# Tag Info

### Unique candidate that fails

The obvious example I immediately thought of is that, if the divergent geometric series $$1 + 2 + 4 + 8 + \dotsb = \sum_{k=0}^\infty 2^k$$ converged, it would converge to $-1$. Proof: If the series ...
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Accepted

### What's a replacement for "married couples" in combinatorics problems?

I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.
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### How to explain Monty Hall problem when they just don't get it

For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me. Rather than talk about probabilities as fractions, I explain it this way: "If you ...
• 1,214
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### How rigorous should high school calculus be?

Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, ...
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### Lecturers "(intentional) mistakes" as a teaching tool

Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas. However, this does not mean that you should come to the lecture ...
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### Why is it possible to teach real numbers before even rigorously defining them?

It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. ...
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There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts ...
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### Are there science-backed effective teaching strategies?

In terms of controlled experiments, then, yes. Note that most are opposite or orthogonal to virtually all pedagogical norms in math education. Active recall. "Put away all your notes and ...
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### Can mathematics be learned by ONLY solving problems?

Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described ...
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