Maths Lectures: Tips from an Audience Member's Point of View

There are two or three big principles of lecturing maths (or any subject). I'd hesitate to advise on how to prepare notes, say, or the more mathematical side of the art; but having been an audience member for years in maths lectures, I do feel I can give some tips on one of the chief overriding principles of lecturing:

PAY ATTENTION TO YOUR AUDIENCE.

A lecture is not just a means of distributing lecture notes - see Tom Körner's essay on Lectures for more on this: on his page, about half-way down. However, neither is its aim a purely verbal explanation of its subject matter; it lives a curious dual life, attempting both to induce understanding at the time, and to create a comprehensible and coherent set of lecture notes for the student to refer to later.

"The ability of lecturers to give good notes is different from the ability to give good oral presentations... as you know from the quality of your notes and of your lecturers" (Prof Grimmett, Lent 1999)

This principle will work itself out in ways like:

Note: The following suggestions are written from the point of view of maths in Cambridge. If your location or subject is different, then bits about blackboards etc may seem irrelevant; please feel free to ignore any such as appropriate.

• Simple examples:
  Check with the audience before cleaning boards

• Check they can see (not too low/far) and read (large&legible enough) what you write!

  "Pure mathematicians don't squiggle" (Dr Hudson, Michaelmas 1997)

• Don't automatically assume they understand:

  • Notation you use!

    "Notation is baaaad. But bad notation is worse!" (Ian Grojnowski, 10.2.00)

  • Why something is 'obvious'/'trivial'

    "Take some time to think about this until you see it's blindingly obvious" (Imre Leader, 31.10.00)

  • What was covered several lectures ago

    "If you don't know this, you're bad! I did teach you! Bad bad bad..." (Ian Grojnowski, 22.2.00)

  • What was covered in previous courses!
  (As in, ask them; if the response is broadly or specifically negative, give a quick verbal or written explanation, then carry on. It doesn't have to take more than 30 seconds.)
• Remember they'll be taking notes, so:
  • Use boards in roughly linear order
  • Make it clear when you're writing things not for noting
  • Make it clear when proofs/examples start/end, which theorem is currently being proved, what's going on generally

  • As far as possible, put in "in-betweeny" words like 'Let', 'So', 'If', 'In the case where', 'such that', 'Assume that', 'As we know from Theorem such-and-such', ...

    "It's not the most useful comment without some words to go with it" (Elizabeth Ayer, Lent 1999)

  • If you want to explain something so want them to be thinking not writing, give them time to catch up writing!

• I know opinions differ on this one, but it helps the student enormously if major points (sections, theorems) can be numbered. (If not, then at least bear in mind the vague structure to the course, and let the audience know it, preferably written down. It's very frustrating when a lecture course seems to be a game of Mornington Crescent, plucking unrelated theorems and definitions out of thin air for no visible purpose - and this has happened several times in my time at Cambridge!)

  [asks audience] "What number shall I give this theorem?" - [student calls out "29!"] - "Right. Theorem 29, then..." (Dr Thomason, 1.3.00)

  That said, it seems that opinion among students is divided on whether, when explaining a deduction or reasoning, saying "the theorem on zeroes of polyomials" is more useful than "Theorem 2.2" or not. I guess the ideal is to say "Theorem 2.2 on zeroes of polynomials", but that's a bit of a mouthful...

  "'Example 5'... are there 5 examples?... 'Yet another example'" (Ian Grojnowski, 20.1.00)

• Think what will be the hardest parts to understand of your course, and take special care with those. A good pair of examples are: consider the "top" and "near-bottom" students in your course. Which part of the course will the "top" have difficulty with? (Maybe some complex definition? A tricky line of reasoning in one of the proofs?) Try to come up with an extra example to help understand that bit, or spend a bit longer explaining it. Now consider the same question for those who're likely to have difficulty fairly early on in the course: which single part is likely to cause them the most trouble? Can you give a handout on that, or go through it slower, or explain it in simpler terms?

  "This idea looks pretty simple. And is pretty simple. But I still don't understand it" "Shall I explain why these are obviously equal?" (Dr Grojnowski, 25.1.00, 15.2.00)

• Too rarely done, always appreciated: stop and take questions!! It's so useful, even though we're not too used to it :)

  "Are there any less highbrow questions?" [silence for 20 seconds] "Oh, well, it's not Cambridge tradition to talk, especially at 9 in the morning" (Tom Leinster, 20.10.00)

I appreciate if a lecturer has a too-large amount of material to get through, the above may not all always be possible; but the principle remains, PAY ATTENTION TO YOUR AUDIENCE.

GOOD LECTURERS

For each of my years in Cambridge, one or two lecturers have struck me as being very clear, attentive to their audience's understanding, and generally very good at the art of lecturing, quite independently of their mathematical skills. Learn from them.

Part IA, 1997-8:	F Kelly (Probability)
Part IB, 1998-9:	JME Hyland (Groups, Rings & Fields); GR Grimmett (Markov Chains)
Part II(B), 1999-2000:	Ian Grojnowski (Representation Theory)
Part III, 2000-01:	Tom Leinster (Category Theory)