I don’t tend to do a lot of proofs anymore. When I think of math, I find it most important to be able to flip back and forth between symbol and referent freely—look at an equation and visualize the solutions, or (to take one example of the reverse) see a curve and think of ways of representing it as an equation. Since when visualizing numbers will often not be available, I tend to think of properties of a Taylor or Fourier series for that graph. I do a visual derivative and integral.
That way, the visual part tells me where to go with the symbolic part. Things grind to a halt when I have trouble piecing that visualization together.
I don’t tend to do a lot of proofs anymore. When I think of math, I find it most important to be able to flip back and forth between symbol and referent freely—look at an equation and visualize the solutions, or (to take one example of the reverse) see a curve and think of ways of representing it as an equation. Since when visualizing numbers will often not be available, I tend to think of properties of a Taylor or Fourier series for that graph. I do a visual derivative and integral.
That way, the visual part tells me where to go with the symbolic part. Things grind to a halt when I have trouble piecing that visualization together.
This appears to be a useful skill that I haven’t practiced enough, especially for non-proof-related thinking. I’ll get right on that.