Mike Shulman
• Member for 7 years, 10 months
• Last seen more than a week ago

The Moore method is used at the University of Chicago in some sections of "Honors Calculus", which is really an introductory real analysis course for top incoming freshmen. I assisted with it a ...

In my experience, the "set of ordered pairs" is a difficult and confusing definition. Moreover, I feel it's almost totally unnecessary because in practice in mathematics we do always treat functions ...

I think it depends on your goals with your calculus class. If your goal is primarily to prepare students to apply calculus in other subjects, then perhaps counterexamples are not so important. And ...

My strongly held opinion is that some exact solutions are conceptually fundamental. Knowing that $\sin 45^\circ = \frac{1}{\sqrt{2}}$ (rather than approximately 0.7071) may not be important for ...

One technique which is fairly obvious, but (at least for some of us) surprisingly difficult to implement consistently, is to just model for them in class what you expect them to write on their own. ...

Here's one that I just thought of, by modifying a problem in the ODEs section of my textbook. Question: In the presence of air resistance, does a thrown ball take longer to go up or to come down? We ...

I'm surprised not to see any mention yet of constructive type theory and computer proof assistants, in which a computer program is literally a proof and vice versa. For instance, if you wrote a ...

I am using a form of "peer instruction" in my university calculus classes for the first time this year. I basically use "clicker questions" such as those available from carroll college. I don't use ...

The other answers may indeed be right, but another thing just occurred to me, namely that when they prove the base case of the induction, the sum on the left-hand side does generally reduce to a ...

One answer is to regard this question as an instance of a more general question about whether to build embedding theorems into the foundations of a subject. Should we define abstract manifolds, or ...

An example that I once started a calculus class with was how to build a reflecting telescope. Imagine the light from a distant object arrives as parallel rays, and you want to build a mirror that ...

There is a long list of definitions of "affine space" here; I don't know whether any of them fits your needs.

This idea is not (yet) supported by experience, but one thought that occurs to me is to emphasize the "infinitesimal" definition of limit: $\lim_{(x,y)\to(0,0)} f(x,y) = L$ if $f(x,y)$ is infinitely ...