Mike Shulman
  • Member for 7 years, 10 months
  • Last seen more than a week ago
What happened to the Moore method?
22 votes

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 ...

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Should we teach functions as sets of ordered pairs?
19 votes

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 ...

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Counterexamples in first year calculus
Accepted answer
18 votes

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 ...

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Answers in exact form (e.g. including radicals) vs. Decimal Approximations
Accepted answer
17 votes

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 ...

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Making standards for "showing work" explicitly clear to students
15 votes

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. ...

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Physical applications of higher terms of Taylor series
12 votes

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 ...

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"A computer program IS a proof": Introducing rigor via programming
9 votes

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 ...

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Is Peer Instruction suited to mathematics classroom?
6 votes

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 ...

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Why do students only see the last term of a sum abbreviated with an ellipsis?
5 votes

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 ...

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Should we teach abstract affine spaces?
4 votes

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 ...

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Natural, rich, calculus questions
3 votes

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 ...

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How to teach affine geometry to future high-school teachers?
2 votes

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

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Should we teach functions as sets of ordered pairs?
2 votes

Here is a comment from a colleague: defining functions as sets of ordered pairs may not be so important on its own, but it's a convenient way to learn that functions are a special kind of binary ...

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Multivariable limits
1 votes

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 ...

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