Batchelor’s law on the transfer of understanding: Why A should ask for help and why B should help if asked.

Did I mention that no Part III student has too much time? The corollary to that is that short cuts to learning are to be taken at every opportunity, particularly for those students coming from outside Cambridge, who may find there are critical gaps in their background knowledge. If you need to know something, ask.  Someone is bound to know it.  Ask first.  Yes, read it up later, but it will be much easier reading if you know the story line first.

It’s pretty obvious, that if A needs to learn something in a hurry, A should ask for help.  But why should B help? I claim that B should, in her own interests. Batchelor’s law on the transfer of understanding says the following.  

Let U(A)_0, U(B)_0 be the initial levels of understanding of students A and B respectively, and suppose that U(A)_0<U(B)_0. A and B then meet and discuss, after which their respective levels of understanding are U(A)_1, \ U(B)_1. Batchelor’s law states that 

(U(B)_1 - U(B)_0) - (U(A)_1 - U(A)_0)>0.

Bluntly put, B gains more from the exchange than A does.  The moral is, if someone asks you for help, offer it with willing heart in your own best interests.

Anecdotal evidence: one very able student was always well up in the firsts, but the one year he was right up in the top ten was the year (Part II) in which his girlfriend was finding the work rather difficult, and was very appreciative of his revisions sessions. (I believe she too got a first.)

This is a law, an observation. I think I can suggest a physiological explanation. Anyone who has tried to learn a second language will be well aware of the necessity of actually using it, trying to express even simple thoughts with unfamiliar words. Use the words, even once or twice, to express an idea that you want to express, and the words become embedded in an active bit of your brain that is readily accessible, so those words come readily to mind.  In a similar way, explaining an idea or a construction embeds it in a different location in your own mind, one more readily accessible than its first position as acquired, possibly used, but unspoken knowledge. Even better if student A is stubborn or a slow learner or both and rejects a first explanation.  That means B must re-examine her own understanding of the matter, turn it around, and try and present it in a different way. That operation, of turning an idea around in one’s mind, looking at it from a different perspective, often results in a huge leap in understanding.  Moreover, it is a process fundamental to creative mathematics.  Practice it. 

In particular make good use of the study groups. You will need to work at them, and they do not always work out well - everyone in the group has to work at it, and if the backgrounds of the group are too diverse, frictions will surface.  If you don't have much experience of collaborative study you might try the following template for an hour’s session:

• One person presents a review of the week’s lectures - no proofs, but illustrative examples. 20 minutes.
• People who have worked out selected examples from examples sheets/lectures explain them.20 minutes.
• Look through the problems arising from the lectures/examples sheets, detail people to give the summary/explain problems at the following session.

Every year we had a few complaints, students saying that these study sessions should be led by graduates students, that we should supply supervisions, but I don’t think so.  Part III is about shifting gears, mathematically speaking, learning to learn from your colleagues rather than from your elders.  One has to learn how to do that, as well as learn the material.

It’s hard, but it is fun. During my first traumatic year at Warwick I shared the bottom desk with another woman who similarly found the work hopelessly difficult. We sat together and sorted through the notes through the afternoon on through dinner and into the night, often to little avail.  The sessions tended to morph into dinner, and maybe the alcohol didn’t help.

But through struggling together we got to know each other, and we did things together. We went walking together in the Cotswolds at the weekends, touring the Devon coast in the spring break, and visiting her native Scotland. Forty-four years on we are still in touch, still good friends. 

Work together.  Engage with it.

————storytime.

The scene: Prof. Dan Kan’s office, ground floor, Building 2, MIT, 6:30 am, on a September morning in 1976. Term has been going on only a few weeks. I am supposed to be well on my way to preparing talks 3 and 4 for the Graduate Topology Seminar, and am having my weekly tete a tete reviewing my state of preparation.

At some point Dan got tired of lecturing algebraic topology to sleepy students, and figured out that the way to stop them yawning was to make them do the talking, and the Graduate Topology Seminar was born. It was the baptism of fire for all MIT topology students of that generation. The ground rules were tolerably simple. He provided a (short) list of papers. Students could choose papers (or find their own, and they would have to later in term) and present them to their colleagues.  

There were instructions on how to prepare: choose a focus for the talk, a statement consisting of a single grammatical sentence, figure out how to explain/illustrate/motivate the ideas, practice it until it fit into 50 minutes, and no more than one index card to be used as notes. The consequences of failing to observe these instructions were painful. Oh yes, everyone else was expected to have read the paper before the talk.  Just to make sure, a half page of written commentary on the paper was required from each student.  

Here’s the rub: term was perhaps 13 weeks long, the class met twice a week (Mondays and Wednesdays at 8am), and the class size was limited to about half a dozen.  In my year it was five. Go figure. The course was allocated double the credits of any other course in the university, and it was judged still to be wildly undervalued at that. It was described to me with a fervent recommendation by a survivor: “take it, take it, take it, you’ll hate it”.

Thus I found myself at 6:30 am having my weekly chat about life and the Graduate Topology Seminar in general. I was, at least, one of the lucky ones, whose body clock naturally took to 5am starts. Those who tended to the late shift found it easiest to spend the night sleeping in the reading room.  Dan was quite willing to brew coffee when he got in and wake up a student with a cup.  He was not willing to reorganise his schedule.

It was the third week of term, I was already hopelessly behind.

“So, how are you getting on with the paper?”

“Well, I’ve read it” I lied. No. Wait a moment. I can’t bluff, it’s all going to go horribly wrong, best out with it.

“No, that’s a lie.  I haven’t read it.  Phil did the paper in the seminar two years ago, I asked Phil and he told me what it’s about.”

Dan leapt up an bounced about the room. “You’re learning, you’re learning, you’re a quick learner, it’s only the third week of term and you know how to read a paper!”

Lesson learnt.