This junt is from a few years back when I was really focused on algebra learning. The answer to which, “You learn by doing.” Just like everything else and, in particular, human occupations for which machines can not do. It appears I may have not finished it but I do not honestly know if it does complete.
I watched one of my students explain a concept to a classmate the other day and realized that I had succeeded in doing what I needed to as a teacher for him. There is plenty more math he can and will learn. There is always more to learn. That was not the source of my relaxed understanding. I could see that he was thinking with a solid mathematical logic and could develop new skills with a new found ease. In short, he had learned how to learn and apply math. This student will be leaving the school soon and is rather busy getting everything together so I am not pushing him as hard as I could in math but I honestly feel that I have achieved my primary mission. Besides, he is more interested in programming and origami now so I have no qualms with letting him study and pursue those skills and the problem solving they entail. He wants to make something, not just solve more equations.
To be fair, this student is naturally a deep thinker and one who I have had to argue almost every logical difference in math with. It can be tiresome but he now approaches the subject with a very solid understanding of its structure and dynamic. Not all students are like this. A part of me always wonders why this is so. Math is a playful and fascinating pursuit that we have turned into thousands of drills and practices. One learns far more figuring out a pattern in the units digit of multiples of 3 than a page of multiplication drills. These patterns that are not so difficult to arrive at are the basis for the laws and proofs that build the subject. Solving the little drills just becomes a matter of mechanics. The solutions make sense for how we get to them and the whole study works out the way it should.
This is a long way around to getting to what many of us already know. Teaching math through drills is not teaching or learning of math. It works in despite of itself. We keep doing it because we keep doing it. Grading is easy and it makes for straightforward but deeply flawed and largely pointless testing.Through all these tedious problems and arrays of rules to follow, we suffocate the joy of math. The subject is so profoundly interesting and creative that young minds still enjoy it through all we do to destroy it.