As someone who has taught university mathematics and also taught mathematics education, I've noticed the huge disconnect between views of teaching held by research mathematicians and by K-12 teachers (and the mathematics education professors who teach the teachers.) Below I list some beliefs that I have found to be widespread in the university community (in italics) followed by contradictory beliefs that are widespread in the mathematics education world (in bold). I think the best thing that could happen to mathematics education in this country would be to open up a dialog between these two groups, since each has information and skills that are critical to improving mathematics education.
Any good research mathematician who is interested in teaching can do a better job teaching mathematics than most public school teachers, at least from grades 5 up. The research mathematician can present mathematics as an exciting intellectual endeavor, and the teacher cannot.
Virtually no one can successfully teach K-12 who does not understand the basic techniques of teaching, including classroom management and performance objectives. A K-12 teacher who understands teaching but whose knowledge of mathematics is limited to a basic understanding of the mathematics to be taught can be an excellent mathematics teacher.
Good teaching is a matter of laying out the material in a clear and elegant manner, and answering student questions when needed.
Good teaching requires the instructor to develop a list of performance objectives for every class, and let the students know what these objectives are. The objectives must be specific and testable: for example, "The student should be able to solve a quadratic equation in standard form with real roots, where the coefficients are integers of absolute value less than 20, within 30 seconds, using the quadratic formula. The lesson is not successfully completed until all students can meet the objectives.
"Teaching to the test" is bad. Every test should contain at least some problems that are non-routine and require the student to synthesize knowledge. These problems are the only way to ensure that students have truly learned the material. A student who cannot solve problems that are somewhat different from what they have seen before does not deserve an A.
Tests should determine whether the student has met the performance objectives. In other words, teaching to the test is the essence of good educational practice.
There are two acceptable ways of assigning grades in a course. The first is to "grade on the curve", that is to use grade cutoff points that make the distribution of letter grades follow a normal distribution, with approximately the same number of F grades as A grades, the same number of D grades as B grades, and C grades the most common. This is the safest way to grade. The other way is to build the grade cutoffs into the test based on the professor's subjective opinion. For example, one says that a student must score 90% on a given test to deserve an A. This method may result in unacceptably high levels of failure.
There is no scientific basis to support the idea that student grades ought to form a normal distribution. In fact, one popular theory states that any student who works at it ought to be able to meet the course objectives and get an A; the smarter students will simply reach that point sooner than the not-so-smart.
I could go on, but I hope the reader gets the point. I've tried to present actual ideas that I've heard expressed. To some extent these dichotomies represent a clash of values, and so may be impossible to resolve completely. But I still think they are ideas that we must talk about if mathematics education is to improve.