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Dynamic visual models: ancient ideas and new technologies. (English) Zbl 1443.00012
Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 189-201 (2020).
Summary: We provide dynamic visual models of the following facts established by ancient mathematicians:
1.
$$\sum_{i=1}^n(2i-1)=n^2$$, $$n\in\mathbb{N}$$,
2.
$$\sum_{i=1}^ni=\frac{n(n+1)}{2}$$, $$n\in \mathbb{N}$$,
3.
$$\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}$$, $$n\in\mathbb{N}$$,
4.
$$\sum_{i=1}^ni^3=\left(\frac{n(n+1)}{2}\right)^2$$, $$n\in \mathbb{N}$$.
We contrast the clarity of the models by outlining formal mathematical proofs based on those timeless ideas. We also reflect about the place that proofs play in the calculus classroom.
For the entire collection see [Zbl 1442.00024].
##### MSC:
 00A35 Methodology of mathematics 97E50 Reasoning and proving in the mathematics classroom
##### Keywords:
visual models; mathematical proof; teaching calculus
##### Software:
Wolfram Demonstrations; GeoGebra
Full Text:
##### References:
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