Understanding the process of extending math theory is essential if you want to be able to tell the difference between being led down a good path and being led down a bad path.
If all you do is memorize theorems without understanding their proofs, you can memorize false theorems and not know the difference until, maybe, a contradiction hits you over the head and demands attention.
Checking a solution is often a lot easier than generating an original solution. If you are not going to be generating original solutions, I would guess that there are better uses of your time than learning how to.
In my schooling in general (at middle school and higher level), I was generally annoyed that most subjects were taught with the assumption that you were going to become a practitioner of that subject, rather than simply a user. Naturally, every teacher thinks their subject is the most important in the world (why else would they have chosen it?), but most of their students do not see it that way. I would have been far better off learning how to communicate effectively rather than psychoanalyzing literature, just as those bound for non-quantitative careers would have been better off learning the math they need to keep their finances in order than constructing geometric proofs.
Most people hate geometry and not only promptly forget it once they’re out of the class but develop an aversion to it.
Also, there’s barely any historical discussion of how the geometrical results were first found, with a few colorful anecdotes as exceptions.
Knowing how to construct a proof has nothing to do with knowing how the ancient Greeks viewed geometry. Knowing how to solve a math problem has nothing to do with knowing how the method was originally worked out.
Most science classes aren’t about practicing science, but memorizing science that’s already done. Most math classes are the same way.
If you’re not going to be extending math theory, is remaining ignorant of the process really a detriment?
Understanding the process of extending math theory is essential if you want to be able to tell the difference between being led down a good path and being led down a bad path.
If all you do is memorize theorems without understanding their proofs, you can memorize false theorems and not know the difference until, maybe, a contradiction hits you over the head and demands attention.
Checking a solution is often a lot easier than generating an original solution. If you are not going to be generating original solutions, I would guess that there are better uses of your time than learning how to. In my schooling in general (at middle school and higher level), I was generally annoyed that most subjects were taught with the assumption that you were going to become a practitioner of that subject, rather than simply a user. Naturally, every teacher thinks their subject is the most important in the world (why else would they have chosen it?), but most of their students do not see it that way. I would have been far better off learning how to communicate effectively rather than psychoanalyzing literature, just as those bound for non-quantitative careers would have been better off learning the math they need to keep their finances in order than constructing geometric proofs.
Even high school geometry requires you to construct your own proofs.
It may be different abroad, but in the UK state system, we were never taught the idea of a proof untill sixth form (age 16-18).
Most people hate geometry and not only promptly forget it once they’re out of the class but develop an aversion to it.
Also, there’s barely any historical discussion of how the geometrical results were first found, with a few colorful anecdotes as exceptions.
Knowing how to construct a proof has nothing to do with knowing how the ancient Greeks viewed geometry. Knowing how to solve a math problem has nothing to do with knowing how the method was originally worked out.