empirical studies explore secondary students' [1], undergraduate students' [2][3][4][5], prospective and inservice secondary teachers' [6][7][8], and experts' [9] algebraic and geometric understanding of complex numbers in mathematics and engineering contexts. Some of these studies cover calculational aspects of complex number fluency we discuss including complex algebra [2] and changing between [8], and appropriately selecting [4], forms of complex numbers.…”

“…Some of these studies cover calculational aspects of complex number fluency we discuss including complex algebra [2] and changing between [8], and appropriately selecting [4], forms of complex numbers. While one study included a physics expert [9], we were unable to identify studies which focus on undergraduate physics students' understanding of complex numbers.…”