I enjoyed this article.
I have taught math. I have even understood math, sometimes.
I agree with you that understanding always trumps rote memorization, with one important caveat.
We learn many things initially, as do most primates, through imitation. There are many aspects to a good baseball swing, distribution of weight, timing, and so on. But all great hitters start with somebody showing them where to put their feet, their shoulders, etc.
Math is not a whole lot different. If teachers are too coy about “how to do it,” it can be frustrating to students who just need one tiny insight to take off and unravel the whole puzzle.
Your point about mental math is well-taken, but I would justify it in a different manner. When computers make errors, they make colossal errors. Students need to know or be able to figure out that the square root of two is somewhere between one and two, so that they reject the answer of 32 they get when they type 2^5 instead of 2^.5 on their calculator. Of course, a typing mistake like that is a student error, but it is the kind of error that can only happen with computers. It would never have happened in my day, where we learned to approximate square roots on paper. And students who don’t know how to do math in their head make errors like that all the time.
Speaking of my day, I’m not sure that getting the math curriculum right is all that it’s cracked up to be. I graduated from high school in 1980. Had my teachers wanted to instruct me in what I would need (they probably did want to instruct me in what I would need), they wouldn’t have been able to. The personal computing revolution was more than a decade off. I was taught to type on a manual typewriter, because my teachers thought that would prepare me for a world in which I would run into one of two things, a) manual typewriters, and b) electric typewriters, which would be easier for me to master coming from manual typewriters than vice versa.
Yet I made the transition to computers and even a little coding easily. I did a lot of statistics programming for my graduate thesis in R. The reason is that my teachers taught me how to think. I was taught to research, question, look for root causes, etc. My teachers emphasized understanding, as you do. I learned skills as an adult which weren’t even dreamed of in the 1970s.
If any teacher could tell what students would need in ten years, they would not only be a fantastic teacher, they might also be wealthy, because they could just invest in what technology they knew was coming along! I’m joking a little, but only a little. We don’t know the future. What we do know is that kids will need to think and adapt to whatever is new.
One final point. Math education never seems to be about concrete objects. That is the big struggle! I think it’s important to make it about concrete objects as often as possible. One way I tried to do that was with apples, oranges, and pennies, or similar stuff. Give kids two apples and three oranges, and tell them it cost nineteen pennies. Give them four apples and one orange, and tell them it cost thirteen pennies. Ask them how much an orange costs. Let them push the apples, oranges, and pennies around the table. Give them hints. They’re learning to solve two equations with two variables. They might get really, really good at that game before it turns into x’s and y’s.