Selling knowledge and selling sex are both forms of prostitution

Where do I start?
I talked to a HS teacher and he told me that he told his students that mathematics was just a bag of tricks. I would tell my students that there are no tricks in mathematics, just understanding.

Perhaps this is one of the great divides in mathematics education.

I suppose that algorithms are useful but as Stein says, "Algorithm stops thought."

Algorithms also make students think they know something that they don't.

I was teaching calculus II and a student said he didn't understand why he was failing. He told me that he did well in calculus I, in particular max min problems. I handed him a calculus book and told him to find a max min problem, his choice, and work it on the blackboard. This occurred during office hours and while I answered questions he worked at the board. After about an hour I had answered the last question of the last student and I looked at how the max min problem was going.

He had chosen to find the dimensions of the biggest open topbox with with a square base that could be made with a given amount of cardboard.

He had made no progress. He had made a slice through the box as if he had some kind of integral in mind. He had no idea how to work the problem.

The student wasn't trying to con me. He really thought he knew how how to work the problem and was dismayed that he didn't.

A High School student told me that he had done well in algebra. I asked him to tell me something he knew. He said he knew the quadratic formula. So I asked him what it was. He knew that there was a square root and that there was a denominator. He was surprised that he didn't know it.

Another High School student complained to me that he could pass the weekly tests with A's but couldn't pass the monthly tests. I told him he could remember a week's worth of algorithms but not a month's worth.

Students like the idea of having to memorize. It gives them something concrete to do. And algorithms are easier than understanding to teach.

I am going out on a limb but most High School, Middle School and Elementary School teachers have little if any understanding of mathematics. Many tell their students how hard mathematics is and how they didn't like it when they were in school, remarks that I have no doubt are true.

But these teachers were taught by people that didn't understand mathematics. I was told by a woman in the Dept. of Education that she didn't really know mathematics but felt qualified to teach students in mathematics education how to teach mathematics. Go figure.

I don't think that these teachers are bad people. They aren't giving their students a bad course out of spite of meanness. They think they are teaching the best course they can and in the absense of understanding mathematics themselves, they are. In general they like their students and wish they could find a way to help their students memorize better. (Although some teachers seem to feel that students fail just to spite them.)

People who can memorize algorithms well become mathematics teachers. They work out tricks to help them memorize and pass these tricks on to their students, telling them that mathematics is just a bag of tricks.

Mathematics as algorithm is passed along from teacher to student, and the student who excells at the algorithm game becomes a teacher and continues the tradition.

Students learn the algorithms and pass courses thinking they know some mathematics and then don't understand why they don't.

Passing a student who doesn't know the course well enough to work a basic problem at the end of it is being lied to. A teacher told me that she sometimes gave a student a break and give them a C when they really deserved an F. She wasn't giving them a break if they didn't know enough to pass the next course. She wasn't giving them a break, she was condemning them to the limbo between a course they didn't want to repeat because they passed it and a course they weren't prepared for. Some break.

If the student payed good money for the course, she is being cheated.

But there aren't good guys and bad guys, only people doing the best they can without understanding.