The fact that someone does not understand calculus, does not imply that they are incapable of understanding calculus. They could simply be unwilling. There are many good reasons not to learn calculus. For one, it takes years of work. Some people may have better things to do. So I suggest that your entire premise is dubious—the variance may not be as large as you imagine.
Personally, I learned a semester worth of calculus in three weeks for college credit at a summer program (the Purdue College Credit Program circa 1989, specifically) when I was 16. Out of 20ish students (pre-selected for academic achievement), about 15% (see note 1) aced it while still goofing around, roughly 60% got college credit but found the experience difficult, and some failed. Two years later, my freshman roommate (note 2) took the same Purdue course over 16 weeks and failed it. The question isn’t “why don’t some people understand calculus”, but “why do some people learn it easily while others struggle, often failing”.
Note 1: This wasn’t a statistically robust sample. “About 15%” means “Chris, Bill, and I”.
Note 2: That roommate wanted to be an engineer and was well aware that he could only achieve that goal by passing calculus. He was often working on his homework at 1:30 am, much to my annoyance. He worked harder on that course than I had, despite being 18 years old and having a (presumably) more mature brain.
I learned a semester worth of calculus in three weeks
I’m assuming this is a response to my “takes years of work” claim, I have a few natural questions:
1. Why start counting time from the start of that summer program? Maybe you had never heard of calculus before that, but you had been learning math for many years already. If you learned calculus in 3 weeks, that simply means that you already had most of the necessary math skills, and you only had to learn a few definitions and do a little practice in applying them. Many people don’t already have those skills, so naturally it takes them a longer time.
2. How much did you learn? Presumably it was very basic, I’m guessing no differential equations and nothing with complex or multi-dimensional functions? Possibly, if you had gone further, your experience might have been different.
3. Why does speed even matter? The fact that someone took longer to learn calculus does not necessarily imply that they end up with less skill. I’m sure there is some correlation but it doesn’t have to be high. Although slow people might get discouraged and give up midway.
My point isn’t that there is no variation in intelligence (or potential for doing calculus), but that there are many reasons why someone would overestimate this variation and few reasons to underestimate it.
1) True, but by the time that roommate took the class he had had comparable math foundations to what I had had when I took the class. Considering the extra years, arguably rather more. (Upon further thought I realized that I had taken the class in 1988 at the age of 15)
2) That was first-semester calc, Purdue’s Math 161 class (for me and the roommate). Intro calc. Over the next two years I took two more semesters of calc, one of differential equations, and one of matrix algebra. By the time I met my freshman roommate (he was a bit older than me) and he started the calc class, I’d had five semesters of college math (which was all I ever took b/c I don’t enjoy math). Also, that roommate was a below-average college student, but there are people in the world with far less talent than he had.
3) Because time is the only thing you can’t buy. Time in college can be bought, but not cheaply even then. I got through school with good grades and went on to grad school as planned; his plans didn’t work out. Of course time marched on and I had failures of my own.
I agree that there’s more to success than one particular kind of intelligence. Persistence, looks, money, luck, and other factors matter. But my roommate’s calculus aptitude was a showstopper for his engineering ambitions, and I don’t think his situation was terribly uncommon.
The fact that someone does not understand calculus, does not imply that they are incapable of understanding calculus. They could simply be unwilling. There are many good reasons not to learn calculus. For one, it takes years of work. Some people may have better things to do. So I suggest that your entire premise is dubious—the variance may not be as large as you imagine.
Personally, I learned a semester worth of calculus in three weeks for college credit at a summer program (the Purdue College Credit Program circa 1989, specifically) when I was 16. Out of 20ish students (pre-selected for academic achievement), about 15% (see note 1) aced it while still goofing around, roughly 60% got college credit but found the experience difficult, and some failed. Two years later, my freshman roommate (note 2) took the same Purdue course over 16 weeks and failed it. The question isn’t “why don’t some people understand calculus”, but “why do some people learn it easily while others struggle, often failing”.
Note 1: This wasn’t a statistically robust sample. “About 15%” means “Chris, Bill, and I”.
Note 2: That roommate wanted to be an engineer and was well aware that he could only achieve that goal by passing calculus. He was often working on his homework at 1:30 am, much to my annoyance. He worked harder on that course than I had, despite being 18 years old and having a (presumably) more mature brain.
I’m assuming this is a response to my “takes years of work” claim, I have a few natural questions:
1. Why start counting time from the start of that summer program? Maybe you had never heard of calculus before that, but you had been learning math for many years already. If you learned calculus in 3 weeks, that simply means that you already had most of the necessary math skills, and you only had to learn a few definitions and do a little practice in applying them. Many people don’t already have those skills, so naturally it takes them a longer time.
2. How much did you learn? Presumably it was very basic, I’m guessing no differential equations and nothing with complex or multi-dimensional functions? Possibly, if you had gone further, your experience might have been different.
3. Why does speed even matter? The fact that someone took longer to learn calculus does not necessarily imply that they end up with less skill. I’m sure there is some correlation but it doesn’t have to be high. Although slow people might get discouraged and give up midway.
My point isn’t that there is no variation in intelligence (or potential for doing calculus), but that there are many reasons why someone would overestimate this variation and few reasons to underestimate it.
1) True, but by the time that roommate took the class he had had comparable math foundations to what I had had when I took the class. Considering the extra years, arguably rather more. (Upon further thought I realized that I had taken the class in 1988 at the age of 15)
2) That was first-semester calc, Purdue’s Math 161 class (for me and the roommate). Intro calc. Over the next two years I took two more semesters of calc, one of differential equations, and one of matrix algebra. By the time I met my freshman roommate (he was a bit older than me) and he started the calc class, I’d had five semesters of college math (which was all I ever took b/c I don’t enjoy math). Also, that roommate was a below-average college student, but there are people in the world with far less talent than he had.
3) Because time is the only thing you can’t buy. Time in college can be bought, but not cheaply even then. I got through school with good grades and went on to grad school as planned; his plans didn’t work out. Of course time marched on and I had failures of my own.
I agree that there’s more to success than one particular kind of intelligence. Persistence, looks, money, luck, and other factors matter. But my roommate’s calculus aptitude was a showstopper for his engineering ambitions, and I don’t think his situation was terribly uncommon.