id: 06639211
dt: j
an: 2016f.01167
au: Soto-Johnson, Hortensia; Hancock, Brent; Oehrtman, Michael
ti: The interplay between mathematicians’ conceptual and ideational
mathematics about continuity of complex-valued functions.
so: Int. J. Res. Undergrad. Math. Educ. 2, No. 3, 362-389 (2016).
py: 2016
pu: Springer US, New York, NY
la: EN
cc: I85 C35
ut: complex-valued functions; conceptual mathematics; continuity; ideational
mathematics; mathematicians
ci: ME 2003d.02820
li: doi:10.1007/s40753-016-0035-0
ab: Summary: Adopting {\it N. Sinclair} and {\it M. Schiralli}’s [Educ. Stud.
Math. 52, No. 1, 79‒91 (2003; ME 2003d.02820)] 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, 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.
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