Professor of Mathematics, Fordham University
Cargo cult education
During World War II in the Pacific Ocean American forces hopped from island to island relentlessly pushing westward toward Japan. Many of these islands in the south Pacific were inhabited by people who had never seen Westerners; maybe their ancestors years before had left legends of large wooden ships. We can only imagine their surprise and shock when large naval vessels arrived and troops set up communication bases and runways. Airplanes and those who flew them seemed like gods. It seemed to the natives that the men in the radio buildings, with their microphones, radios and large antennas, had the power to call in the gods. All of the things brought by the navy, radios, buildings, food, weapons, furniture, etc. were collectively referred to as "cargo".
Then suddenly the war ended and the Westerners left. No more ships. No more airplanes. All that was left were some abandoned buildings and rusting furniture. But a curious thing happened. The natives on some islands figured that they, too, could call in the gods. They would simply do what the Americans had done. They entered the abandoned buildings, erected a large bamboo pole to be the "antenna", found some old boxes to be the "radio", used a coconut shell to be the "microphone." They spoke into the "microphone" and implored the airplanes to land. But of course nothing came. (except, eventually, some anthropologists!) The practice came to be known as a "Cargo Cult."
The story may seem sad, amusing, or pathetic, but what does that have to do with mathematics education? Unfortunately a great deal. The south Pacific natives were unable to discern between the superficial outer appearence of what was happening and the deeper reality. They had no understanding that there even exists such a thing as electricity, much less radio waves or aerodynamic theory. They imitated what they saw, and they saw only the superficial.
Sadly, the same thing has happened in far too many high schools in the United States in the last twenty five years or so in mathematics education. Well meaning "educators" who have no conception of the true nature of mathematics see only its outer shell and imitate it. The result is cargo cult mathematics. They call for the gods, but nothing happens. The cure is not louder calling, it is not more bamboo antennas (i.e. glossy ten pound text books and fancy calculators). The only cure is genuine understanding of authenic mathematics.
What Math?
For more than two thousand years, mathematics has been a part of the human search for understanding. Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning. These remain fruitful and important motivations for mathematical thinking, but in the last century mathematics has been successfully applied to many other aspects of the human world: voting trends in politics, the dating of ancient artifacts, the analysis of automobile traffic patterns, and long-term strategies for the sustainable harvest of deciduous forests, to mention a few. Today, mathematics as a mode of thought and expression is more valuable than ever before. Learning to think in mathematical terms is an essential part of becoming a liberally educated person."An essential part of becoming a liberally educated person?" Sadly, many people in America, indeed, I would have to say very many people in America, would find that a difficult and puzzling concept. The majority of educated Americans do not think of Mathematics when they think of a liberal education. Mathematics as essential for science, yes, for business and accounting, sure, but for a liberal education?
-- Kenyon College Math Department Web Page
Why do so many people have such misconceptions about Mathematics?
The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick, what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x +1." There are no such employers.
What is mathematics really like?
Mathematics is not about answers, it's about processes. Let me give a series of parables to try to get to the root of the misconceptions and to try to illuminate what mathematics IS all about. None of these analogies is perfect, but all provide insight.
Scaffolding.
When a new building is made, a skeleton of steel struts called the scaffolding is put up first. The workers walk on the scaffolding and use it to hold equipment as they begin the real task of constructing the building. The scaffolding has no use by itself. It would be absurd to just build the scaffolding and then walk away, thinking that something of value has been accomplished.
Yet this is what seems to occur in all too many mathematics classes in high schools. Students learn formulas and how to plug into them. They learn mechanical techniques for solving certain equations or taking derivatives. But all of these things are just the scaffolding. They are necessary and useful, sure, but by themselves they are useless. Doing only the superficial and then thinking that something important has happened is like building only the scaffolding.
The real "building" in the mathematics sense is the true mathematical understanding, the true ability to think, perceive, and analyze mathematically.
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