## A response to “Is Algebra Necessary?”

As a mathematician, I did not even reach the second page of Andrew Hacker’s opinion piece “Is Algebra Necessary” in today’s New York Times before reaching for my mouse to record some of his more egregious howlers as quotations for this post.  The entirety of his opinion reflects a truly dangerous thread of thought in American education: that the actual contents of an academic discipline are accountable to the people on the grounds of future value, and that every moment of teaching should be justified as providing specific means to a particular vocational end.

The entire article must be understood in the light of the following sentence:

But for most adults, it [mathematics] is more feared or revered than understood.

Mathematics in school is advanced as being a subject of import and mystery: basic algebra is dragged out over many years with “prealgebra” and still apparently fails to result in being learned by many students. What does result is that these people come to wonder what is so great about math, what is the point of learning it, and this attitude affects even people such as Professor Hacker, who can claim some experience that would actually yield a positive answer to those questions.  It is a malaise that is familiar to research mathematicians: the feeling that “I don’t use that field and it’s therefore boring and pointless”.  I will return to this later, but for now, let it set the tone for the rest of this post.

Professor Hacker speaks of a “nation’s shame” that mathematics classes are preventing 25% or more students from finishing high school, with South Carolina and Nevada being given as examples of the “more”.  So when he observes that:

It’s true that students in Finland, South Korea and Canada score better on mathematics tests. But it’s their perseverance, not their classroom algebra, that fits them for demanding jobs.

it must mean that our nation’s shame is a lack of perseverance, and that the demands of today’s favorite jobs should direct that perseverance through the curriculum.  But if pure doggedness is all it takes to succeed, why go to school at all?  Professor Hacker suggests that there is a minimum level of mathematical competency still to be desired:

Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic.

This common reduction of mathematics to the purely numerical is a position taken in ignorance.  It is one informed by the idea that these aspects are the real insights of mathematics into the everyday world, which is all that matters.  What about the investigation of process and algorithm (computer science)?  of systematic analysis (algebra and logic)?  of the imprecise and changing (calculus)?  of visualization, similarity, and even analogy (geometry)?  What about the use of complex numbers in engineering, if the practical is what is to be sought?  This subject can barely be understood without basic algebra.  Professor Hacker’s generous allowance that numeracy is the basis of mathematics is a position abandoned by the world three or four centuries hence.  But he answers my objections:

But there’s no evidence that being able to prove $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ leads to more credible political opinions or social analysis.

This sentence is nothing more or less than “I don’t use that field and it’s therefore boring and pointless.”  I would not expect a political scientist to place a lower priority on the formation of political opinions than on the verification of an algebraic identity, but perhaps he has not considered what goes into doing either.  First, a discussion of $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$: this is a special case of the more general and more famous identity

$(x^2 + y^2)(z^2 + w^2) = (xz - yw)^2 + (xw + yz)^2$

in which $z = x$ and $w = y$.  Both can be proven by performing the operations indicated on both sides and rearranging terms until they are equal.  Both can also be proven using a clever interpretation of the sum of squares as the squared length of a complex number and using the fact that lengths multiply in products of numbers.  Each method has its virtues:

1. The purely computational method requires the ability to understand an inexpressive spread of symbols as indicating a process, namely the process of squaring some expressions.  It requires the ability to follow a logical sequence, namely the sequence of applying the rules to perform the process.  And most crucially, it requires the ability to recognize patterns in the result: namely, the pattern that $(x^2 + y^2)^2 = x^4+ 2x^2y^2 + y^4$ and $(x^2 - y^2)^2 = x^4 - 2x^2y^2 + y^4$ share the terms $x^4 + y^4$ and therefore one can be converted to the other by altering only the middle term, which is what appears as $(2xy)^2$.  Completing this proof—really completing it—is an exercise of several kinds of insight.
2. The complex numbers method is simply that $(x + iy)(z + iw) = (xz - yw) + i(xw + yz)$, and that the operation of replacing an expression $a + ib$ with $a^2 + b^2$ leaves the equation valid.  Understanding this proof is understanding that while there may be an ordinary reason for an equation to be true, coming just from comparing both sides, there can and perhaps should also be an extraordinary reason, one that has its roots in something larger and more significant.

The first method alone contains valuable lessons that, frankly, I do not see in many of those who are supposed to lead the United States or of those who are supposed to make informed decisions in electing them.  Politics is full of superficially logical lies and disingenuous half-truths; it should be within the capabilities of every citizen to figure out if some claim that “once the government requires us all to buy health insurance, it will be able to say who lives or dies” is derived logically from anything factual, and if so, from what facts.

The second method is a rare gift in social policy.  It is the gift of seeing past the details of who should pay for health insurance (or perhaps, what particular crumbs of mathematics should be taught in school) and having a vision where everyone gets it at all.

For these reasons I can only revile the following statement:

Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.

After all, a subject that is of no use but is “feared and revered” by most people can only be fakery.  The imagery of mathematics as idolatry pervades Professor Hacker’s conception of the subject.  Given what I wrote above I trust the reader to understand why I will not respond further.

Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”

I am wholly in favor of replacing the mathematics curriculum.  Currently, school math is a limited and repetitive sequence of ill-chosen subjects drawn from the 19th century curriculum and presented as representative of a subject that has grown enormously both in its pure and its applied aspects.  But the proposal that algebra (and presumably geometry and calculus, not to mention poor outdated trigonometry) be replaced by the equally limited offering of what I could call “statistics for newspaper readers” is not sufficient.  This is one thing—one thing—that one must be able to do after making a claim to have a mathematics education.  Just like being able to formulate and execute simple algebraic procedures is one thing to do.  Just like trigonometry was one thing that was highly applicable to employment in the Renaissance if one wanted to be a sailor, though it is of no practical use now.  Professor Hacker claims that this is not “dumbing down” the subject but it is: it is reducing the literal universe of mathematics to a single application that can be taught to attentive students in a year or two, but will no doubt be spread across twelve and watered down with soft goals such as “numeracy” that can never be attained without a complete immersion in a variety of ideas.  This idea is the idea that there is a magic bullet that will kill the fear of mathematics and raise our students to a higher standard of understanding.  My view is that if we are to expect more, we should teach more, and not less.

Why not mathematics in art and music — even poetry — along with its role in assorted sciences?

Any research mathematician will probably volunteer the opinion, unsolicited, that mathematics is an art rather than a science.  In the sciences it is one step away from any direct application, that step being the experimental or interpretive layer furnished by the science itself.  As an independent practice, math is the science of ideas.  This very fact is why we should not pursue a curriculum that strips math of all its ideas and variety, all the art, and teaches only a short-sighted distillation of what the current trends in our economy need for day-to-day work.

Nor should we say, though, that “math is just an art” and put it on the same footing as the arts.  For one, “just an art” is a denigration reflecting the place occupied by the various unserious subjects in schools now, subjects forgotten after the fifth grade when they are replaced by ones that are “of use”.  It is certainly the case that new math was first discovered in art, such as projective geometry, but to make that lesson a mathematics lesson requires more than description or observation; it requires analysis, and that is something that takes place at a higher level than simply learning the facts of how math is used in art.

But there is no reason to force them to grasp vectorial angles and discontinuous functions.

Professor Hacker chose this sentence as his conclusion, and I will make the response mine.  There is every reason to force our students to grapple with, and ultimately grasp vectorial angles and discontinuous functions—two examples that I could not imagine being more poorly chosen for his purposes, as linear algebra (the study of vectors) is the single most applied portion of algebra, including in Professor Hacker’s beloved statistics, and discontinuous functions are the kind of function that occurs most in measurements, any kind of engineering being an example.

But applications are of passing importance.  These concepts are elements of the act of learning something new: inhabiting its world of ideas, combining them and dissecting them, investigating their consequences and following the implications wherever they may lead.  If mathematics is indeed a liberal art, and if it is ever to give our students practicable skills, then this is what it can give them, and if they can’t be expected to “grasp” that idea, they will certainly never grasp the more vague and deceptive ideas that pervade the world for which they are supposedly being prepared.