Rosetta stones

September, 1973. Mathematics Institute, University of Warwick

“You don’t need to start taking notes just yet.”  Well, that was reassuring, I suppose.  I had never taken notes in a maths class before.  There were books where things, proofs definitions and the like, were written down, right?   You just listen, and think about what you are being told, understand it, and then there’s no need to remember it is there? Why write anything down? How can you forget what you understand? Something understood doesn’t need to be remembered, not as such.

“A representation is a homomorphism from a group into a group of invertible matrices.”  Cool.  Yes. Got that. A few handwavy remarks. “Now you may take notes.  We will begin with a review of semi-simple Artinian rings.” Ice settled in my stomach. Fear paralysed my arm. I had never before been lost in a maths class*, and here I didn’t even know the words. Well, ok, I knew what a ring was, review and begin were familiar words too, as were the one syllable ones.  This was clearly going to be a new experience.  An unwelcome one.  I took notes. Very careful notes, being quite unable to do anything else.

By the end of the first fortnight I had notes, but nothing to speak of by way of understanding. It appeared that the course was going to be excellent training for a career as a stenographer required to take dictation in foreign languages for which she didn’t even know the alphabet. It was grim. Something had to be done, so I approached one of the Phd students and proposed a trade: I would cook him a dinner if he would explain representations to me.

We decided to sit down to the mathematics before attempting dinner. He explained again what a representation was.  I sat there, jaws glued shut, mute. The words floated past me. “So now write down the matrices.” I sat there, jaws glued shut, mute. “Ok” he said, “so I’m going to cook dinner, and if you have written down those six matrices you can have some.”

It was a very acceptable dinner. I carried those six matrices about in my pocket, as it were, for the duration of the course, testing every definition, every proposition on them, translating bald definitions and statements into meaningful ideas. Representation theory yielded up its secrets. 

As a methodology for learning too, the penny had dropped. For the other courses too, I hunted out and found rosetta stones, on which the hieroglyphs of mathematics were translated into simple examples carrying meaning, which could be carried about in one’s pocket, which held the clue and gave context to the bald statements so carefully copied.  I can usually find a torus or a symmetric space in my pockets still. And many fine suppers came into it too. 

I was still way behind the coursework; I repeat, in spite of those examples in my pocket I did fail the exams that first year - I had yet to learn that understanding alone did not suffice for passing the exams. I will probably tell you more about that later, and what to do about it. The important thing was that the fun had come back, the intense pleasure of playing with the ideas. If I change this, does this still work? can I do that? do they all behave this way? whatever this, that and all of them may be. I had toys again to play with, toys that illuminated the bleak sheets of notes and made mathematics dance again.

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*This is not quite literally true.  There was an incident when I was eleven too. We had moved south from Massachusetts to Maryland, and I was not at all pleased with the relocation in any way. Losing friends was bad enough, but the new school was bad news. My previous elementary school had taken the enlightened view that students who could be given a set of guided exercises, be it in arithmetic hand-writing, spelling or reading, and would get on with it quietly on their own could be allowed to do so, with the result that I had never had the experience of being sat in rows and expected to listen to a teacher actually teaching me new material. Move south, all change. 

I did not take kindly to being sat in rows or told what to think about and when to think about it. In this setting I was introduced to decimals. “Just as you have the units, the tens, the hundreds,” the teacher said “with decimals you have the tenths, the hundredths, and so on, only it’s backwards..” 

Interesting. Have to think about that. Hmm, 1, yes, I can do that, 2, a bit problematic, but with a bit of practice, 3, somewhat easier, 4, oh phooey!

“Marjorie!” a sharp voice interrupted my considerations.

Oh, ah, yes, what were we supposed to be doing - ah, reading out decimals as fractions, .8, phew! that one really is easy.  “Eight tenths.” I returned to my thoughts, absorbed, oblivious to what was going on round me, with the challenge of writing the numerals backwards.

The next time round I was not so lucky, .66. What’s a 6 backwards? A d? Can’t be. I know! She didn’t mean reflect, she meant rotate by 180 degrees. “Ninety-nine hundredths.”

There was silence, and a stare. She corrected me quietly and proceeded round the class. It was only then that I became convinced that if she really wanted us to write all the numerals backwards she would have begun with a penmanship lesson. It was surprising, even to the eleven year old me, how long I remained suspicious of decimals, regarding them as untrustworthy. Numbers had always been friends. I was deeply shaken.