> Are we so sure that filters and skills are all that bad? As a middle
> aged eighth grade algebra teacher recently certified, I learned the value
> both of the "filter" of abstract (modern) algebra and the algebra
> manipulation skills that I still lacked even though I could get through
> the calc series with "Bs" at U. Missouri St. Louis. I bumped up against
> the filter of abstract algebra a couple of times before oozing through,
> not so much because I can't handle the abstractions as because I lacked
> sufficient symbol manipulation skills and experience to be able to see
> the patterns that get you to proof/solutions.
Seems to me that there's a lot of difference between, on the one hand, a course that's designed to prepare students to use calculus, and, on the other hand, a course that's called a calculus course but is in fact designed to filter or teach algebra. By the same token, it is legitimate for us to say that we want high school teachers to have some knowledge of abstract algebra because, e.g., abstract algebra is one of the major subdivisions of mathematics (or any of several other very good reasons based on the nature of abstract algebra and its value in the natural history of a mathematics teacher's mind). It is not legitimate for us to place abstract algebra in our curricula if we can justify it only by saying "Students who can't pass this don't deserve licensure (or a degree, or a job, or...)."