Introduction
Introduction
Some Philosophical Notes
In a conventional math course, you might be confronted with a question like the following:
Find the roots of [latex]x[/latex] in the expression:
[latex]4x^2 - 13x + 6=0.[/latex]
You may be instructed or implicitly expected to apply an algorithm to this problem and provide the two possible roots. Most likely this would be an opportunity to use the trusty quadratic formula:
[latex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/latex]
If you weren’t already turned off, you might plug the values 4, -13, and 6 in for [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] in the quadratic formula, perform some arithmetic and find the roots to be 0.56 and 2.69.
Alternatively, if you’re like me, you’d let a computer program like Geogebra apply this algorithm, since that is what computers are for.In any case, with these two roots in hand, you have an answer, isn’t that satisfying!? OK, maybe it is for some of you, but this has never been satisfying for me. No meaning was ever assigned to the variables or constants, nor was it claimed that the result had any context or significance. I don’t really know what to do with the answer, now that I have it. I’ve learned very little, except perhaps that I can enter the proper numbers into an algorithm. That is not to say that learning the algorithm is without value – indeed it is very valuable. But for most of us, the algorithm itself is not an end in itself, it is a means to an end. It is a useful tool that allows us a shortcut to a result when an equation presents itself in a quadratic form.
Though the problems in a conventional math class may look arbitrary, they are often designed to be “well-behaved”. You wouldn’t often see equations exactly like the example above, because the roots turn out to be icky decimal numbers rather than nice, clean integers. Furthermore, things get complex (literally!) if the numerical coefficients on the left-hand side of the equation are such that the term under the square-root in the quadratic formula turn out to be negative. Such a poorly-behaved case belongs to a completely different subject in the mathematics curriculum (complex analysis), and so cannot be imposed upon an unsuspecting algebra student. However, in the “real world”, there is no more reason to suspect a real result than a complex one, in those rare practical instances when one needs to find the roots of a second-order polynomial. Thus, in this approach we learn a very strict set of rules applicable only to an idealized subset of problems that may or may not have any significance outside of abstract trivia.
The approach we use in this course is to encounter math and statistics in the process of finding solutions to real—or at least plausible—problems in the natural sciences. Sometimes these real problems are messier than those out of a textbook. Often they will be open-ended and will require multiple steps and a variety of techniques. We’ll need to decide for ourselves