Objectives for Differential Geometry

Regular readers (all three of you) are aware that

  1. I am scheduled to teach differential geometry this coming term.
  2. I am very excited about this.
A minimal surface

a minimal surface

Less well known is that I am also terrified of this experience. I taught the course in the Spring of 2010 and the experience was not all warm and fuzzy. The biggest problem was the disconnect between what my students knew coming in, and what I thought they would know. Suffice it to say that I have a much better understanding of what it means to have a student who passed linear algebra and multivariable calculus at UNI.

Let’s take it as given that this iteration of the course will also contain surprises, but I hope they will be much smaller ones. And I am at a place now as an instructor where I am much more alert for this kind of trouble, so that can only help. (Oh, please, please let that be a true statement.)

So, now I have spent a fair amount of time in the last two weeks worrying about how this course will all work out when I should have been properly enjoying time with my family. I started by trying to write a set of learning goals for the students.

First Missing Piece:

I started by making a list of content goals for the semester. I figured this is the easiest way to make some progress and get rolling. Uh… I didn’t get very far. I can easily make long lists of what I hope students might learn about some beautiful mathematics. Apparently I can not effectively pare down that list to something manageable.

But I realized the problem, at least. I need a guiding principle. Ahem, a

Guiding Principle.

(Is that an echo?) What I mean by this is that I need some major meta-mathematical goal for the student’s educational progress so that I can keep moving in a coherent direction. For my Euclidean Geometry course, everything is about definitions: understanding the making, use, abuse, and structure of a mathematical definition (as opposed to a conventional dictionary definition) is a key step to understanding the axiomatic method that underpins modern mathematics. For Linear Algebra, the main point is to learn to make your own examples and work toward the concept of abstraction as a verb.

Differential Geometry is an advanced topics course, and students who enroll have gone a lot farther in their study of mathematics than the average bear, and there is not much time left in their undergraduate studies. I have decided to aim high and make the course about the process of finding your own way when learning. I will explicitly ask the students to make a plan for what to learn during the semester, and we will engage in the process of frequently revising our plan for learning. Students will have to struggle with making summaries of literature they don’t quite understand, picking pieces to figure out, and then revising their map of the “differential geometry landscape” based on what they have learned so far and what seems like the right thing to work on next.

I will have to let them in on what is going on and why, and I will likely have to repeat my justifications for this choice many times with a cheerful and supportive look on my face.

I should mention that I am inspired by the general goals of a liberal arts education, but more specifically by Robert Talbert’s recent writing on using Pintrich’s concept of self-regulated learning as a touchstone.

The General Plan

The first assignment will be to find some literature and make a bird’s-eye map of the general layout of classical differential geometry. I will ask students to start designing their own list of content goals for the semester. We will then have a big class discussion and I will lead a negotiation of what should be on these lists.

I also plan to let each student keep their own personal list. I will require more of the graduate students than the undergraduates, and I will allow students to pick topics which seem to interest them more than others. It is likely that they will not all be studying the same thing, but they will all be studying related things. I think that this will be conducted in some sort of “grant proposal” format, with revisions allowed after comment.

For assessment, I plan to give them a little speech which amounts to “convince me you’ve learned something.” I will allow the students to design their own demonstrations of competence. I think these will go everywhere from a simple quiz on paper, through oral exams in the form of “come talk with me” all the way out to some creative/exploratory project with a tangible outcome.

We will take our class time to discuss what they want to discuss. They can work in small teams. They can talk with me. If they ask, I will prepare little “mini-lectures” on crucial pieces they are missing. I will find a way to bring my books to class so we will have a little library available. (Regular readers, who now number only two, certainly know that I have many differential geometry books.) Sometimes we will set aside time for student presentations or student-led discussions.

Things I will offer to “review”

Thanks to my last run-through with this material, I have the following list of things I am sure that students need to know more (or better) about, even though they aren’t differential geometry exactly. These I will happily give mini-lectures on at the drop of a hat.

  • Taylor series for functions f: \mathbb{R}^n \rightarrow \mathbb{R}^m. Mostly we need second order Taylor polynomials, but more might be better, especially if a student wants to take up the subject of “higher order contact” of objects. Of course, we mostly need the cases where m, n are no greater than 3.
  • Quadratic surfaces
  • points, lines, circles, planes, and spheres in \mathbb{R}^2 and \mathbb{R}^3 (especially comfort with using vectors to describe these things).
  • Eigendata and diagonalization for 2 by 2 and 3 by 3 symmetric matrices
  • Isometries of the plane and of three-space as affine maps x \mapsto Ax + v
  • basic ODE’s and the existence and uniqueness theorem for reasonable equations
  • the change of variables formula for integrals over planar regions
  • the inverse and implicit function theorems

And this item is usually covered in differential geometry courses these days, I suppose, but not well enough. Some students just get it, and others need a lot of help:

  • Curves and surfaces as parametrized objects. (Their nature as functions is really difficult for some students.)

Big Project Ideas

Students will need interesting things to do as vehicles for demonstrating their learning. I don’t expect that all of them will just have wonderful ideas for projects. I have been trying to brainstorm some to help get them started. Here is what I have so far:

  • Design a single class lecture organizing and presenting the main ideas of the topic you have in mind.
  • Make some 3D printed objects and write a paper discussing their principle interesting properties.
  • Make some animations of interesting curves or surfaces and write a short paper discussion the interesting mathematics.
  • Investigate the recently announced “non-reversing mirror” and report back on how it works.
  • Write some Sage/Python code that handles some interesting computations from the subject.
  • Pick an interesting theorem “off the main path” and pick it apart. Find a good way to share it with the class and/or the world. (Struik is full of references to neat results that I have never really taken time to learn. Lots of them would work great.)

I would like to have more of these. If you have an idea, drop it in a comment.

Things I Need to get ready before the term starts?

This needs a itemized list of things to prepare so that the first two weeks runs smoothly. I will work this out later and report again.

Oh, man. Tell me about the traps.

I am quite certain that I am jumping into the deep end here. Has anyone tried something so open-ended before? I think I am really going out into the dark with the hope there is a torch somewhere in the front yard.

I should mention that I plan at least one day’s worth of lecture on mangled and mixed metaphors. But this will likely occur in one minute pieces spread out over the term.


9 thoughts on “Objectives for Differential Geometry

  1. Here’s the webpage from when I taught differential geometry a couple of years ago: http://www.math.sunysb.edu/~bowman/teaching/mat362-spring2011.html
    If any part of that is likely to be helpful, it’s probably the “Handouts” section. All my students had to do final projects, and there’s a list of possible options there. (These projects were, in my opinion, extremely successful.) The rest of the handouts are additional material from outside the textbook that I wrote for those who would be interested, without requiring knowledge of it for assessment purposes.

    When I taught the course, I mostly lectured, so I don’t have any thoughts to add to the self-regulated learning setup you have, but it certainly looks both exciting and terrifying. And because differential geometry is such a broad subject, you’re likely to get several different initial plans from the students. (Does that count as a “trap”?)

    One question to consider: are you sticking with 2- and 3-dimensional geometry, or allowing exploration of higher dimensions? I gave definitions in arbitrary dimension as often as possible, but when proving theorems focused on the classical theory of surfaces in 3d. This could also be an easy way to track undergraduates and grad students (# of dimensions they deal with).

  2. Thanks, Josh! I will check those out.

    I plan to stick to two and three dimensions exclusively. Even the grad students could use learning this stuff in greater depth before wading into deeper waters.

    If I need stuff to keep grad students occupied, there is certainly enough stuff about surfaces that is genuinely hard to chew on.

    Students working on several disparate projects is a possible trap for me. The main problem being that I could misjudge a project and have it turn out way too hard…

  3. This sounds awesome. I am both scared and excited for you.

    Keep us posted. I am already trying to figure out a way to hybridize this to work on non-topics courses (the Guiding Principle is great for everyone; can we do this while also teaching specific content?).

  4. I think that having some sort of guiding principle is important for each course. I don’t pretend to know what it is for each one, but in a way, it is (nearly) independent from the actual content, as it is more about student development of meta-mathematical skills. You know, their actual education.

    I think of it as depending more on what stage of development most students are at. I have not yet thought about what it might be for each course in a “standard program”…

  5. This relates to one of my goals for this summer: I am hoping to determine what my goals are for my students’ general education (e.g. I want them to expect and to use evidence for any claim in any aspect of life), and then to use the mathematics course as a tool for achieving this (rather than being an end in itself). So I am particularly interested in your class, since it seems you will be doing exactly this!

  6. This sounds like a really fun, exciting, and challenging way to run a course. You have created a lot of different paths for students–what to learn, how to learn, how to demonstrate it–which should make for a very rich experience for everyone. Your deep knowledge of the content will be crucial when it comes to putting everything in context, i.e., that all the different things learned by the different groups and individuals are all connected to the same ideas.

    In my experience implementing open-ended projects and courses like this, I’d watch out for two things. First, students will likely not be ready for this kind of autonomy and responsibility right away. One way I combat this is by creating simple “Warm-Up” projects to get groups active and involved immediately. This could be something like “In your group of four, pick a review topic and prepare (1) a twenty-minute overview presentation; (2) a collection of traditional exercises, (3) an enrichment “investigative” activity, and (4) an original tech piece that highlights something about the topic”. It gets everyone involved on day one and starts building the necessary collaborative spirit.

    The other suggestion: make sure there is a regular and meaningful “check-in” process for the big projects. Inevitably, the least satisfying student work arises for me when I don’t invest the time along the way to make sure things are headed in the right direction.

    Look forward to following along!

  7. Thanks! I had thought of the second one (because I have had underwhelming projects before), but that first comment sounds like good advice. I will think about that carefully.

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