Technology in Mathematics Education - Panel

In a few hours I'll be in a panel discussion about technology in mathematics education. I might say something like the following.

Math and technology in general: To me mathematics is about solving problems by understanding them more deeply, finding ways to explore them, finding elegant proofs, looking for ways to visualize and compute with deep structure, etc. Math software is simply another tool or technique that one can use to _massively_ enhance ones ability to solve mathematical problems. Asking whether or not to use computer technology in mathematics is no different than asking whether or not to use "logical reasoning" or "linear algebra" or any other major tool in the mathematician's toolchest.

Math and technology in teaching:

A. Open Source -- I personally feel it is terrible to *train* students mainly to use closed source commercial mathematics software. This is analogous to teaching students some weird version of linear algebra or calculus where they have to pay a license fee each time they use the fundamental theorem of calculus or compute a determinant. Using closed software is also analogous to teaching those enthusiastic students who want to learn the proofs behind theorems that it is illegal to do so (just as it is literally illegal to learn *exactly* how Maple and Mathematica work!). From a purely practical perspective, getting access to commercial math software is very frustrating for many students. It should be clear that I am against teaching mathematics using closed source commercial software.

B. Problem solving -- Many students are interested in mathematical software mostly because it allows them to solve *a lot* of problems that are simply impossible to do by hand. For some (at least me) it transforms math from a tedious, error prone, and frustrating (but addictive) exercise into an exciting exploration of an amazing "virtual" world of interesting problems, ideas, and techniques. For most people, even before computers, anything but fairly trivial algebra or calculus is nearly impossible to do correctly by hand. Gauss (1777-1855) spent years counting the 216816 primes up to 3 million... and wrote in a letter in 1849 that there are 216745 such primes. Any undergrad can type prime_pi(3*10^6) into Sage and get the right answer instantly.

C. Regarding student skills being lost due to the existence of computers -- If you really want students to be able to do something, e.g., compute integrals without using a computer, give them in class exams on that material. By far the main complaint I have and hear from both students and professors is that students have too limited skills at using mathematics software. This situation is ridiculous. Would we give a degree in Software Engineering to a student who has never written a computer program? Why give a degree in techniques of "Mathematical Problem Solving" to a student who can't even use a computer to solve mathematical problems, given that computers are one of the most important basic tools in the mathematician's toolchest.