I think there are many useful advances in teaching number concepts and arithmetic. To me, the issues in teaching math, beginning in particular with algebra and then trigonometry, revolve around a mistaken notion that kids need to “understand” and the ideas that we teach “understanding” by testing for what kids don’t know and, in particular, by testing with “word problems” or their equivalent that require kids to identify concepts, label and then manipulate them symbolically. Other countries don’t do this; they believe most people can’t “get” math but that many, many can learn how to do math. These countries, which include our Asian competitors, test what you know, which in most cases consists of demonstrating that you in fact can do the problems assigned to you or demonstrated in class. I think - and I’ve mentioned this before - that testing for what kids don’t know, for how well they can “apply” concepts is extremely demoralizing. And our society compounds the problem by labeling all other methods as mere “rote”. Well, mere rote works better for nearly everyone.
I also wonder - and I may have mentioned this other times as well - why we don’t teach arithmetic and then algebra, etc. with more history and with more tricks. One of my favorite things to demonstrate to a kid is the story of Gauss as a kid when the teacher asked the class to add the first 100 numbers and he did it by realizing that 1 + 100 = 101 and 99 + 2 = 101 and that’s true for every pair and there are 50 pairs so 5050. (As a note, the usual way to write this out is to say 100 pairs and then divide by 2, meaning you write out all the numbers from 1 to 100 and pair them. That makes it easier to figure out the general equation.) And a neat thing is if you sum up to an odd, just start at 0 and make pairs. Cool trick. And as I noted, you can show how it becomes an equation you can memorize - rote! - and apply all over. I love when a kid sees that an equation means it’s the same thing done every time and this is just how we write that method down. Then they get the concept of algorithm: it’s just a method you write down so when people say “equation” they’re really saying a bunch of operations you string together. Kind of like how you get through your day at school can be described as an algorithm: you go here, do this for this long, etc. The more you do this, the more math and numbers becomes a “language” used to describe stuff that happens and just not something you fail at. You can be fairly bad or fairly good at Spanish and the same is true of math.
Heck, when I was a kid and we had to memorize times tables I thought it was such a bore that I figured out how to multiply any 2 digit numbers. At the time I was fascinated by numbers and took a sheet of paper and wrote out the list of numbers and their squares and saw the difference always rises by 2 and then it became obvious that, well, for 92 squared, you square 90, which is 8100 and then figure where you are on that ladder which goes up by 2. That was just (2 x 90) x 2 - or 360 - and then 4 or 8464 . If it were 94 squared, the process is just 8100 + (2 x 90) x 4 + 16. There are many tricks like this. I figured the same thing out for cubes but it took way more steps because the difference between cubes doesn’t reduce to 3 as quickly. You can see the only “realization” was that you double whatever number you’re starting with and multiply it by the number of steps you need. You can also go backwards from any number and it works for longer numbers but they get hard to keep in your head. My point - and I was never all that good at math - is that tricks are really useful and they reveal a lot about numbers and they help you overcome the basic fears about calculation. Write 77 squared on a board and that’s intimidating. And so they teach you all these estimation/guesstimation stuff rather than actual tools for actually solving things. I got away with estimations for years because I knew how to go from 70 squared up and 80 squared down to 77 squared and could give the exact answer in a few seconds with a pencil (or my head if I concentrated - which I’m not good at and I can’t manipulate things in my head like real math whizzes can). I always thought they’d be better off teaching basic stuff out of one of those 101 Math Trick books, just as I thought the best way to learn physics was from a book of physics tips and tricks.
