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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” was. By going to seminars, reading papers, and talking to other graduate students, I gradually began to catch on.
Within any field, there are certain theorems ...
18
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 the problems you mention. I highly recommend it, some of the articles are even available online if you google for them.
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It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.
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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:
Dickman, B. (2017). Enriching Divisibility: Multiple Proofs and Generalizations. Mathematics Teacher, 110(6), 416-423. Pre-Print (no pay-wall).
(Adapted from a ...
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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 of boredom, lost motivation, exciting learning, disappointment and enthusiastic crankery. I can tell you that taking more interesting courses and finding good ...
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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 are a few from
that list:
Are there an infinite number of primes $p$ such that the repeating
part of the decimal expansion of $1/p$ has length $p-1$?
Link.
...
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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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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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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 term.
Students often flounder when they don't understand quite what they should be doing. Research is hard to figure out, and students often don't know how to ...
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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 examples.
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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 convince your reader that what your reasoning is correct (and not bore them to tears in the process), and (hopefully) convince them that it is relevant. It very much ...
9
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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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 engineering. A sampling of other topics includes:
Boolean algebra (digital electronics)
Graph theory and algorithms (networks)
Probability and statistics (...
8
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 these words/terms and axiomatizing them would make for a nice textbook, but I digress...
Back to the question: Some examples.
What should "bigger than" mean?
(...
8
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 tangent circle instead of a tangent line?"
They started by finding a formula for the center and radius of a circle given three non-colinear points:
https://www....
7
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 "$\infty=\infty+1\Rightarrow 0=1$" dilemma. Let him resolve the apparent contradictions on his own; he'll either conclude that infinity isn't a number, or make up some other algebraic ...
7
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 institution). Obviously not all have the same quality, but a lot of them have high-quality coursework, with assistantships of various kinds. I regularly see ...
6
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 provide a counterexample.
If $K_1$ and $K_2$ are isomorphic subgroups of $G$, then $G/K_1$ is isomorphic to $G/K_2$.
If $\varphi:G\rightarrow H$ is a group ...
6
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 exercises/test them" cycle isn't for everyone. But it is the only accepted way to select the few whith the abilities to go farther. Yes, there have been exceptions, ...
6
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 true craftsmen start off by using existing tools before they design their own, and they're probably more competent for all that as well.
So there should be no ...
6
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 someone up to speed" mean that you want the student to learn the background, then I believe this would be very hard unless your student is very bright and ...
6
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, Guy.
The fact that games can be solved by mathematics, and that mathematics can be built out of games, is the perfect kind of ridiculous abstract concept for ...
6
Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.
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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 that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience.
Now, I'm not entirely ...
6
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 Mathematics,
Students who lack understanding of a topic may rely on procedures too heavily. Without
a flexible base from which to work, they may be less likely ...
6
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, my personal opinion has no value. I need academic articles/books written by researchers of mathematical education supporting this idea.
Is there an agreed ...
5
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 him or her with someone who can spare a little time?
In the US we have a set of Research Experience for Undergraduates Programs. That might be a way to go. If ...
5
In line with your identification of graph theory, I suggest you might look into
what is now known as "Discrete and Computational Geometry."
Although there is much to learn (there is a 1500-page Handbook of Discrete and Computational Geometry, CRC link, soon to be expanded), one can reach frontiers in narrow areas
without mastering the whole field. In ...
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