In the United States there are long-running "math wars" over how to educate children in mathematics. CNN had a recent article about the latest skirmishes in these math wars. Now, I could spend an entire entry ranting about how people abuse the word "war" and I intend to do that sometime in the future. I could also write an entry about how the CNN article has no real news content and is hard to see as newsworthy. Instead I'm going to focus on the issues that the article raises.
The article focuses on how in certain schools teachers are deliberately not teaching the children how to do long-division and multiplication. However, frustrated parents are teaching their children how to do it, rather than focus on the "conceptual" understanding.
As someone who has tutored kids in math and has worked as a counselor at a summer program which teaches number theory to high school students, I have to side with the parents.
First, long-division is conceptual. It isn't that hard to understand what is going in the algorithm. And if a teacher cannot explain conceptually why long-division works they are not a very good teacher.
Second, conceptual in this sort of context often is a disguise for ad hoc methods that sometimes are shorter but in general will not be efficient. For example, to quote from the article "When a parent is asked to multiply 88 by 5, we'll do it with pen and paper, multiplying 8 by 5 and carrying over the 4, etc. But a child today might reason that 5 is half of 10, and 88 times 10 is 880, so 88 times 5 is half of that, 440 -- poof, no pen, no paper." This works fine, but what if I asked you to multiply two 3 digit numbers? Or 4 digit numbers? Can you easily use such methods then? If you cannot do that, then you do not have a true conceptual understanding of the content in question.
Third, there's an issue of causation v. correlation. People who are good at math use such short-cuts all the time when they are available. But they do that because they are good at math. That does not mean knowing how to use those tricks will make you good at math.
Fourth, many more advanced ideas depend on long-division and multiplication. For example, what if you want to understand how to multiple or divide in another base? Or what if you want to divide polynomials, or to do arithmetic in more abstract rings like Z7[x]? The "conceptual" understanding will not help much there. But having a good understanding of how to do division and multiplication the "old-fashioned way" and understanding why they work will help a lot.
There's also one problem that bothers me which has little to do directly with the math at hand: Some parents in the article report that they feel like they are being "rebels" for teaching their children. There is something seriously wrong when parents providing additional education to children are made to feel unwanted or to feel like they are doing something wrong. We have serious problems with parents not being involved in their kids' education. We don't want to discourage the good parents who are willing to help out simply over pedagogical disagreements.
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Another way to look at it is children who are motivated or interested will go beyond the basic instrucion. So if taught just the algorithm they will learn to use shortcuts and if taught just some fuzzy concept will be able to come up with and apply some mehod.
Something I appreciate after teaching intro college physics to mostly engineers is how willing students are to not question any formulae. Students will calculate the acceleration of a ball to be 100m/s^2 or the mass of the Earth to be 400kg and think nothing of it. This is all to say that I disagree with your claim that the procedure of long divison is inherently conceptual. A student can quite successfully go through an algorithm without ever looking beyond the next step.
The subject of math education is quite near and dear to me, and I consider myself the world's leading expert. Something that is not being emphasized enough in the big debates is that the arithmetic young kids are asked to understand and perform is completely trivial. Now, it is not up for dispute that it is essential for people to have facility with arithmetic. The wonderful thing is that there happens to be a procedure, say, for multiplication of integers, which reduces any computation to the instant recall of a few facts which need to be memorized only in the sense that one memorizes his own name, and then once you've done one multiplication, you've effectively done them all, and can move on. What conceptual understanding are the math reformers talking about? That we use a base-10 labeling system for numbers and multiplication distributes over addition? Understanding means only one thing: knowledge of definitions, theorems, and proofs, and (perhaps) the ability to seek out deeper, more precise definitions and theorems, and prove them.
When I am in charge of education, children will be made to gain a complete facility with arithmetic in minimum time, and as soon as that is done (first grade?), we will make the essential move towards abstraction: We will make a preliminary discussion of sets, define the set of integers, and begin math properly with number theory. Force children to prove everything from the very beginning (somewhat like at PROMYS, but with considerably more guidance); that will give them conceptual understanding. This will fix the problem I perceive of students needing to learn a lot more math a lot earlier in order to study physics properly, with precision and rigor, and the other problem of religion. When one of my students is taken by his ignorant parents to some institution for religious instruction, he or she will know to demand precise definitions and proofs, which will obviously not be forthcoming, because the religion is wrong. The bogus ideologies and thought systems of the world will then crumble and die before me. That is the solution to the problem of how to begin a child's mathematical education.
You've hit on something with the way that American school systems make parents feel like rebels for simply teaching their kids. It's more common than many people realize. A slightly off-topic example I can't resist adding: my parents were yelled at by my (Alabama) school system for teaching me to read before first grade . They were told it would stunt my social and psychological development.
Well put Mr. Zeweenskie. I have enjoyed many of your former entries. This perhaps is the most uplifting, educational piece that you have put together. It involves the reader in a complex network of facts and interesting information. You realize Mr. Zeweenskie, that one may come to the realization that this is an educational piece. Todays modern economy does not accept education the way it did forty years ago. Perhaps you should write less intricate pieces Mr. Zeweenskie?
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