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Thursday, September 25, 2008

Math Magic

I believe it was Dr. Piekoff who said that the most critical need today is education. So in keeping with that value I repost an email I received from the VanDamme Academy today.


Pedagogically Correct Volume 3, Issue 2
September 24, 2008

"Pedagogy": The art and science of teaching.
:: Calling All LifeLong Learners: Learn Science the VanDamme Academy Way!
:: Recommend Pedagogically Correct to five friends, get Lisa VanDamme's e-book, "Reclaiming Education," for free!
:: Announcement: Pedagogically Correct Blog


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.




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Recommend Pedagogically Correct to five friends, get Lisa VanDamme's e-book, "Reclaiming Education," for free!
Lisa VanDamme's educational career began when a group of parents, disillusioned with standard public and private schools, hired her to educate their children. In 1998, she chronicled her successes homeschooling and explained the methods that made them possible in a lecture, "Reclaiming Education." The audience, fascinated by her insights about education, and inspired by the stories she told, gave her a standing ovation. In 1999, she made "Reclaiming Education" available in written form, to the delight of thousands of readers. Since 1999, the essay version of "Reclaiming Education" has been unavailable. Until now.

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3 comments:

Anonymous said...

Many things in math — such as the divide algorithm, or even the formula for the area of a circle — are validated by means of more advanced math. For example, in order to explain why the area of a circle is pi times the square of the radius, you need to be able to do integral calculus.

Similarly, in order to be able to prove the correctness of the divide algorithm, you need algebra. Donald Knuth hints at this in his chapter “The Classical Algorithms” in The Art of Computer Programming.

Similarly, digital signal processing requires some advanced math such as Fourier transforms, but when you implement it in a computer, it typically boils down to add, subtract, multiply, and divide. And that’s it. Even sines and cosines can be reduced to arithmetic (e.g., by Taylor polynomials).

This is not an inversion of hierarchy so much as a reflection of the fact that it takes more brilliance to create these algorithms than it does to use them. When people study higher math, they create a lot of techniques that are easy to use and that work, but that require more sophisticated techniques for their proof than are needed just to state them or use them. That's why it makes sense to value so highly the people who did create them.

It is also fun to study higher math because you finally learn the abstract principles that make the lower-level math make sense.

Memorization is not enough to yield true understanding — but it cannot be neglected, either. You will never be able to prove the divide algorithm correct if you don’t know what it is.

The same phenomenon is at work in the case where children learn to speak in sentences before they study how to diagram sentences in high school.

You have to learn the concretes before you can learn the abstractions. Once the abstractions are learned, they give you a much better grasp of the concretes.

You form some abstractions yourself as you go over the material.

Division must be taught as a step-by-step process. I know no other way to do it. After the student runs through the process a few times, she will begin to form abstract concepts of how it works. These will be solidified and reinforced in more advanced math courses.

Mike N said...

Anon:
Great comments. You're exactly right about:

"You have to learn the concretes before you can learn the abstractions. Once the abstractions are learned, they give you a much better grasp of the concretes."

Ayn Rand pointed out that only concretes exist and that it is man's mind that must infer the principles regarding the nature of that existence.

michaeledlavitch said...

Have you tried the online Math Games at Hooda Math yet?
hoodamath.com