What is your opinion of people who memorize math like history?

My Answer to a question on Quora: What is your opinion of people who memorize math like history?

These unfortunate people were perhaps misled by their teachers (maybe inadvertently): in history as well as in mathematics, understanding is more important than memorisation.

I got this lesson when I was 7 years old, from my big brother, who was 12.

My paper Calling a spade a spade: Mathematics in the new pattern of division of labour, clearly shows my obsession with the concept of division of labour. I can trace its almost Freudian roots to my brother’s Year 6 textbook History of the Middle Ages (1962). Recently I have found on the Internet its electronic copy and checked that the book indeed contained an explanation of the emergence of towns and a quote from old chronicles about the origins of Brugge – the stuff that I talked about with my big brother, who was in Year 6 – and used this textbook – when I was in Year 1 (that is, I was 7 years old and my brother 12 years old).

So our conversation went that way. My brother discovered that I was reading his textbook – and asked, with some scepticism, whether I was finding it interesting.

“Yes – said I – I had just read how towns appeared.”

“And how did they?” – retorted my brother.

I started the story from the chronicles: there was a castle on a bank of a river, and a bridge next to the castle, and peasants were bringing their goods to the castle, and craftsmen were peddling their services first to the castle, and then to other people who were coming there, and inns were built, bla-bla, and the conclusion was that that town that grew up was called Brugge, which, in the local language, meant “Bridge”.

“Silly you – said my brother – towns were built because of separation of crafts from agriculture.”

I was dumbstruck. I was deeply pained because my brother again proved to me his intellectual superiority – but also astonished by the discovery that a few words:

“separation of crafts from agriculture”

explained and summarised everything that was in the story. I was even more pained by the realisation that I had actually seen these words in the text, but had not paid attention to them. I had taken the lesson in immediately: I decided, from that point on, always look for these kind of special words which summarised everything.

With the benefit of hindsight I can say that this approach was really helpful indeed in study of history- less memorisation, more understanding.

I can say now that, in studying history, it is useful to focus on three principal points:

  • how people of yestertimes saw and explained their worlds
  • what were their motives for their actions and their justification of them
  • what were the tectonic forces of socio-economic change that moved the history.

And, of course , the principle:

Down with memorisation, long live understanding !

helps in mathematics. For example, I honestly tell my students that I do not remember trigonometric formulae beyond a few basic ones, of  \(\cos^2 x+\sin^2 x = 1\)  kind — but unlike my students, I can recover, from first principle, formulation of any trigonometric formula that I may need — and prove it.

[I cannibalised, in this my answer, another my paper, Comments on “Stop Ruining Math! Reasons and Remedies for the Maladies of Mathematics Education” by Rachel Steinig. ]