Alternatives to University Lectures: Non-lecture Mathematics Classes

Question

I am looking for resources for designing undergraduate mathematics classes that are not lecture-based. (Bonus points if the design is for an introduction to proof course).

For example, Robert Talbert blogs about flipped classes (calculus, and intro-to-proof). Do you know of others?

Answer 1

I think you will definitely want to check out the The Journal of Inquiry-Based Learning in Mathematics. It is dedicated to being a "resource for designing undergraduate mathematics classes that are not lecture-based."

Another place to start that is overwhelmingly huge and mostly aimed at the K-12 level, but definitely has a lot of good ideas that are applicable to this question, is the huge community of math teacher bloggers. Here is one place to start that exploration.

I also second Michael Pershan's recommendation to examine the PCMI problem sets. They provide a magnificent model for what it looks like to develop content through problems, and it's a little different from much of the stuff you'll find at the Journal of Inquiry-Based Learning, which is coming out of the Moore Method tradition.

Answer 2

There are a few ways to do this. I recently did it in Carnegie Mellon's introductory proofs course. I'll try to outline here a few different solutions that various places have tried.

Carnegie Mellon (15-151): We alternated introducing new topics (usually via lecture) with what I called workshops. A workshop is during normal lecture time. Students are given 3-5 questions that range from conceptual questions to difficult proofs. We created the groups randomly, and we changed them every few workshops. We gave them whiteboards (you can buy big sheets of whiteboards at home depot for like $5 and have them cut them down for you) and markers, and asked them to work together on the proofs in order. As the lesson progressed, the TAs and I would walk around the room answering questions and reviewing proofs. Depending on how many resources you have this could be very difficult, but we decided to split the large lectures down into smaller ones so this was feasible.

MIT: Albert Meyer in 6.042 takes a different approach. The classes are still smaller, and I believe there are still "lab assistants" running around the room helping out, but he does the flipped classroom every lecture. He's done it in several different ways over the years. I believe originally, he gave mini lectures for the first 20ish minutes, and then the students worked for the rest. He might have moved to asking the students to watch the lectures at home--in a true flipped classroom format--now. One other thing to note is that the book Albert wrote is absolutely fantastic and completely open. It's a great resource for good problems, good explanations, and expository material for students to read.

Stanford: I can't remember the course number or name right now. I will do my best to follow up later by adding a link. Stanford takes a different approach than either of the other two. They have a standard lecture series, and sometimes, they offer workshops in the evening (which I believe are optional). This approach is particularly nice in the college setting, because it requires fewer resources and less contention for classroom space. Both of the CMU and MIT courses used a "special room" which allows for collaboration. At CMU, 15-151 is held in the CTC, and at MIT, 6.042 is held in the TEAL rooms.

There are other various strategies to do this, but these are the ones I've seen most commonly (and successfully) used for introductory proofs courses. Albert's resources are very useful, and I would be happy to share some of mine as well.

Answer 3

Jim Fowler and I created a hybrid linear algebra/multivariable calculus/multilinear algebra course at

https://github.com/kisonecat/m2o2c2

The structure of the course was:

1. $\mathbb{R}^n$ as an inner product space.
2. Linear maps $\mathbb{R}^n \to \mathbb{R}^m$
3. The derivative of a function from $\mathbb{R}^n \to \mathbb{R}^m$ as a linear map of tangent spaces.
4. The derivative of a function $\mathbb{R}^n \to \mathbb{R}$ as a covector valued function (differential 1 form). The relationship of this covector to the gradient.
5. Matrix with respect to a basis other than the standard basis and Eigenstuff.
6. Understanding bilinear functions $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ and the associated matrix. Connection with adjoints when considering inner product structure.
7. Understanding the second derivative of a function from $\mathbb{R^n} \to \mathbb{R}$ as a symmetric bilinear form. 2nd degree Taylor theorem, with applications to max/min (eigenvalues give definiteness of the form).
8. General multilinear forms and the full Taylor theorem.

This course was based on problem solving. Students largely discovered the mathematics through fill in the blank computational problems and also free response items for proofs which they could have conversations with each other about, and compare to the "official solutions".

Unfortunately even compiling the source files seems really difficult now. I would have to match up the state of different github repos (the content source code with the processing system source code). So I think this work is basically lost forever now. I might find time to sort it out someday. In the mean time, if you are really interested, you can just read the tex files for each topic.

Answer 4

I really think that the Park City Math Institute's courses are fantastic, both as resources and as models for what it could mean to have students spend their time in class solving interesting problems.

They run 3-week courses in the summer, often on a topic relating to number theory. For example, check this set of problems out.

Or, more generally, here.

Answer 5

This is simple, but effective. Consider doing a weekly, period-long test, every Friday. I remember having that in HS for pre-calc and calc. At first was surprised by the frequency. But ended up admiring the pedagogic effect. Tests are some of the most effective drill (because of the drive for prep and the higher stakes...and the learning from the results). In addition, I find the "midterm plus a final" practice of many college courses is fraught with issues (cramming, etc.) and ends up being too high stakes, too limited in feedback. Weekly is a better frequency.

And keep it on FRI. Weekend free for the students. And regularity is a powerful social tool. I've even used it to drive behavior by multi billion dollar CEOs. Pavlov did good shit...don't dismiss it.

This needn't keep you from introducing more drill, less dulcet tones, into other parts of the classes. But right away, it is replacing several hours of sage time with practice.

Answer 6

Dave:

You might consider programmed instruction. (To answer your question, but maybe not helpful for making money as a teacher!)

https://en.wikipedia.org/wiki/Programmed_learning

It's sort of a step-by-step way of learning, self studying. I have found such texts to be extremely efficient as a self studier, much better structure than just reading lessons and working problems in a normal textbook. Of course, preparing such materials is a significant effort. And it has less market in academia, since it is designed for use without a teacher.

Ken Stroud was a famous programmed learning math textbook writer, can check out some of his texts. Just look at them overall as well as the prefaces, where he talks about the methodology. [HE covers a fair amount of engineering math and below, but not proofs, I don't think.]

https://en.wikipedia.org/wiki/Ken_Stroud

Note, the system is not restricted to math. I have used programmed texts in basic accounting and in nautical rules of the road. In both cases, they were extremely time efficient, easy to follow and not give up or get bored, etc.

P.s. Apologies for other answer. I probably had two tabs open, you know how that is. Cookie mouse request for a moderator to cut the 29APR21 answer...