Concluding Remarks

The model of "applied mathematics" teaching that says "learn the mathematics, then apply it" is a powerful one because it enables mathematical ideas to be taught independently of a particular application and therefore to feed into a number of such applications. However, it runs foul of the well-known difficulty that students have in linking different parts of their knowledge.

Powerful computer software is helpful if you want to link mathematics to a scientific application without having to trivialize the science. If you have access to such software, you're a good deal less constrained by the narrow range of pen-and-paper techniques in which students may be confident and proficient at any particular time. For instance, there are some chemically interesting systems that can be modeled using the types of differential equations that students can solve exactly, but there aren't many, and the one in Lab 6, in particular, would be off limits without computer power.

If we deploy that computer power in an open way (which Mathematica lends itself to), students can come to wield it themselves, and even use it in creative, original, and unforeseen ways. It's natural to want to protect novice students from Mathematica's syntactical complexities, but there is such a thing as being overprotective. Where Mathematica, without modification, comes into its own is in those situations where we want our computational representations to reflect a) the key concepts of the problem, b) the underlying mathematics, and c) the connections between the two.