There are many on-line resources for math students these days. There are college/university level courses, such as those offered by MIT and Stanford. There are YouTube videos suitable for high school level study, such as Khan Academy and others. There are also many downloadable textbooks and on-line learning aids. A conscientious parent might ask, “What is the benefit worth paying for in having a live but on-line instructor?”
The question is both easy and hard for me to answer, because of my varied experience. I have taken classes from teachers in person, via live video links, and on-line. I have also learned material on my own with no teacher to interact with. In previous blog entries, the Drs. McMenomy have discussed the virtues of finding things out on one’s own (“Freedom to fail” and “Failure is not an option” below). So, as a student, why not tackle calculus on your own?
In an ideal context, you would have all the time you needed to explore all of the nooks and crannies, all the dead-ends, of a mathematical subject until you reached the same conclusions as previous generations of mathematicians. Most likely, you’d be a good mathematician then, too – math certainly requires practice to do well, and you would have had a lot of practice. But independent exploration takes a long while – unless you are an amazing genius, much more time than a year’s worth of classes.
So there’s the first reason to have a math teacher: to shorten the time it takes to reach mastery, by pointing out unprofitable dead ends in thought.
As you learn a subject, you must make mistakes. (This is a lesson I personally resisted for a long time – I wanted to be perfect the first time through.) Because mathematics has a definite sense of “correct and incorrect”, of “perfect and imperfect”, it’s not so bad as with Greek or history, where there are matters of style and personal orientation. Proofs, which can convince any sceptic, are not only possible, but expected, at least at times. However, you have to avoid learning the mistakes you make during learning as if they were true.
So there’s the second reason to have a math teacher: to point out the errors in your reasoning and understanding, so that you don’t have to un-learn and re-learn the material involved. This is especially true when there’s very careful reasoning required – and calculus certainly has areas where it has taken centuries for some very bright people to reach an adequate level of care in reasoning!
Anyone who has read a textbook (or any other nonfiction book) realizes that not all the material in them is equally important. That seems pretty blatantly obvious, but not everyone sees that. I’ve been known to grouse about professors’ pet theories which, while true, aren’t useful, using terms like “academic fantasies”.
So there’s the third reason to have a math teacher: someone to point out the crucially important parts, and differentiate them from the merely interesting parts. (OK, I’m not perfect there – but I try!)
But perhaps the most critical reason for an outsider’s presence in the learning activity is embodied in the statement, “you don’t know what you don’t know.” It sounds tautological, until you realize that you can know what you don’t know in some contexts (“I don’t know anything about the aorist in Greek, except that there’s something knowable under that label”), and there are so many situations possibly contrary to fact that you can’t even know them all. (A recent example: “Lifetime warranty” – I knew that sometimes it refers to the buyer’s lifetime, sometimes to the useful life of the product – but I didn’t know, in the sense that I really believed it in a way that I could act on it, that it might also refer to the lifetime of the company offering the product…) There can be holes in your knowledge that you’re unaware of: intellectual blind spots.
So there’s the fourth reason to have a math teacher: someone to make sure that your knowledge is reliably complete.
All that said, no one can practice doing math for you, just as no one can do physical exercise for you. Learning is the goal, and teaching one of many means to the goal.