id: 05533946
dt: j
an: 2009b.00191
au: Doorman, Michiel; van Maanen, Jan
ti: A historical perspective on teaching and learning calculus.
so: Aust. Sr. Math. J. 22, No. 2, 4-14 (2008).
py: 2008
pu: Australian Association of Mathematics Teachers (AAMT), Adelaide, SA
la: EN
cc: D35 I45 I55 D45
ut: calculus; symbols; concept formation; teaching methods; thinking skills;
higher education
ci:
li:
ab: Summary: Calculus is one of those topics in mathematics where the
algorithmic manipulation of symbols is easier than understanding the
underlying concepts. Around 1680 Leibniz invented a symbol system for
calculus that codifies and simplifies the essential elements of
reasoning. The calculus of Leibniz brings within the reach of an
ordinary student problems that once required the ingenuity of an
Archimedes or a Newton. One can mechanically "ride" the syntax of the
notation without needing to think through the semantics. Calculus
education typically has a strong routine aspect, focusing on methods
for differentiation and integration without justifying these methods,
since current teaching practice barely has time to discuss the
underlying concepts. A question for the design of a teaching trajectory
that focuses on ways to support the understanding of the underlying
concepts is: How can students invent this? It is useful to look at the
history of a topic to gain insight into this issue, to investigate
concept development, and to analyze how and why people tried to
organize certain phenomena without having any notion yet about the
basic principles of calculus. In this paper, the authors will first
review some highlights in the history of calculus. This review will
lead into recommendations for an instructional sequence on calculus.
The authors conclude with a plea for historical reflections in
mathematics education as a method for changing routine-oriented
practices. (Contains 5 figures.) (ERIC)
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