In my last post, commenter AllaninPortland said, of my Math Support students, “Their brains are wired a little too literally for modern life.”
James Flynn, of the Flynn Effect:
A century ago, people mostly used their minds to manipulate the concrete world for advantage. They wore what I call “utilitarian spectacles.” Our minds now tend toward logical analysis of abstract symbols—what I call “scientific spectacles.” Today we tend to classify things rather than to be obsessed with their differences. We take the hypothetical seriously and easily discern symbolic relationships.
Yesterday I gave my math support kids a handout on single step equations similar to the one in the link.
“Oh, I know how to do this,” said Dewayne. “Just subtract six from both sides.”
“You could do that,” I said. “But here’s what I want people to try. I want everyone to read the first equation as a sentence. What is it saying?”
“Some number added to six gets fourteen,” came from Andy.
“You mean, you don’t want us to subtract, add, do things to get x by itself?” asked Jose.
“That’s called ‘isolation’. You are ‘isolating’ x, getting it all by itself as you put it. Who knows how to do that?” Over half the class raised their hands. “Great. You can do that if you want to, but I’d like you to try seeing each equation just as Andy described it. Put the equation you see into words. This will help make it real, and will often give you the answer right away. For example, what number do I add to six to get 14?”
“Eight.” chorused most of the room.
“There you go. Now, remember, what did I say a fraction was?”
“So instead of saying ‘x over 5’, you’re going to say….”
“X divided by 5″ came back a number of students.
“Off you go.”
This worked for most of the students, but one student, Gerry, sat at the back of the room drawing, as he often does. After watching him do no work for 10 minutes, I called him up front. (Normally, I am wandering the room, but every so often I call them up for conversations instead.)
“So you aren’t working.”
“Yeah. I can’t do this.”
“Remember yesterday, when we were doing those PEMDAS problems? You were on fire!”
“Yeah, but it didn’t have the letters in it. I can do math when it doesn’t have letters. And yesterday, when you showed us how to just draw pictures for the word problems? That was cool. I think I can do those now.”
“You need to look at these problems from a different part of your brain.”
“A different what?”
“This is a really, really easy problem. Way easier than the math problems you solved in your head yesterday. But you don’t see this as the same kind of problem, so we have to fool your brain.”
“How do we do that?”
“Read the first problem aloud.”
“X + 6 = 14. This is when you have to do stuff to both sides, right? I can’t do that.”
“Read it again. But instead of saying x, say ‘what’.”
“Definitely. Try it.”
“What plus 6 = 14? 8.”
“There you go.”
He was sitting in one of my wheeled chairs, pushing it back and forth with his feet. This stopped him cold.
“Eight’s the answer? Holy sh**.”
“Try another. Without the language.”
“What minus 3 = 7. That’s nine…no, 10. Ten? Really? No f**k….no way.”
“And this one?”
“Oh, that’s a fraction. I can’t do those.”
“What did I tell you fractions were?”
“Division. Oh. What divided by 5 is 9? Forty five? No way?”
“So. I want to see you do this whole handout, 1-26, and every time you see an x, call it ‘what’. Remember to sketch out subtraction questions on a numberline and think about direction.”
“Okay. Man, I can’t believe this.”
Fifteen minutes later, Gerry was done with the entire set. Only three minor errors, all involving negative numbers.
“I feel like a math genius,” he said with a wry grin.
I sat down next to him. “It’s like I said. We have to ask your brain a different question. So instead of tuning me out, next time I come up with some goofy idea using pictures or tiles or different words, give it a shot. And tell me if it works to give your brain the right question. Some of my ideas will work, some won’t. And some things, we won’t be able to fool your brain to answer a different way. But you know a lot more math than you think you do. You just have to figure out how to ask the question in a way your brain understands.”
Back to Flynn:
A greater pool of those capable of understanding abstractions, more contact with people who enjoy playing with ideas, the enhancement of leisure—all of these developments have benefited society. And they have come about without upgrading the human brain genetically or physiologically. Our mental abilities have grown, simply enough, through a wider acquaintance with the world’s possibilities.
But not everyone is capable of understanding abstractions to the same degree. Some people do better learning the names of capitals and Presidents and the planets in the solar system. They’d learn confidence and competence through interesting, concrete math word problems and situations, and enjoy reading and writing about specific historic events, news, or scientific inventions that helped society. Instead, we shovel them into algebra, chemistry, and literature analysis and make them feel stupid.
Students’ names have been changed. They are all awesome kids. Do not say mean things about them in the comments, which I can control, or other blogs, which I cannot.
October 8th, 2012 at 4:48 pm
Nice story. Math teacher gets his fix, shows up for work next day. Says next time he says he’s going to quit he’ll really mean it. 😉
Seriously though, this reminds me of one of the parent seminars they have at the snowflake school my son goes to. They had one on teaching math in 1-4 grades. The method is heavy on narrative and introducing topics from multiple angles hoping each kid will have one way “stick” for them. I’d known it for a while, but seeing so many smiling parents was one of those instances where you’re experiencing a cliche first-person it stands out, because it reminded me how high achievers automatically make so many of these connections on their own, they don’t even think about it. That’s why they’re high-achievers.
In fact, they get bored going through every retrograde permutation and being forced to jump through hoops like a trained seal. AKA show your work, as you mentioned in the next post, which becomes offensive when it’s additionally “in this particular arbitrary manner we’re learning this week.” But yeah, that is a different topic.
October 10th, 2012 at 11:58 am
I had a very similar experience in a single-semester Algebra II course, only without the help of a teacher. Algrebra I was easy, but in Algebra II I started getting lost. I was solving problems by rote and superficial pattern recognition. Then one day it hit me; I started seeing seeing the x-y-and-z’s as sort of mystery quantity containers and the logic fell in place. Too bad the insight came too late for me to make an A. I just edged in a B- in fact. I’m so ashamed. :,(
But what I want to know is, what does everybody else do when they “understand abstractions” if not make connections between the symbols and something more concrete, something you can see (baskets of apples, graphs, geometry, anything)? At least by my own assessment I have a very analytic, abstraction-oriented mind and that’s what I do, and visualization isn’t my strong suit. You could mean having a handle on formal “typographical” systems a la Godel, Escher, Bach but grade-school mathematics doesn’t give you a taste of those, and that sort of reasoning doesn’t show up much outside of pure math, computer science, some physics, some chemistry because of sheer incomprehensibility, and economics when it’s utterly detached from reality.
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December 7th, 2012 at 10:17 pm
I stumbled across this because of the Jom Flynn quote (one of my heroes), but I just wanted to thank you for the rest of the post. I teach psychology, and am always on the lookout for clearer ways to teach them maths without them noticing 😉
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