Math Skills versus Math Concepts (Pt. 1)
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There seems to be a big “fight” about “which should you teach first, math skills or math concepts.” A popular example is the “multiplication tables” versus the concept of multiplication (as repeated addition, for example).
It’s a pretty good bet to say that when memorizing things it’s easier if you can relate the objects. Like if you went shopping and had to get toothpaste, a toothbrush and dental floss, that would be easier to remember than if you had to get shoe polish, armadillo meat and an f-string for a lute (do lutes even have f-strings?)
So if you can have the items related to each other, it’s easier to remember them. It’s certainly easier to make sense out of them.
Now, when you consider the following numbers from the point of view of a child learning them:
- 9, 18, 27, 36, 45, 54, 63
they seem to have nothing in common, except that they are on a chart of “the multiplication tables” that the child is told to “shut up and memorize!”
On the other hand, if you show the child groups of objects, which have nine in each group, and you put them next to each other, they can readily see how nine grows into 18 if you add two groups, then into 27 if you add another group, and so on.
This increases the child’s appreciation for what he is learning, and decreases resistance to that learning. So showing the concept definitely helps learn the skill.
The school of thought that says, “Teach the skill and the concept will be easier for the child to understand later” is based on what we want the child to learn, not how it’s best for the child to learn it. This is utterly misguided.
It’s the “authoritarian” way. It’s the same school of thought that says, “You will do as I say whether you like it or not, because I know what’s best for you.” That is the most counterproductive way that was ever thought of. It is the default, brain-stem, neanderthal way. And it doesn’t work to produce intelligent children.
It may produce obedient drones if that’s your goal. But I’m pretty sure that if you read this blog, that’s not exactly what you’re after.
Actually, the “authoritarian” way doesn’t even create obedience. But it’s a great way to produce anti-social rebels who will revolt against the thing you are trying to teach them. I don’t mean that they’ll become the creative-rebellious types who produce great things for society, I mean they’ll become the destructive little shits who only complain, and then grow up to be exactly the kind of disappointed authoritarian types as the dunces who tried to inculcate them with “facts” and “skills” that became neither to them.
A Theory of Relativity…
Imagine giving your child a list of names, like: Mark, Simone, Leonora, Derrick, Romulus and Matthilde. Now tell them what each looks like and acts like, and that Mark and Simone are married, and Mark is their father’s brother, and Leonora and Derrick are their children, and Romulus and Matthilde are Derricks friends from school. You can even make a chart of all their relationships and characteristics.
Or, you an invite them all over for a barbeque and letting the child play with them and get to know them. They gradually learn how each is related to whom, and they get a personal relationship with each. Those people become meaningful to the child (imagine that!)
Which do you think will stick with the child more? Which would he rather do, sit and stare at a list and a chart while you grade him on his progress, or play and interact?
I hope this little analogy, imperfect as it is, nailed the difference between understanding and drilling for you.
Respect your Mind
When we teach concepts first, and skills later, we are reinforcing the Idea of respecting the minds of the children we teach. It may take a while, but when you do that, the child slowly gets the Idea that it’s okay to trust your brain. They’ll see that when you don’t expect them to memorize something and then use it quick (because, say, you have to “cover material”, and it’s in this “module” of the “curriculum”, and that “it will be on the test soon”), that they can actually grasp it, and understand it more deeply. That way, everyone gets more satisfaction from the teaching/learning endeavor.
Tomorrow I’d like to mention a kind of caveat to all of this, concerning a bit of Zen philosophy.
Apropos of nothing:
I’ve recently gotten into the “Twitter” thing. Not sure if I like it, or what it will bring, but it is new, interesting, and a great distraction from work. I’m supposed to be using it as an organizational and motivational tool. We’ll see.
Recently I read a “tweet” from http://twitter.com/bcubbison. He asked, “What kind of people would name their planet after dirt?” It made me wonder what the people who named their planet “Uranus” are like.
BTW, do you use Twitter? If you do, check me out at twitter.com/Prof_homunculus, and follow me.
Addendum
On Thursday, July 31, 2008 I read an interesting blog post at The Amazing Mind blog, entitled Learning Mastery 5 - Back to the Basics. You might want to check it out. The blog is very interesting in general as well. He bring up many topics and points about them that everyone should ponder now and then. Have fun.
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[...] dolcevitaacademy, which talked a bit about something that is close to my heart, which is: should you focus on teaching concrete skills first, or rather on concepts? (I’ve blogged about that on the post about Math Skills v. Math Concepts.) [...]
Comment by Trent
In my class I always attempt to teach the concept first. This is largely a result of how I learn math myself. I am not a rote learner, I learn a concept and then apply it to problems. However, I don’t think it can be discounted that students learn differently. My wife (much smarter than me) tends to be a rote learner first. She would want all the information about the kids first and then she would use that information to put the puzzle together of who is who. Me, I would gain nothing from the initial information.
Basically, I think both are required in a positive math classroom. I like to use scaffolding to inroduce a concept, but I always, at some point, will give the students a very specific set of procedures and rules for solving a problem. The best thing that can happen is for the students to understand the concept and then have them lay out the logical procedure for me(with a little guidance from me of course).
If I had to choose I think a student needs the concept more than the skill. Mainly because when a student understands a concept it has more staying power than when he has memorized a skill/procedure.
Professor Homunculus sez:
Right on, Trent! That is basically how I feel, as well. By the way, readers, Trent has his own very good websites, one about Teaching and Learning Math and another about How to Teach Math, which covers algebra to pre-calculus. They look very good. Check ‘em out.