Sunday, July 29, 2012

Algebra and the Intellect

The NY Times[1] in its inimitable fashion has a cover article in its weekly section, the lead so to speak, bemoaning the necessity to learn Algebra, at all, and perforce I suspect Geometry, and subsequently any other form of math higher than perhaps pushing keys on an iPhone.

The article states:

A TYPICAL American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

Now, a word of history. When I first took algebra in the 9th grade, some 60 years ago or more, I frankly did not do that well. I had no idea what my tutor was saying and frankly I suspect neither did he. Geometry fared slightly better. But between reading Men of Mathematics at the end of my sophomore year and intermediate algebra I saw all the pieces come together, thus managing to complete that and trig in some six weeks, then onto advanced algebra, calculus, probability and even solid geometry. I may have actually been the last student to take the solid geometry Regents exam in New York State.

Now that some sixty plus years later my two grandsons take algebra in the eighth grade. Now that is a year earlier, and the one in New York already has his Regents prep book.

“Struggling”, well yes if your instructor does not have a clue and you are being taught by rote. But “subject” is a rather strong word. In fact it is reflective of the arrogance of the ignorant. If you cannot do it then it must be undoable. I cannot learn Czech, no problem with Russian, and I lived in Prague, it was just that no matter how I tried it never stuck. Same for Portuguese, but no problem with Spanish. That is my problem not a reason to doing away with two countries.

The author continues:

This debate matters. Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources.

To be educated means that one has a minimal set of skills. Making change, understanding data, taking measurements, predicting the future. So take global warming. This can be examined by a simple look at data. I have data for certain daylily species going back 30 years. The date of the first bloom is measured and then I plot that data on a graph. Now I seek to get a simple linear regression on the days from the first of the year that the first bloom occurs. I use a linear regression and plot the data. The result, warming! Voila. But that simple step assumes a de minimus level of understanding of algebra. Not much, really! But some, and if we want informed decisions rather than just religious followings then we need an educated electorate, at least some of them.

Or take ratios, for example, the cost per unit, the price per unit, etc. We then look at multiples of these ratios to get other ratios. People use these all the time. They rely on algebra.

But on the other hand many people are just dumb. Why waste any time and money at all. So we can avoid wasting resources by just not educating them at all. Then we can have a dual class society, the dumb ones without jobs, and the educated ones supporting everybody!

The write continues:

California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.

It is a University, you are hopefully expected to be educated, and that means mathematics. If an eighth grader can learn this in New York and West Virginia, what is wrong with students elsewhere. Consider the draftee in WW II. The Navy had a problem, it needed students who understood math quite well. Torpedoes, 6” guns, radar, radio, fire control systems, and the like demanded algebra and geometry and trigonometry. Many draftees had it and thus it opened their career path after the War. Those who did not were just seamen, doomed to stay below petty officer rated sailors. The officers, including those with history degrees had to have even more.

As for high Math SAT scores, there is the old MIT tale, somewhat true, when the Freshman Calculus instructor states:, “Well you all got 800 in your SATs, but you still have to earn you’re A here.”. The response to this was to look around at the 100 or so students and think that all 800s were here. Then the Instructor noted, “Oh, the other 100 are at Cal Tech.”

Even jobs such as an electrician or carpenter require a modicum of algebra and geometry. The thought process is essential. Trade schools teach algebra, geometry, otherwise getting licensed to practice one's trade would be impossible. Ohms law and Pythagoras' triangle would be a sine qua non. Understanding nature is predicated on measuring and predicting, in reality. Is algebra too hard, possibly with the wrong teacher, I had one. But with the right teachers it provides insight. Perhaps the problem is our educational system.

The again the world uses spreadsheets, all the time, from budgets to understanding any business structure. What is a spread sheet, a set of algebraic instructions, unless of course you use it just for entering numbers. But to learn one needs understanding of relationships. That is algebra. Word problems must become visceral, one must "feel" the system, understand the implications. If one thing gets larger does the other thing get larger or smaller, and how quickly. Judgement results.

In today’s world we would have to add all the IITs in India, and Tsing Hua in Beijing for starters. And by the way, how many historians do we really need?