FIRST, MATH TEACHERS VS. TEACHERS OF MATH
Teachers in math departments generally are two types. One is what I would call a Math Teacher.
This Math Teacher is a brilliant mathematician. However, when explaining math to a class, and shows steps to solving a problem, usually skips many steps.... these skipped steps are so intuitive to the Math Teacher. When a student asks how he got from step two to step three, the math teacher can't explain it. Since the step to so intuitive, the math teacher can't explain it.... because he can't see it.
This is why so many students hate math.
I had a calculus teacher in college. First course in calculus and the students had some questions. Our teacher went into a rant, "I don't understand why you can't understand calculus; because calculus is so simple!"
Well, yea to someone who has done post-PhD work in math.
Another of my calculus's teacher's favorite statements was: "in order to solve this differential equation; you look at it, till the solution occurs to you." I used to add, "is that after one or two beers?"
Then there is that teacher of math... probably a little slower at math. Math was a struggle for him. When I student asks a question; he probably asked the same question when he took the class. Only, the teacher of math can answer the question. When he does steps to solve an equation; he does all the steps. Students can follow this.
And the teacher of math can answer the question.
Unfortunately, most people involved in math education are the first type - the brilliant mathematician. This is as true in high school as it is in college. To teach in high school you must past the Texas Math Tests, which almost eliminates the teachers of math.
This is why, so many students hate math and don't want to take any more courses than they have to. This is why our students do so poorly on the international math tests.
This is why only fifty present (others drop out, usually in 9th grade (after the first Algebra class)) of the high school students actually passed all the state tests and graduate. Of this 50% who graduate from high school; nearly 50% of these in college must take high school math classes again before they are allowed to take real college classes. Others are required to take high school English classes again.