Comment by openrisk
11 hours ago
There is this element of abstract mathematical thinking that many young people get exposed to at some point in the educational system but just never "get it" and they disconnect. This is where it goes awry as the gap only widens later on and its a pity.
Working with symbols, equations etc. feels like it should be more widely accessible. Its almost a game-like pursuit, it should not be alienating.
It might be a failure of educators recognizing what are the pathways to get the brain to adopt these more abstract modes of representing and operating.
NB: mathematicians are not particularly interested in solving this, many seem to derive a silly pleasure of making math as exclusive as possible. Typical example is to refuse to use visual representation, which is imprecise but helps build intuition.
Lots of people seem to get permanently lost right around when operations on fractions are introduced. Other places, too, but that seems like the earliest one where a lot of people get lost and never really find their way back.
Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.
I don't know how widespread this phenomenon is but in the book Do Not Erase [0] it seems that there are quite a few prominent mathematicians who do use visual representations in their work.
[0] https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_a...
There is this long-running (and quite fascinating I think) debate about the different "types" of mathematical thinking. Logical vs Intuitive, Geometric vs Algebraic etc. Can't recall where exactly but I remember reading about a 19th century mathematician that crowed they had not a single figure in their masterpiece.
Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.