Category: Full Article

How many sizes of infinity are there?

“O God, I could be bound in a nutshell, and count myself a king of infinite space – were it not that I have bad dreams.”

Hamlet, Act 2, Scene 2. Lines 252-254.

Recently, a colleague asked me:

I know that there are different sizes of infinity, but what I want to know is how many different sizes of infinity there are?

-Curious Colleague

This is a great question! I tried to explain my answer at the time, but it came out garbled and I think I confused him more than I helped. So this post is an effort to remedy that and answer his question.

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Review of Ungrading (Blum 2020) for use in UTM math courses

My brief, unfair summary

At its best, the book Ungrading (Blum, 2020) and its authors

  • Provide many blueprints, implementations, and reflections on ungrading practices.
  • Address many practical difficulties of implementing these practices.
  • Repeatedly make calls for revolution, and provide tools and recipes for making that happen.
  • Encourage critical pedagogy.
  • Come from a variety of disciplines and settings (K-12 vs post-secondary).

At its worst, the book and its authors

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Aligned teaching and punishing student success

This term I’m teaching (and coordinating) a 1200 person Integral Calculus course at UTM. Term Test 1 went surprisingly well for students, and Term Test 2 was an unqualified success (very high average, lots of people got 100% on the test).

This lead to the following (paraphrased) question on our course message board:

Is the exam going to be a lot more difficult because the averages on the term tests were so high?

I took the time to think about exactly why I don’t like adjusting the difficulty of exams based on students doing well previously. Here’s the answer I posted on our message board.

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How to organize an equitable pre-CUMC conference for students

Every summer, Canadian undergraduate students in mathematics meet at the Canadian Undergraduate Mathematics Conference (CUMC). Hundreds of students attend, and it gives them a chance to meet other people excited by mathematics. Students are also encouraged to give a short presentation on a math topic that interests them.

I attended the 2007 CUMC at Simon Fraser university and the 2008 CUMC at the University of Toronto (where I would go on to complete my PhD and then eventually work at).

In the summer of 2018, while I was a Post Doc at the University of Calgary, we hosted a “mini pre-CUMC conference” for undergrads to give their presentations ahead of time. It was so successful that I ran an expanded version of this at the University of Toronto for CUMC 2019.

I think these events and workshops are important for all students, but in particular it helps break down barriers to entry for marginalized students. With that in mind, I’m sharing my resources, thoughts and experiences about our pre-CUMC conference with the hope that other universities and colleges in Canada will benefit.

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My teaching advice and resources (so far)

As I general rule I find thinking about math pedagogy deeply rewarding. Teaching a technical and beautiful discipline like math is difficult to do well. Students come from all sorts of backgrounds, the material can be challenging, and there are tons of moving parts in a course. It’s a challenge that I find exhilarating.

On the other hand, I find the act of reading the scholarship of math education to be dreadful and unpleasant. It is filled with jargon and hero-worship.

That being said, I’ve been extremely lucky to have great mentors and colleagues to bounce ideas off of. I’ve collected some of this advice in a Reddit post, which I’ll recreate here.

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How does modern AI work? – Math for my mom

This is part of a series of posts aimed at helping my mom, who is not a scientist, understand what I’m up to as a mathematician.

 

Lately, Artificial Intelligence (AI) has made some remarkable milestones. There are computers that are better than humans at the strategy board game GO and at Poker. Computers can turn pictures into short moving clips and can “enhance” blurry pictures as in television crime shows. They can also produce new music in the style of Bach or customized to your tastes. It’s all very exciting, and it feels pretty surreal; remember back when Skype video calling felt like the future?

I’m going to give you a broad overview for how these types of AI work, and how they learn. There won’t be any equations or algebra.

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Stepping up your Math Game – 35 Summer Projects for Post-Secondary Math Students

Build your math muscles!

This summer invest in your math education.

Undergraduate students often ask me for advice for how to improve themselves as mathematicians. There are many answers to this question: you can focus on learning new mathematics, you can work on your programming skills, you can improve your career prospects by working on professional development, …

This post is the result of many conversations and coffees with undergrads; It is the advice that I would have given myself as a second-year undergrad at the University of Winnipeg. These 35 projects will also provide necessary skills for completing a Ph.D. in Mathematics and will increase your employability.

Some of these projects will take an afternoon (“Learn how to make a bibliography in LaTeX”) and some might take a month (“Make a Predator-Prey Visualizer”). There is no time limit and no test, so take your time and make it your own.

Table of Contents

  1. Math Skills
  2. Reading
  3. Math Projects
  4. Teaching
  5. Professional Development
  6. Other Skills

[B] Beginners

Some projects are marked with a [B]. These are especially well-suited for beginners.

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Future of Set Theory – Panel Discussion

The following is a (somewhat complete) transcription of a panel discussion held at the Fields Institute on April 1, 2015 as part of the “Forcing and its Applications Retrospective” workshop.

The speakers were

  • Stevo Todorcevic (University of Toronto)
  • Jindrich Zapletal (University of Florida)
  • Christina Brech (University of São Paulo)
  • Assaf Rinot (Bar-Ilan University)
  • Matteo Viale (University of Torino)
  • Justin Moore (Cornell University, Moderator)

I took notes, so please bear in mind that some of this is paraphrased. I apologize if I misrepresented anyone, and I am happy to make corrections if I am emailed.

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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.

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Measuring the (Seemingly) Impossible

(This talk was given as part of the Adventures in the Classroom lecture series at the University of Toronto on August 7, 2014. It aims to connect math educators with researchers in the academic math community. A video of this talk will be available later.)

We will introduce two fun projects involving Geodesy, which is devoted to measuring things to do with the Earth.

From Harold Llyod’s “Safety Last!”

“Without doing anything dangerous, measure the height of your school”

I love this question. It is a bit affronting (how could I possibly do that?!), but there’s something magical about it that draws you in and gets you using your imagination. If you have never thought about this question before, please take the time now. (Seriously, go for a walk and think about it.)

I have posed this question to many people and have received many different solutions. When I talk to people about this problem I am struck by what solution they think is the most “obvious” or “natural” solution. Think about that as we examine this problem.

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