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.

Delta-Epsilon Magic

2022 update: I’ve added a couple of sentences about the context of this event to emphasize how positive this experience was.

In my second year of undergrad I had a formative experience with Delta-Epsilon proofs that stuck with me for a long time. Last week I was able to provide a similar experience for some first year calculus students.

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?

“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?“.

Is Algebra Necessary? – Andrew Hacker

Today on The Sunday Edition, on CBC Radio 1, host Michael Enright interviewed Andrew Hacker, about his controversial views on mathematics education. Basically, he thinks that demanding that all high school students master Algebra is “an onerous stumbling block for all kinds of students”. Listening to the interview this morning made me angry. I don’t get riled up easily, but my blood was boiling this morning.

Here is the link to the radio interview. (It starts around the 6:30 mark and is about 24 minutes long.) Here is the article Andrew Hacker wrote for the New York Times.

I intend to write a detailed response to the claims of Andrew Hacker later this week, but first let me share some reckless assertions that pissed me off. (These are paraphrases.)

“Should we be learning mathematics for learning’s sake? Don’t get me wrong, I’m a liberal-arts professor; I teach multi-disciplinary courses. However, I think that we should be focusing on learning about mathematics: the history of mathematics, the philosophy of mathematics. Not the dry stuff.”

“Hacker: 1.7 million Americans enter American universities every year, all required to have some high school mathematics. How many do you think choose to major in mathematics?

Micheal Enright: I don’t even want to guess.

Hacker: 15 Thousand. This tells me that even though teachers have had the chance to show the students the ‘beauty of mathematics’ they just aren’t getting it. [Goes on to imply that it’s because this is too hard to do.] Why are we training all of these students in high school when so few are going into STEM [science, technology engineering and mechanics] careers?”

“Do people need to learn about Fermat’s Theorem [sic] or Rye-man’s Hypothesis [sic]?”

I agree that something is broken with the way we learn, teach and perceive mathematics, especially in middle years and high school. This is a conversation we must have. However, it is troubling when the social scientists try to use their irrelevant authority to persuade the masses, leveraging the common man’s bias against mathematics.

I’m mad. I don’t like when people piss on what I love, and I especially dislike when they do it while tricking the public. As people we need emotions: love, anger, joy, contempt; but as a people we need thought, analysis, reason and mathematics to universally prevail.

Again, I plan on writing a cool-headed, but strong reply, and this is just me venting. Until then, let me leave you with this, taken from JFK’s famous speech:

“But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic? Why does Rice play Texas? We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.”

Every Day I’m Simulating

Have you ever tried to explain/inflict forcing on your non-set theory friends? Let me tell you it is hard. In my ongoing effort to try to explain everything to everyone here is my attempt at explaining the idea of forcing, without explaining forcing.

First, I direct you to this post which explains the simulation argument.

tl; dr – Imagine that our civilization could simulate (artificially) intelligent civilizations. They wouldn’t know that they are in a simulation and could also run their own simulations.

Reading this post reminded me about forcing. Here is my reply to the OP (who happens to be my brother-in-law), and I would appreciate feedback on this so that I can refine my analogy:

Cool. This illustrates a key observation that underlies a lot of the mathematics (set theory) that I do. It goes like this for the interested parties:

We run a simulation as you’ve described, but we make sure that the simulation only has “a small number of things in it”. Perhaps we have some sort of minimal simulation, like we don’t include the letter Z in their languages or something (Call this SIM1). Now we check that the simulation can come up with its own simulations. It can? Great! So now we know that “having the letter Z in your language” is not a requirement for coming up with simulations. Or we could add a whole bunch of new crazy letters (in say SIM2) and see if they can still run simulations. Lets say they can still come up with simulations. Continue reading Every Day I’m Simulating