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I'm attempting to put together a list of topics for a general education course centered around "great ideas in mathematics," and I'm looking for input. Ultimately, I would like topics to be approachable for those with a background only up through precalculus.

Things I've thought of so far include the following:

  1. Numeral systems
  2. Graph theory
  3. Infinite sums
  4. Noneuclidean geometry
  5. And much more!

Note: I'm interested in especially influential or revolutionary ideas that tend to fall on the purer side of math. Measures of fun and/or practicality are of secondary importance to me, as I tend to be better at convincing students that something is interesting, rather than entertaining or useful.

Something like RSA cryptography is great, because it's somewhat fun/practical while also having a really robust theoretical underpinning that allows me to discuss primes, modular arithmetic, etc.

What do you all think? I'd love to hear any additional ideas.

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Two very different curricula which are very much along these lines are

There are plenty of others which cover 2 and RSA but finding ones with 1, 3, and 4 that are available for non-majors (especially once you get beyond the "highly selective" colleges) is a bit more challenging. I suspect that The Heart of Mathematics will have most of what you are looking for buried in it as well.

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    $\begingroup$ I was also impressed by The Heart of Mathematics when I surveyed it (I was about to add it to my answer). Turned out to be too much for our community college students, but still. $\endgroup$ – Daniel R. Collins Oct 11 '16 at 2:38
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    $\begingroup$ I really like the look of The Heart of Mathematics. I'll definitely have to check that out along with the other things that have been suggested here. $\endgroup$ – AegisCruiser Oct 11 '16 at 4:51
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    $\begingroup$ There are positives and negatives to Heart. It has some really cool stuff and covers a nice wide range of interesting math; but it can also have questions that are (intentionally) very ill-defined for students used to memorizing algorithms. I've also heard it called "coffee-table mathematics" but to be fair that was from someone at a very high-level research university who normally suggested teaching dihedral groups to non-majors (which happened to work there). You'll want to look at the details of the sections you choose to see if it is good for you. $\endgroup$ – kcrisman Oct 11 '16 at 13:51
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You should probably read Eves' Great Moments in Mathematics series? (Two volumes, about 40 lectures/moments, and an appendix of 20 more that were left out for space.)

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A few things that are interesting and a bit real world:

  1. the formula for the area of an irregular polygon in terms of its vertices. For example, this website, but, it's a standard calculus III homework problem you'll find in many text books.

  2. counting via Burnside. There are some nice examples in Fraleigh's text or Rotman's basic abstract algebra text. For example, how many nonstandard dice you can design using the usual shape and dots. The possibilities are endless in terms of weird counting problems which are best understood via the group theory. I think the basic ideas can be described without getting overly technical.

  3. group representation in quantum mechanics. Ok, this is harder, but if you could find a toy example this could be very interesting to those students curious about the role symmetry plays in our modern physics.

  4. fourier analysis and the analogy to finite dimensional vectors. Maybe begin by explaining how any vector in three (or two) dimensions can be decomposed into vector components then... for music we need infinitely many components.

I must stop. There is so much you could try. I wish we had such a course, where math was approached as an art as opposed to a task.

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  • $\begingroup$ "...as an art as opposed to a task." Yes, a hundred times over. I've always thought that the "practical" math taught in non-major courses is almost entirely dissatisfying. Even without this, one can argue that a non-major course in "beautiful" math is a better fit for a liberal arts college anyway. $\endgroup$ – AegisCruiser Oct 11 '16 at 4:51
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Here are some of the topics from a great ideas in mathematics class that I am teaching this semester:

  1. Modular arithmetic: parity/UPC/ISBN codes, periodic phenomena like days of the week (E.g. 365 mod 7 = 1)
  2. Rational/irrational numbers: What their decimal expansions look like. Why we know $\sqrt{2}$ is irrational.
  3. Cardinality: Countable sets & the Hilbert Hotel, Uncountable sets.
  4. Tesselations and Platonic solids (plus Penrose tilings, etc)
  5. Graph theory: Bridges of Koenigsburg, Hamiltonian & Euler paths

Our textbook is "The Heart of Mathematics: An invitation to effective thinking, E. B. Burger and M. Starbird" but personally I don't make a lot of use of it in my class.

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As a department chair, I think this is the most difficult and expensive course to staff, unlike college algebra, elementary statistics, calculus, etc. I want the best faculty who can bring their own mathematical interests to teach this class. This course requires an instructor with exceptional content knowledge and teaching skills.

What topic should you include? Find that mathematical interest that truly interests you, and struggle to find a way to pitch it to this audience. From your question, I think you have a good start already!

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I teach, essentially, a "math course for non-math students who need to fulfill a college requirement" and have found great success with topics that don't seem like math to the students but certainly are math. Each semester, I teach 3ish units of self-contained material, and I have listed some of those topics below. We use the text For all Practical Purposes, which contains enough material to do each of the units below, although I often supplement the text with extra examples from "the real world".

I find this works well because the students are already apprehensive about having to take a math course. Indeed, I presume that many of them had lackluster experiences in high school, either due to poor teaching or lack of confidence/ability. So, I like these topics because (1) I cannot actually assume that any of them have even basic algebra skills or decent numeracy, and (2) I want to, overall, allay their fears and show them that there is a lot more to math than number-crunching, timed tests, and anxiety.

  1. Graph theory in management: We learn about Euler and Hamilton circuits in graphs and see how they can be used in city planning. As part of a homework assignment, I have them draft a letter to our city's Department of Public Works to recommend rethinking the way they route their snow plows. We also talk about the Traveling Salesman Problem, and I supplement the text with some information about computational complexity and "the limits of human knowledge". I then touch on minimal spanning trees, showing them Kruskal's and Prim's Algorithms. (In general, graph theory is a great topic for a course like this because you can go as in depth into one topic as you'd like, depending on your students' backgrounds.)
  2. Voting systems and elections: We learn about different ways to take individual rankings from voters and make a collective choice: Condorcet's Method, Borda count, Instant-runoff, Plurality, etc. We touch upon Arrow's Theorem and we work through the logic of a proof of a weaker version of it. This leads to lots of discussions, as well, about how no one in the U.S. seems to even question how we vote (because everyone is fixated on who to vote for). (Although that may change this year if ballot initiative #5 passes in Maine!) I believe this could fit your course, as well, because Arrow's Theorem is a truly revolutionary result in that field and it's decidedly, purely mathematical in nature.
  3. Cryptography and coding schemes: We learn about the ISBN coding scheme and other similar examples. This introduces them to modular arithmetic and, depending on the background of your students, you could go more in depth about algebraic structures (e.g. "For which $n$ is $\mathbb{Z}$ modulo $n$ a group?") We also learn about using matrix multiplication to send coded messages, which can be used to introduce them to the idea of inverses in a new context (other than simple arithmetic).
  4. Statistics & data: I feel like the students in this course, if it will be the only course they take in college, should learn some skills about how to interpret "margin of error" of a poll, and to understand the difference between correlation and causation. So, we learn about that kind of stuff, normal distributions and standard deviations and the 68-95-99.7 rule, etc. Again, you can go as in depth into this as you'd like.
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I'll just add one more little cute topic that can make a 2-hour lecture: Secretary problem (probability and random processes). If you're interested, I can share my old power point slides with you.

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