bv Yves Nievergelt, Eastern Washington University
The absence--worse than a lack--of articulation has led to a mathematics curriculum that bears no relationship with mathematical research and applications. One obstacle particularly hinders the articulation of educators, employers, researchers, and the public: the resistance of members of each constituency to take into account published and verifiable findings from the other sectors.
For example, researchers had already found in the 1950's, that to solve quadratic equations with digital computers, an algorithm published in 1750 produces provably accurate solutions, whereas the rounding inaccuracies inherent to computers may cause the 2000-year old quadratic formula to fail. Nevertheless, the mathematical literature directed at students and instructors in high schools and community colleges still demonstrates how to program the quadratic formula into calculators, without a trace of the provably more accurate algorithm.
Understandably, technical employers consider the computational use of the quadratic formula unprofessional. Unfortunately, the public remains unaware of the effects of such a lack of articulation. Such an example helps explain why educators and the public resent the abstraction imposed upon students in higher-level courses, while employers and researchers complain about the lack of preparation provided to students in lower-level courses.
The obstacles to articulation increases as the concepts under consideration become more abstract. For instance, the speed and accuracy of programs for the computer-aided geometric design of aircraft, automobiles, and animiation depend critically upon such mathematical tools as Stokes's Theorem and Betti's numbers in algebraic topology. Without such tools, the programs would run thousands of times slower and be more prone to errors. Moreover, proper use of such tools requires a familiarity at the level of proofs rather than mere statements and applications.
For abstract mathematical tools, articulation has been effective mainly between employers and researchers. This is due, in no small measure, to common background from graduate schools, professional conferences and symposia, and many informal relations that develop privately but function extremely effectively in communicating mathematical aspects of work, research, and higher education.
Thus information from employers passes easily and quickly to students in higher-level courses, but, because of its technical nature, not to students and instructors in lower-level courses. Unfortunately, this situation is invisible to the public, so very few people can readily assess how well (or poorly) the current educational system fits either the current state of science or industry.
I recommend, therefore, the following four actions to greatly improve the articulation among educators, employers, researchers, and the public: