Monday, August 11, 2014

low-threshold exercises for analysis

This fall, one of my courses will be Introduction to Analysis. At my school, this has been taught using a modified-Moore method for the last few years, and I will be largely adopting the structure and content of these previous years. In this IBL implementation, students work in groups on one problem per week. Each week has three assigned problems (so generally multiple groups are working on the same problem) that are loosely related. At the end of the week one class period is devoted to presentations: for each problem, one group is selected to present their solution in about 20 minutes, and the rest of the class is expected to be engaged in discussion with the presenters. Many of the problems were developed by David Cohen (now professor emeritus), who described the method in an article for the American Mathematical Monthly. Further developments were made by Christophe Golé, with whom I co-taught the course two years ago. From my first exposure to the materials for this class, I have been impressed by the clever way students are led through standard material by a non-standard path.

As with many introductory analysis courses, one goal of this class is to help students transition to more formal mathematics, giving them experience with absorbing definitions and writing proofs. The problems themselves guide students through much of this process. I felt, however, that at times students could benefit from having exercises that allow them to interact more rapidly and immediately with new definitions. So one aspect I’m adding this year is a collection of “Warm-up exercises”, one per week. These are intended to be “low-threshold” activities, in the sense that a student should be able to work on them and produce results even with just a superficial understanding of the definitions involved. My hope is that by interacting with the definitions in a meaningful and productive way, they will feel more prepared to grapple with the assigned problems.

Here is a list of the exercises I’ve written, together with a rough description of the corresponding week’s topic. In addition to being “low-threshold”, several of these are also “high-ceiling”, meaning that immediate extensions and generalizations are evident. (For most of the course, however, the “high ceiling” is provided by the main set of problems.)

  • (Counting and cardinality) Prove that the sets {1,2,3} and {4,5,6} have the same cardinality. Prove that {1,2} and {1,2,3} do not.
  • (Balls in metric spaces) Recall |x|=x if x≥0 and |x|=−x if x < 0. Prove |x+y|≤|x|+|y| for any real numbers x,y.
  • (Topology of real numbers) Prove that if x is isolated from a set TR, then x cannot be an accumulation point of T.
  • (Topological properties) In R, is a set that contains just one point compact? (A bit of clarification here: in this course, the definition given for “compact” is a variant of sequential compactness, namely, that every infinite subset has an accumulation point.)
  • (Continuity) Prove that x^n is continuous at zero for any nN.
  • (Properties of functions) Prove that x^n is differentiable at zero for any nN.
  • (Sequences of functions) Use the algebraic identity (1–r)(1+r+r^2+…+r^n) = 1–r^(n+1) to prove that the series 1+r+r^2+r^3+… converges to 1/(1–r) if |r| < 1. (I keep finding that students have forgotten the sum of a geometric series in classes after calculus, so I figured it made sense to remind them of this fact while also suggesting they prove it.)
  • (Uniform convergence and degrees of differentiability) For any kN, give an example of a function that is C^k but not C^(k+1).
  • (Borel sets) Suppose X is any set and A is the power set of X, i.e., the collection of all subsets of X (including ∅ and X itself). Show that A is a countably complete Boolean algebra.
  • (Lebesgue integration) Show that a sum of simple functions is a simple function.
There’s one more week’s worth of problems—all focused on properties of the Cantor set—which don’t require any new definitions.

I’m not quite sure what role to give these in the course. I don’t want them to be required, and I definitely don’t want to make them “extra credit”. I do want them to provide a useful entry into playing around with definitions and not seem like extra work. Thoughts?


Bret Benesh said...

I have a couple of ideas on how to integrate these into your classes, although I do not love them:

1. Tell the students to work on these if they like, and give students an opportunity to present at the beginning of class. This would require you to ask if students would like to present at the beginning of class each day, so that the students were constantly reminded that this is something that they could do.
2. Make it a requirement that a student do N of these problems if they want to get at least a B for the class (or A, or whatever).

Again, I don't love either of these. I particularly do not like my second suggestion because that will attract the students who do not need the low-threshold problems.

Really, it seems to me like you might want to replace some of the existing problems with these, since low-threshold problems benefit the most students. But I have no idea what your constraints are, and it sounds like this might not be feasible.

Joshua Bowman said...

Bret: Eventually I might want to design my own set of analysis problems and have them more scaffolded, providing a wide range of possible levels of exploration for the students. At the moment you're right that it's not feasible for me to make more substantial changes, since I'm also preparing to teach two other classes (one of them for the first time).

Here's how the rest of the weekly schedule goes: In the class meetings after presentations, I meet with the groups individually to see how they're progressing. The next class period is spent with the groups drafting written proofs and preparing to present the following period (all groups are expected to be prepared to present). Could be interesting at one or the other of those meetings to ask someone to present a solution to the warm-up exercise, as you suggest.

I do plan to meet with individual students a couple of times during the semester, and I could take stock of how helpful the warm-ups are in those private meetings. I find that students rarely take advantage of "additional resources" that are provided as links or extra reading, but maybe easy exercises included with the main assignments will be more likely to be used.