For a while I tutored middle school students in algebra. Very frequently, I heard things like this from my students:
“I’m terrible at math.”
“I hate math class.”
“I’m just dumb.”
That attitude had to go. All of my students successfully learned algebra; not one of them learned algebra before she came to believe herself good at math. One strategy I used to convince them otherwise was giving out easy homework assignments—very small inferential gaps, no “trick questions”.
Now, the “I’m terrible at math” attitude was, in some sense, correct. You could look at their grades and their standardized test scores and see that they were in the lowest quartile of their class. But when my students started seeing A’s on their homework papers—when they started to believe that maybe they were good at math, after all—the difference in their confidence and effort was night and day. It was the false belief that enabled them to “take the first steps.”
I think this phenomenon illustrates a very widespread misunderstanding of what math is and how ones becomes good at it. Consider the following two anecdotes:
1) Sammy walks into advanced Greek class on the first day of school, eager and ready to learn. He is crushed when, about 15 minutes after the class begins, he realizes he has no idea what the teacher is talking about. Despairing, he concludes that he is “terrible at Greek” and “just dumb”.
2) Sammy walks into advanced algebra on the first day of school, eager and ready to learn. He is crushed when, about 15 minutes after the class begins, he realizes that he has no idea what the teacher is talking about. Despairing, he concludes that he is “terrible at math” and “just dumb”.
Anecdote 1) just seems ridiculous. Of course if you walk into a language class that’s out of your depth, you’re going to be lost, everyone knows that. Every normal person can learn every natural language; there’s no such thing as someone who’s intrinsically “terrible at Greek”. The solution is just to swallow your pride and go back to an earlier class. But it seems like anecdote 2) is not only plausible but probably happens rather often. There is some irrational belief that skill at mathematics is some kind of unrefinable Gift: some people can do it and others just can’t. This idea seems absurd to me: there is no “math gene”; there are no other examples of skills that some people can get and others not.
It’s actually anecdote 1 that seems plausible to me and anecdote 2 that does not.
I happen to have been a language teacher, teaching adult hobbyists. It seemed to me that lots of my students had very unrealistic ideas about how easy it would be for them to learn a foreign language. They really did come to class expecting one thing and 15 minutes later finding out quite another thing. They typically brushed off their past bad experience with learning, say, Spanish back in school, on the theory that they’d never really been motivated to learn Spanish but they were really truly motivated to learn language X which I was teaching. Then they realized that learning language X involved a lot of the same boring grammar talk and memorization which they’d found so hard/boring when learning Spanish. (Of course it’s also possible that my classes just sucked.)
By contrast, no-one walks into advanced algebra classes having no idea what math is about. People who think they’re terrible at math usually infer this from having spent 10 years in a school system where they consistently had trouble with math assignments, performed poorly on math tests and had trouble understanding what math teachers were talking about. Most people who think they’re bad at math probably are actually bad at math. Sure, in some other universe they might have become good at math if some early stimulus had swung another way—maybe another teaching style would have helped, or a role model or whatever. But it also seems very reasonable to think that some people are born with more aptitude for math than others. General intelligence certainly has a large heritable component and I’m sure that holds for the special case of mathematical aptitude.
I spent many years operating under the assumption that everyone was about as smart and constructing elaborate explanations for why the reality I was confronted with seemed to be in so much conflict with that theory. Sure, if you add enough epicycles you can do it—but there’s a far simpler theory which has more explanatory power: Some people are “just dumb”. I personally find that a liberating theory to operate under. A lot of my “aha moments” seem to involve either the realization that “yes, people really are that stupid” or the realization that “yes, I really am that stupid”.
Unlike most other subjects, math is cumulative: students are taught one technique, they practice it for a while, and then they’re taught a second technique that builds on the previous. So there are two skills required:
The discipline to study and practice a technique until you understand it and can apply it easily.
The ability to close the inferential gap between one technique and the next.
The second is the source of trouble. I can (and have) sat in on a single day’s instruction of a language class and learned something about that language. But if a student misses just one jump in math class, the rest of the year will be incomprehensible. No wonder people become convinced they’re “terrible at math” after an experience like that!
Unlike most other subjects, math is cumulative: students are taught one technique, they practice it for a while, and then they’re taught a second technique that builds on the previous.
How is that unlike other subjects? Seems pretty universal.
An example of dark arts used for a good cause. The problem is that the children weren’t strong enough to understand the concept of being potentially better at math, of it being true that enthusiasm will improve their results.
They can’t feel the truth of the complicated fact of [improving in the future if they work towards it], and so you deceive them into thinking that they are [good already], a simpler alternative.
Vladimir, the problem has nothing to do with strength—some of these students did very well in other classes. Nor is it about effort—some students had already given up and weren’t bothering, others were trying futilely for hours a night. Even closing the initial inferential gap that caused them to fall behind (see my reply to Daniel_Burfoot above) didn’t solve the problem.
The problem was simply that they believed “math” was impossible for them. The best way to get rid of that belief—maybe the only effective way—was to give them the experience of succeeding at math. A pep talk or verbal explanation of their problems wouldn’t suffice.
If your definition of “the dark arts” is so general that it includes giving an easy homework assignment, especially when it’s the best solution to a problem, I think you’ve diluted the term beyond usefulness.
For a while I tutored middle school students in algebra. Very frequently, I heard things like this from my students:
“I’m terrible at math.”
“I hate math class.”
“I’m just dumb.”
That attitude had to go. All of my students successfully learned algebra; not one of them learned algebra before she came to believe herself good at math. One strategy I used to convince them otherwise was giving out easy homework assignments—very small inferential gaps, no “trick questions”.
Now, the “I’m terrible at math” attitude was, in some sense, correct. You could look at their grades and their standardized test scores and see that they were in the lowest quartile of their class. But when my students started seeing A’s on their homework papers—when they started to believe that maybe they were good at math, after all—the difference in their confidence and effort was night and day. It was the false belief that enabled them to “take the first steps.”
I think this phenomenon illustrates a very widespread misunderstanding of what math is and how ones becomes good at it. Consider the following two anecdotes:
1) Sammy walks into advanced Greek class on the first day of school, eager and ready to learn. He is crushed when, about 15 minutes after the class begins, he realizes he has no idea what the teacher is talking about. Despairing, he concludes that he is “terrible at Greek” and “just dumb”.
2) Sammy walks into advanced algebra on the first day of school, eager and ready to learn. He is crushed when, about 15 minutes after the class begins, he realizes that he has no idea what the teacher is talking about. Despairing, he concludes that he is “terrible at math” and “just dumb”.
Anecdote 1) just seems ridiculous. Of course if you walk into a language class that’s out of your depth, you’re going to be lost, everyone knows that. Every normal person can learn every natural language; there’s no such thing as someone who’s intrinsically “terrible at Greek”. The solution is just to swallow your pride and go back to an earlier class. But it seems like anecdote 2) is not only plausible but probably happens rather often. There is some irrational belief that skill at mathematics is some kind of unrefinable Gift: some people can do it and others just can’t. This idea seems absurd to me: there is no “math gene”; there are no other examples of skills that some people can get and others not.
It’s actually anecdote 1 that seems plausible to me and anecdote 2 that does not.
I happen to have been a language teacher, teaching adult hobbyists. It seemed to me that lots of my students had very unrealistic ideas about how easy it would be for them to learn a foreign language. They really did come to class expecting one thing and 15 minutes later finding out quite another thing. They typically brushed off their past bad experience with learning, say, Spanish back in school, on the theory that they’d never really been motivated to learn Spanish but they were really truly motivated to learn language X which I was teaching. Then they realized that learning language X involved a lot of the same boring grammar talk and memorization which they’d found so hard/boring when learning Spanish. (Of course it’s also possible that my classes just sucked.)
By contrast, no-one walks into advanced algebra classes having no idea what math is about. People who think they’re terrible at math usually infer this from having spent 10 years in a school system where they consistently had trouble with math assignments, performed poorly on math tests and had trouble understanding what math teachers were talking about. Most people who think they’re bad at math probably are actually bad at math. Sure, in some other universe they might have become good at math if some early stimulus had swung another way—maybe another teaching style would have helped, or a role model or whatever. But it also seems very reasonable to think that some people are born with more aptitude for math than others. General intelligence certainly has a large heritable component and I’m sure that holds for the special case of mathematical aptitude.
I spent many years operating under the assumption that everyone was about as smart and constructing elaborate explanations for why the reality I was confronted with seemed to be in so much conflict with that theory. Sure, if you add enough epicycles you can do it—but there’s a far simpler theory which has more explanatory power: Some people are “just dumb”. I personally find that a liberating theory to operate under. A lot of my “aha moments” seem to involve either the realization that “yes, people really are that stupid” or the realization that “yes, I really am that stupid”.
Unlike most other subjects, math is cumulative: students are taught one technique, they practice it for a while, and then they’re taught a second technique that builds on the previous. So there are two skills required:
The discipline to study and practice a technique until you understand it and can apply it easily. The ability to close the inferential gap between one technique and the next.
The second is the source of trouble. I can (and have) sat in on a single day’s instruction of a language class and learned something about that language. But if a student misses just one jump in math class, the rest of the year will be incomprehensible. No wonder people become convinced they’re “terrible at math” after an experience like that!
How is that unlike other subjects? Seems pretty universal.
An example of dark arts used for a good cause. The problem is that the children weren’t strong enough to understand the concept of being potentially better at math, of it being true that enthusiasm will improve their results.
They can’t feel the truth of the complicated fact of [improving in the future if they work towards it], and so you deceive them into thinking that they are [good already], a simpler alternative.
Vladimir, the problem has nothing to do with strength—some of these students did very well in other classes. Nor is it about effort—some students had already given up and weren’t bothering, others were trying futilely for hours a night. Even closing the initial inferential gap that caused them to fall behind (see my reply to Daniel_Burfoot above) didn’t solve the problem.
The problem was simply that they believed “math” was impossible for them. The best way to get rid of that belief—maybe the only effective way—was to give them the experience of succeeding at math. A pep talk or verbal explanation of their problems wouldn’t suffice.
If your definition of “the dark arts” is so general that it includes giving an easy homework assignment, especially when it’s the best solution to a problem, I think you’ve diluted the term beyond usefulness.