Attention is a Finite Resource
Date Written: June 14, 2024; Last Modified: June 14, 2024In my first few proof-based math classes, my notes consisted of nothing but definitions, propositions, lemmas, theorems, and proofs. I realised that these notes were more or less worthless — all of this information was written down more neatly, more correctly, and with more detail in textbooks.
But my lectures were not live recitations or readings of textbooks; in fact, there’s a lot of insight a (good) lecturer can present that’s usually absent from many textbooks. These include heuristic arguments, guiding intuitions, and poorly drawn diagrams.
On one hand, it costs attention to squint and the board and copy down a theorem statement from the board.. On the other hand, it also costs attention to listen to and process these pieces of verbal information that aren’t written down on the board. At any given point in time, the amount of attention available to you is limited, and a decision must be made on where to spend it.
When the lecturer is describing some heuristic or intuition about a particular theorem or proof, I think it’s important to spend one’s attention there. This may come at the cost of not being able to write down some details of a proof, or perhaps you can’t finish copying something down quickly enough. However, this is a small opportunity cost, as the precise boardwork and details of a proof can typically be found in a textbook or on the internet in finite time. On the other hand, spending one’s attention on writing down readily available information comes at the cost of not fully processing the fleeting verbalisations of the lecturer.
At times, the lecturer will pause to write something down on the board, or perhaps they are writing out a lengthy computation. These are good times to spend one’s attention on writing down the notes themselves, as the cost of moving one’s attention away from the lecturer becomes small.