[Question] When do you visualize (or not) while doing math?

I’ve been doing a lot more math lately, and, of course, also thinking about how I can do it faster and with deeper understanding. When I was in college, I remember that I found visualizing immensely helpful, to the point where it’s almost what “doing math” is for me.

The most obviously useful place to visualize was in multivariable and vector calculus. Especially because so many things were set in three dimensions, visualizing expressions made it so much easier to see why they were true.

I also visualize all the way down to doing algebra. When I symbolically manipulate an equation, I visualize things like bringing the letters across, or them merging into each other.

But it has its downsides. One is that, since I’m visualizing, I’m visualizing something specific; a specific function, a specific type of set. It’s not necessarily high-resolution, but it don’t usually cover the full range of what is represented by the symbols on the paper. And so by visualizing something specific, it’s likely that I’m not keeping in mind all the types of cases.

The most obvious downside is that visualization is computationally intensive. When I’m rendering a big scene in my mind, I can feel that it’s hard. Sometimes I have to close my eyes and block out the light. Things can slow down, and it can take several tries to get the visual just right, or to maintain it all the way through the manipulation. It seems plausible that this gets me more tired than is necessary, and causes me to stop doing the task earlier.

So, it has at least occurred to me that I could “try” doing something else. I put try it quotes because visualizing is so natural that it would be effortful for me to even think of an alternative. It’s as if someone said “you know, you don’t have to shovel that snow by hand” and I’m thinking, do they mean I could do it with my feet? What they actually mean is using a snow-blower.

Are there fundamentally different ways that you do math? Are there particular tasks you’ve tried that go faster with one method or another? Are there books that talk about this type of strategy?