# What should be offered in undergraduate mathematics that's currently not (or isn't usually)? [closed]

What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.

Ex: Just to throw some ideas out there Mathematical Physics (for math students, not for physics students) Complexity Theory

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## closed as no longer relevant by Felipe Voloch, Bill Johnson, Andrés Caicedo, Mark Meckes, Yemon ChoiJan 8 '12 at 20:11

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I like the idea of promoting mathematics to students, i.e. more explanation of the contribution of mathematics to civilization, so that students have some language and background to justify their subject. See other articles on my web page.

On specific subjects, I have enjoyed showing first year UK students the power of the symbolic algebra packages on dealing with Grobner bases, i.e. solving polynomial equations in more than one variable. To more advanced students one can give exercises like:

Find a polynomial in x,y which has more than 5 critical points, classify them as max, min, saddle, and use the computer algebra package to draw your function and display or indicate the critical points. Verify with the package that the points you find are critical points.

(The last part gives useful lessons in rounding accuracy.) The whole exercise gives students a nice sense of power, as the machine manipulates vast expressions!

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Computing with Feynman diagrams in zero dimension, i.e., a graphical calculus for tensor contractions. It is very elementary yet can lead quickly into rather deep mathematics. It would make later studies in say mathematical physics or low dimensional topology much more congenial. Possible applications could be

• redoing a good portion of linear algebra see e.g.: http://arxiv.org/abs/0910.1362

• doing some basic representation theory following the formalism in the book by Cvitanovic: http://birdtracks.eu/

• projective geometry on the line and on the plane and some elimination theory, Bezout's theorem is very easy to understand in this language.

• computer graphics in the spirit of J. F. Blinn see, e.g., the account given in: http://www-m10.ma.tum.de/foswiki/pub/Lehrstuhl/PublikationenJRG/52_TensorDiagrams.pdf

• since many answers in this post are about asymptotics, another application is to compute the higher order terms of the Laplace method.

• combinatorial enumeration, e.g., a proof and examples of application of Lagrange inversion, explicit forms of the implicit function theorem, etc. etc.

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To have a hard time : not all the problems are easy to solve.

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The original question wanted a specific example or suggestion of a topic, not a general theme – Yemon Choi Jan 8 '12 at 20:09

Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc.

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But, please, with motivation. Understanding manifolds is THE hardest part of "elementary" advanced mathematics. Just introducing them by an abstract definition and then doing proofs by "local-global" handwaving doesn't do the job; the students will neither have an idea what the definition signifies, nor why the handwaving is allowed. Maybe some non-Euclidean geometry as a nontrivial motivating example would be useful... – darij grinberg Mar 15 '10 at 11:18
Read John and Barbara Hubbard's VECTOR CALCULUS,LINEAR ALGEBRA AND DIFFERENTIAL FORMS:A UNIFIED APPROACH,2nd edition,to see how it's done,Jacques. – The Mathemagician May 7 '10 at 4:56
I actually would love to see non-euclidean geometry (from a more elementary perspective) and differential geometry unified into one course. You always see elementary books which give you bits and pieces of the idea that there are other geometries, and that the sophisticated way of doing this is differential geometry - then much later in life you get thrown into a course where you pick up from multivariable calculus and define manifolds, tangent spaces, tensor fields, etc... There should be an attempt to show the connection between the two. – David Corwin Jun 16 '10 at 8:12
I agree, David. Even after reading tomes like Spivak's first couple volumes, it took me decades to realize that the classical euclidean and non euclidean geometries were just those Riemannian surfaces which were simply connected and of constant curvature. Thus the natural progression would have been to learn Euclidean geometry as flat geometry, then spherical/projective and hyperbolic geometry as universal constant non zero curvature geometry, then quotients of these as other constant curvature geometries, and finally more generally curved surfaces. Nikulin/Shafarevich is a good source. – roy smith Jun 23 '13 at 19:57

What should one teach to liberal-arts students who will take only one math course, and that because it's required of them?

The conventional answer: Partial fractions. And various useless clerical skills that they'll need if they take second-year calculus, although they'll never take first-year calculus. Et cetera.

E.g. in third grade you were told that $$3+3+3+3+3 = 5+5+5$$ and so on. Why should that be true? Assign that as a homework problem. At this point they may think that means there's some formula to plug this information into to get the answer. They've been taught that memorizing algorithms and applying them is what math is. That's a lie. We should stop lying and level with them.

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So why the negative vote? – Michael Hardy Jun 14 '10 at 20:16
I posed the question and asked for a course, not a rant on the substance of mathematics education for non-mathematicians: -1 – Michael Hoffman Jun 15 '10 at 23:48
There is no "the truth". It's much easier to blindly criticize than to offer constructive answers: -1. – André Henriques May 11 '11 at 18:02
+1, if only for displaying the eternal problem about why is $3+3+3+3+3=5+5+5$. I've asked that of many bright undergrad before, and rarely got and answer better than "that's the law" (the commutative law, they mean, which we all abide by except when we do powers). – Dror Bar-Natan May 11 '11 at 18:56
Michael: answering the questions you want to pose is surely something better done on a blog. Redefining the original question to one you can then be righteous about does not strike me as very courteous – Yemon Choi May 12 '11 at 23:37

First, statistics, indeed, is not taught enough. I studied statistics in a good school, but when it came to actually using it, found that I don't understand it. Second, motivation: they have to show the student how much and how urgently (s)he will need these concepts while on the workplace, with good real examples.

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Quantum mechanics, as in understanding the mathematics behind its foundational issues, and not just as in computing the spectrum of the Hydrogen atom (though that's good too).

It's hard to think of a topic that shakes one's image of the physical world harder than quantum mechanics. General relativity is easy to digest once you are not scared of things like manifolds. Quantum mechanics remains a challenge to one's worldview no matter how hard one tries to get used to it. You cannot count yourself scientifically literate if you were not exposed to the foundational issues of quantum mechanics.

And it's a math course at least as much as a physics course. The pre-requisites are basic probability and logic and complex numbers and basic Hilbert space theory, and the content is philosophy and non-commutative probability theory and (may as well, at the end) some spectra of some differential operators.

Mermin article "Is the moon there when nobody looks? Reality and the quantum theory" was an eye opener for me, the year after I finished my undergraduate studies.

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I would love to see more differential geometry offered in undergrad. The course I envision would start with a review of vector calculus, move to studying hypersurfaces in $\mathbb{R}^n$, and then move into a study of manifolds. You could tie all these subjects together via the Fundamental Theorem of Curves, the Fundamental Theorem of Hypersurfaces, and the Fundamental Theorem of Riemannian Geometry. I feel such a course would help bridge the gap between undergrad and grad school; simultaneously reviewing the key ideas of calculus at a high level while also giving a solid foundation from which to study manifolds in grad school.

Here are some topics that could be covered:

• Curves in $\mathbb{R}^n$
• $k$-frames and curvature, leading to the Fundamental Theorem of Curves

• Hypersurfaces in $\mathbb{R}^n$

• Tangent spaces and curvature
• First/second fundamental forms
• Moving frames, Christoffel Symbols, Gauss Equations, Codazzi-Mainardi Equations.
• Fundamental Theorem of Hypersurfaces, a word on curvature tensors

• (Real) Manifolds, charts, multiple definitions of tangent vectors

• Mention Lie Bracket and Lie Algebras (after doing derivations for tangent vectors)
• Affine Connection, leading to the Fundamental Theorem of Riemannian Geometry

P.S. I am not a differential geometer. I study homotopy theory. I just thought a course like this was missing from the curriculum. Any feedback on other topics that could be covered or tangents that could be mentioned leading off from this material would be welcome.

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Personally, I think the answer to this question is largely going to depend on one's particularly interests (whether they lie in algebra, analysis, topology, or whatever). This can be seen from many of the previous posts.

That being said, I do think that more number theory would be a great addition to the undergraduate curriculum. Many students take an introductory number theory course (or skip it because they learned it all in high school) and then don't do any more. There are lots of great areas of number theory which don't require too much background. P-adics would be great (Gouvea even laments in his book that p-adics aren't taught earlier - so maybe such a course should use his book). One could teach a basic semester of algebraic number theory, or a course in elliptic curves (following Silverman and Tate, for example). Both of these require no more than a basic course in undergraduate algebra. You can probably find these courses at many top universities, but they usually aren't emphasized as much to undergraduates. The reason why I think that these would be good is because number theory is a particularly beautiful area of math, and by getting glimpses of modern number theory early on, students get to see how beautiful is the math that's ahead of them. (Another possibility is to have a course on Ireland and Rosen's book A Classical Introduction to Modern Number Theory. Princeton had a junior seminar on this book, for example.)

I also think Riemann surfaces are a very beautiful topic which should be taught early on and aren't too complicated in their most basic form. For, you get to see the deep geometrical theory lying behind the $e^{2 i \pi}=1$ and the ambiguity of complex square roots which you learned about when you were younger. It shows the student that there can be very deep ideas lying behind a simple observation, and it shows the beauty and deep understanding that modern mathematics can lead you to.

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As a college student myself, I wish to study these classes when I was in my college, but they are not offered(I took most of these in Moscow instead):

1. Algebraic topology.
4. Measure theory, geometric measure theory.
5. Commutative algebra and homological algebra (at least Ext, Tor, etc)
6. Riemann Surfaces
7. An intro course in algebraic geometry
8. Algebraic number theory.
9. Classical Mathematical Physics
10. Some intro course in ODE, dynamical systems (like Smale's horseshoe), and PDE.
11. Combinatorical game theory.
12. Elliptic curves.
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"What's one class that mathematics that should be offered to undergraduates that isn't usually?"

OK, I'll rephrase my earlier answer. A class that should be offered to undergraduates that usually isn't is a "what is mathematics" course for those liberal-arts majors who will take only one math course in their post-secondary schooling. It would be a truthful course that would avoid telling them that mathematics consists memorizing algorithms whose utility can be seen only by taking later courses that they won't take. It would acquaint them with the fact that mathematics, like physics, is a subject in which new discoveries are constantly being made. It would tell them that one doesn't generally do math by taking a problem and feeding it into an algorithm that was given to one by a prophet who came down from Mount Sinai. It would tell them that mathematics is a subject that, like music, relies heavily on technical skills but does not consist of those alone. Among the goals would be that a student who takes only that course and becomes a professor of some liberal arts subject would not be among the many such professors who don't suspect the existence of such a field as mathematics.

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DESIDERATA:

• asymptotical analysis and its applications
• analytic combinatorics
• analytic number theory
• complex analysis with focus on transforms (i.e laplace, inverse laplace, saddle-point method, stationary phase...)

WE NEED MORE OF THOSE BEFORE THE UGRADS THINK THAT CATEGORY THEORY IS THE PINNACLE OF MATH!!! (the last sentence is a joke, please do not get offended anybody out there).

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Combinatorics seems like one of those subjects that you really have to have a natural affinity for. – Harry Gindi Dec 7 '09 at 10:15
IMHO there is already too much unmotivated analysis in undergrad education. I rarely meet undergrads who think categories are the pinnacle of math, but I do meet more than enough undergrads who think that two-lines bounds involving epsilons, deltas, absolute values (as commonly seen in stochastics, diff. equations and asymptotics) are the pinnacle of maths. – darij grinberg Mar 15 '10 at 11:06
@Harry: I used to think so too. But then I saw analytic combinatorics, 'done right', and I learned to really enjoy it, while I hated my 2nd year combinatorics class. – Jacques Carette May 20 '10 at 2:31
@Jacques, which book did you use? or did you go from lecture notes? It always seems that combinatorics is almost never taught in an understandable way... – Michael Hoffman May 20 '10 at 21:14

Computer Science. I know programming has been said already, but computer science isn't programming. (There's the famous Dijkstra quote: “Computer science is no more about computers than astronomy is about telescopes.”)

There is a vast and beautiful field of computer science out there that draws on algebra, category theory, topology, order theory, logic and other areas and that doesn't get much of a mention in mathematics courses (AFAIK). Example subjects are areas like F-(co)algebras for (co)recursive data structures, the Curry-Howard isomorphism and intuitionistic logic and computable analysis.

When I did programming as part of my mathematics course I gave it up. It was merely error analysis for a bunch of numerical methods. I had no idea that concepts I learnt in algebraic topology could help me reason about lists and trees (eg. functors), or that transfinite ordinals aren't just playthings for set theorists and can be immediately applied to termination proofs for programs, or that if my proof didn't use the law of the excluded middle then maybe I could automatically extract a program from it, or that there's a deep connection between computability and continuity, and so on.

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+1 for "Computer science isn't programming" and the great Dijkstra quote. – Gabe Cunningham May 20 '10 at 1:17
Numerical methods should be considered a very advanced topic, even though it was historically 'early'. The beauty of computer science, as far as I am concerned, generally lies in those parts where everything is exact. The connections between a multitude of areas of mathematics and computer science seem to be exploding in the last few years - and generally with no numbers in sight! – Jacques Carette May 20 '10 at 2:23
I'm one of a very large and growing number of graduate students whose training didn't require any serious computer science-and who deeply regrets it now. In many ways,computer science is one vast interrelated set of applications of mathematics to engineering.It DEFINITELY should be required of mathematics students and as early as possible in thier training.There's very good reason to unite the CS and mathematics departments as many universities do. – The Mathemagician Jun 12 '10 at 1:57
I agree wholeheartedly. I feel that Pure Maths is more relevant to, and has more in common with, CS than Stats, and yet my department is in the middle of combining with the Stats school. – ADL Jun 14 '10 at 13:36
@Alan Stats will make the joint department a lot more money then theoretical computer science will. It's all about the bottom line. – The Mathemagician Jun 14 '10 at 18:11

I am actually thinking about functional analysis and modern Fourier analysis.

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I was a graduate student at Yale, and all but a few courses had single digit enrollment. I distinctly remember a course where I was the other student, and a course where our exchange student from Germany got stuck, being the only one attending and not wishing to offend the professor by dropping. – Victor Protsak May 20 '10 at 7:39

What about "Mathematics with Computers"? Having modern computer algebra and symbolic computation tools available, one can use them to present and explore nontrivial examples in various fields of mathematics. Part of the course could also present basic algorithms and other techniques used.

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I think you could fudge a bit and get a good idea of how the algorithms work, assuming they have a basic background in abstract algebra – Michael Hoffman May 20 '10 at 21:10

Caveat: My undergraduate & graduate studies were both not in Math but in Engineering. But I would love to have taken a course on the history of mathematics and I think this isn't a commonly offered course. There are lots of compelling stories here and it also gives a great perspective on how the different areas of math came to be born (non-Euclidean geometry from work on the parallel postulate to name one). Knowing how a subject evolved historically can give a nice perspective on the subject especially when a formal course on the subject might not necessarily follow the same order of ideas.

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With the proviso that much of the older "folk history" is, I'm told, not accurate, or is misleading. I went to several entertaining history of mathematics lectures where this was pointed out vehemently. Also, the history of ideas is really tricky, because we have to try and understand how e.g. the Greeks thought, not how we would think about what they appear to describe. – Yemon Choi Mar 15 '10 at 6:40

Motivation.

I have seen many students dropping out of math because they didn't get an answer to the question "why should I learn this?". Of course, one could say, a good student should have intrinsic motivation and/or figure out the motivations by himself, but this seems to me like wasting potential.

I don't want to say that mathematics courses don't provide any motivation, but in undergraduate courses (and even textbooks), especially in linear algebra and calculus (when there isn't so much time), I haven't seen enough motivation.

This motivation should go beyond "we want to model the physical world" and/or "with this theorem you can calculate the Eigenvalues". Students need the story between these two extremal answers, they need to know how calculation of Eigenvalues is really applied in modelling the physical world. (This is just an example, I would appreciate to see more motivation for abstract, non-applied mathematics, too)

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When teaching eigenvalues the textbook I used gave a simple predator prey example and then looked at the diagrams of the phase space . (I think it was woodrats and spotted owls (oops that is a prey-predator problem!)) The oscillations in population levels could be seen to be interesting as a simple explanation of things students have heard about. That lead o to how on earth can you calculate these eigenvalues and eigen vectors and to some numerical methods. The point was that the example chosen was simple enough to comprehend without knowing a whole lot of some other subject area. – Tim Porter Mar 14 '10 at 19:38

A course that just attempts to define the current research areas of maths. If the landscape is so complex, why can't undergraduates be provided with a map, so to speak, in order to begin to decipher this subject?

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I was never offered a geometry course as an undergraduate, and there's so much lovely geometry, from Euclidean and non-Euclidean geometries, to algebraic and differential geometry, and the rest. So...geometry.

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Courses aimed towards applied, rather than pure, mathematics. Like a modeling course related to environmental sciences, perhaps. Most math majors prepare the students for graduate school in pure mathematics, but offer less support for applied tracks, and there be some good careers there.

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I did a course with Atomic and Nuclear Physics making up a third of the time. Apart from beautiful functional analysis and topology courses, one of my favorite courses was in Quantum Mechanics taught by an excellent Physics prof. I there understood about orthogonal functions, etc. He talked sense! The earlier courses on these subjects went straight past me. I could do the problems but did not grok them. – Tim Porter Mar 14 '10 at 18:27

How to write on a blackboard! At the very least, how to write so that the chalk doesn't squeak.

(Declaration of interests: this was inspired by Kim Greene's answer to Tyler Lawson's question about getting fonts right on a blackboard.)

Slightly more seriously, we should teach our students how to present their ideas well.

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It won't be much longer ... classrooms won't have chalkboards at all. – Gerald Edgar Mar 14 '10 at 18:30
WHY NOT?!? Low tech is good sometimes. – The Mathemagician May 7 '10 at 5:01
Blackboards are threatened in many places (I'm aware of Oxford, and some places in Australia). The mathematicians I know in those places are very unhappy with this situation. – André Henriques May 11 '11 at 18:05

Following Greg's lead, I wish that undergrads who want to become math majors didn't "skip out" of differential equations classes (in their eagerness to get to the good stuff like Drinfeld chtoukas, or whatever). I can't think of a more important foundational subject that tends to be systematically avoided by the "best" undergraduate mathematics students.

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I completely agree!!! Lots of places have "rigorous linear algebra" type classes (I'm teaching one right now) -- I wonder why more places don't have a differential equations class geared for math majors. There are even nice books available (like Hurewitz's beautiful little book, or Arnold's). – Andy Putman Nov 12 '09 at 19:35
I think part of the problem here is the types of course offerings universities have. For example, when I was an undergrad at U.Alberta, there was an honours intro ODEs course, but it was taught like a service course and was largely non-rigorous. It's difficult to attract students to material when the curricula is presented in such an unflattering light. – Ryan Budney Aug 31 '11 at 15:21

When I was in my first year, I always missed order theory. I teached it to myself then and thought many times "Why didn't they teach us this - we would understand everything so much better!"

And I still think so today. Order theory starts off easy, when you lern about relations, preorders, lattices and so on, and then you get into Zorns Lemma, Schörder-Bernstein Theorem and stuff like that.

But I'm also on the category theory track. I think the notion of category will become, just like the notion of a group, more common sense, not just in mathematics but also in computer science, physics, chemistry and maybe even more.

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That should cover that in an axiomatic set theory course-which ALL math majors should have as a required course. – The Mathemagician May 7 '10 at 5:05

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Igor Rivin has a whole diatribe on this topic!

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A calculus class that goes very slowly.

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If you don't teach calculus well, you can't have a decent math major. – Charlie Frohman Mar 15 '10 at 12:44

I'm going to change the question slightly. What topics do we all think are taught in undergraduate mathematics, but often leak out of the curriculum so that students see too little of them? I have in mind the standard situation at large research universities, where there is a mix of good and not-so-good students.

My pet peeves:

1. Complex numbers as they should appear in standard calculus and linear algebra. They tend to be postponed to upper division courses. Complex numbers greatly simplify both trig identities and partial fractions, but calculus students aren't told.
2. Complex analysis. It tends to float to the top of upper division and disappear.
3. Full multivariate calculus: The Jacobian of a general change of coordinates, the derivative of a multivariate inverse function, maybe also the multivariate Newton's method. The calculus sequence often chickens out and just does special cases of the first of these.
4. Higher-dimensional Euclidean geometry. Like, the definition of an n-cube and the fact that it has 2n vertices.
5. Multivariate probability, especially with both discrete and continuous features.
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Linear algebra. At least, it's not taught to math majors. They tend to see little bits of it in the calculus sequence (in order to solve systems of ODEs, for example) and then again in an abstract algebra class (matrices being a nice source of examples of groups) but never see a coherent treatment of it. At least this is true at the universities I'm familiar with. In both cases there is a linear algebra course mostly taken by nonmajors, but it's not possible for majors to get credit for both linear algebra and first-semester abstract algebra (basically the group theory course). – Michael Lugo Nov 4 '09 at 13:31
Adding emphasis to your comment (3), I find especially in the States there's a rather artificial distinction between "analysis" and "calculus". To the point that calculus is seen as mindless symbol manipulation, but in analysis thought is allowed. John Hubbard's textbook does a great job of blurring those boundaries, especially when it comes to things like the inverse function theorem and Newton's method. – Ryan Budney Nov 6 '09 at 22:32
My son took a really good AP calculus course at the local high school, using the most standard textbook and in some sense the most standard syllabus. So I've seen it from the other side. I don't think it's fair to call it "mindless symbol manipulation". But it is true that American universities sell calculus short in various ways. One reflection of that is that a good AP calculus course can be harder than the university product. – Greg Kuperberg Nov 6 '09 at 22:59
In Germany, linear algebra is emphasized very much. Every university I know has a compulsory one-year linear algebra course along with the one-year analysis course (whose second semester is devoted to multivariate calculus) in the first year. Complex numbers are usually one of the first things which are introduced. Sometimes, they are even introduced in the first week of both courses since the professors talk less to each other than they should ;). But then, in most courses in analysis one is not really told what to do to solve an integral. – Lennart Meier Jun 16 '10 at 8:35

I think a great class for undergrads (in particular, for seniors planning on grad school) would be a capstone "Comparative Mathematics" course. In my imagining, this would be a mix of math history, the "greatest hits", contrasting the fundamental objects of study and proof techniques, and an introduction to the map of modern mathematics. Think the Princeton Companion to Mathematics distilled into a semester.

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Maybe this is an overbroad answer, but I'd like to see more specialized subjects that are just really fun. Computational geometry (in the classical/Euclidean sense, not the computational algebraic geometry sense) is the example that leaps to mind -- I'm not aware of anywhere that offers it as an explicitly undergrad-level course, despite the fact that it's amazingly fun, quite simple (I suspect that bright undergrads could get to Arora's PTAS for Euclidean TSP within a semester, and certainly Christofides' algorithm is within the reach of anyone who's taken basic algorithms), and practically useful, although I guess this is more (T)CS than straight math...

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I would have loved a class on how to write mathematical papers and what goes into mathematics research. Everything from neat Latex tricks to how to organize and structure ideas, theorems, etc, going over bad vs good papers, even perhaps discussing what makes good math books. As well, an overview of what goes into a PhD thesis would have been extremely useful. It's a shame that one usually has to pick up these various bits of info on their own. The class could coincide with a current undergraduate senior project for example and act as a supplement. I took a similar class like this in the physics department and it helped me immensely with my work.

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There's not a whole lot of magic when it comes to writing good papers or books. Usually the magic is an immense amount of hidden effort. Allen Hatcher's "Algebraic Topology" is considered a pretty good book. I have dot-matrix printouts of some of the first versions, dating back to about when Hatcher arrived at Cornell. So the book represents about 30 years of teaching algebraic topology, and many iterates of revising the lecture notes. – Ryan Budney Nov 6 '09 at 22:43
I now wish I had such a class as an undergrad, incidentally, although I think some of the undergrads actually in the class view it (incorrectly!) as "fluff." (Many students take 18.100B, which gives the analysis without the writing.) I disagree with Ryan Budney: just like giving presentations, or teaching, or even conducting research, there are lots of specific, teachable skills that can improve one's writing. Before helping to teach this class, I could diagnose a paragraph as unclear, and with enough work I could fix it. ... – JBL Mar 18 '10 at 15:50
The content of this class includes both a language to describe certain types of bad writing, and tools for quickly identifying and correcting what about it loses the reader. In other words, there are techniques to make good writing easier, and people who are good writers typically either know these techniques or have a strong intuitive feel for them. Just as is the case with any other endeavor, teaching the techniques makes it easier to do it well, and having really good heuristics lets one write better without having to work any harder at it. – JBL Mar 18 '10 at 15:58