Flipping

Here are some very explicit learning outcomes I would expect from a student coming out of a four-year mathematics degree:

By the end of this degree, you should be able to:

1. quickly and correctly evaluate which parts of a proof are easy or routine and which require more thought or new ideas.
2. identify gaps in mathematical arguments (your own and those of others) and evaluate whether they are easily fixed or not.
3. extract and summarise the key new idea or construction at the heart of a proof.
4. devise detailed, step-by-step strategies for proving previously unseen results.

Of course, we hope that all students will pick these skills up by osmosis in analysis courses, but think about it from a student's perspective:

• they are confronted with totally unfamiliar notation and ideas,
• they struggle to see why they are required to prove these abstract or banal statements,
• with some effort, they can follow the lecturer's proofs, but if they cannot follow some steps then they assume they are stupid,
• they attempt to prove new results in homework questions, unsure where to start, and often the only feedback they receive is a numerical mark and a model solution.

Whereabouts in all this do they get the chance to achieve any of the learning outcomes listed before? If it happens, it happens by accident.

First, it means that much of the standard lecture content must be delivered in another format: either in books/lecture notes or online videos. Indeed, this aspect is what most people think of as lecture-flipping.

More importantly, it means that you have to think of something else to do in lecture time. You have to design meaningful learning activities for the students to do to help them to achieve the learning outcomes you want.

I have already written a comprehensive set of lecture notes for the module I'm planning to flip. My plan is the following:

1. Restructure and divide the notes into 10-20 minute chunks and decide which part of the notes I should just tell them (in a video) and which parts they should be able to figure out for themselves (maybe in groups in class).
2. For each chunk, produce either:
   1. a video, along with a set of comprehension questions and a set of notes for them to read alongside the video. The comprehension questions should be relatively quick and easy (the kind of things you should be able to do instantaneously/mechanically if you understood the video). They might involve
      • filling in details of a proof from the video,
      • correcting a slightly incorrect statement that was made in the video,
      • doing a slightly longer computation similar to one that was covered in the video.
   2. a lesson-plan for a face-to-face session in which we cover the content, for example:
      • 0:00-0:10 At the beginning of the class, state the result that the deck group is isomorphic to the quotient \(N_H/H\) where \(N_H\) is the normaliser of \(H\) and \(H\) is the subgroup associated to the quotient space. Get the class to remind you of the definitions of all terms involved.
      • 0:10-0:25 In pairs, work out what this means for some explicit covering spaces (need to prepare handouts: one covering space for each pair).
      • 0:25-0:30 Share your conclusions with another pair.
      • 0:30-0:40 As a whole class, formulate a strategy for proving the theorem (lead them to the idea that if \(\beta\in N_H\) then there is a deck transformation \(F_\beta\) taking \(y\) to \(\sigma_\beta(y)\) and that you need to show this assignment \(\beta\mapsto F_\beta\) is (a) a homomorphism, (b) surjective, (c) injective).
      • 0:40-0:55 In pairs, try to implement that strategy. (Homework: complete the proof. Next session, someone will present their solution).

Of course, you end up covering less content in class this way, but as I tried to emphasise earlier, the kind of skills they are developing are perhaps more important than just learning the content. For example, they are learning how to approach proofs for themselves, and how to talk constructively about maths with, and explain stuff to, their peers.

The last week of term I was not in UCL and wasn't able to give my lectures in person. Having already settled upon flipping next time around, I decided to record the week's lectures as videos and set some learning activities as homework. This doesn't count as flipped learning, because it wasn't backed up with face-to-face contact time. Nonetheless, I learned a lot from doing it. Here you can see the online learning materials I provided.

• The videos are still not great quality. I used Explain Everything on my iPad, but I find writing on an iPad makes my writing look (even more) like that of a 5-year old (than usual). I'm still trying to figure out a good solution.
• I embedded the videos into webpages which contained HTML lecture notes and exercises. The students' feedback was that having lecture notes on the same page as the videos was very helpful: when speaking out loud, I tend to be slightly imprecise and try to convey the sense of an argument; when writing I am usually more precise, and having both media together helped mitigate the occasional wooliness of the videos.