the first two parts of Theorem 3 doesn’t require almost any algebra for me, and so of course I could do it in my head—I’m confused why you think they do.
I honestly don’t remember why I thought they required multiple steps of algebra. I might have just mixed them up with the third part or some other problems that I did later that day. This does slightly update me against my hypothesis though as this might indicate that I’m overestimating the difficulty of the problems I’m solving when I do them in my head.
I am also a pacer and there is definitely a continuum from writing every single step down to doing the whole thing in your head. I think maybe we agree there is a continuum but you claim the optimal point is pace and think, then write when needed. I claim it’s much further than that and that most people including experienced mathematicians haven’t pushed anywhere near it because they’ve never deliberately practiced it.
One way to practice this is to do problems which are hard but doable completely in your head. I could imagine another way would be to simply try extending how many steps you do in your head before writing.
I also understand that paper allows you to offload and frees up your working memory. There is a trade off though and there is definitely a point where you can just easily think through the problem without writing anything down. For example, the first 2 parts of theorem 3.17 for you. A model where paper improves everyone’s ability on difficult problems is certainly wrong because it would predict that Euler’s productivity would drop after going blind.
I think our crux is one or both of
1. Whether doing entire problems in your head is a good way of training this skill. 2. Whether most people can get good enough at doing problems in their head for it to start mattering on meaningfully difficult problems.
I honestly don’t remember why I thought they required multiple steps of algebra. I might have just mixed them up with the third part or some other problems that I did later that day. This does slightly update me against my hypothesis though as this might indicate that I’m overestimating the difficulty of the problems I’m solving when I do them in my head.
I am also a pacer and there is definitely a continuum from writing every single step down to doing the whole thing in your head. I think maybe we agree there is a continuum but you claim the optimal point is pace and think, then write when needed. I claim it’s much further than that and that most people including experienced mathematicians haven’t pushed anywhere near it because they’ve never deliberately practiced it.
One way to practice this is to do problems which are hard but doable completely in your head. I could imagine another way would be to simply try extending how many steps you do in your head before writing.
I also understand that paper allows you to offload and frees up your working memory. There is a trade off though and there is definitely a point where you can just easily think through the problem without writing anything down. For example, the first 2 parts of theorem 3.17 for you. A model where paper improves everyone’s ability on difficult problems is certainly wrong because it would predict that Euler’s productivity would drop after going blind.
I think our crux is one or both of
1. Whether doing entire problems in your head is a good way of training this skill.
2. Whether most people can get good enough at doing problems in their head for it to start mattering on meaningfully difficult problems.