I just had a successful weekend. I just tutored my first student in a while, and it re-affirmed something that I've seen for a long time: People don't know how to teach math. No, that's not right, there are many people that do know how to teach math, but they're drowned out by the "experts." These experts think that they know how to teach math because it worked for them and it worked for others and that's the way it's always been done. The simple fact that it's always been done that way doesn't mean there isn't a better way though. That way is so obvious to me that it's absolutely infuriating that no one else sees it! It's so simple. Actually, that's it. Simplify.
Simplicity. For math, there really is no other concept that's required. Elegance, synchronicity, beauty, wonder, magic, perfection. All of these are attributes of the numeric system of natural philosophy of mathematics. Integers, rationals, irrationals, primes, factors, multiples, all these concepts are well defined and interact in a wondrous dance, an epic waltz without limit and with symmetrical mystery, perfection unattainable yet always pursued. Then I take a look at these texts and teaching methods and it's a jarring, like nails on a blackboard, that jolts me off of my cloud of amazement and brings me crashing down to earth for a most painful fall. Function manipulation taught with one set of rules for 'y' and another set for 'x' except for this case where the rules for 'y' are the same for 'x' and so on. Statistics taught like it's all a big secret, that there's actually something other than "enumerate the possibilities and multiply by each probability" and the cheap parlor tricks used to simplify calculation. Trigonometric rules being taught for memorization, without the reason for why each works. Calculus being taught without the emphasis on identity transformation (from the earlier function manipulation in algebra) that simplifies the subject of rates and the downfall of Descartes. It's frustrating, seeing math taught without thought, just the recipes without invention, the methods without innovation, the gritty meat without the joy and brilliance of the only perfect subject.
Actually, there is one other thing. Teaching without joy leads to formulaic methods. Now I'm sure these teaching methods work with most students, but the problem with teaching math by formulas is that when only 90% get each subject that remaining 10% falls behind. The next subject then builds on each prior one in a way that's not seen in English or history. Once you get 7 subjects down the line in a year of math you then have half of your class that's either struggling to keep up or has been struggling to keep up because of some subject taught 2 months ago. Do you slow down for the 10% who couldn't keep up, or do you continue to press on for the 90% that are doing fine? Then there's the question of what do you do for the 10% who understand the subject naturally and keep wanting to push on ahead? Mathematics is the one subject that is most vulnerable to this problem, even more than physics or anything else. Therefore, it's the biggest issue in mathematics. My parents opted to keep me with the class. I wished then that I had been allowed to push ahead. I still wish that today, 20 years later.
My point is that teaching with joy, with innovative and multiple methods, can allow you to successfully instruct the bottom 10% and invigorate the top 10%, as well as teaching the middle 80%. It requires more work and greater devotion, a love of the subject that many consider unnatural. Special treatment for both the top and bottom 10% would also help. My wife told me, on seeing the look on my face after tutoring just this one student, that I should become a high school math teacher. She might be right. Maybe I should do that instead of getting my Ph.D. in physics and completing Einstein's unfinished Theory of General Relativity. Maybe I should do that and learn tensor calculus and complete Einstein's theory anyway.