Revision tactics

When I was doing things for Part III I started giving a talk on revision strategy.  My reason for doing so was part of my humiliating experience on the first occasion of attempting the MSc exams at Warwick. One contributing factor to that disaster was a complete failure to understand the style of exam I was up against, and complete absence of any strategy to prepare for it. 

Perhaps I should explain. My undergraduate training compressed four years worth of courses at Smith College, a woman's liberal arts college in western Massachusetts, into three years. At the end, my level of knowledge was not up to Part IB standards, and significant parts of that had been self-taught. Don't get me wrong, I learned some stunningly important things at Smith; I learned that I liked doing mathematics research, and that I was good at it, and I learned to read. And I resolved that no one was going to bully me out of being a mathematician, and I had excellent support in that decision, but in the words of Jo of Bleak House, I "didn't know nuffink".

More seriously, I imagined that all mathematics study would be qualitatively like my experience in courses at Smith, where concepts were introduced at such a stately pace that I assimilated them without effort sufficiently quickly that I never imagined I would ever need to work at getting my head round a concept. Furthermore, while exams were sometimes challenging (the group theory exam, that read "this is the definition of a ring; now prove the following 45 statements about rings.  You have 24 hours") bookwork was never a feature, and most exams required only routine applications of understanding to routine problems.  An able student would blush if she get less than 90%. 

My adviser that first year at Warwick did vaguely suggest that one strategy was to memorise all significant proofs, but I found that so extraordinary that I dismissed it straightaway. Unfortunately, my own strategy, to start at page one of my notes, and try to get my head round each definition, lemma, proposition, theorem in turn tended to founder at about page seven, or something less than a week's worth of lectures. In the one case that I really felt I understood the course (representation theory - I told you about that) it turned out that in the exam, yes, I did (really) understand about the orthogonality of characters, and could prove it as required, but sadly, that effort to reconstruct the proof took the best part of the three hours, and it was only half of the first question.

This is a long-winded way of explaining why I felt that I ought to say something about strategies for revising, in case there were any others coming from outside who might not believe in tripos exams (the Warwick MSc exams of that day were essentially Part III tripos exams, so many of the staff then having been brought up on Part III).

Initially, I used to give the talk at the beginning of the Easter term, and then realised that it would be much better given before the end of the Lent term, and have since decided that actually, it would be better to give those of you who are first-timers in the mysteries of tripos exams some guidelines for revision now, and then more specific strategies towards the end of Lent term.

Therefore: if you have no established revision strategy, or if your revision strategy seems not to be working, you might try this.

Step 1. Graph the outline of the course on a large piece of paper. Start at the bottom of the sheet. Go through your notes, adding a node for each definition, result, important example, and connect a node to one below it if the definition/result/example depends on a node below it.  This will give you some idea of the shape of the course.  It doesn't matter at this stage whether you understand what the definition says or what the statement means, just write them down and draw the graph.  You might need an A3 piece of paper.

This exercise can be in itself revealing. Seeing the structure of the course this way, sometimes some of the lecturer's more mysterious diversions can actually be seen to be actually sensible and even essential.

It reminds me somewhat of the experience of climbing a mountain.  When I am climbing a mountain I see the rock immediately before my foot.  Particularly if the terrain is treacherous I do not let my eyes wander far beyond the next step.  Once on top however, I can look back, review the track by which I have come, see it snaking through the rock falls, winding underneath the crags, avoiding the worst of the scree, and finally I understand why the track followed the path it did*. So too do mystifying definitions, apparently unnecessary conditions and bewildering statements come clear and seem sensible once one stands back admires the topography of the graph, and considers the course as a whole. That view is not available to you while you are sitting in lectures or copying notes.

Step 2.  Start trying to figure out what the words, the definitions, the statements actually mean. Perhaps start with the definitions.  Make up examples.  Make up more examples.  If the definition is a property you need two examples, one of something that has the property, one of something that doesn't. Start low down on the page, work up. Then go on to propositions, statements, equations.  What does the proposition mean in stupidly simple examples?  In less stupidly simple examples? With an equation, simplify it. Set constants equal to 0, 1, or whatever simplifies the equation. What does it reduce to?

Don't bother with proofs until the words and the statements really really mean something, until your pockets are full of useful rosetta stones. I don't think you need to bother with proofs at all this early unless you are taking the exam at the beginning of January.

Quick strategy if you are taking the exams in January:  

• Having spent perhaps half the available revision time making your graph, understanding what the words mean and what the statements say, then prune your graph: decide which results you are not going to put any effort into revising. Choose a pleasing distribution of a few significant results that you think you will be able to get your head round - results you will prepare in case they appear as "bookwork" questions.
• Practice those results. Start with the statements.  Make sure that you can state it correctly. At the beginning this might not be at all trivial. Then go through the proof looking for the critical steps. Reduce the proof to a list of no more than four or five critical steps (where the track takes a sharp bend to the left under the crag, where you need to cross the stream to avoid the unpleasant scree slope)- intermediate steps being algebraic "yada yada", or the "obvious" next thing to try. Then practice writing it out, (blackboards are good for this) until you can write it out without hesitation. Keep the little list - it is your last minute revision notes for that result. The process is not essentially different from preparing a talk.  

(Yes, I did exactly this the year after I failed the exams, and that year I did pass.  It can work.) The decision when to start practicing bookwork questions, is a tricky one, the point being that bookwork questions are far far easier to prepare if you actually understand the result well to begin with.

Finally, if you possibly can, ink in a block of five days for a real holiday sometime between now and the beginning of the Lent term starts. A real holiday, not just sitting at home watching telly and worrying.  Go travelling with friends, go walking, go skiing, get out of the house and leave the books behind. You need to pace yourself.  Or else the mist may well set in.

*Unless of course the weather has taken a turn for the worse and the mist has rolled in... I think some courses remain shrouded in mist for eternity, possibly the fault of the lecturer, possibly not.