Teaching

Whenever I award a student a course grade of A+, I write a congratulatory email and ask for advice on how to succeed in my classes, which I distribute without modification to students the next time I teach that course.

If you find it difficult to relate to advice from A+ students, here are some unsolicited thoughts from a student in Fall 2021: “First and foremost I'd like to to thank you for a wonderful first semester at university, Math 157 was a great way to kick off the year for me. I've never been good at math and as a result I never really liked the subject in high school but your passion for mathematics proved to be nothing less than contagious. Heading into this course, I admit grades were my main (if not only) focus but it quickly became second to working towards figuring out the intuition behind concepts and really understanding the content. It gradually made sense to me why you kept encouraging us to focus on the underlying concepts instead of solving a specific kind of question in a specific way. It wasn't entirely inspiration that drove me, however, the 40% on my first midterm was motivating to say the least. Watching myself improve over the semester and becoming increasingly confident with the course material was a very rewarding experience and really made the studying worth it. I still am very surprised I got a B- because I was seriously expecting something in the C to D range, even with adjustments. I held off on writing this email until after the letter grade came back for fear of it seeming disingenuous so thank you for filling in the gaps in my learning caused by the pandemic as well as teaching me to think about math in a new and drastically less tedious way.”

A calculator is broken so that the only keys that still work are the sin, cos, tan, sin^-1, cos^-1, and tan^-1 buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
Sal the Magician asks you to pick any five cards from a standard deck. You do so, and then show them to Sal's assistant Pat, who places one of the five cards back in the deck and then puts the remaining four cards into a pile. Sal is blindfolded, and does not witness any of this. Then Sal takes off the blindfold, takes the pile of four cards, reads the four cards that Pat has arranged, and is able to find the fifth card in the deck (even if you shuffle the deck after Pat puts the card in the deck). Assume that neither Sal nor Pat has supernatural powers, and that the deck of cards is not marked. How is the trick done?

Six years after completing MATH 480W, a student wrote to me: “Hope you're well! I'm an old student from a few years back, I took your Art and Craft of Problem Solving course and the Putnam a few years back. I'm doing my PhD now in Germany, and was just thinking about your course, while making a diagram of a complicated problem I'm trying to solve. Honestly, I think it was my favourite course I ever took at university, and I still use the strategies I learned all the time. It also gave me a lot of confidence in my problem-solving abilities which really contributed to me being brave enough to take on a doctoral position. So I just wanted to say thanks. Thank you!”