In Defense of Algebra

In algebra we learn to handle abstractions that are not part of visceral human experience. We learn not only to be comfortable with such external unknowns but how to master them.
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An apocryphal joke has a medical student failing physics and questioning why he should ever have to solve useless mechanics problems that he will never again see in his life. The physics professor reassures the hapless student.

"These problems are terribly important: They save lives."

"How?" cries the student.

"They keep thousands of idiots like you out of medical school."

There are many reasons why we teach various parts of Physics and Mathematics but not all of them are obvious. Physics centers around finding a simple set of universal laws that govern the universe at the most basic level. The skill set that physics is trying to teach medical students is the ability to disassemble a complicated problem into smaller, more easily solvable component parts, use some of those laws to understand the parts and then reassemble the pieces into a whole. The human body is one of the most complicated machines we have ever studied and, if a doctor cannot understand the workings of a simple mechanics problem, then he really will kill people.

Andrew Hacker recently argued with some force, in the New York Times, that we should not be torturing the minds of high-school students with algebra. The primary burden of the piece is to catalog the abysmal performance of US students and to evoke sympathies of the vast majority of people who never use algebra after high school. While recognizing the need for an intellectual elite that can do algebra, Dr. Hacker goes on to argue that something that is so useless should not be holding back students who might be able to make remarkable contributions elsewhere in our culture. He advocates that students should be taught things that are more real, like how the CPI is constructed and the meaning of statistics. "This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry." So presumably another set of differently-abled people will be held back and many more useless (and annoyingly difficult) things like Shakespeare, French and Astronomy can safely be dropped from the curriculum. I sympathize with the critical need for everyone to know what statistics and margins of error mean and for them to be able to compute their mortgage payments, but I also believe it is crucial that high-school students also learn very basic algebra.

One of the less obvious goals in algebra is to get people to think more abstractly. Very elementary mathematics is all about "real things" and initially employs realia to help us add, subtract and multiply. From this experience we learn the language and some of the basic rules of mathematics. We abstract and generalize the experience and learn that, when we manipulate one side of an equals sign then the equality is only true if we do the same thing to the other side. Algebra makes a major intellectual leap: It names and labels things that we do not immediately know and that sometimes lie outside our direct experience. There are certainly other studies that involve abstractions like love, empathy and ethics, but in algebra we learn to handle abstractions that are not part of visceral human experience. We learn not only to be comfortable with such external unknowns but how to master them.

In algebra we develop essential life skills. We learn dispassionate analysis of external realities: how to simplify the things that we know and reduce the things that we do not; to see that some problems are unsolvable as presented; to identify exactly what data is needed to solve a problem entirely; to recognize extraneous data that is irrelevant to our problem; to identify data that conflicts with what we already know about a problem. By learning algebra we all become far better thinkers and even the majority who never use algebra again will still have enriched their life experience and expertise by grappling with difficult abstraction.

A limited understanding of one's passions and of the real things that can be manipulated by hand were sufficient to the needs of peasantry in medieval times. Today's society requires us to think in abstractions, to understand why an invisible, odorless gas that we breathe out every moment of our lives might be killing us all through climate change. We need to manipulate these abstractions to reasonably determine whether something is a fad or whether we must change our life-style. Does vaccination cause an unacceptable risk of autism? Does your body mass index affect your long-term health? What further data do we need to make an informed decision? What are the unknowns we should try to corral and eliminate before we make a critical decision or before we vote?

Algebra was developed by the Arab cultures as Western Europe was emerging from the Dark Ages. Algebra is not just the language of mathematical elites, it is one of the cornerstones by which we have emerged from a peasant society, ruled by the small elites sometimes capable of abstract thought, to become a complex, vibrant democracy. Algebra has helped us to rise beyond the simple understanding of immediate, tangible experiences and frame questions and look for the essential data that will give us deeper understanding. Only authoritarian and reactionary politicians benefit from a population for whom abstractions have no meaning. Such a population will be satisfied by sound bites and flag waving and will be placated by bread and circuses while their economy is subverted and their democracy implodes. Like mechanics problems in physics, the study of algebra, and the skills it develops, are not just critical to our long-term health individually but to our survival as a society.

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