Posted By Christina M on April 7, 2011
Conceptual learning is all the rage. Sure, it has been ever since they introduced the first “new math” that didn’t do a very good job of teaching it; but the idea was there: method is fine, but it should never be taught in place of concept. And tricks should be avoided because then kids will just use the tricks and never learn (or understand) the concept.
The only problem is that line of thinking doesn’t work.
Kids need a combination of approaches, and one “great idea” does not a curriculum — or an education — make. They definitely should be taught, for instance, what the concept of multiplication means before they begin to learn how to do it. And they should have frequent reinforcement of the concept as they continue to learn it. It should be tied in with their other subjects, and tied in with money, and tied in with geometry, and tied in with home life. But that isn’t exactly going to help them pass the No Child Left Alone tests, now, is it? Once they understand what the concept means, they then have to learn a number of other related subjects or the conceptual knowledge isn’t very useful. They have to learn when to use the function. And when not to. (Is it multiplication or division? Or is it subtraction?) They have to learn how to apply it to story problems. (Now that I know it’s a multiplication problem, which numbers in the question do I multiply? What does the answer mean?) And they have to learn — no matter how much it rubs the wrong way — rote facts.
When a child first learns what multiplication is, it’s helpful to have them figure out the answer instead of calling up memorized facts. If they have to figure out that three times three is nine by counting the cherries in each bowl, they will understand the concept much more clearly than if they are simply told. But eventually one hopes they already understand the concept, and they need ciphering skills. Concept without skill is useless. That’s where memorization comes in. They should learn that 3×3=9, every time. They should not have to figure it out again and again. Do you have to look up your phone number every time someone asks you for it? Of course not. Even though looking up numbers is a useful skill, eventually your time becomes more valuable than that, and you memorize the phone number. The same goes with math facts. If a child has to figure out three times three every time, he is going to be slowed so much in his ciphering that he will get behind in his seat work and homework. Eventually, he will be behind in his math learning.
Which leads to the next point. Teaching “tricks” may not aid conceptual knowledge, and therefore may not be a good way to introduce the material. But once they understand a concept, mathematical ciphering tricks can be a very effective way of mastering facts. If you refuse to teach a child the “tricks” the smartest kids will figure them out anyway, but those who don’t give it much thought, or those who don’t naturally look for alternative angles may not. All it does when you withhold the tricks is widen the gap between the “smart” kids and the “average” kids. It hurts those who most need the help. So yes, teach the concept, and teach it well. Cement it. Review it. Practice it. Review it some more.
And then tell them that the digits of a multiple of 9 always add up to 9, and that every other multiple of six rhymes. Give them every help you can. Set them up to succeed, however possible. And then review the concept some more, just to reassure yourself that you haven’t harmed them by rote memorization.