Reading science, math, and philosophy one hour per day will likely put you at the upper echelon of human success within seven years.
Of course, it’s no use reading these things if you don’t do anything with the information. This is why I write here: to integrate what I learn.
The case for using computer algebra
Many academics and researchers get annoyed when students use computer algebra programs such as Mathematica to evaluate simple integrals that they maintain should be done by hand. The question I ask is “At what point do you expect your students to switch over to using a computer?”. Most mathematical examples are artificial in that closed form expressions exist. However, in nearly any real problem, this is not the case.
I learnt mathematics using slide rules and tables, then calculators, then computers, then symbolic algebra. To me, this is a valid progression — but not one that everyone should have to go through. I feel that the only way true progress can be made is if we don’t have to learn a whole set of rules. If we had to do calculus using Newton’s geometrical constructs then progress would be very slow. The real question is what are the essential tools and lessons. To me, knowing what a derivative and integral mean “physically” is far more important than knowing how to compute a specific integral.
Many people feel that reliance on computer algebra means that students can’t do calculus by hand and hence they really don’t understand what’s going on, just how to get the answer by computer. Calculus concepts are subtle. However, just knowing the mechanics of computing an integral or derivative does not imply understanding. I believe that it is possible to have true understanding without computation.
From a list of pet peeves of Paul Abbot, physics professor at University of West Alabama.
Comparing apples and oranges
In 1995, Scott A. Sandford, a researcher at the NASA Ames Center in Mountain View, California, took up a landmark study that proves, once and for all, that one can compare apples and oranges. In fact, they’re quite similar.