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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 E. Caicedo, Mark Meckes, Yemon Choi Jan 8 '12 at 20:11

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Can someone here explain why my first answer below got a negative vote? I raised a more serious and important question than whether calculus on manifolds or asymptotic enumeration, etc. etc. etc., should be taught. Is a truthful statement that there is a crucially important part of the curriculum that mathematicians should be ashamed of unwelcome within math overflow? Or if someone disagrees with it or thinks it's not important, can they say so in words in a dignified manner? – Michael Hardy Jun 15 '10 at 2:38
Your answer is polemical and not too the point, it does not answer the question. – Michael Hoffman Jun 15 '10 at 23:46
I liked the question originally, but in view of the latest answer I fear it has reached the point where it is not productive (signal/noise ratio), and so I am casting the final vote to close. – Yemon Choi Jan 8 '12 at 20:11
My undergraduate mathematics classes involved zero critical thinking and zero play. I don't think which subject matters, but typical areas could be approached so that students ask & answer questions like "Why do we want it to be this way?", "What would happen if we did it another way?", and "What would be the pro's and con's of doing it that way instead?". Time for this could come at the expense of covering more material. – isomorphismes Sep 14 '15 at 19:58

42 Answers 42

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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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
Why should one study axiomatic set theory in a separate course? I disagree that all math majors need an axiomatic set theory course, but rather need a brief intro that you can get in an introduction to proofs class. – Michael Hoffman May 20 '10 at 21:12
Could be because I'm in my last year of undergraduate and haven't taken an axiomatic set theory course – Michael Hoffman May 20 '10 at 21:13

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

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.:

  • doing some basic representation theory following the formalism in the book by Cvitanovic:

  • 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:

  • 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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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.

My answer: the truth.

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

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
on the beginning missing duality of finite dimensional vector spaces (preferable over the real field) garanties from minor to almost no understanding :P – janmarqz Aug 30 '12 at 18:24
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 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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We do offer a course like that using MATHEMATICA at Queens College,Tomaz. I've been after the department for awhile to offer it in the summertime to increase enrollment and get a cash cow. – The Mathemagician May 7 '10 at 4:44
Do use symbolic computation to present and explore nontrivial examples from mathematics. I don't think you'll get as much traction from the basic algorithms, and the algorithms that CASes actually use really need a lot more background [but then, they are rather interesting]. – Jacques Carette May 20 '10 at 2:29
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

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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Good suggestions,Davidac-but again,the students need to be very well prepared (such as at Princeton) for these kind of courses.They need to be pretty comfortable with basic algebra and rigorous calculus.Topology would help,but isn't really necessary. – The Mathemagician Jun 16 '10 at 17:20
Ok, but this could work as long as the student takes a basic abstract algebra course in junior year, which is common at a lot of places. – David Corwin Jun 17 '10 at 13:31
That is, all you need is an introductory abstract algebra course for p-adics, elliptic curves, and algebraic number theory. The mention of class field theory was just to appease the person who said algebraic number theory at its most basic level doesn't get anywhere interesting. You do need basic rigorous calculus for p-adics, but a typical undergraduate senior has done that. – David Corwin Jun 17 '10 at 13:33

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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I agree that there could be much improvement in how statistics is used as well as presented. However, most undergraduate curricula that (in my opinion) deserve the title have an offering in statistics, however elementary it may be. Was there a particular theorem or application you feel is not offered? Gerhard "Ask Me About System Design" Paseman, 2011.08.25 – Gerhard Paseman Aug 26 '11 at 3:44
For example, how to compute confidence intervals when estimating many parameters? What if regularization is used? How to pick/estimate a prior? Some basic intuition about stochastic processes; an introduction to statistical machine learning; graphical models; how to model temporal data; how to estimate if you don't have enough data for asymptotic results to be valid; robust statistics; modeling interaction between multiple features, etc. Above all, some intuition that would make me feel how to proceed, what may work and how it may fail. – Alex Aug 26 '11 at 4:28
Alex: shouldn't this be in a statistics degree rather than a math one? – Thierry Zell Aug 26 '11 at 11:32
Where I studied there was just math degree, no special statistics degree. And I didn't realize I'd do much statistics later. One thing about students is that they misplace their priorities getting fascinated by weird, fashionable or super-general stuff, and don't pay enough attention to the basics. That's where some guidance by the elders would be most appreciated. BTW, I'm not sure why one should separate math from statistics. To learn one but not the other is very dangerous for the future career. And in pure math randomness gives a different kind of intuition, just like geometry. – Alex Aug 26 '11 at 21:20
I see. It is certainly the case in the US that students in non-math programs (certain business, social sciences or information technology) end up learning a lot more statistics (at least in advanced degrees) than is offered to math students. The funny thing is that many such programs don't advertise it too openly, giving courses neutral names like "research methods". But of course, the statistics make students very employable in other contexts, see e.g. this New York Times article: – Thierry Zell Aug 26 '11 at 23:47

I had a course on "Asymptotic Enumeration" that was an advanced graduate level course that was fun and wish had an undergrad form.

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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.
  2. Real analysis (graduate level)
  3. Complex analysis (graduate level)
  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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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

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