I have been following the debate over the importance of basic algebra vs calculators.
In the best of all possible worlds our students would be good at hard core algebra and be able to use machines when necessary.
But at CSUN we are far from the best of all possible worlds.
We have 400 fewer math majors than we did a decade ago and our average calculus student, regardless of major, knows next to no math, does not study but can deal with a calculator and a computer.
This person would stand no chance on a job which required some creativity and learning something new.
Last year we had a Mark Jankins - a senior software engineer from Disney Feature Animation - come give a lecture on how math is used to simulate camera motion in their animated features. It is all advanced calculus. Dr. Jankins is a person who spends his life working with computers and is completely unimpressed with this aspect of a student's college training. He told me that if a Disney hire wants to be more than a thirty grand a year hacker, they will need to be able to learn math and theoretical computer science from texts which have proofs. He saw no value in using calculators and/or computers in the teaching of mathematics.
The chairwoman of our CS dept. at CSUN told me that she was teaching a junior level CS programming class and her students were very good at high level programming but could not follow the most elementary logical math arguments. She expressed concern that this will ultimately harm them in their future CS careers.
I can go on and on with such examples. The argument concerning the importance of the rational root test and other algebraic ideas is the not the primary issue. While these things certainly have application, their main value is that students who have learned these things have developed their mathematical sophistication to a certain point where they are able to apply concepts in various settings. Furthermore, such students have developed the important ability to work independently and know how to devote time to learning.
It might be that Richard's student's are good at math and can excel in real college courses. If this is the case I say good job to Richard and am reasonably sure more is going on in his classes than just button punching. But at CSUN, where we admit only the top third of the graduating HS class, we are getting hundreds of very poorly trained students who can't do anything other than button punching. These students have received much of their training under the California Framework which advocates calculators be used in kindergarten. Hundreds of my students have attended HS's where calculators were prominent and these are almost always the weakest students.
Many of my friends who I went to public school with in NYC (in the '60's and early '70's) learned math with no calculators and computers. They had no trouble with math and science in college and those who needed to learn something about machines had no problem. These were slightly above average students who would be in the genius category compared to our current crop of HS graduates.
I agree with David that it is perfectly absurd to think that students can excel in college math without a solid background in the fundamentals. Of course, we can re-norm our courses, but this doesn't change the fact that such a student will have a gaping whole in their education and in my experience will have a very poor chance of doing well in a college math course which has genuine assessment and content.