… an infinitely tall ladder where the rungs get spaced further and further apart the higher you climb.
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Learning math is like climbing an infinitely tall ladder where the rungs get spaced further and further apart the higher you climb.
Douglas Hofstadter has a really interesting reflection on his own personal experience climbing that ladder of math.
He describes it as “being very high on a mountain where the atmosphere grows so thin that one suddenly is having trouble breathing and even walking any longer.”
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*“I am a ‘mathematical person’, that’s for sure, having grown up profoundly in love with math and having thought about things mathematical for essentially all of my life (all the way up to today), but in my early twenties there came a point where I suddenly realized that I simply was incapable of thinking clearly at a sufficiently abstract level to be able to make major contributions to contemporary mathematics.
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I had never suspected for an instant that there was such a thing as an ‘abstraction ceiling’ in my head. I always took it for granted that my ability to absorb abstract ideas in math would continue to increase as I acquired more knowledge and more experience with math, just as it had in high school and in college.
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I found out a couple of years later, when I was in math graduate school, that I simply was not able to absorb ideas that were crucial for becoming a high-quality professional mathematician. Or rather, if I was able to absorb them, it was only at a snail’s pace, and even then, my understanding was always blurry and vague, and I constantly had to go back and review and refresh my feeble understandings. Things at that rarefied level of abstraction… simply didn’t stick in my head in the same way that the more concrete topics in undergraduate math had… It was like being very high on a mountain where the atmosphere grows so thin that one suddenly is having trouble breathing and even walking any longer.
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To put it in terms of another down-home analogy, I was like a kid who is a big baseball star in high school and who is consequently convinced beyond a shadow of a doubt that they are destined to go on and become a huge major-league star, but who, a few years down the pike, winds up instead being merely a reasonably good player on some minor league team in some random podunk town, and never even gets to play one single game in the majors… Sure, they have oodles of baseball talent compared to most other people — there’s no doubt that they are highly gifted in baseball, maybe 1 in 1000 or even 1 in 10000 — but their gifts are still way, way below those of even an average major leaguer, not to mention major-league superstars!
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On the other hand, I think that most people are probably capable of understanding such things as addition and multiplication of fractions, how to solve linear and quadratic equations, some Euclidean geometry, and maybe a tiny bit about functions and some inklings of what calculus is about.”— Douglas Hofstadter (2012) in “Some Reflections on Mathematics from a Mathematical Non-mathematician.”*
Some people might feel like this is pessimistic take, but it’s just realistic.
Yes, you can improve pedagogy to better scaffold (i.e., move the rungs closer) even at higher levels, and doing so can drastically increase the level of math that a student is able to reach (i.e., how high they can climb up the ladder).
But ultimately you’re on a smooth transition in which scaffolding gradually disappears as you get closer and closer to the edge of human knowledge, where there is no scaffolding. (I described this in more detail in stages 3/4 here.)
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