Sunday, April 15, 2007
Mathematics education and students -- what math is good for
On a substitute teaching assignment in math recently, there was a test and there was going to be a question on the geometric mean. I went over a very simple example: what is the geometric mean of 2 and 8? You set up the equation 2 / x = x / 8, or x**2 = 16, and get 4. (The generalization is discussed in Wikipedia here. So on the test, there is a question something like, “9 is the geometric mean of 3 and what other number”. It’s easy to set up as 3/9 = 9/x and get x=27. But I noticed when the papers were handed in that many students did something like 3 / x = x / 9. In other words, they parroted what I did on the board rather than reading the question carefully and thinking for themselves. Standardized tests (SOL’s in Virginia) do a lot of this, posing simple problems that have to be read properly exactly as asked.
Another day, a team of teachers come in to help kids with conic sections. These are parabolas, circles, ellipses (a circle is a special case of the ellipse) and hyperbolas. These entities from analytic geometry have various attributes like vertices, diameters, foci, directrixes, etc. The formulas for these in terms of the coefficients of the various functions are well known and available on wikipedia (just search) and in all standard algebra textbooks. In practically any textbook, they are clearly stated, with sample problems. The student needs to learn how to match the components of the function to the elements of the formula. It’s a very straightforward process. Maybe a bit tricky (as whether a parabola opens up, down, or “sideways”, yes, as in the movie by the name.; the same for a hyperbola). And yes, it may be necessary to memorize a few formulas for tests. In trigonometry, you have to know the definitions and it helps to memorize a few identities. In calculus, it helps to memorize a few basic formulas for derivatives and integrals.
But skill in math is mostly practice and mental agility. Compared to other academic subjects (especially in social studies) there are relatively few “facts” to learn, so it cannot be crammed at the last minute the night before an exam. It is learning to think a certain way, according to the rules of logic. That’s especially true in plane geometry, when the student is introduced to the idea of a mathematical proof in statements and reasons. Built up properly, the proof of something like the Pythagorean Theorem becomes very simple.
In one work session, a kid asked, “what do we need this for? The cash register at MacDonalds does the math for you.” Yes, “out of the mouths of babes.” He literally said that. I wondered if he expects to spend a lifetime in the fast food business. But the manager of a Burger King has to plan a budget, determine how many employees to hire, meet a payroll, deal with work schedules, a lot of mathematic concepts that are like the notorious word problems (“story problems”) in Algebra I. In fact, a job where you have to balance the register every day or get fired is no picnic. In the recent hot film “The Lookout” the character played by Joseph Gordon-Levitt says that a good bank teller balances day after day for years.
There seem to be two things that are “hard” about math. One is the tedium and attention to accuracy, and that has gotten easier in the era of computers and graphing calculators. I can recall the strain of getting through Algebra I tests without “careless errors.” I particularly remember one test of seven polynomial long division problems that took an entire 50 minute period. I recall the tedium of arithmetic with logarithms (in the days before calculators) and particularly (in trigonometry) solving triangles with logarithms (in grade school, arithmetic had sometimes been called "number work").
But what seems to befuddle many people is the abstraction. They don’t, as that one student said, see the use for it, since “it” is not physical. Rather, math is a paradigm, a map for how to think and reason, to solve any problem. It says that given a set of facts and postulates, you can draw certain conclusions about consequences, or (in a statistical setting) predict them. On an objective level, this is very important to understanding right and wrong, especially what libertarians call “enlightened self interest.”
Students will say that they can’t work a particular kind of problem because that problem hasn’t been “taught” yet. This excuse seems to come from deference to authoritarian thinking. One does what one is permitted to do, or been told to do. But in a free society, one has to go to information sources, and apply the information to solve new problems. So reading a textbook (or web) discussion of the attributes of conic sections and then parsing the formulas to solve problems is a step in self-sufficiency.
I certainly got a practical lesson in that during the last two years of my traditional information technology career. After the Y2K exercise, I switched from mainframe to “client server” at the beginning of 2000 and was in a position maintaining the midtier (java) and graphical user interface (Powerbuilder) in the “customer service workbench” for a major insurance company. There would be problems with no obvious method for attack. Object oriented code (more so with powerbuilder than java) would be hard to read and follow because it is not procedural (lending itself to “bird’s eye thinking”), but rather a descriptive model of a “real world” of which one cannot see much of (not beyond the “horizon” or over the next “hill”) from any one ground point. Doing this kind of work required an entirely new style of thinking, and accepting a loss of comfort and certainty in one’s work world. The high school math student may feel the same way, that she is expected to work blind, without the usual input from sensory feedback. She will want the security of specific directions and will feel “lazy” about shifting gears to look at the same simple but abstract problem from a different point of view. But to solve real world problems, that is what we must do, look at the same facts, from a different view. In politics, it’s from a different constituent’s point of view.
As for the practical use of these things, these give a pretty good start on the conic sections. In chemistry, you study the gas laws, and find out that they are based on a hyperbola rotated 45 degrees. The first website shows the derivation. The next is the wiki site.
As for the bigger significance of math, some of it can come out of playing with fractals. A favorite geometry test problem is the infinite coil of triangles, with successive applications of the Pythagorean Theorem. In nature, that structure is used by some animals, especially mollusks like the chambered nautilus. The Mandelbrot set (related to Julia and Mantou sets), which comes out of complex variables, generates “buds” infinitely into borders that remind one of biological (or, particularly, viral or retroviral reproduction (or zoological structures like the seahorse tail, or cusps on tentacles). Much of biology follows fractal mathematics, so consistently that it gives rise to the notion that life (or self-replication) is likely, maybe inevitable, on any world with the sufficient elements, perhaps even Titan, even given its cold.
The following link has an applet for displaying Mandelbrot set views. This.
If you go back to the 1950 World Book Encyclopedia (and maybe more recent editions) there is a long article "Mathematics for Fun" with games like how to prove 1 = 2 (by a hidden illegal division by zero, a S0CB on the IBM mainframe!) or how to make a Mobius strip.
As far learning this, it's always hard to be forced to perform at something one is not "good" at. Sometimes, whether in school or at work, this means tackling problems for which one has no reliable crutch, no reliable precedent for solving. If one is doing what one has chosen, that's great; if it's to satisfy someone else's prerequisite in order to earn adult freedoms, it's not good, because it puts one at risk. Being forced to learn unwanted skills and fit into other people's agendas is something I have taken up in other blogs and will take up again.
But, meanwhile, mathematics drives everything that we see in our universe.