Needing "school math" without using it

I was thinking about my previous posts about the UW College of Education's (CoE's) recent political polemic about so-called reform math. One of their major points is that engineers don't use what they call "school math": they just use computers. Please allow me to outline my own work, which is highly compute intensive and rarely involves what they would call school math, but which nevertheless I could not do without a healthy dose of school math -- not just in my education, but also in my work.

My research is in the area of computational neuroscience, in which I build mathematical models of individual nerve cells (neuron)s and groups of neurons, develop simulation software for single computers and clusters of computers, and analyze data from simulations and also from experiments on actual living tissue. This sort of work is very much like that done by anyone simulating physical systems, be they biological, chemical, mechanical, electrical, etc.

Like the engineers described in the CoE's publication, my work is heavily computational, as it isn't feasible to do this work with pencil and paper, as in "school math". The basics of the mathematical models involve a number of differential equations: equations that describe how some part of the system changes in response to other parts of the system. Now, it turns out that differential equations is covered by a pretty much standard college sophomore mathematics course. So, why isn't the stuff I do "school math"? It's simple:

1. We only cover the mast basic type of differential equations in that class, linear equations. These are actually quite good for describing simple systems: electrical circuits made up of resistors and capacitors, mechanical systems with springs, etc. The advantage of these equations is that we can solve them on paper and they're easy to learn. The disadvantage is that they aren't very good descriptions of complicated systems like neurons (and many other, nonlinear systems). Once we move to nonlinear systems, we almost certainly need to use computers to do numerical simulations.
2. We mostly just solve single equations in that class (there are other classes where we learn to solve groups of differential equations, later on in the curriculum). The systems I'm interested in can have hundreds or thousands of differential equations, and so I have no choice but use computer simulation.

If you were to watch me work, you would see the following (between the long periods of time in meetings, in class, preparing for said things): I decide on a question I'd like to answer, such as how the behavior of a network of neurons changes as some parameter (think: "tuning knob") is changed. I set up the parameters for a simulation or maybe bunch of simulations and, anytime from a few minutes to a few days later, I have some results. I load those results into MATLAB (numerical mathematics software) and plot the results. I then either exclaim, "Wow!", and hurriedly start writing a summary and thinking of what else I need to do to finish telling the story for a publication (rare), or I say, "Nuts!" and think again why the system either displays uninteresting behavior (Who knows; maybe its lack of interest is in itself something noteworthy? Or is that just wishful thinking?) or doesn't behave like the living nervous system. So, the observer sees that I don't "do" "school math". End of story?

Well, not quite. Because the observer doesn't see what's going on "behind the scenes" (i.e., in my mind). First of all, I would have no hope of even being able to start understanding what simulations I need to run without a very firm and extensive "school math" background. For instance, I work with a number of bright undergraduate students in my research. Some of them have math backgrounds that include differential equations and beyond and some don't. This has nothing to do with how smart they are; math beyond calculus isn't required for computer science and so only those students who come to us via a "nonstandard" pathway (e.g., changed major, previous degree/career) will have the more advanced math. Though all of these students can help out in my research, only those with more advanced "school math" are able to understand the underlying mathematical model well enough to mess with that aspect of the project (unless I teach a student some of the required "school math"). After all, unless you want to resort to randomly poking something just to see what it does, you really have to understand what's going on inside it; that's the only way you can intelligently select what kind of "poking" is likely to tell you something interesting. In fact, it's the only way you can begin to ask questions about the system, let alone start formulating experiments to answer those questions.

Even after the simulations are over, I still need to interpret the results, and this requires yet more "school math" running around in my head. What kind of result did I get? What relationship does it have to previous results I've gotten, or for that matter, results others may have gotten? What does this result mean in the overall context of the system in question and the thinks I'd like to know about it/do with it? And so on.

In other words, it is emphatically not the case that the computer has relieved me of the need to know math. All the computer has done is take over the grunt work: it has become an additional tool in my mathematical arsenal. But the computer can't think, and that thinking is where all the "school math" is. It just isn't apparent to the observer because I know it well enough that it happens in my brain automatically. This is no different than the automaticity with grammar that we use in everyday life. Just because we don't carefully label each of our utterances with "subject", "verb", "object" doesn't mean that grammar isn't necessary.

Finally, does this apply to "everyday" engineers, or just people doing research? Of course it applies to engineers (at least those who haven't "moved up" to management)! That's why businesses hire engineers: they need people who can think about solutions to problems and have the depth of background to understand the interrelationships among parts of solutions from "first principles" on up to final product. Some tasks may become routine and thus almost automatic or thoughtless, but its important to have someone who can look at a problem (or a solution proposed by some software) and say, "Wait a minute; something's fishy here." And, in the final analysis, that's the most important contribution of "school math": it is the language of creativity.