(This was the basis for a talk I gave at the Toronto Student Set Theory and Topology seminar on January 15, 2013. The assumed knowledge is an undergraduate course in general topology. This is only a draft, and will be updated soon.)
There are many questions in mathematics and sciences in general whose main object of study is the topological group. These objects are very versatile and can represent many of the structures we encounter. One question that I’ve been working on examines the (extreme) dynamical properties of topologies on the integers. On the recommendation of Vladimir Pestov (one of my advisors) I have been learning about T-sequences, which provide a rich method for producing topological groups with extreme behaviour. Here I will present two techniques involving T-sequences that help to answer two different questions about topological groups; one is about dynamics and the other is about combinatorial properties of . These results all come from “Topologies on the Integers” by Protasov and Zelenyuk.
A study was done at the University of Chicago looking at whether math can cause physical pain. (The rather laughable title of the article is “Doing Math Really Does Make Your Head Hurt, Says Science”. Here‘s the actual published study.)
The idea is that for people with high levels of math anxiety, just knowing that they are about to do a math problem is enough to make the pain part of their brain go off. The actual solving of the problem wasn’t such a big deal. This suggests that presenting math is an important part of teaching it and discussing it.
Some other anecdotal examples are that people are notoriously good at calculating retail discounts – “It will be 12.50 for that 25 dollar shirt that is half off”. Compare this to the more difficult “What is 25 divided by 2?” or even worse “What is 25 x 0.5?”.
Similarly compare these two questions:
After the half-hour 6 o’clock news you watch an episode of Jeopardy, then Survivor, then two episodes of Wheel of Fortune. You read until it’s a quarter to 10 and then go to bed. How long did you read before you went to bed?
Here’s an easy question about fractions: Solve 6 + 1/2 + 1/2 + 1 + 2 (1/2) + R = 10 – 1/4.
I got a little bit anxious looking at that second question.
What experiences do you have about teaching people with math-anxiety?
The idea is that if someone has only ever taken high-school math, they might think that mathematicians spend their time multiplying and dividing very large numbers; (“Are you even allowed to use calculators in your work?“). If someone has been lucky enough to take a first-year calculus course they might imagine that mathematicians solve harder and harder related rates questions. Going a step further, telling a mathematician that I study set theory often invokes brow-furrowing and questions like: “Don’t we already know everything about ordinals and cardinal arithmetic?“.