I taught college freshman math at a big state university during a time period in between other jobs. My students were certainly bright -- they all had good high school grades and test scores -- but were the ones who didn't get placed into calculus.

I learned that my students came to me with a wide variety of interesting beliefs about math, such as:

1. Math is subjective! The teacher solves the problem by knowing the "trick" that gets them the answer they want to see. They can't explain where the trick came from, or why your answer is better than theirs.

2. They were explicitly taught "test taking skills," such as the "guess and try" method. (This method was described in a hand-out for parents, that one of my kids brought home from school).

3. They will never use their college math after college. Even students who are destined to become engineers believe this. They heard it from somewhere, not from their math teachers, meaning that culture plays a role in math education.

In addition, I perceived that the curriculum itself reinforced a problem solving method that is unrealistic: 1) Identify the "form" of the problem, which is one of the "forms" studied in the most recent chapter. 2) Extract the parameters from the problem statement. 3) Perform the algorithm associated with that form, to produce an answer.

For instance, they only learned one function with an extremum: The parabola. So in the chapter on "maxima and minima," every single problem boiled down to finding the parabolic equation representing the problem statement, and then finding the vertex of the parabola. This bears virtually no relationship to any useful or even theoretical math topic.

I decided that with as much ** as had been handed to them, the least I could do was help them get good grades that were needed to get into some of their intended majors such as Business. But not wanting to completely abandon my integrity, I adopted a compromise: Instead of test taking skills, I would teach math learning skills. Imagine you are working on a homework problem, and have not been paying attention in class at all, but you have the textbook. How do you approach the problem? It taught them how to learn from the textbook, which is at least kind of analogous to how most of us do math today -- by looking things up. And I promised them that if they added just one more ingredient -- repetition -- they would get good grades on the exams.

I avoided pulling up facts from memory. Instead I would say "I know the answer, but let's find it in the textbook."

There were dozens of sections of the course, and one great big exam, that was graded by a team of graders. So I got to see how my kids scores stacked up against the "competition," and they actually did quite well. I also had kids from other sections dropping in to my lectures. So I can't claim to have some foolproof method, but it at least got me through my teaching stint with my conscience intact.

As a pet peeve, school math isn't even real. It doesn't represent how anybody does math: Mathematicians, STEMmies, or laypeople. Nobody does anything with math, without a computer in front of them. In my perfect world, I would place a much greater emphasis on computation when teaching math.