My father, who had a PhD in history, had exactly the kind of math skills the study describes in Indian children. With ordinary arithmetic, he was fast and accurate. Money and baseball statistics were no problem. Algebra? No fucking way. As soon as x and y were involved, he struggled. Strike that: he wasn't able to struggle. He wasn't able to engage the gears that would allow him to apply effort.
One time, when I was in high school and already contemplating majoring in math in college, he told me that a math professor had told him that in modern mathematics you didn't have to know what you were talking about. All their theories could apply to anything. Like you could pick up a paper, and they're talking about X, and you could decide X was Donald Duck! He told me this like it was an exotic glimpse into another culture -- he knew that it looked ridiculous to him, but he also knew that it probably looked ridiculous because he didn't understand what he was looking at. You could tell that one part of his brain felt like it was a gotcha moment for the mathematicians, but another part of his brain could see that they weren't embarrassed about it, and he taught WWII every year so he knew that Donald Duck could also be artillery shells or atoms. He had that last defense of common sense that stops people from embracing crank theories about other fields of study.
This was a guy who taught recent history and accepted the abstract ideological struggles of the 20th century without blinking, but when you told him that someone could write an entire doctoral thesis about X without knowing concretely what X was, it was such an alien idea that it was out of range of his curiosity.
From this I would be confused about how he got through high school math, except that my sister, who also has a PhD in history, explained how she got through calculus: she studied all the homework problems and all the solutions over and over until she had memorized them, and she reproduced the patterns on the tests. At our high school, that was good enough for As. In college, it was good enough for Bs.
(It makes me feel a tiny bit more empathy for the condescending mathphobes who denigrate virtually all school mathematics work as pointless, deadening rote learning. For many of them it might be a sincere belief. They might have actually experienced it that way and never experienced any of the worthwhile aspects of it. But, on the other hand, they should have the grace my dad did to stop short of declaring it worthless just because they didn't get it.)
It is very sad to me that so many people can't enjoy that aspect of math. I was lucky, pbs used to show math stuff to kids, so it was fun and interesting before it was a school thing. Of course a huge part of math learning is just hatd work for most if us. But kids should taste the delight first, it motivates them to do the less delightful practice.
Unfortunately, the pleasure or displeasure of doing math compounds quickly. A kid who doesn't enjoy it is going to do the minimum required, and if that isn't enough, it will become ever harder and less enjoyable in the future. You need practice, and practice is a dirty word in pedagogy.
Someday education is going to catch up with music and sport in its attitude towards practice. As adults we all understand that in music or sports, the most elite of the elite, the top 0.01% of human beings in attainment, are never too good to need more practice and polish of basic skills. But when we look at children learning math, repetition becomes anathema. My question is, how could anybody ever enjoy math without repetition? You need to make the boring stuff easy and then keep it easy. How does that happen without repetition? If you don't practice, the boring stuff becomes hard again, and you don't have brains to spare for the interesting stuff.
I can’t relate to this at all. In music, you practice until you get it, and then you stop. It’s important not to over-practice, in fact. In high school math classes, I was assigned an order of magnitude more practice problems than I actually needed. There was no “practice until you get it, then quit.” It was “do all the problems whether you understand them or not, or you get a bad grade.” Repetition is absolutely anathema when you’ve ceased to learn anything from it.
I agree. Kids need to eat their vegetables, but we can make vegetables quite delicious if we try. Also, the music metaphor is imperfect because some musicians are performers others composers. I prefer sport as the metaphor: a mix of short laregely repituous training and longer term strategy, different styles for different athletes, influence of genetics and talent heterogeneity acknowledged at the elite end of the spectrum.
One time, when I was in high school and already contemplating majoring in math in college, he told me that a math professor had told him that in modern mathematics you didn't have to know what you were talking about. All their theories could apply to anything. Like you could pick up a paper, and they're talking about X, and you could decide X was Donald Duck! He told me this like it was an exotic glimpse into another culture -- he knew that it looked ridiculous to him, but he also knew that it probably looked ridiculous because he didn't understand what he was looking at. You could tell that one part of his brain felt like it was a gotcha moment for the mathematicians, but another part of his brain could see that they weren't embarrassed about it, and he taught WWII every year so he knew that Donald Duck could also be artillery shells or atoms. He had that last defense of common sense that stops people from embracing crank theories about other fields of study.
This was a guy who taught recent history and accepted the abstract ideological struggles of the 20th century without blinking, but when you told him that someone could write an entire doctoral thesis about X without knowing concretely what X was, it was such an alien idea that it was out of range of his curiosity.
From this I would be confused about how he got through high school math, except that my sister, who also has a PhD in history, explained how she got through calculus: she studied all the homework problems and all the solutions over and over until she had memorized them, and she reproduced the patterns on the tests. At our high school, that was good enough for As. In college, it was good enough for Bs.
(It makes me feel a tiny bit more empathy for the condescending mathphobes who denigrate virtually all school mathematics work as pointless, deadening rote learning. For many of them it might be a sincere belief. They might have actually experienced it that way and never experienced any of the worthwhile aspects of it. But, on the other hand, they should have the grace my dad did to stop short of declaring it worthless just because they didn't get it.)