There is no shortcut to insight in pure math research. You pay attention, follow where it leads, and fail, often enough that failing becomes familiar. I’ve come to think this is how I learn everywhere, not just in math.

If I had to condense the power of the marginal into one sentence, it would be: just try hacking something together. The phrase means deciding what to do as you’re doing it, rather than executing someone else’s vision. It also means accepting that the result won’t be pretty, because it’s made quickly from inadequate materials. It may barely solve the problem, or not solve it at all and surface a different problem along the way. That’s fine. The value of the first version isn’t the thing itself but what it leads to. Insiders who won’t walk through the mud in their nice clothes never reach the solid ground on the other side. ~ Paul Graham

Small, weird experiments are what expand my mental model. Everyone knows you shouldn’t parse HTML with regular expressions, but seeing Postgres do it, and seeing what that syntax looks like, changed my sense of what makes sense to try. That’s the case for tinkering over goal-driven work: the payoff is a wider sense of possibility, not the artifact. Much of the skill in building things is knowing what’s possible and roughly how, and every answer you collect makes you likelier to spot a problem others haven’t thought to attack.

This style of working allows you to be opportunistic and build software quickly on demand when you need to and when it economically makes sense to do so.

So this is how I work: build small, scrappy tools, learn my environment as I go, and push until I hit constraints, which then shape how I plan the rest of the project.