To extend these findings we explored the relationships between mathematicians’ use of standard forms of mathematical communication and their personal reasoning about continuity of complex-valued functions, from C to C or equivalently from R 2 to R 2. ![]() Some empirical studies have explored students’ understanding of continuity of functions from R to R, and a few linguistic and theoretical investigations have explored mathematicians’ reasoning about these ideas. Given the pedagogical intent of many of the participants’ domain-first IM examples, we recommend that care be taken during instruction to deliberately elucidate where the IM is incomplete or fails to encapsulate the intricacies of the CM at hand. 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 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. 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. 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. 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. While CM centers on formal mathematics as a discipline, IM focuses on how an individual perceives formal mathematics. Adopting Schiralli and Sinclair’s notions of conceptual mathematics (CM) and ideational mathematics (IM), we investigated mathematicians’ reasoning about continuity of complex-valued functions.
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