I guess I have to agree with Dan that there are some basic Ideas that have to be developed prior to any real practical applications of mathematics. I think I have an example close to home for most of us...this weeks POW on the Geometry forum. The problem is a simple triangle problem given one side and two perpendicular segments to the other two sides. The reason it relates, IMO, to this discussion, is that to solve it most of my students had to recognize opportunities to use the Pythagorean Thm, the law of Sines, and if they wanted an exact answer (and they really do see that as more pleasing now) they also had to recognize an opportunity to use the expansion of Sin(A+B). None of this is directed in the problem, and without the depth of experience that comes from the "drudgery" of practicing these skills I don't think they would have recognized the opportunity to use them.. One of the things I think makes these skills "available" on call is regular practice with a wide range of "short, quick learning experiences focusing on each of these as a particular goal". I often use the term "homework problems" as a general description of these practice sessions. I don't feel any software I can imagine would replace all the analytic thinking that had to go into this particular problem, and I don't yet know of a better way of developing the capacity to make those decisions than the repetitive practice mentioned above. I am thrilled when my students can show me a new way to solve a question on the sketchpad, or ti-85 or derive or .... but what thrills me more is that the software and technology serves both to reinforce the ideas learned by "drudgery", because they are the tools to open up additional ideas which the students discover for themselves. They serve both to validate the hard work that the students have done before, but reward them with the excitement of new discoveries.
I'm off to the states in three days so if I do not respond.....QED
Pat Ballew "lovin' what I do" Edgren HS, Misawa, Japan firstname.lastname@example.org