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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 ...
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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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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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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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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 ...
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?
(...
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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....
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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
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 (...
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 ...
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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 ...
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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 ...
5
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 ...
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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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I think the term research causes a lot of confusion here. When mathematicians think about research, we immediately think about proving new theorems.
But that doesn't seem to be what's asked here. As I read the outline, the end product is supposed to be a research proposal, not research itself. That is, students are asked to propose a suitable question ...
5
Perhaps a research project could be replaced by an independent study project. The idea is to learn more math, in a more independent way. The student would still have to produce something to show what they've learned (although if the teacher found it easier, they could instead require an oral explanation of the content learned).
I learned a lot on the ...
4
I believe Discrete Mathematics, Logic, and Calculus to Electrical Engineers.
How much has to be studied in Calculus, and how the curriculum is to be arranged, however, is a tough question.
The current educational system in my country teaches Electrical Engineers the following:
Year 1: Linear Algebra, Matrix Theory, Differentiation and Integration, Logic
...
4
There will be an application form on the website during the application season. It gets removed during the rest of the year to make sure nobody wastes time filling it out when there's no hiring going on, but this ought to be clarified on the website. (I'll look into fixing this. Full disclosure: I've been involved with PROMYS in different ways over the ...
4
Yes, however there aren't lots of great resources. For instance, I have a friend who taught a combinatorial game theory course that turned into research for many students, but I don't think he wrote anything up as a resource for "how to do it".
So I don't have any great insight into this, but there are a some papers in PRIMUS you may find useful if you can ...
3
There's an important distinction to make. It's not clear to me if your friend is failing to really learn at all (reading about abstract topics, feeling satisfied and not doing any problems at all, but not really grasping the subject---I know I'm prone to this mistake), or if your friend is in fact learning in his preferred way, but is struggling with ...
3
Some local (or not) businesses may have fairly mathematical internships. For instance, a biotech company may have nontrivial graph theory or other discrete computational problems to solve that will require something analogous to math research, though without proofs. (I'm not making this example up.)
Certain computer science internships would be heavily ...
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Assuming that one of the disciplines involved was mathematics and you have some novel insights or ideas about how to create such a course, PRIMUS is a good generalist option, especially if you can "aim" the article at other undergraduate faculty who might be interested in creating such a course but aren't sure where to start.
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Student Research Projects in Calculus
Cameos For Calculus
I particularly like the first one because the authors include with each project a description of how long it may take a student, any issues they've encountered, and (sometimes) how it can be explored further.
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