Most math curricula are an absolute pedagogical mess.
I have long known that math programs treat children like human calculators, programming them with processes they use to input numbers and churn out results. But this became poignantly clear to me when I tried to teach my daughter long division this summer.
Confronted with a problem such as 2,832 divided by 8, I began my "explanation," hearkening back to the process that had been drilled into me in third grade. "8 goes into 28 how many times? 3. So you write a 3 above the 8. 8 times 3 is 24. Subtract 24 from 28 and you get 4. Then bring down the 3. 8 goes into 43 how many times?..." and so on. At the conclusion of my presentation, she said something simple but telling: "That is going to be a lot for me to remember."
Indeed, it is a lot for her to remember, because she is remembering, and not understanding.
If you want to grasp the poverty of your own education in math, I offer you the following challenge: explain long division. Explain it to a child, to an adult, to yourself—but really explain it. Use words to describe not the process, but the reason for the process: why each number goes where it does; why you subtract, or divide, or bring down; why the process works. It won't be easy. I maintain that if you had been educated properly in math, it would be.
One of the defining principles of the VanDamme method is a concerted effort to ensure that every item of knowledge possessed by the child is true knowledge, to ensure that he understands it thoroughly, independently, conceptually. To realize this goal in math will require a total overhaul of the standard curriculum. It will require that someone strip the program down to essentials, arrange the material with total faithfulness to hierarchy, and design assessments that are true tests of the child's understanding.
Meanwhile, we can take moderate steps in that direction, by requiring, for example, that the children give complete, verbal explanations for all that they do in math.
Mr. Steele, VanDamme Academy math teacher for a group of 7 & 8-year-olds, demands of his students that they not just blurt out answers, or crank through mechanical processes. He makes them explain the processes using the proper terminology and demonstrating that they understand what they are doing and why.
If, for example, he is teaching subtraction with borrowing, and puts a problem on the board such as 2700 – 350, someone in the class will invariably ask, "Can I just tell you the answer?" Mr. Steele's answers are charming—and pedagogically correct.
Sometimes he says, "I don't want you to do 'magic math.' I don't want you stare up at the sky, come up with a number, and blurt it out to the class. That doesn't help us understand, and that doesn't show me that you understand. I want you to explain how you arrived at your answer."
At other times, he says, "Let's play a game called 'Mr. Steele bumped his head and can't remember math.' Don't just give me the answer, teach me the process by which you arrived at your answer."
The students proceed with explanations that demand, among other things, that they use concepts of place value (if they begin the problem above by saying, "0 minus 0 is 0," he says, "That's true," and waits for them to tell him that you put a 0 in the ones' place before he writes a 0 on the board), and that they explain what they are doing when they borrow (if they say, "Cross out the 5 and put a 4, and put a 10 in the tens' place," he will ask, "What does that 10 represent? 10 what? 10 monkeys?" which will make them giggle and offer the correction, "10 tens silly!").
These children are not treated like human calculators, they are treated like thinking beings. And when they truly grasp the concepts they are using, when they can explain them fully and articulately, when they retain them because they are not memorizing, but understanding—that is real math magic.