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.
Tomi Lahren Fired by Glenn Beck
1 hour ago