24 votes

What do mathematicians call a proof?

From Bill Thurston: When I started as a graduate student at Berkeley, I had trouble imagining how I could “prove” a new and interesting mathematical theorem. I didn’t really understand what a “proof” ...
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18 votes
Accepted

What do mathematicians call a proof?

This collection of essays: Reuben Hersh: Experiencing Mathematics: What do we do, when we do mathematics?, Amer. Math. Soc., 2014 contains, among other topics, also lots of excellent discussion of ...
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16 votes

What are some of the open problems that can be suitably introduced in a calculus course?

It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.
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14 votes

Good problems that uncover difficult points in a theory

Edit (Dec 2016): Encouraged by a few comments on SE, and a few direct emails about this post, I wrote up the ideas below for a journal of math education. The citation, and linked pre-print, are: ...
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13 votes

How to nurture a good student?

I don't see my thoughts expressed by anyone here so perhaps I can chip in! I'm currently a graduate student but I was once that kind of bright bored student you are talking about. I had my fair share ...
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12 votes

What are some of the open problems that can be suitably introduced in a calculus course?

It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here ...
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11 votes

What are some of the open problems that can be suitably introduced in a calculus course?

You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.
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  • 6,171
10 votes

Examples of Research Level Math Discoveries Done by Undergraduate Students

You can point them to Involve, a journal all of whose articles involve substantial contributions by undergraduates. [Disclaimer: I am on its editorial board]
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10 votes
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How does an advisor effectively motivate progress on an independent project?

I've mentored roughly a dozen year-long undergraduate senior research projects, and I've always used a mix of the following techiques to keep students motivated. Set clear goals, both short and long ...
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  • 7,930
10 votes

What are some of the open problems that can be suitably introduced in a calculus course?

This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the ...
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  • 19.1k
9 votes

What do mathematicians call a proof?

I'd go first for the literature on mathematical writing, e.g. Knuth, Larrabee and Roberts' "Mathematical writing" (MAA, 1989). Mathematics writing is not just "proofs," in the end, you want to ...
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  • 12k
9 votes

How to get better at proofs

A great start to doing proofs is working through Daniel Velleman's How to Prove it: A Structured Approach, 2nd Edition.. I've used it many times in teaching, usually as a supplementary text.
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  • 2,008
8 votes
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What math courses should be taught to undergrad electrical engineers: a 40 years update

In my experience teaching undergraduate engineering students, key topics include at least some calculus, linear algebra and differential equations. Exactly how much depends on the field/subfield of ...
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  • 4,308
8 votes

Good problems that uncover difficult points in a theory

I think there are several good problems that can be explored (e.g., using the Moore method) by beginning with a word or term and trying to axiomatize it. I happen to think that assembling several of ...
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8 votes

Doing research projects when one's knowledge is limited: is it preferable?

I think it is very possible to have real, meaningful research projects at all levels. As an example, I had 2 of my students in Calc 2 work on a project which started with the idea "Can we define a ...
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7 votes

How to nurture a good student?

In defense of enthusiastic crankery, I feel it's very valuable to try to define "infinity" as a number. You can ask the student critical questions, like how to deal with the "$\infty=\infty+1\...
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7 votes

Grad school after doing an online bachelor's degree without support for undergraduate research

It seems like some of the other answers are aiming at PhD programs. I would suggest (as your question on academia.sx suggests) that you may wish to look at a Master's program (at a non-PhD-granting ...
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  • 5,752
7 votes

Seeking references for why it is good that students understand why mathematical rules work

The key term you are interested in is "conceptual knowledge" (more specifically, "conceptual understanding"). According to this document from the National Council of Teachers of ...
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7 votes

Seeking references for why it is good that students understand why mathematical rules work

I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view,...
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6 votes

Good problems that uncover difficult points in a theory

Here are a couple problems to shatter misconceptions about homomorphisms, while introducing the student to constructive thinking in group theory. Are the following statements true? Prove them or ...
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6 votes

Problem of a talented student who doesn't like to solve too much problems

Perhaps he should hang out in MSE a bit, and ask for confirmation/refutation/"this is well known, check ..." on results. It is a fact that the standard "tell people of some results/have them do ...
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  • 12k
6 votes

Problem of a talented student who doesn't like to solve too much problems

If He wants to be judged by his works , perhaps point out that being able to replicate other people's proofs isn't intrinsically any more original than just using the results of the proofs. Most ...
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  • 1,214
6 votes
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How to bring an undergraduate researcher up to speed on a brand new topic

If your student would like more background, then there is only one way to get that: Study a lot. There is a reason that it often takes years to get a research degree (Ph.D.). So, if you by "bringing ...
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  • 1,754
6 votes

How to nurture a good student?

The book that I found as a young student that propelled my mathematical career forward was On Numbers and Games by Conway, and then later Winning Ways for your Mathematical Plays by Conway, Berlekamp, ...
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  • 19.1k
6 votes

How to get better at proofs

Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.
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  • 6,171
6 votes

How to get better at proofs

Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is ...
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6 votes
Accepted

Ball rolling down curve simulator

The Energy Skate Park by PhET seems to come quite close: In the "Playground" section you can draw arbitrary curves and add a stopwatch. The simulation has to be manually stopped at the end ...
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  • 2,107
5 votes

How to nurture a good student?

The undergraduate-research tag suggests you know the answer already. Get this student doing something engaging and interesting. It doesn't have to be you that mentors this person. Can you help match ...
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