I don’t think this is compelling, at least if extended beyond “Alternate between pacing and writing on a whiteboard”, which is what I actually do. But it’s possible that I’ve already got whatever skill you are trying to train—for example, 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 just think “Apply linearity and divide by n, and scaling squares variance”. If I struggled with it, I don’t think I would be helped much by writing things down except for the problem statement.
The third part was trickier because it took me a little to figure out how to evaluate the covariance, and also I made some errors that would’ve been hard to diagnose had I not decided to write it down when the problem popped up. I would’ve done it faster had I just written it down, and I don’t feel that I understand it more for having done most of it in my head.
Paper lowers overhead, where by overhead I mean mental overhead i.e. the overhead that matters most. The point is to expand my working memory so I can focus on what matters, and make better visualizations in some respects. This is very important because math often strains my working memory or visualization, so offloading that allows me to do harder stuff and understand more. If I have to keep X in my head, then I can’t throw it out to visualize Y, especially if Y turns out to be a bad idea and so I need to go back to X in order to think of what to try next.
Remembering the current state often simply isn’t enough, as I’ll want to explore what the previous steps were, what they imply, how they relate to the current step, etc.
I write down shorthand English for various ideas of what to do, or what the problem is, and then stare at them while I think. It is somehow easier to think when the words are on the wall.
**The actual hard steps happen in my head**, as you can tell from the fact that I’m usually pacing around thinking and only come back to write when I’ve had a good idea or need to do some algebraic calculations or whatever, just like using a computer to do numerical experiments. I definitely recommend it, but I would’ve guess most pacers would already have found themselves pacing and thinking. Guess I was wrong there!
For longer periods of time, writing or typing lets me remember. I don’t have a problem coming back to something I’ve worked on when it’s been on my mind or if it hasn’t been that long, but if it’s been a week and I’ve thought about many other things then the details have gone right out the gutter.
For physics, it’s definitely useful to do Fermi estimates in your head, by memorizing the values of certain things and being good at approximate arithmetic. I’ve also heard Oliver Habryka mention how quick Fermi estimates are useful to him constantly for things like checking contractor projections or planning how hard it’ll be to hire a specialist in a city.
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 don’t think this is compelling, at least if extended beyond “Alternate between pacing and writing on a whiteboard”, which is what I actually do. But it’s possible that I’ve already got whatever skill you are trying to train—for example, 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 just think “Apply linearity and divide by n, and scaling squares variance”. If I struggled with it, I don’t think I would be helped much by writing things down except for the problem statement.
The third part was trickier because it took me a little to figure out how to evaluate the covariance, and also I made some errors that would’ve been hard to diagnose had I not decided to write it down when the problem popped up. I would’ve done it faster had I just written it down, and I don’t feel that I understand it more for having done most of it in my head.
Paper lowers overhead, where by overhead I mean mental overhead i.e. the overhead that matters most. The point is to expand my working memory so I can focus on what matters, and make better visualizations in some respects. This is very important because math often strains my working memory or visualization, so offloading that allows me to do harder stuff and understand more. If I have to keep X in my head, then I can’t throw it out to visualize Y, especially if Y turns out to be a bad idea and so I need to go back to X in order to think of what to try next.
Remembering the current state often simply isn’t enough, as I’ll want to explore what the previous steps were, what they imply, how they relate to the current step, etc.
I write down shorthand English for various ideas of what to do, or what the problem is, and then stare at them while I think. It is somehow easier to think when the words are on the wall.
**The actual hard steps happen in my head**, as you can tell from the fact that I’m usually pacing around thinking and only come back to write when I’ve had a good idea or need to do some algebraic calculations or whatever, just like using a computer to do numerical experiments. I definitely recommend it, but I would’ve guess most pacers would already have found themselves pacing and thinking. Guess I was wrong there!
For longer periods of time, writing or typing lets me remember. I don’t have a problem coming back to something I’ve worked on when it’s been on my mind or if it hasn’t been that long, but if it’s been a week and I’ve thought about many other things then the details have gone right out the gutter.
For physics, it’s definitely useful to do Fermi estimates in your head, by memorizing the values of certain things and being good at approximate arithmetic. I’ve also heard Oliver Habryka mention how quick Fermi estimates are useful to him constantly for things like checking contractor projections or planning how hard it’ll be to hire a specialist in a city.
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.