Thought on learning mathematics

Methods of Learning

My approach to learning how to use a computer was to ignore it until I had a need for it and then figure out how to make the computer fulfill that need. I used this technique to learn about motorcycles; when something broke, I learned enough so that I could fix it.
I know people who as soon as they got their computer started fooling around with it, doing this, doing that, and this was how they became familiar with the beast.
I know people who took a course of instruction on the use of the computer. They have left the decision of their possible needs to someone else. Those who like the very idea of computers and enjoy doing anything on them, find the lessons a pleasant way to spend their time.
I used the "fool around technique" to learn about photography. I started taking pictures, saw what kind of pictures I could take and then tried to improve them.
Many guitarists I know started when they heard a tune they liked and wanted to play it. They learn the tunes they like.
I know a guy whose dad had traded a motorcycle for a drum set. It was just sitting around the house so he started fooling around with it and became quite an accomplished drummer.
I started to learn classical guitar by taking lessons from a former student, letting him decide what exercises and tunes I should use to master the instrument. Since I liked the sound of all the exercises and tunes it didn't make much difference where I started and the lessons were partially successful.
What about learning mathematics?
If a student, mirable dictu, is interested in mathematics then any of the learning techniques, need, fooling around or lessons can have some success.
Unfortunately, most students are in a mathematics course because they are required to be there. They see no need satisfied by mathematics and they would rather spend their time doing anything else than fooling around with mathematics so mathematics education falls back on an extreme form of lessons to teach the students.
It is like being forced to take guitar lessons without ever having heard a guitar, without knowing what the end result is supposed to sound like, indeed, without knowing what music is.
"Just put your finger here and pluck this string, then put your finger here and pluck this string..." and before you know it, voila, "Classical Gas". I had a friend who was absolutely tone deaf and claimed that he was going to use this technique to learn the guitar; he wanted to be the life of the party and felt that the guitar was the path to this desired end. I never knew how it turned out.
"Just follow these rules of mathematics, memorize these formulas and before you know it, voila, differential equations."
I am always surprised, although I suppose I shouldn't be, that when I ask a student who has passed a course in differential equations if they know what a differential equation is, they have no reply. Some can even solve one without knowing what it is.
A guitarist pointed out to me that the end result of the fretting and plucking is sound and that's the ball you want to keep your eye on.