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There are some general questions about potential topics for undergraduate seminars:

I am looking for topics for a 15-hour undergraduate mathematics seminar, to be attended by both Master-level mathematics educators or Bachelor-level mathematics undergraduates.

Prerequisites would ideally be limited to linear algebra, real analysis and elementary geometry (some Euclidean and hyperbolic geometry), which everyone will have seen.

For this question I would like to ask about topics which are directly relevant to the "life" of a mathematics educator, for example

  • deepening understanding of topics that are usually tacitly assumed or brushed over when entering university because "everyone knows this from high school" (e.g. how to really construct the real numbers, what is a set really)
  • relation to high school curriculum (calculus, Euclidean geometry, probability theory, ...)
  • relation to daily life (voting theory, mental arithmetic, ...)
  • relation to recreational ese topics would be imathematics (origami mathematics, ...)
  • etc.

(This eliminates quite a few fun topics — for example I don't see how knot theory, or fractals, or singular value decomposition (and compression of images, say) would be directly relevant to teaching high school mathematics — of course I don't mean to imply that that these topics shouldn't be learned by mathematics educators!)

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  • $\begingroup$ You said I don't see how... would be relevant to teaching.... Well, it is a common misconception that what will not be taught should not be learned (see the discussion here). $\endgroup$ – Pedro Mar 22 at 11:02
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    $\begingroup$ @Pedro I fully agree (and maybe should have said so in my question) — just like the education in high school is not just about learning mathematics for tests or exams (immediate relevance), the curriculum for mathematics educators should not just be about teaching mathematics. I hope it's still a valid question to ask for what kind of topics are more or less directly relevant to teaching (or relevant to mathematics educators in the slightly wider context in my question). $\endgroup$ – Earthliŋ Mar 22 at 11:31
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There are deep mathematical questions everywhere. I am fairly certain that, given enough time to prepare, I could take any topic in mathematics, no matter how elementary, and present a 60 minute talk that is both (1) accessible to a lay-audience and (2) concludes with several fairly deep—possibly even research-level—questions. By way of example:

My go-to talk when I am asked to speak to high school students, non-math major undergraduates, or a general lay audience is on dimensions. It starts with a series of fairly simple examples, which are built up slowly:

  • If a one-dimensional object of length $L$ is scaled by a factor of $s$, the scaled object has length $Ls$. This can be justified non-rigorously by appealing to "obviousness" for a line segment, then generalizing to the perimeter of a polygon. If one wants to take a digression into a little bit of calculus, the more general result follows by approximating a (rectifiable) curve with segments. Otherwise, appeal to intuition.

  • If a two-dimensional object of area $A$ is scaled by a factor of $s$, the scaled object has area $As^2$. This is "obvious" for a square, and any "reasonable" region can be approximated by squares. One can also appeal to formulae for the ares of standard examples (triangles, disks, etc).

  • If a three-dimensional object of volume $V$ is scaled by a factor of $s$, the scaled object has volume $Vs^3$. Again, start with the "obvious" example of a cube, and work from there. I like to use Minecraft as an example of how three-dimensional objects can be approximated by cubes (voxels, in that context).

  • From these examples, we might conclude that the dimension of an object is the exponent that shows up when that object is scaled. If an object has measure $M$ (whatever "measure" means: non-rigorously, the measure of a one-dimensional object is length, of a two-dimensional object is area, and so one—this hints, however, at measure theory, which is a deep mathematical idea already) and is scaled by a factor of $s$, then the scaled object will have measure $Ms^n$, where $n$ is the dimension of the object.

At this point, we can introduce the Sierpinski carpet. After showing students how the Sierpinski carpet is constructed, we can ask about the dimension of the carpet. Students typically assert that it must be two-dimensional, but we can show them that it isn't! Suppose that we start with a carpet of measure $M=1$ unit (again, whatever "measure" means—it is a little unclear in this context). If we scale this carpet up by a factor of $s=3$, it can be precisely covered by $8$ copies of the original carpet. This means that the scaled carpet has measure $8$ units. But remember that the measure of the scaled carpet should be $Ms^n = 3^n$, where $n$ is the dimension of the original carpet. Now for the scary part: a logarithm! A little bit of computation reveals that $$ 8 = 3^n \implies n = \log_3(8) \approx 1.893. $$ Hence the Sierpinski carpet is approximately $1.893$-dimensional!

Now the talk can be concluded in a number of ways, generally by asking questions. Because it is my area of research, I typically start asking about whether or not this notion of dimension is reasonable. For example:

  • Most shapes cannot be covered by smaller copies of themselves (they are not "self-similar"), so can the computation for the Sierpinski carpet be generalized at all?

  • The notion of "measure" is quite nebulous and inexact—can we pin it down at all? (Welcome to measure theory!)

  • The Sierpinski carpet example shows that dimensions need not be integer valued. Given a real number $x$, can we make a shape which is $x$-dimensional? What if $x$ is negative? What if $x$ isn't real at all, but is complex?!

Notice that any one of these questions could, with a little more background, form the foundation of a PhD thesis project.

I would suggest that the main thrust of a seminar of the type described in the question here should be to expose potential secondary instructors to this kind of thinking. These instructors should understand that very simple ideas can lead to deep questions—they need to know that the material taught in high school fits into a larger context, and they need to have enough knowledge of that context in order to answer difficult questions posed by really clever students.

I would also suggest that the actual topics taught in such a seminar are utterly irrelevant. The goal should be to get the seminar attendees to think mathematically. My example is fractals, 'cause that's what I know. But the other topics dismissed by the question-asker are also fertile ground: a few years ago, I attended a great talk by Gilbert Strang on singular value decomposition which fairly neatly falls into the category of lectures I've described. I think that knot theory is a great way to engage in mathematical reasoning, as it provides a path for asking the fairly deep question of how we can tell things apart. Thus I would be careful about dismissing any particular topic as "not relevant".

My recommendation would be to pick several topics which are of interest to either the student in the seminar, or the organizers. Pick a couple of research topics, and spend some time getting at the deep questions. Pick an example which could benefit from some coding (SVD? the Collatz conjecture?). Mine the content of, for example 3Blue1Brown for interesting topics. Have the organizers talk about their own mathematical research. Any topic is fair game, if it is approached correctly.

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  • $\begingroup$ You didn't mention it, but my way (haven't given such a talk, however) would be to work in the square-cube law somewhere early, with applications to biology (insects don't need blood vessels but we do), engineering (has to be accounted for in scale models, when scaling things up), economics, etc. I learned about this several decades ago (I think it was something written by Isaac Asimov), where the principle was used to explain why the giant monsters in the 1950s movies couldn't exist. $\endgroup$ – Dave L Renfro Mar 23 at 9:00
  • $\begingroup$ @DaveLRenfro In the precalculus curriculum that a colleague and I have been developing over the last several years, these kinds of scaling relations (the square-cube law in biology, inverse square laws in physics, etc) are a major part of the class. Unfortunately, in 60 minutes, there is only so much that I can talk about, and my preference is to get to sexy fractals as quickly as I can. ;) That said, I love the idea of building a math-bio talk around why Kaiju don't work. $\endgroup$ – Xander Henderson Mar 23 at 13:58
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I would do something on point group symmetry or 2-D space groups. (Allows to bring in some Escher and the like.) Keep it a little "art-y" and emphasize examples, description, and recognition. Can bring to a close with some remarks about group theory, but I wouldn't even try to definitively describe the topic in this abstract manner (not enough time and too hard). But at least letting them know it exists. (Sort of how we can tell people that only 4th and lower polynomials are all soluble to radicals...without proving it, but just saying somebody did prove it.)

Resist any tendency to assume these math teachers have had/understand real analysis and want to go deeper down that hole, getting even "better" at it.

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