Neat. Some of us can't see things in our heads at all (aphantasia), so we definitely can't do things this way.
Although now that I think about it there is still some element of what's described in this article. There's no visual shape involved in the way I model numbers, but it resonates to think of 7 as "10 with a 3 missing", but also as "5 with a 2 on it". The concepts are built in reference to their closest multiple of 5, and slide between different equivalent forms as necessary in calculations.
By the way, the way I do mental math without images feels like it is using sounds and words for the short-term storage and recall. The language brain seems good at putting something aside for a minute and then bringing it back afterwards with a low chance of error, like repeating something someone just said back to them verbatim even though you weren't really listening.
The one method I am sure _doesn't_ work well for mental math is picturing the grade-school algorithms on an imaginary sheet of paper. For whatever reason it is very error-prone. I once did an informal (definitely unscientific) survey on this (30 or so people IRL plus like 100 reddit users) and iirc there was a strong correlation between "imagining the pen-and-paper algorithm", "being bad at mental math", and "not liking math". Wish I still had the data from that -- all I remember is roughly confirming my hunch that those were related. I also wrote a blog post about this a few years ago (https://alexkritchevsky.com/2019/09/15/mental-math.html) but I wish I had included the survey information in there, it would have been much more interesting.
While writing this article, I learned that Ed Catmull has aphantasia. It's amazing to me that someone with a Turing award for work on computer graphics can't mentally "see" those graphics when he closes his eyes. It'd be really eye-opening to somehow get his (or anyone else's) mental state into my own brain, just to try it out for a little.
Interesting that we share some conceptual similarities in how we think about numbers, but they're expressed through different pathways (language vs. visual.) I wonder if the people who imagine pen-and-paper stuff when doing mental math just don't have these pathways set up, and instead recall memories of math-adjacent experiences in lieu of another internal representation of numbers.
Although now that I think about it there is still some element of what's described in this article. There's no visual shape involved in the way I model numbers, but it resonates to think of 7 as "10 with a 3 missing", but also as "5 with a 2 on it". The concepts are built in reference to their closest multiple of 5, and slide between different equivalent forms as necessary in calculations.
By the way, the way I do mental math without images feels like it is using sounds and words for the short-term storage and recall. The language brain seems good at putting something aside for a minute and then bringing it back afterwards with a low chance of error, like repeating something someone just said back to them verbatim even though you weren't really listening.
The one method I am sure _doesn't_ work well for mental math is picturing the grade-school algorithms on an imaginary sheet of paper. For whatever reason it is very error-prone. I once did an informal (definitely unscientific) survey on this (30 or so people IRL plus like 100 reddit users) and iirc there was a strong correlation between "imagining the pen-and-paper algorithm", "being bad at mental math", and "not liking math". Wish I still had the data from that -- all I remember is roughly confirming my hunch that those were related. I also wrote a blog post about this a few years ago (https://alexkritchevsky.com/2019/09/15/mental-math.html) but I wish I had included the survey information in there, it would have been much more interesting.