Intuiting and formalizing: coextensive processes

Luis Moreno Armella

Abstract: There is a force that goes through the teaching of Calculus: The tension between
the intuitive and the formal. Calculus continues to be taught as if it were natural
to introduce the study of change and accumulation by means of formalized concepts
known as the mathematics of e and d. It is frequently considered as a failure that
“students still seem to conceptualize limits via the imagination of motion.” This kind of
assertions shows the tension created by traditional teaching between students’ intuitions
and a misdirected formalization. The internal connections of the intuition of change and
accumulation are not correctly translated into that arithmetical approach of e and d.

Keywords: intuition, formalization, infinitesimal, analog cognition, symbol, metaphor,
embodied knowledge, digital medium, mediation.



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