Not necessarily. One of the theories I read had it that practice was the single most important factor for mastery; and that the apparent difference in aptitude between people was actually a difference in interest: If you enjoy something, you're much more likely to drive yourself to practice more, and thus achieve a higher mastery; whereas if you actively dislike something, you're more likely to do the minimum amount you can get away with.
So it certainly may be the case that if someone says they dislike math, that they simply haven't been exposed to the right kinds of math, or the right kinds of problems; and that trying different kinds of presentations will allow them to overcome a dislike of it, and practice more than they were. But maybe there are people who will dislike math no matter what form it takes; for those people, doing math will always be a motivational struggle. And in any case, even if you can move someone from "I hate math" to "math can be cool", will that person ever have the drive, the thirst for math which would be required to be a top performer? Seems very unlikely.
IOW, even if in theory everyone could reach the same level of expertise by practicing the same amount, it's still not possible in practice for everyone to practice the same amount.
(Just as, in theory, everyone can lose weight by simple diet and exercise; but in practice, actually maintaining the "simple diet and exercise" routine to lose weight and keep it off isn't something most people have shown themselves capable of.)
If practice is the single most important factor, then e.g. exam scores should be a function of time spent studying, and not of the test taker.
I know people from college that worked a lot harder than me and spent a lot more time studying than me, and they got worse grades. I never "practiced" and aced all my exams.
To be pedantic, you're proposing that practice studying should result in improved exam scores, which doesn't seem to follow as well as you're proposing.
Practice shooting basketballs leads to better results at shooting basketballs.
Practice playing piano leads to better results at playing piano.
Practice studying leads to better results at studying.
Practice scoring well on exams leads to better results at scoring well on exams.
That's how the parallels work. And it's clear true: however one defines studying, it gets easier as one practices it. It might or might not be effective in improve exam scores, for many reasons, but it definitely improves studying itself.
>> If practice is the single most important factor, then e.g. exam scores should be a function of time spent studying, and not of the test taker.
> you're proposing that practice studying should result in improved exam scores
I'm not sure that he is. Me and my kids tended to ace tests without studying, often with the barest understanding of the material (that vanished soon after).
I'd guess his solution might be to have a complex representative problem to solve as proof of knowledge. Have it ready at the beginning of the course and as soon as kids can solve it, they're done.
Unfortunately, this doesn't line up with predetermined schedules which tend to dominate every mm of schooling.
You're just playing word games. If the goal is "mastery of material", then studying is "practicing" that goal and exams are a measure of how well you attained that goal.
But lets not get bogged down in the weeds. OP's theory is that "practice", or time invested, is the single most important factor in mastery. However, it is clear that some people can reach a significantly higher level of mastery in less time than others. Some people walk out of lecture understanding the concepts clearly, whereas some need to spend hours struggling with it to get that same level of understanding.
> Some people walk out of lecture understanding the concepts clearly, whereas some need to spend hours struggling with it to get that same level of understanding.
So suppose there were three people who received the same lecture:
A's mind naturally and enthusiastically is drawn to considering the concepts discussed in class; without any conscious prompting or effort, for the pure joy of it, their brain spends a few hours in the day after the lecture chewing over the content and assimilating it.
B's mind has no such natural draw; their brain only goes over the material of consciously prompted -- but at least when this conscious review is being done, their brain dutifully considers and chews over the material.
C's mind actively dislikes the material. They have to expend willpower every moment that they're studying; and their brain is always looking for excuses and other things to do.
Now, it will appear that A has spent very little looking at the material, because they have spent no conscious study time looking at it; but in fact, if you could measure actual brain engagement time, A has spent hours and hours engaging with the material.
Similarly, it will appear that C has spent hours and hours looking at the material, because they have spent hours trying to study it. But because they're working against their brain's inclination, their brain has actually only spent a fraction of that time actually engaging with the material.
Thus it appears that A achieves mastery without practice, B achieves mastery with practice, and C fails to achieve mastery even with practice. But in fact, A has practiced the most -- driven to because of their natural interest in the subject; while C has practiced the least -- impeded by their natural aversion to the subject.
ETA: I should note this is no moral judgement on either A or C: far from it. In high school almost all subjects were interesting, and thus easy for me; when I got to university, I began to find certain subjects less interesting, the result was that I avoided taking classes in those subjects if I could.
> I know people from college that worked a lot harder than me and spent a lot more time studying than me, and they got worse grades. I never "practiced" and aced all my exams.
I've got similar anecdotes. In high school, the way I studied for history exams was to come in to class early and listen to two of my classmates quiz each other with flashcards. In my head: "Yup, knew that one. Yup, knew that one. Oh, didn't remember that one. Hmm, didn't know that one either." My classmates got 100% on the flashcards, then got C's on the exam; I got an A.
My sophomore year of university, at the beginning of the year, there were three of us -- myself, K, and C, who got along well and sat together in lectures. C eventually got bored and stopped coming to class. The morning of the final exam, K was basically giving C a crash-course for the entire semester -- patiently explaining all the concepts and techniques that C had completely missed by not coming to class. Then they took the exam, and K got a 'C', and C got an 'A'. (So did I.)
The thing about both of these anecdotes is that in both cases, as far as I can tell, the people who got C's actually knew the material. I didn't grade any of their exams, so I don't know exactly what went wrong; but whatever's happening is more complicated than simply, "They weren't smart enough to master the material."
So it certainly may be the case that if someone says they dislike math, that they simply haven't been exposed to the right kinds of math, or the right kinds of problems; and that trying different kinds of presentations will allow them to overcome a dislike of it, and practice more than they were. But maybe there are people who will dislike math no matter what form it takes; for those people, doing math will always be a motivational struggle. And in any case, even if you can move someone from "I hate math" to "math can be cool", will that person ever have the drive, the thirst for math which would be required to be a top performer? Seems very unlikely.
IOW, even if in theory everyone could reach the same level of expertise by practicing the same amount, it's still not possible in practice for everyone to practice the same amount.
(Just as, in theory, everyone can lose weight by simple diet and exercise; but in practice, actually maintaining the "simple diet and exercise" routine to lose weight and keep it off isn't something most people have shown themselves capable of.)