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For questions about teaching students at the undergraduate (university) level.

6 votes

How do I become a Scarer?

I do not think you can change the opinions of professors who hold these views. All you can do is be a human: interact with undergrads, and let them know that you do not think inherent ability is nee …
34 votes

Epsilons and deltas in a first calculus course

For some reason there is a wide spread view that the $\epsilon-\delta$ definition of a limit is an obscure thing, relevant only to mathematicians, and that the only reason to care about them is to mak …
8 votes

Should we teach functions as sets of ordered pairs?

I think as long as students know that: A function $f:A \to B$ assigns one and only one element of the codomain $B$ to each element of the domain. We write $f(a) = b$ if $a \in A$ is assigned to $b …
12 votes

Should we teach abstract affine spaces?

I am sure you know this, but the Euclidean plane is a prime example. The Euclidean plane starts as just a collection of points together with a group of isometries. There is no natural origin. Howev …
31 votes

Why is the concept of injective functions difficult for my students?

I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty. The following three statement …
10 votes

How to teach if calculations and algebraic manipulations are off limits

I highly recommend looking at the Calculus exams from the University of Michigan. These problems tend to focus on extracting relevant information for sol …
2 votes

Where can I find hard exercises for logic?

Here are all of the textbook chapters and assignments related to logic which I wrote for my Discrete Mathematics course last semester. We spent about half of the course time (8 weeks) on this content …
2 votes

Motivating students to take homework seriously without grades

I think my system solves your problem Assign homework due by date $X_1$ and time $t_1$. The submission is accepted electronically until that date and time. The solutions to this homework are automa …
7 votes

Alternatives to University Lectures: Non-lecture Mathematics Classes

Jim Fowler and I created a hybrid linear algebra/multivariable calculus/multilinear algebra course at The structure of the course was: $\mathbb{R}^n$ as an inner p …
9 votes

What's the most effective way to introduce/motivate the anti-derivative of $\sec x$?

In my opinion, a good story to tell about techniques of integration in general goes something like this: We know that some functions (like $x^2$) have elementary antiderivatives, and some (like $e^{ …
10 votes

What's the most effective way to introduce/motivate the anti-derivative of $\sec x$?

To expand on Gerald Edgar's comments above: Assume that you have already covered partial fractions. Develop a general strategy for integrals of powers of trig functions: If a power is odd, you can …
14 votes

Do you mention the continuity and the differentiability of the empty function

I think that this would be too much of a detour in a regular Calculus class. You would need to first establish that the empty function is even a function. This requires a really pedantic reading of t …
7 votes

What are the essential mathematics skills that uni based STEM math educators want high schoo...

This is entirely opinion based, but I don't really care what content is "covered" at all. I care that students are engaged with thinking about problems which involve quantitative and spatial reasonin …
4 votes

Unique steps leading to a non-unique answer

A "non-mathematical" answer: If you are in need of a doctor, then one unambiguous algorithm for finding one is to call everyone in the phone book, in order, and ask them if they are a doctor. Eventu …
2 votes

Encouraging students to see value in things that can't be measured

Take this advice with a grain of salt, as I have not implemented anything like it yet. I have been thinking a lot about the responsibility that we have, as mathematics educators, to not only give stud …

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