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According to your experience as students and professors, what are (and why) the courses that should be part of a math undergraduate degree, but that are missing in most institutions?

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    $\begingroup$ I wonder how much a course in geometry is a generic part of an undergraduate degree in mathematics (a) in the U.S., and (b) elsewhere. I simply do not know. At my college, it is not; but that may be an idiosyncratic lacuna. $\endgroup$ – Joseph O'Rourke Nov 5 '14 at 0:01
  • $\begingroup$ @JosephO'Rourke: I took courses in topology and differential topology: these could be considered to be hybrids of geometry and analysis. Trigonometry, a hybrid of geometry and algebra, is also taught in many colleges. $\endgroup$ – Rory Daulton Nov 5 '14 at 0:04
  • $\begingroup$ The basic areas of maths are: number theory, algebra, analysis, and geometry. These areas rest on a foundation of set theory and topology. These 5 areas of maths should be equally represented in any undergraduate program. $\endgroup$ – mdg Nov 15 '14 at 19:53
  • $\begingroup$ How are the program's alumni expected to use what they learn in their undergraduate math degree? As grade-school teachers? As algebra and geometry teachers? As calculus teachers? As statisticians? As "data analysts" (informal statisticians)? As "quants" (securities traders who develop computer trading algorithms)? As computer programmers? As applied mathematicians? As orbital physicists? As preparation for a graduate mathematics program? As preparation for a graduate program in a different field? People going into different fields may have different needs. $\endgroup$ – Jasper Feb 1 '17 at 23:40
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Before I answer your question, I want to preface my answer with the statement that, what courses are available at any one particular university are contingent on the faculty. If there is nobody appropriate to teach a branch of mathematics, then it is difficult to run such a course!

Discrete Mathematics

In America at least, I feel as though there is such a heavy emphasis on the continuous track of mathematics. The discrete track, I feel, is underrepresented. Courses such as Graph Theory, Game Theory, and Combinatorics are classes I have taken but gained so much value from, and are more difficult to come by. I've been blessed by attending two schools (undergraduate and graduate) that have had a wide variety of courses to choose from. Even then, these courses were frequently only offered every few semesters. There are a wide number of sub-genres of topics that could be developed into a full class, if the right person were available to create the course.


I want to answer another question that seems relevant: What courses should be a part of a mathematics education degree program?

I have long yearned for a course that focuses specifically on the methodology of a particular commonly taught mathematics course. A course on teaching high school algebra, teaching high school geometry, teaching college group theory or teaching college analysis would do wonders, I think. To my knowledge, no such courses exist.

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    $\begingroup$ I believe that courses called "Discrete Mathematics" are common these days in universities, often taken by computer science majors (such as my son). Sometimes there are variations such as Finite Mathematics, though that is more common in high schools. The other classes you mention, such as Combinatorics, are more difficult to find. $\endgroup$ – Rory Daulton Nov 5 '14 at 1:46
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    $\begingroup$ @RoryDaulton, I agree. I think finding a "Discrete Mathematics" course isn't that uncommon. However, I think having a course labeled "Discrete Mathematics" is akin to having a course labeled "Continuous Mathematics" that covers all of analysis. Yet, analysis is too large a topic for one course, so it is split into Real and Complex (usually with 2 semesters for Real). I'm making the same claim here: That a single course for all of discrete mathematics is not sufficient. $\endgroup$ – Andrew Sanfratello Nov 5 '14 at 4:33
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    $\begingroup$ math education majors are taking too much edu-babble to have time to actually study math education. As always, the real danger of education (and assessment) is not so much what is it. Rather, it is what is necessarily displaces in education (and departmental discussion). $\endgroup$ – James S. Cook Nov 8 '14 at 1:57
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A course in number theory.

Number theory is one of the "big four" topics in mathematics (with algebra, geometry, and calculus/analysis). It has a long history, equal to that of geometry. It is foundational to algebra and calculus, inspiring and giving many examples to those topics. It is the first application of set theory to the "big four." It has strong links to non-big-four topics such as set theory, probability, statistics, and combinatorics. It is simple enough to be understood by elementary school students but deep enough to provide many of our famous unsolved problems (Goldbach conjecture, twin-prime conjecture, etc.). It forms many puzzles that can be understood by non-mathematicians. It inspired the most famous current unsolved problem, the Riemann hypothesis. Finally, it has important real-world applications in topics such as cryptography.

I was not able to take a number theory class when I was a math undergrad at Cornell University. A professor tried to get a class together but could not get the minimum number of students. My main exposure to number theory was in "math camp" during a high school summer, and it was the only required topic since it has practically no prerequisites but is studied in university.

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    $\begingroup$ Yes! Number theory is of a different flavour to other areas of maths, being in my mind more playful and grounded in the most accessible maths concept: whole numbers. As such it serves as a nice counterpoint to the furrowed brows of analysis and the wild abstraction of algebra. It also has more-or-less no prerequisites and can be taken by students outside the maths degree. We used to have Number Theory at my uni (Uni of Adelaide, Australia) but it was cut. $\endgroup$ – DavidButlerUofA Nov 4 '14 at 21:25
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    $\begingroup$ It also has applications in cryptography, which can be a very lucrative field. $\endgroup$ – rbp Nov 5 '14 at 18:28
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At large research schools, or moderately large state universities there is often an undergraduate or graduate course which deals with the following topics. However, at a small liberal arts school, too often one, two or all three of the following are either missing or under-emphasized:

  • elementary topology
  • differential geometry
  • complex analysis

Actually, topology is very common, but, I list it for my own reason. More important than all of these:

  • a second course in linear algebra including modules and cannonical forms, multilinear algebra and other fun things we dare not cover in the linear course for the masses.
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    $\begingroup$ +1 for the second course in Linear Algebra. $\endgroup$ – Benjamin Dickman Nov 8 '14 at 2:22
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A course in experimental mathematics could be valuable so that students are introduced to making conjectures and testing their plausibility. This would give them a taste of an aspect of research mathematics and might help counter the notion that all the mathematics there is has already been discovered. Appropriate subject matter, depending on level and interests, could include combinatorics, dynamical systems, discrete geometry and/or number theory.

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