I listened to episode two of this podcast: https://www.benbenandblue.com
I would like to share my thoughts. I apologise in advance if I appear over-confident in my views — I welcome disagreement and am open to being corrected!
How I intend to use lectures next academic year. I teach maths to 18–19 year olds who are hoping to go onto a STEM degree. The approach I had planned to use in the next academic year is this:
1pm-2pm: Highly interactive lecture. I will talk to them for a few minutes (introduce a new concept or idea or go through an example) then pose a single question for 10 seconds to a minute or two, depending on difficulty. I will walk around and maybe talk to one or students about their thought process. Then I will come back to the front and go through the answer. Then rinse and repeat for 50 minutes. There will be variations on this, depending on what is being taught. Lecture notes with gaps in them will be provided at the start, along with a problems sheet.
2pm-3pm: I will recommend that students take a short break in this time, then review the lecture and/or attempt some problems, then maybe take another short break break.
3pm-4pm: Problems class. Students will attempt the problems, with 3 or 4 members of staff on hand to help.
I welcome any thoughts you have on this plan!
Concerns I have about 100% flipped-classroom approach
- There is a difference between live explanations and textbooks or videos. The advantages over a textbook are obvious: there is no pacing in a textbook, textbook is static and without flow, textbooks require everything to be precisely defined whereas in a lecture I can point to stuff. Advantages over a video are less clear but include: stage presence, use of persona, audience interaction, the possibility of informal remarks which you wouldn’t want to include in a recorded video, live demos (“Give me a number between 1 and 100…”). These views are heavily influenced by https://www.dpmms.cam.ac.uk/~twk/Lecture.pdf
- I inevitably make mistakes in a lecture. This is a valuable thing to witness for students.
- In a lecture-setting, a student is more likely to be focussed and less distracted. If they had to pre-read the same material I present in my interactive lecture, it would probably take them much longer to reach the same level of knowledge and/or understanding.
- In a lecture-setting, a student will be more disciplined. If an easy way to get the answer is available, it is difficult to resist (e.g. using a calculator for a mental calculation, looking at the back of the book.)
- There will be students who do not do the pre-reading. Now either I do or do not include a mini-lecture. If I do, those students learn that the pre-reading is unnecessary. If I don’t, then those students will fall behind, which is particularly problematic for mathematics.
- I do not know what I expect them to learn in their pre-reading. What can I expect them to learn? Unless the pre-reading is well-designed, includes some form of active engagement and some form of personalised instant feedback , I suspect many students will forget what they read and so I would have to give a mini-lecture (or let them fall behind).
- I believe most of the students I teach do not know how to read mathematics; they do not know that they should be checking every symbol from one line of calculation to the next, and making sure they know why any changes from one line to the next are valid changes.
- I believe I overall explain mathematics in a better, more intuitive way than any single resource. For a less bold (and less pretentious?) claim, I believe I can explain things better than the textbook that my students are given. I illustrate with a few examples (only comparing with the textbook). (a) Fractions: the textbook does not have a single picture or diagram when discussing fractions — it just describes all the rules for manipulating them without any intuition for why they ought to be true. (b) BODMAS: this is plainly wrong (or at best, misleading) — e.g. using BODMAS I’d say you should add before you subtract, but in 10–5+2 you should subtract first. (c) Functions: functions are presented as boxes or machines in which a number enters and then exits — I prefer presenting functions as a bunch of numbers on the left, a bunch of numbers on the right, and arrows going from the left to the right. This is more intuitive, for example “invert the function” just means “reverse the arrows” as opposed to “make the machine run backwards”. (d) Composite functions: the textbook only uses f(g(x)) for general x, it never illustrates what f(g(x)) is actually doing with explicit numerical inputs. (e) I explain the gradient of a line as the amount the height increases if you move across 1, instead of just giving the formula to calculate the gradient.
Do teachers know their subject matter? I agree with the sentiments in the podcast and I have little doubt that the majority of maths teachers do not have enough expertise to teach in a highly interactive, flexible way. I also believe that improving university teaching is a necessary step to improving the situation. Here are some anecdotes to support this:
- I did my PhD at the University of Leeds and I taught undergraduates there. Without exaggerating, 80–90% (maybe more??) of the students do not understand the notion of proof at the end of their degree. Leeds is ranked 12th for maths in the UK, so I believe this 80–90% figure is representative of a large proportion of the universities in the UK. This means that the majority of school teachers do not understand the notion of proof, which in my opinion means they have a limited understanding of mathematics.
- At Leeds, I helped in ‘Maths Drop-In’, where any student of the university could drop-in and ask for help with their maths. It was organised by a PhD graduate in (Applied) Mathematics. Once, they struggled to help a student, and asked me if I could help. The first part of the question was to work out the square roots of a complex number. The second part was to solve a quadratic with complex coefficients. I saw it and said “Oh, I assume that the square roots you found in part (a) will be the square roots of the discriminant for part (b).” The organiser replied “Oh, so you can just use the quadratic equation?”. Maybe I am mis-reading the situation, but the only conclusion I can think for this is that they do not understand where the quadratic equation comes from. This belief is further supported by various small conversations. Once they said that doing a research in Mathematics is just about applying some known tools to a new situation. Another time they said they never understood the pure mathematics stuff, so just focussed on applied mathematics for their degree.
- When applying for jobs, I once saw in a university’s syllabus the notion of a “direct proof” (as one of the three or four “types” of proof, others being induction, contradiction and perhaps “proof by contraposition”). I cannot put my finger on it exactly, but my gut instinct is that anybody who uses the term “direct proof” does not fully understand or appreciate what proofs are , and thus it is unlikely that their students will. (As far as I know no working mathematician says or writes “To prove Lemma 1.3 we can simply use a direct proof”. Also, I do not even know what kind of proofs count as being “direct proofs”.)
Oxford tutorials. I did my undergraduate at the University of Cambridge, which has the same system: one-to-few un-assessed tutorials in which you discuss your work. Sounds dreamy! What is worth adding, given the main focus of the podcast, is that the only other form of contact time (for maths, at least) was traditional lectures. Furthermore, the ratio of lectures to tutorials (again, for maths) was 6-to-1: we had 3 hours of lectures per week per module, and one tutorial every two weeks per module. (And there are four modules at a time.) So imagine this: you watch 24 hours of MIT OpenCourseWare lectures in two weeks, attempt four sets of problems and then only have four hours to discuss. It no longer sounds so good! Overall it works, but only because Cambridge and Oxford have the luxury of choosing the students who are able to teach themselves and who only need a nudge in the right direction once every two weeks. Of course, it does not work for everyone — if it turns out you have little motivation or have poor organisational skills or your ability is not quite high enough or you are highly anxious about being put on the spot, you don’t have a good learning experience.
Mysteriously, bad lectures are not zero value. I have two anecdotes from my undergraduate days which indicates that even bad lectures provide value. First anecdote: if typed lecture notes were available for a module (they frequently were available for 2nd and 3rd year modules, but never for 1st year modules), I would not go to lectures. During a tutorial I mentioned this. The teacher was surprised, as I produced good work. In their experience, students who don’t go to lectures don’t do well.
The second anecdote is very similar. I was involved in the Maths student society, and we made available first-year printed notes. A few weeks later, I got an email from a senior academic (Imre Leader, who is renowned for their excellent teaching), asking us to make the notes unavailable, as they had noticed a drop in the quality of their students’ work!
I find this mysterious. I rarely learnt anything in lectures — the pace of the lectures was so fast that I could not write notes and engage with the mathematics. I do not know how best to explain these two anecdotes.
Reflection a day after writing the rest of this. Walking to work, I realised my concerns with pre-reading can be summarised by the idea that students need guidance or practice pre-reading before they can pre-read. Furthermore, the lectures I am planning are essentially guided pre-reading sessions: instead of them reading they listen to me and instead of them getting their answers from a book or by clicking “Show Answer” they get the answer from me.
This realisation has led me to the idea that maybe my lectures should actually be guided pre-reading sessions — not just in essence but by design. As of now, I do not have the time to re-design all the lectures, but I could try including snippets of pre-reading exercises: “For next 5 minutes, read page 7 and attempt the questions on page 8.”
This is nice — even if nobody reads this, at least I have gained this new idea from the experience of writing it!