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
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
Programming. I think it varies a lot from department to department but some places seem to do a bad job of teaching programming and it can be a really important skill.
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.
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)
Approximation/asymptotics. It amazes me how many otherwise good students don't have a sense for which parts of an expression are large and which are small.
Inequalities!
I don't think I've ever seen a course on inequalities, and there's certainly enough elementary material to cover in a one-semester course. Very few undergrads know much about inequalities.
Category theory!
I think undergraduates should take problem-solving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problem-solving in high school would definitely get a lot out of them.
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:
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.
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.
Computational Algebraic Geometry. Something like the book "Ideals, Varieties and Algorithms" by Cox, Little and O'Shea serves as a good bridge from high school algebra with lots of computations and polynomials to modern algebra with rings and groups, without assuming knowledge of the latter.
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.
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.
Basic logic. Coming into university we all start from different backgrounds and some of us have been taught poorly in the past and haven't had the opportunity to learn the fundamentals of conditional statements or if and only ifs, etc. For example, knowing that proving the contrapositive is the same as proving the original statement is worthy knowledge indeed! Try proving that if x^2 is even then x is even without knowing this trick.
Courses in physics and complexity theory were certainly available in my undergrad days, and were mandatory for some undergrads. I guess it depends on the country, and possibly the college...?
One glaring omission that was prevalent in my time (and place) was number theory. It was typical for math graduates to never have seen even the statement of quadratic reciprocity, which I find crazy.
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?
Bayesian Statistics. I think it's more useful in many practical situations than traditional statistics.
Basics of numerical methods: What computers can and can't do and how they operate in general.
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.
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...
I am actually thinking about functional analysis and modern Fourier analysis.
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.
Traditional Statistics. Many biology majors end up knowing more statistics than many mathematics majors which I think is a weird state of affairs.
A calculus class that goes very slowly.
"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.
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:
$k$-frames and curvature, leading to the Fundamental Theorem of Curves
Hypersurfaces in $\mathbb{R}^n$
Fundamental Theorem of Hypersurfaces, a word on curvature tensors
(Real) Manifolds, charts, multiple definitions of tangent vectors
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.
Having been offered a not-all-that typical undergraduate curriculum, and having then proceeded to miss a lot of it through over-sleeping, I'm not sure what is or isn't usually offered up. Does Ramsey theory (or even just Ramsey's theorem) get a mention in undergrad-level combinatorics? If not, that'd be my suggestion: about the only mathematics I've succeeded in explaining to non-scientists in the pub, from R(3,3) to the idea of lower bounds via random colourings.
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.
DESIDERATA:
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).