Polytropy

This essay was provoked in part by a New York Times opinion piece by Andew Hacker (July 28, 2012) called “Is Algebra Necessary?” (the suggested answer being No):

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.

My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus. State regents and legislators—and much of the public—take it as self-evident that every young person should be made to master polynomial functions and parametric equations.

There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong—unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)

A provocation from a different direction was an article (by Anne Ryman, The Republic | azcentral.com, July 28, 2012) reporting on increasing high-school mathematics requirements in Arizona:

Students in the Class of 2013 and all classes that follow will need an extra year of math and science to graduate, even if they have no college plans. The fourth year of math is causing the most anxiety among the many students who struggle with the subject.

The increased requirements are part of a state and national push by education reformers to better prepare students for college and the workforce. Last school year, 13 states required four years of math. For the 2012-13 school year, five more states, including Arizona, will have similar requirements, according to the Education Commission of the States, a group that tracks state policy trends.

Myself, I do not propose to ask students to learn more mathematics than they can handle. And yet it misses the point to argue that most people do not need this or that piece of mathematics in life.

Among all subjects taught in school, mathematics may be considered as that subject in which two lessons can best be learned:

1. The power of the student’s own mind.

2. The possibility of harmonious resolution of differences between people.

The correctness of a piece of mathematics is handed down by no external authority, be it a teacher, a textbook, or God. The correctness must be recognized freely by the individual student, by use of her or his own reason. Mathematics can thus be distinguished:

• from history, where the student must accept that past events happened more or less as the textbook says;

• from science, where the textbook will present the distilled results of generations of experiments that the student cannot repeat.

It may well be that the more advanced student of history will learn to be skeptical of all books; and the more advanced student of science may repeat certain experimental developments. But in mathematics, from the very beginning, students have the power to confirm that their solutions to problems are correct. They need not consult books or teachers for this.

In the earliest grades, the tables of sums and products of single-digit numerals can be confirmed by counting. The algorithms for adding, subtracting, multiplying, and dividing multiple-digit numbers may simply be memorized; but their efficacy can be demonstrated to the interested student.

Algebra students should memorize the “quadratic formula”: the rule that the solutions of the general quadratic equation

are

(assuming a ≠ 0). I think students should also be able to derive this formula by the method called completing the square. But even if they cannot do this, they can verify by substitution that the rule does indeed give solutions to the quadratic equation.

When I asked my 10th-grade English teacher Paul Piazza what the point of our course was, he said it was to find books that we liked to read. I have always appreciated this response. It gives all the justification an English course needs. And yet it may have students reading a lot of books that perhaps they do not like. (I myself could never appreciate the Dickens I read at school. When Evelyn Waugh needed to create a vision of hell, he could hardly do better than he did in A Handful of Dust, where Tony Last is condemned to spend the rest of his life reading Dickens to his illiterate captor.)

Students may not like mathematics either. But the subject is like, or should be like, physical education: it develops one’s own resources. It should enable the student to say: “This statement is correct, because I have proved it.”

And yet mathematics is not individualistic. If your classmate or your teacher disagrees with your argument, you cannot say, “That’s tough; it works for me.” Agreement must be reached, and this agreement is not like a Treaty of Versailles: it cannot be the result of fighting.

People say they don’t like mathematics because if you’re wrong, you’re wrong, and you can’t argue about it. But then if you’re right, you’re right, and everybody who cares about the matter must accept this. Unfortunately it would seem that many students have trouble being right about mathematics. I don’t know why this is or what can be done about it. I do want people to learn the confidence that can come with mathematics: not the confidence of the fanatic who avers, “God said it, I believe it, that settles it,” but the confidence of the rational person who knows that they are fallible, but knows too that they have the right to have their mistakes pointed out in such a way that they can understand them.

Note added, May 6, 2013. I have edited this article today, technically, simply because, by changing the visual “theme” of my blog, I seemed to cause this article to be treated as an “aside” without a title, to be displayed in a font much too large.

Note added, August 9, 2016. I did further editing of the format of this article.

Note added, June 10, 2020. Gave ambiguous gender to the pronoun in the last sentence. Thought I should note that the teacher of mine that I referred to was Paul Piazza, whom I have also referred to elsewhere.

Note added, October 30, 2020. Most recently in “Pacifism,” I have developed or refined the ideas of this post into two antitheses. Mathematics is

1. both personal and universal;

2. both a right and a responsibility.

These abstractions may not clarify the matter any better than the present post does with its examples.

Since for this early post I have been conscientious about noting changes, let me continue to be.

• The underlying html file is now created by pandoc from a text file, edited from the one that I obtained by applying pandoc to the old html file. I talk about pandoc in “LaTeX to HTML.”

• I have done some editing for style, clarity, and correctness:

  • title in title case;

  • New York Times italicized;

  • “handed down by no external authority” instead of “not handed down by any external authority”;

  • “they can verify by substitution that the rule does indeed give solutions” instead of “they can verify that the rule does indeed give solutions”;

  • Mr Piazza’s name inserted in the original text;

  • a comma now inside quotation marks;

  • a closing quotation mark, once wrongly single, now double.