Thursday, November 1, 2012

Necessary Algebra

Paul ZornSaint Olaf College

I remember vividly the moment—and the room decor, the time of night, and the LP on the stereo—when my cousin Jon taught me algebra.

He and I, then seventh-graders, enjoyed those hoary old story problems (Al is twice as old as Betty; in seven years . . .) that once appeared in magazines such as Life and Look. I had concocted a simple strategy that one might charitably call iterative: Make any old integer guesses and tweak them as the errors suggest. What Jon first saw, and memorably pointed out, was that an unknown, say A, for Al’s age, can be manipulated as though it were a known quantity like one of my guesses.

What thrilled me then was the prospect of zipping through an entire genre of contrived puzzles. What amazes me still is the power of one simple idea: You can manipulate unknowns and knowns to solve equations.

That prescription seems a decent nine-word summary of what algebra does, even beyond the seventh grade. Jon and I got a preview, however dim, of an idea bigger and better than we could have suspected. Every student should encounter, and eventually own, an idea so simple and powerful. I’m convinced that almost every student has a fighting chance.

Is algebra necessary? 

So asked a provocative New York Times op-ed last July. In fact, the title is slightly misleading. Author Andrew Hacker, professor emeritus of political science at Queens College, doesn’t question algebra’s larger importance. He notes cheerfully that “mathematics, both pure and applied, is integral [Hacker’s good word] to our civilization, whether the realm is aesthetic or electronic.”

Hacker’s different but equally provocative question is how much “algebra,” that “onerous stumbling block for all kinds of students,” should be required in high school and college. His answer: Much less. And less of other mathematics, too.

Here “algebra” is in quotes because Hacker’s beef is not really with that subject in particular. Indeed, Hacker sees both “algebra” and existing curricula idiosyncratically. His examples of supposedly superfluous material—“vectorial angles” and “discontinuous functions”—are unlikely examples of “algebra” and even less representative of what is typically taught. And Hacker’s en passant endorsement of teaching long division (right up there with reading and writing) surprised me. He doesn’t acknowledge, or seem aware of, creative efforts to improve school teaching of “algebra” by teachers like those supported and mentored by, say, Math for America.

Hacker’s real curricular concern is broader than algebra: It’s the curricular complement of quantitative literacy (QL). He refers generally to “the toll mathematics takes” (my emphasis), not just to difficulties posed by algebra. In this sense Hacker’s three Rs proposal—require QR, but not “mathematics”—is more radical, and Philistine, than the article’s title suggests. But let’s concentrate on algebra.

Where he’s right, and wrong. 

Some of Hacker’s rhetorical targets are legitimate. Algebra can indeed be taught rigidly and applied ineffectively. (I remember the joy of solving algebra puzzles but also tedious hours of FOIL-ing quadratics.) Hustling high school students toward calculus sometimes pushes them too rapidly for effective mastery through prerequisite courses—including algebra. And Hacker, keen to avoid “dumbing down,” suggests some interesting applications of QL methods to such topics as the Affordable Care Act, cost/benefit analysis of environmental regulation, and climate change. (Whether such topics can really be approached without algebra is another question.)

As Hacker observes, few workers use algebra explicitly in daily life. (We all use it implicitly.) To infer that algebra can therefore vanish from required curricula is mistaken. Similar arguments might be made against history, the humanities, and the sciences generally, none of which is widely practiced in daily life. More important in curricular design than eventual daily use are broader intellectual values, which algebra clearly serves: learning to learn, detecting and exploiting structure, exposure to the best human ideas, and—the educational Holy Grail—transferability to novel contexts.

Transferability is undeniably difficult, as Hacker duly notes. The National Research Council agrees (see Education for Life and Work Report (pdf)) and indeed stresses the value of “deeper learning,” of which a key element is the detection of structure.

“Transfer is supported,” says the NRC, when learners master general principles that underlie techniques and operations.

Algebra is a poster child for deeper instruction. We should teach it. Students can learn it. 

Paul Zorn is a professor of mathematics at Saint Olaf College and currently serving as president of the Mathematical Association of America.

This article was published in the November 2012 issue of Math Horizons.

Saturday, September 1, 2012

Measuring Women’s Progress in Mathematics

Linda Becerra and Ron BarnesUniversity of HoustonDowntown

Many believe that residual effects of past hindrances and discrimination against women in mathematics are being overcome. Studies by the American Mathematical Society and National Science Foundation on women in mathematics appear to reinforce this belief. Conventional wisdom suggests it is only a matter of time before women achieve parity. Julia Robinson (instrumental in the solution of Hilbert’s tenth problem) suggested that one measure of parity would be when male mathematicians no longer consider female mathematicians to be unusual.

Unfortunately, a close reading of AMS and NSF data suggests that significant progress is not being made. One can be deceived by looking only at raw numbers without considering the related percentages.

Among entering students at U.S. institutions, data for the years 2000 to 2008 indicate the number of female and male freshmen expressing interest in a major in mathematics went from 44,500 and 49,500 in 2000, to 66,000 and 66,600 in 2008. These figures indicate the gap between female and male interest in majoring in math narrowed from 5,000 in the year 2000 to 600 in 2008. However, among all undergraduates, the percentage of females and males interested in a math major went from 0.6 percent (female) and 0.8 percent (male) in 2000, to 0.7 percent and 1 percent, respectively, in 2008. Hence, the percentage gap between the sexes increased during this time. This is because there was considerably higher growth in the overall female undergraduate population during this period.

Total graduate enrollment in the mathematical sciences increased from about 9,600 in 2000 to 22,200 in 2009 (131 percent), while female graduate enrollment increased from 3,670 to 7,979 (117 percent). However, the percentage of female graduate enrollment in the mathematical sciences remained relatively static in the decade—38 percent in 2000 and 36 percent in 2009.

AMS mathematics data noted that the number of Ph.D.s awarded to U.S. citizens in the mathematical sciences increased from 494 in 2000 to 669 in 2008, and the number of Ph.D.s awarded to women grew from 148 to 200. However, the percentage of Ph.D.s earned by women in 2000 and 2008 were both approximately 30 percent, with some variation in the intervening years. Also, the percentage of bachelor’s degrees awarded to females during this time varied little from 41 percent.

The NSF used a different data set, and the conclusions are even less encouraging. It indicates that women’s percentage of bachelor’s degrees in mathematics from 2002 to 2009 steadily decreased from 48 percent to 43 percent.

In mathematics, the number of doctoral full-time tenure/tenure-track (T/TT) positions held by women at U.S. institutions increased from about 2,850 in 2001 to 4,000 in 2009 (a 40 percent increase). However, the percentage of T/TT positions held by females increased only from 18 percent to 23 percent during this time. This smaller difference is explained by the fact that significantly more males also obtained T/TT positions in this period.

There are reasons to believe that women’s progress in mathematics should be much better by now. Since 1982, considering all fields, women have annually earned more bachelor’s degrees than men. By 2011, more women than men had earned advanced degrees. Yet, the statistics cited show that in mathematics, women’s participation at advanced levels is still unusually low and either improving slowly or, in some cases, making no progress whatsoever.

The real question is: How can meaningful progress be effected? Evidently the present strategies are not working.

A few ideas for consideration:

• Engage in a rigorous, sustained intervention with girls throughout school-level mathematics and in universities—not a few small programs, but a broad, concentrated, and sustained effort to integrate girls into mathematics, its culture, and its relevance. This effort must involve all the professional mathematical societies.

• Reengineer the culture in the mathematics professoriate with an eye toward more flexibility in the tenure and promotion process. The standards need not be watered down in any way, but the process should allow for a variety of pathways to meet them.

As Julia Robinson observed, “If we don’t change anything, then nothing will change.”

Linda Becerra and Ron Barnes are professors of mathematics at the University of HoustonDowntown.

Monday, April 2, 2012

My Conversion to Tauism

Stephen Abbott—Middlebury College, Math Horizons Co-Editor
There was no identifiable moment when I said, yes, I believe. My conversion must have come on silently and unexpectedly. I do, however, remember the moment when I realized something had inalterably changed...

Read the full article PDF

About the Author: Stephen Abbott is a professor of mathematics at Middlebury College and currently co-editor of Math HorizonsEmail: 

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.

Wednesday, February 1, 2012

Unduly Noted

Tommy Ratliff—Wheaton College
When I opened the MathFest program in Lexington last summer, I took one look at the first page and nearly yelled out loud “NO! NO! NO!” The inside cover to the program contained an advertisement for an online homework system with the following example:

Find the derivative of y = 2 cos(3x − π) with respect to x.

The assertion is that a competing system marked this answer as wrong, but the advertised system identified the expression as correct, demonstrating its superiority. I assume that the intent is to show that the system can recognize equivalent, but not identical, algebraic expressions. What caused me to react so strongly, however, is that I would have also marked the given answer as wrong. The answer should have been

The parentheses matter! The expression sin(x) represents a function!

I should be clear: My irritation is not directed at this particular homework system as much as at the entire mathematics community for the sloppiness in notation that we tolerate, and even encourage, when dealing with trigonometric functions. You can pick up almost any calculus text, peek into almost any math classroom, or attend any number of talks at various MAA events to find a plethora of examples of trig functions lacking their parentheses.

Why do I think the parentheses matter so much? This is not just a pedantic preference on my part. The lack of parentheses represents an irregularity in notation that obscures the meaning of the mathematics. We often use a space to indicate multiplication, as in or 3 sin(x), so leaving off the parentheses hides the fact that we are using a trigonometric function. The confusion is compounded when we say that the derivative of “sine” is “cosine.” If we were to be consistent, this would lead to applying a distorted product rule to get

I have seen students struggle with this, even when they understand the intent of the original notation. They correctly apply the chain rule only to confuse the order of operations at the end because they did not put the constant multiple of 3 at the beginning of the expression:

After all, why should you apply the cosine function to the first 3 in the 3x but not to the trailing 3? If we always used parentheses to enclose a function’s argument, then there would be no confusion.

An even worse abuse of notation occurs in the location of the exponent when a trig function is raised to a power. I will never write sin2(x) for sin(x)2 because the first notation leads to ambiguity when discussing the inverse trig functions. Since f− 1 (x) is the standard, consistent notation for the inverse function of f(x) , we also use sin− 1(x) for arcsin(x). If we were consistent with notation, a perfectly reasonable calculation would be

Notice that I had to make a choice about the meaning of sin−2 (x) in simplifying the expression—an impossible task! Should it be sin2(x)-1 = 1/sin2(x) or sin -1(x)2 = arcsin (x)2? The point is that we should never have to make this choice! We should be taking the derivative of the inverse sine function. This is horrific. The bad notation allows at least three different interpretations of the expression

We in the mathematics community pride ourselves on the consistency and deterministic nature of our discipline. I think we do ourselves a genuine disservice when we use sloppy notation that requires another layer of interpretation to understand the intended meaning. The purpose of mathematical notation is to provide clarity and, ideally, to provide insight into the mathematics being notated.

Therefore, I implore you: The next time you use a trig function, please remember the parentheses, put the exponent on the outside, and never, ever write anything like sin3x2 cos-25x.

Tommy Ratliff is a professor of mathematics at Wheaton College in Massachusetts where he enjoys thinking about voting theory, building new science centers, and being precise in his notation.