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Math is not the most popular subject among students, especially in America. I have recently been thinking about giving a talk to undergraduate students encouraging them to do mathematics not because it is useful, but because it is beautiful/interesting/surprising etc. Since I feel most confident in topology and topology related fields, I would like the topic to revolve around topology/algebra, but other ideas for topics in other fields are also appreciated. The specifics are the following:

The audience: undegraduate students, a lot of them with only basic understanding of mathematics

Length: flexible, anything between $15$ minutes to $1$ hour

Purpose:

  • showing the beauty of math
  • overcoming students' fear of mathematics
  • getting people interested in math

What to avoid: I want to avoid the argument "math is useful." I want them to appreciate math the way they appreciate music, art, or a good book. The argument "math is useful" is a very dangerous one (for example, for some it puts an equal sign between the words "good" and "useful") and I want to avoid that (The question "should math be useful" is a topic for a different discussion)

There is a similar question already posted on here:

Link

and some of the ideas there are worth considering. However, my question is a little different since:

  • I do not have the time constraint as the author of the previous post
  • My audience may not necessarily consist of very intelligent students

Moreover, getting new answers to this question can help us all to get new ideas on how to persuade our students that math is beautiful and does not have to be dreaded.

Any help will be greatly appreciated.

Pawel
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    I gave a talk about the basics of Category Theory and realized I was being too ambitious. Since students are attracted to the fun-and-games aspect of mathematics, I would recommend a talk connecting basic combinatorics to some ideas in Algebra. – Yunus Syed Feb 01 '18 at 19:15
  • A little number theory, a little modular arithmetic, cryptography. Beautiful and even useful (despite Hardy's comment). – Ethan Bolker Feb 01 '18 at 19:25

3 Answers3

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Possibly tell the story about solving equations in radicals, starting with the quadratic equation, cubic and quartic (including the dispute between Tartaglia and Cardano), and eventually wrapping up with the story of life (and death) of Galois.

Try to sneak in some mathematics, but also liberal doses of human relationship, passion, heroism and tragedy - that is what might just do the trick, if you are a good storyteller.

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Maybe show something not only beautiful (like an idea can be beautiful), but also pleasing to the eye. Like aperiodic tilings (tilings encyclopedia), or any neat geometric construction.

EDIT: Regarding story telling - Kleenex getting a writ for using the pattern Penrose discovered is just great.

kai-dj
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Explaining to them how unrigorous but clever ideas can be used (such as Euler did to show $\sum_{k \ge 1}k^{-2}=\frac{1}{6}\pi^2$) can be used to sum series? There are many such tricks, including those mentioned in the article `Six ways to sum a series' by Dan Kalman. Regarding $\sum_{k \ge 1}k^{-2}=\frac{1}{6}\pi^2$, the basic idea is to use the fact sine has zeros at $x=\pm \pi$, $\pm 2\pi, \cdots$ and so on. This is not perfectly valid, but Euler then claimed or stated $\frac{\sin x}{x}=(1-\frac{x^2}{\pi^2})(1-\frac{x^2}{4\pi^2})(1-\frac{x^2}{9\pi^2})\cdots$. Then he used the Maclaurin series (for $\sin$) to conclude $-\frac{1}{6}=-\frac{1}{\pi^2}(\sum_{k \ge 1}k^{-2})$. More information, regarding this proof, should be available online.

Surajit131
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