Teach How To Learn
I was troubled by what I taught in elementary courses. It seemed like I was teaching stuff and I didn’t know why I was teaching it.
A hundred years ago students could put the mathematics they learned to use right away. Now that is not so.
It used to be that the student was looking for a job for life. It took generations to build a cathedral. Now technology moves so quickly that a student can look forward to many different jobs and each one has to be learned.
Consider a radio repairman. I doubt if there are any now, they had to learn new skills. From tube radios to transistor radios and TVs to VCRs to DVDs to Blueray. Every few years there is a new technology to learn.
The appropriate skill to teach is not how to, say, differentiate but how to learn how to differentiate. This should be started in pre-school.
And it doesn’t make any difference what the skill is. They should start learning something that appeals to them. The goal is for the student to have confidence in their ability to learn, in their ability to face an unknown skill without fear.
Memorizing is not learning. The confusion between the two is probably the biggest obstacle to learning. It turns out that it is easier to test memorization than learning and since the student has been memorizing from the beginning it is a preferred way to take classes. In an unholy alliance the students and the teachers take the easy way out.
I would have my students take an oral in my office where they would have to prove a significant theorem, say, that the sum of the angles in a triangle is less than or equal to 180 degrees in hyperbolic geometry. It was too long to memorize, they had to see the flow of the theorem and see how the parts fit together naturally. And I gave them multiple tries.
I found that students didn’t know if they knew something or not. A student came in for help on implicit differentiation. It turned out that he didn’t understand what the derivative of a function was. And then it turned out that he didn’t understand what a function was. He wasn’t trying to con me; he thought he knew these things and was surprised when he didn’t.
I have many stories like this. A student sat through an entire semester of business calculus and through the semester he thought it was beginning statistics. And he thought he was doing ok in the calculus although his highest grade had been a 25. A student thought he understood max-min problems and didn’t.
Why is it that students believe they know things when they don’t?
For one thing, when you learn something, barring serious head injury, you don’t forget it. You forget things you’ve memorized. You don’t memorize how to ride a bike, you learn how to ride a bike”¦and you never forget.
Some people learn things easier than others. I was with a guy as he drove right to the Harley shop in Kansas City, Kansas and he had been there once three years previous. Arlo Guthrie never learned how to read music. When he started piano lessons, his teacher always played the piece he was supposed to learn from the sheet music. Once he heard the tune, he didn’t need the sheet music and never looked at it. Some people don’t forget tunes they’ve learned.
I was playing the guitar chords to “Don’t Think Twice, It’s Alright” while a friend played lead. At the point where it goes, “When the rooster crows at the break of dawn” he stopped us.
I had played an A instead of an A minor or something like that. He said, as if he was pointing out that the sky was blue, “If you play the A there it ruins the tune.”
I realized that he was hearing something that I wasn’t. I had memorized the chords, he understood the sequence of chords. He would always play the right chord.
I have heard that the piano prodigy and the piano non-prodigy have about the same proficiency when they are in their twenties so if a person isn’t a prodigy they can still learn to play the piano.
So how do you teach a student to know what learning is?
First they have to understand what “knowing” is. They have to have some idea of when they have learned something.
I think you have to start when they are young. Really young kids seem to be into acquiring skills and knowledge.