
06639211
j
2016f.01167
SotoJohnson, Hortensia
Hancock, Brent
Oehrtman, Michael
The interplay between mathematicians' conceptual and ideational mathematics about continuity of complexvalued functions.
Int. J. Res. Undergrad. Math. Educ. 2, No. 3, 362389 (2016).
2016
Springer US, New York, NY
EN
I85
C35
complexvalued functions
conceptual mathematics
continuity
ideational mathematics
mathematicians
ME 2003d.02820
doi:10.1007/s4075301600350
Summary: Adopting {\it N. Sinclair} and {\it M. Schiralli}'s [Educ. Stud. Math. 52, No. 1, 7991 (2003; ME 2003d.02820)] notions of conceptual mathematics (CM) and ideational mathematics (IM), we investigated mathematicians' reasoning about continuity of complexvalued 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 complexvalued 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 domainfirst 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 epsilondelta 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 complexvalued 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' domainfirst 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.