## Succeeding in First-Year Calculus: My guides to MAT135 and MAT137

Last year I wrote a guide for students taking the University of Toronto’s big (2000 student) first year calculus class MAT135. It was so successful that I wrote another guide to MAT137, the more specialized first year calculus class. Let me share them with you:

They are both links to Reddit, but you don’t need an account there to read them.

## Van der Waerden’s Theorem is false for $latex omega_1$

One of my main research problems involves something I think is related to arithmetic progressions in $mathbb{Z}$. After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on $omega_1$. We answered it, and the example was sufficiently nice that I would like to share it.

## Van der Waerden’s Theorem is false for $latex \omega_1$

One of my main research problems involves something I think is related to arithmetic progressions in $\mathbb{Z}$. After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on $\omega_1$. We answered it, and the example was sufficiently nice that I would like to share it.

## Secret Santa 4: The Surprise

After a long hiatus (7 months!) I am finally back to writing. This week I revisited an old problem that Sam Coskey told me a couple of years ago. Some of you will remember this “Secret Santa problem”, which I wrote about before. The problem is:

Sam’s Problem. Is it possible for two people to each choose a natural number so that both numbers are exactly 1 apart and neither person knows who has the larger of the two numbers?

## Math Anxiety

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:

1. 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?
2. 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?

## Perceptions about Mathematics

“When someone asks you what you do as a mathematician, they take the most advanced training they have in mathematics and assume that you do a much harder version of that.”

This is a paraphrase of something I will attribute to Leo Goldmakher.

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?“.