TLDR: Doing math quickly in your head is an underrated and undertrained skill. You can practice this skill easily by trying to do math problems in your head before writing them down. It’s actually fun, and you can go on walks and do this.
Chess grandmasters are able to visualize long sequences of moves extremely quickly. Is there a similar skill in doing math? In trying to answer this question, I discovered a form of math practice that seems promising and was quite enjoyable.
We have research on how chess grandmasters visualize board position, and we know that they can very quickly memorize a board position from a real game; however, when shown a position of randomly arranged pieces, they’re no better than average at memorizing it. This is because they memorize by translating the board position into familiar patterns, a process known as “chunking”. I don’t know how much chess cognition and math cognition have in common, but it seems plausible that a similar kind of chunking occurs when working with mathematical expressions. For example, seeing ²² as ² , or recognizing a sum as a geometric series, rather than processing each term individually. This got me interested in the idea of training the ability to do long sequences of math calculations quickly in my head.
A few mathematicians in particular stand out for having this sort of ability. Von Neumann was known for being able to do math calculations extremely quickly in his head that would have required pages to write down. In one particular anecdote, Von Neumann was presented with the “fly and two bicycles” problem. This problem involves a fly flying back and forth between two bicycles approaching each other. There is a simple solution which just involves multiplying the speed of the fly by the time for the bicycles to reach each other. And there is a much more complicated solution that involves calculating how far the fly travels at each step and calculating an infinite geometric series. Von Neumann, when posed this question, gave the correct answer instantly. The questioner, assuming he knew the trick, said, “Ah, you’ve obviously heard it before, most people try to add up the infinite series!” and Von Neumann, looking puzzled, answered, “That’s what I did!”
The fly and two bicycles problem
Euler was essentially blind for the final 17 years of his life, and yet his output arguably increased during this time. He produced his second theory of lunar motion during this period, doing the calculations in his head and dictating to his colleagues and family members. He could also stop mid-calculation when interrupted by his children and restart without losing progress.
Euler's blindness period is significant not just because he kept working, but because his output increased. This is an existence proof that the human ceiling of mental math ability is close to the ceiling of pen and paper math ability. The question is whether this is a skill that an ordinary person can build with deliberate practice. My hypothesis is yes and I aim to test this. Here is my model:
Cognitive Offloading
Using a pen and paper is a form of cognitive offloading. It removes the cognitive effort of holding the equations in your head as you manipulate them at the cost of speed. By removing the pen and paper, you train your ability to maintain and manipulate mathematical expressions in your head. Standard education excessively emphasizes writing down your steps so that your work can be graded. Most people can't do as much math in their head as on paper, and I think this is largely because they've never practiced it.
Writing each step down trades off long-term practice for short-term performance and represents a local optimum in math performance. If you practice doing math in your head, at first you will be worse than on paper, but if you keep practicing, you will build the ability to maintain and manipulate equations in your head and train your pattern recognition. Once you get good at this, you won’t want to use pen and paper anymore, as it will slow you down.
Speed
Writing things down serves as error-checking and frees up working memory for the next step. But it comes at a cost to speed. You can think much faster than you can write. This means the further you can get before you have to write down your work, the faster you can do problems.
Time to solve a problem using pen and paper (top) vs in your head (bottom)
When you do a math problem, most people do one or two steps, and then they write them down, then they do a step or two, and so on. My theory is that Von Neumann and Euler did many steps, often the whole problem, and they wrote it down only occasionally or when they were finished. I don’t claim every mathematician works like this. But some do, and it seems like a skill worth training. Doing it in your head also allows you to try different approaches quickly without much overhead. If you try a lot of approaches on paper, you have to write and start over. The overhead might prevent you from even trying these approaches in the first place. Having a good sense of whether an approach is good from the start helps, but it’s still better to be able to try things quickly.
Trying It Out
So I had a hypothesis. Rather than speculating further, I decided to test it. I’m currently working through “All of Statistics” by Wasserman. I had just finished reading chapter 3 on expectation and moments and decided to try doing the chapter exercises in my head.
The first 5 exercises were pretty easy and I was able to complete all of them without ever stopping to write anything down. It definitely felt a bit slower than doing them with pen and paper but not drastically slower. The slowdown mainly came from me struggling to retrieve the equations back into my working memory. After this I skipped a few and chose some of the remaining exercises that seemed worth doing. At this point I had to leave to get lunch with some friends and I decided to take a problem with me. This problem was a bit harder than the others and was a 3 parter as well. As I was walking to the lunch I was able to do the first 2 parts in about 5 minutes which surprised me a bit because they involved a decent amount of algebra and I didn’t expect to be able to keep track of all of the steps. However, while doing the problems I realized that I didn’t actually have to keep track of the steps, I only had to keep track of the current state which was much more manageable.
For reference, the 3-part problem was to prove this theorem
The final part was a bit trickier, and I hit a sticking point where I wasn’t immediately sure which approach would work, which meant that I had to backtrack a couple of times. It took a bit more mental effort to recall my “checkpoint” after trying an approach, but it was still doable. By the time I had reached my destination, I hadn’t quite solved the problem yet, but I had an approach I thought would work and just needed to solve a specific subproblem. While at lunch, I mostly didn’t think about the problem, but towards the end I found myself returning to it, and as I left, I was eager to work on it. Here, I was again surprised at how I was able to pick up right where I left off. I definitely already do quite a bit of chunking, piecing together equations as common patterns (, sum over probability distribution, etc.). This helps with remembering the equations. On the way home, I was able to solve the problem, and although one part of the solution felt a bit vague, I was pretty confident that when I wrote it out, it would be correct. Afterwards, I felt pretty good. It had felt more fun than it normally feels sitting down and doing a problem on paper. When I got home, I wrote out the solutions, and although I didn’t totally remember the solutions, I was able to reconstruct the first two pretty much instantly. When I went to reconstruct part 3, I realized I had skipped some steps in the vague part and had to fill in the solution. Still, it was pretty easy to fill in the gap, and the solution was generally correct. This is a common failure mode, as I further discovered later on. Working in your head, you can skip steps and use a bit more intuition, which can help you get through algebra-heavy problems without actually doing all of the algebra; however, you still need to fill in the gaps in the end. This is fine as long as your intuition is correct, but it’s probably best to try and do all of the algebra in your head if you can manage.
Overall, my experience suggests that mental math is more accessible than expected. A key realization is that I didn’t need to track every step; I only needed to track the current state and a few checkpoints. Chunking seems to play a role in compressing the equations to reduce the load on working memory. The main failure mode of skipping steps is manageable, as long as I write up and verify the solution afterwards.
This small experiment makes me more confident in the broader hypothesis. My mental math ability is already closer to my general math ability than I thought, suggesting my own ceiling is higher than I previously thought. The fact that Euler’s math output didn’t decrease after going blind is evidence that the ceiling is generally higher than people assume.
The main question is to what degree it will impact my general math ability. There are two separate questions I want to answer
Does practicing mental math make me better at mental math?
Does it improve my overall math ability more than equivalent time spent practicing with pen and paper?
For (1) I am going to periodically time myself on comparable problems done mentally and track whether I get faster or can handle harder problems. This will require finding a source of problems with consistent difficulty and not too much variance in time to solve (see theorem 3.17). Ideally I’ll be able to increase the level of difficulty over time as well. I will detail the set-up and my baseline results in a follow up.
(2) seems a bit hard to measure in a controlled fashion, especially with only one person. But I can still measure my overall math ability over time using a similar method of periodically testing my speed and capabilities with pen and paper allowed. And beyond controlled measurements I will continue to observe and see what patterns emerge.
Arguably the most important finding is that walking math is more enjoyable for me, meaning I’ll likely do more of it. The low level physical activity in the background makes it less boring and I am able to focus for longer without stopping. Sitting down and doing math problems feels like a chore. Walking math is fun and I’m excited to do more of it.
TLDR: Doing math quickly in your head is an underrated and undertrained skill. You can practice this skill easily by trying to do math problems in your head before writing them down. It’s actually fun, and you can go on walks and do this.
Chess grandmasters are able to visualize long sequences of moves extremely quickly. Is there a similar skill in doing math? In trying to answer this question, I discovered a form of math practice that seems promising and was quite enjoyable.
We have research on how chess grandmasters visualize board position, and we know that they can very quickly memorize a board position from a real game; however, when shown a position of randomly arranged pieces, they’re no better than average at memorizing it. This is because they memorize by translating the board position into familiar patterns, a process known as “chunking”. I don’t know how much chess cognition and math cognition have in common, but it seems plausible that a similar kind of chunking occurs when working with mathematical expressions. For example, seeing² ² ²
A few mathematicians in particular stand out for having this sort of ability. Von Neumann was known for being able to do math calculations extremely quickly in his head that would have required pages to write down. In one particular anecdote, Von Neumann was presented with the “fly and two bicycles” problem. This problem involves a fly flying back and forth between two bicycles approaching each other. There is a simple solution which just involves multiplying the speed of the fly by the time for the bicycles to reach each other. And there is a much more complicated solution that involves calculating how far the fly travels at each step and calculating an infinite geometric series. Von Neumann, when posed this question, gave the correct answer instantly. The questioner, assuming he knew the trick, said, “Ah, you’ve obviously heard it before, most people try to add up the infinite series!” and Von Neumann, looking puzzled, answered, “That’s what I did!”
Euler was essentially blind for the final 17 years of his life, and yet his output arguably increased during this time. He produced his second theory of lunar motion during this period, doing the calculations in his head and dictating to his colleagues and family members. He could also stop mid-calculation when interrupted by his children and restart without losing progress.
Euler's blindness period is significant not just because he kept working, but because his output increased. This is an existence proof that the human ceiling of mental math ability is close to the ceiling of pen and paper math ability. The question is whether this is a skill that an ordinary person can build with deliberate practice. My hypothesis is yes and I aim to test this. Here is my model:
Cognitive Offloading
Using a pen and paper is a form of cognitive offloading. It removes the cognitive effort of holding the equations in your head as you manipulate them at the cost of speed. By removing the pen and paper, you train your ability to maintain and manipulate mathematical expressions in your head. Standard education excessively emphasizes writing down your steps so that your work can be graded. Most people can't do as much math in their head as on paper, and I think this is largely because they've never practiced it.
Writing each step down trades off long-term practice for short-term performance and represents a local optimum in math performance. If you practice doing math in your head, at first you will be worse than on paper, but if you keep practicing, you will build the ability to maintain and manipulate equations in your head and train your pattern recognition. Once you get good at this, you won’t want to use pen and paper anymore, as it will slow you down.
Speed
Writing things down serves as error-checking and frees up working memory for the next step. But it comes at a cost to speed. You can think much faster than you can write. This means the further you can get before you have to write down your work, the faster you can do problems.
When you do a math problem, most people do one or two steps, and then they write them down, then they do a step or two, and so on. My theory is that Von Neumann and Euler did many steps, often the whole problem, and they wrote it down only occasionally or when they were finished. I don’t claim every mathematician works like this. But some do, and it seems like a skill worth training. Doing it in your head also allows you to try different approaches quickly without much overhead. If you try a lot of approaches on paper, you have to write and start over. The overhead might prevent you from even trying these approaches in the first place. Having a good sense of whether an approach is good from the start helps, but it’s still better to be able to try things quickly.
Trying It Out
So I had a hypothesis. Rather than speculating further, I decided to test it. I’m currently working through “All of Statistics” by Wasserman. I had just finished reading chapter 3 on expectation and moments and decided to try doing the chapter exercises in my head.
The first 5 exercises were pretty easy and I was able to complete all of them without ever stopping to write anything down. It definitely felt a bit slower than doing them with pen and paper but not drastically slower. The slowdown mainly came from me struggling to retrieve the equations back into my working memory. After this I skipped a few and chose some of the remaining exercises that seemed worth doing. At this point I had to leave to get lunch with some friends and I decided to take a problem with me. This problem was a bit harder than the others and was a 3 parter as well. As I was walking to the lunch I was able to do the first 2 parts in about 5 minutes which surprised me a bit because they involved a decent amount of algebra and I didn’t expect to be able to keep track of all of the steps. However, while doing the problems I realized that I didn’t actually have to keep track of the steps, I only had to keep track of the current state which was much more manageable.
The final part was a bit trickier, and I hit a sticking point where I wasn’t immediately sure which approach would work, which meant that I had to backtrack a couple of times. It took a bit more mental effort to recall my “checkpoint” after trying an approach, but it was still doable. By the time I had reached my destination, I hadn’t quite solved the problem yet, but I had an approach I thought would work and just needed to solve a specific subproblem. While at lunch, I mostly didn’t think about the problem, but towards the end I found myself returning to it, and as I left, I was eager to work on it. Here, I was again surprised at how I was able to pick up right where I left off. I definitely already do quite a bit of chunking, piecing together equations as common patterns (
Overall, my experience suggests that mental math is more accessible than expected. A key realization is that I didn’t need to track every step; I only needed to track the current state and a few checkpoints. Chunking seems to play a role in compressing the equations to reduce the load on working memory. The main failure mode of skipping steps is manageable, as long as I write up and verify the solution afterwards.
This small experiment makes me more confident in the broader hypothesis. My mental math ability is already closer to my general math ability than I thought, suggesting my own ceiling is higher than I previously thought. The fact that Euler’s math output didn’t decrease after going blind is evidence that the ceiling is generally higher than people assume.
The main question is to what degree it will impact my general math ability. There are two separate questions I want to answer
For (1) I am going to periodically time myself on comparable problems done mentally and track whether I get faster or can handle harder problems. This will require finding a source of problems with consistent difficulty and not too much variance in time to solve (see theorem 3.17). Ideally I’ll be able to increase the level of difficulty over time as well. I will detail the set-up and my baseline results in a follow up.
(2) seems a bit hard to measure in a controlled fashion, especially with only one person. But I can still measure my overall math ability over time using a similar method of periodically testing my speed and capabilities with pen and paper allowed. And beyond controlled measurements I will continue to observe and see what patterns emerge.
Arguably the most important finding is that walking math is more enjoyable for me, meaning I’ll likely do more of it. The low level physical activity in the background makes it less boring and I am able to focus for longer without stopping. Sitting down and doing math problems feels like a chore. Walking math is fun and I’m excited to do more of it.