Using spaced repetition systems to see through a piece of mathematics

Michael Nielsen explores a method for deepening one’s understanding of a mathematical result, to the point that one becomes so familiar with the pieces that you can almost *see* the result directly. The method is expressed in terms of writing Spaced repetition memory system prompts, but as he notes near the end of the piece, the SRS format isn’t really necessary to the process. It just provides some pressure to piece apart the ideas into fine-grained atoms. (Note-writing does something similar for me, at a coarser grain: Evergreen notes should be atomic) But the advantage of writing these pieces as SRS prompts, of course, is that one will remember the results.

He suggests two rough phases: first, understand the proof through iterative grazing. You’ll make repeated passes, writing prompts which represent individual elements of the proof. You’ll try to make these as concise and focused as possible, and you’ll look for “multiple different ways of thinking about the ‘same’ mathematical ideas”—finding connections between the pieces, coming up with variations, explaining in multiple ways, etc.

Michael aspires to end this phase with Forcing questions: i.e. distilling the whole proof to one simple prompt—articulating its essence.

Studying another person’s spaced repetition memory prompts is usually ineffective, and Michael argues that this is particularly true here: for him, most of the value is not really in the remembering, but rather in the process of picking apart and understanding more deeply. In fact, the more deeply he understands, the less legible that understanding is likely to be to others; many of his cards have blank or inscrutable reverses because they’re referencing representations he can’t externalize.

Michael has some notes about understanding via “Chunks” in human cognition: Learning increasingly complex ideas may amount to forming larger effective chunk sizes

Michael’s second phase is “pushing the boundaries”: changing the assumptions of the proof, exploring what happens when you weaken its conditions, finding alternative proofs, “poking at it in various ways and seeing what happens”.