Mathematics

Math education deserves support and attention (essay)

Andrew Hacker’s The Math Myth and Other STEM Delusions simply continues to promote the misguided path he got on several years ago, and it’s difficult to see how it could lead us anywhere productive. Hacker started the business of attacking school mathematics in a New York Times op-ed where he argued, in sync with gimmicky T-shirts claiming the same, that algebra was unnecessary, or perhaps even detrimental to our future. In a national scene where mathphobia is rampant and most people’s memories of school mathematics remain unpleasant at best, he struck a chord. Then, of course, come book contracts and even more adulation.

Thoughtful people have already responded authoritatively to the various errors in Hacker’s argument -- see here for another scathing review. A short and quick reply is here. For this audience of college and university educators, some of whom might be tempted by Hacker’s bravado and wonder about implications for higher education, I’d like to also point out that Hacker seems to forget why we educate our young. Even if as students years ago we may have had difficulty in certain subjects, as parents we want to ensure that our children go beyond what we ourselves have achieved. We expect that what they learn will be beneficial to their growth and future opportunities. We also hope that they will gain certain personal characteristics that, together with their knowledge and skills, will help them build a better future for our society and the world.

The Western tradition starts this conversation in ancient Greece with Socrates arguing that virtue is central to the education of the young. Aristotle teaches us that the ultimate goal of education should be happiness -- the durable contentment of a creative and intellectual life. St. Augustine shows us that we should not depend on teachers to teach us everything, that there is much to be learned from the internal wisdom of the heart, which itself is cultivated by our moral compass. Rousseau argues that children need to be exposed to the world as they grow to learn to live within the society to which they belong. Locke and Mill teach us that education should be well-rounded, cultivating an intellectually capable mind aware of the complexities of the world.

Mathematics educators agree. We know that in mathematics, as in any other knowledge system that builds on itself, the procedures that work so well are only part of the package. That in the center is the student, but always situated in the midst of a society that is constantly evolving. That students learn best when encouraged and supported by knowledgeable teachers who help them explore and understand underlying concepts. That intellectual stimulation and growth are possible and enjoyable for all children. That in our classrooms, we can help students sharpen their ability to persist in the face of apparent failure. That today’s students need to learn to tackle complex and ill-defined problems requiring both individual and collaborative effort.

And to these ends, we have been working to improve what we do. Mathematics teachers, mathematics education researchers and mathematicians are working together in classrooms, in math circles, in conferences and workshops, in curricular reform efforts and in policy discussions. We are working to create meaningful mathematical experiences for students to encourage critical thinking, foster creative reasoning and enhance problem-solving abilities. (See here and here for two collections of mathematics lesson plans and modules that were developed by or in collaboration with researchers. See here for a college-level initiative for revamping the mathematics curriculum.)

We are working to engender the sense of wonder and accomplishment that mathematics -- when done right -- naturally inspires. We are working to develop and support a coherent set of curricular standards that will help tomorrow’s adults live up to the expectations of this nation from its children. We are working to discover and share with parents, teachers and educators what works well in the classroom even if it is not typical, and what doesn’t work even if it “just makes sense” and “it’s the way I learned things.” (How many people believe that the point of multiplication tables is to torture students till they can recite them at the speed of light? Linda Gojak, past president of the National Council of Teachers of Mathematics, is one among many educators speaking out about fluency in mathematics and how it is no longer acceptable to equate it with “fast and accurate.”)

Admittedly, we mathematics instructors don’t always help our own cause. People remember how their middle school math teacher made them feel, and I don’t need to tell you that it’s generally not a good memory. (I was lucky -- mine made me feel like there wasn’t a problem I couldn’t solve if I put in the time and effort.) But dropping mathematics from the required K-12 curriculum would be a perfect example of the cliché of throwing out the baby with the bathwater. (While I myself have argued elsewhere that we might just do that, my tongue was decidedly set in my cheek, and my concern was the essence of what is lost in most mainstream experiences of school mathematics. Now, can I get a book contract, too?)

The current state of mathematics education in the United States is certainly not ideal. Yet the fact is that teachers, parents, mathematicians, mathematics education researchers and policy makers are working on it. Furthermore, this is definitely a problem worth working on. It is tough, it is messy and there are many nuances to the issue and many implications to any avenue of resolution.

Hacker writes of high school graduates unable to perform simple numeracy tasks. I’d venture to guess that more than a handful of high school grads are also incapable of understanding IRS publications, may occasionally be unable to interpret correctly the user manuals of their DVD players and can’t foresee all repercussions of a ballot measure they are willing to vote for or against. But we do not blame all of these insufficiencies on K-12 English teachers. Nor do we suggest replacing English courses with courses on reading ballot measures or user manuals. What do we do? We demand that English Language Arts curricula be developed that are more sensitive to the range of literacy demands of our daily lives.

Hacker gets it right at least in one instance; quantitative literacy is crucial in today’s society. And it should be one of the essential outcomes we expect from our education system, as I argue elsewhere. However, the role of mathematics in our education system goes beyond quantitative literacy. (And conversely, quantitative literacy as a goal itself should not be limited to the mathematics classroom. Most science and social studies classrooms offer excellent contexts for quantitative literacy.)

During this election year, I offer you another analogy. Today there are many, including some reading this, who worry that the American democratic machine is not producing the results they would like. So shall we give up on democracy? I’d like to believe that the overwhelming majority would agree with me when I say no. Instead, we continue working to improve our system; we continue to fight for broader access; we continue to work to further political and social justice.

Mathematics education is perhaps not on the same level of importance and urgency, but the solutions are the same. We must work to improve the system. We must fight for broader access. And we must work to further political and social justice.

Today mathematics acts as a gateway (or a gatekeeper, depending on your perspective) in terms of who has access to the lucrative STEM jobs that many aspire to. Students who learn mathematics as far as their school contexts allow have many more opportunities open to them when they graduate from high school. Knowing the fundamental building blocks of mathematics today leads well-prepared high school graduates to a range of rigorous paths of college-level study in many disciplines. And those are also the students who will become the adults who will create the new mathematical, statistical and computational tools we will need in the future.

What would happen if we dropped mathematics? Which schools and school districts would not be offering those “now optional” advanced mathematics courses? Which students would be deprived of the opportunity to learn, and, can I suggest, find meaning, confidence and opportunity through advanced mathematics? And which students would be able to move forward with those STEM careers that many parents dream of?

People can succeed without mathematics in their lives. You can also choose to never try sushi, to vacation only within the continental United States despite being able to afford international travel, to never wear flip-flops or learn to ride a bike, and still lead a happy and productive life. But nobody’s job prospects are affected by their decision to avoid sushi (unless you want to be a sushi chef, which would be odd if you didn’t like sushi to start with). And having the choice to decline comes out of privilege. Can this nation afford to make such a decision for all its children? When people choose to drop mathematics later in their academic paths, we can say they made a decision knowing their options and the opportunities they are letting go. But do we want to make these decisions ahead of time for all students?

The American education system differs from many of the nations that are touted as high performers. In most of those countries, students are channeled into various tracks early on. This nation does not regiment its schoolchildren, because we believe that all children have potential and that they can make choices once they are old enough to know what is out there.

And the American education system is still one of the best in the world. I know the international test scores and rankings, but I also know to read the fine print. Therein you learn that once you restrict to schools where less than 50 percent of the class is in the free lunch program, the performance of students is in par with those high-performing nations. The schools that are “failing” are the ones that have 75 percent or more of their students in free lunch programs. So our schools are not failing our students; it is our society that is failing them. As most education researchers (and teachers in classrooms across the nation) will agree, the problem of public education in the United States is one of poverty. And that problem is not going to get solved by dropping the mathematics requirement in the K-12 curriculum.

In fact mathematics can help. Here is where Plato’s virtue and St. Augustine’s moral compass come back into school mathematics. Brazilian mathematician Ubiratan D’Ambrosio has been telling us for years that it is mathematics that will help our children solve the varied problems of today and tomorrow -- if we can teach them to see the inherent mathematics involved. Mathematics, historian Judith Grabiner points out, has evolved precisely to describe social, environmental and political, as well as industrial and scientific, problems that a society happens to confront. And it remains, to this day, our most successful method to seek out creative and productive solutions for them. (Readers perplexed by my inclusion of social, environmental and political problems above might like to google “mathematics for social justice” or “mathematics of sustainability.”)

I write this with the hope that some good may come out of Hacker’s simplistic recommendations. Students reciting their multiplication tables as fast as a bullet train are not the desired outcome of mathematics education. We want students to understand the power and limitations of the mathematics they are learning. We want students to move flexibly from one specific model of a situation to another. We want students to be able to find unexpected and novel solutions to problems that are ever-growing in their complexity.

Mathematics is where we can train young minds to do all these things. Mathematics is where we can teach that critical ability to reason analytically. Mathematics is also where we can encourage creative exploration of the multitude of options a problem solver invariably has. As college and university educators, these are points we must not forget when the next cycle of general education debates begins to shake things up on our campuses.

Gizem Karaali is an associate professor of mathematics at Pomona College, editor of the Journal of Humanistic Mathematics, associate editor of Mathematical Intelligencer and associate editor of Numeracy. Follow her on Twitter @GizemKaraali_.

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Essay on "The Best Writing on Mathematics 2012"

Writing about music, the saying goes, is like dancing about architecture. The implication is that even trying is futile and likely to make the person doing so look absurd.

The line has been attributed to various musicians over the years – wrongly, as it happens, though understandably, given how little of what they do while playing can be communicated in words to people who don’t know their way around an instrument. I don’t know if mathematicians have an equivalent proverb, but the same principle applies. Even more strictly, perhaps, since most nonmathematicians can’t even tell when things go out of tune. And in many of the higher realms, math drifts far from any meaning that could ever be expressed outside whatever latticework of symbols has been improvised for the occasion. (Kind of like if Sun Ra went back to performing on his home planet.)

Against all odds, however, there is good writing about music – as well as The Best Writing on Mathematics 2012, the third anthology that Mircea Pitici has edited for Princeton University Press in as many years. He is working toward a Ph.D. in mathematical education at Cornell University, and teaches math and writing there and at Ithaca College. A majority of the pieces come from journals such as Science, Nature, The Bulletin of the American Mathematical Society and The South African Journal of Philosophy, or from volumes of scholarly papers. But among the outliers is an article from The Huffington Post, as well as a chapter reprinted from an anthology called Dating Philosophy for Everyone: Flirting With the Big Ideas (Wiley-Blackwell, 2011)

There’s also a paper from the proceedings of the Fifth International Meeting of Origami Science, Mathematics, and Education, which opens with the sentence: “The field of origami tessellations has seen explosive growth over the past 20 years.”

Chances are you did not know that. It came as news to me, anyway, and I cannot claim to have followed every step of the presentation, which concerns the algorithms for creating fantastically intricate designs (resembling woven cloth) out of single flat, uncut sheet of paper.

The author, Robert J. Lang, is a retired specialist in lasers and optoelectronics; his standards of numeracy are a few miles above the national average, even if the math he’s using is anything but stratospheric. But Lang is also an internationally exhibited origami artist. The images of his work accompanying the article offer more than proof of what his formulas and diagrams can produce; they are elegant in a way that hints at the satisfaction the math itself must have yielded as he worked it out.

Other papers make similar connections between mathematics and photography, dance, and (of course) music. But one of the themes turning up in various selections throughout the book is the specificity of what could be called mathematical pleasure itself, which can’t really be compared to other kinds of aesthetic experience.

In his essay “Why is There Philosophy of Mathematics at All?" Ian Hacking -- retired from a university professorship at the University of Toronto – considers the hold that math has had on the imagination and arguments of (some) philosophers. Not all have been susceptible, of course. Among humans, “a high degree of linguistic competence is [almost] universally acquired early in life,” the ability “for even modestly creative uses of mathematics is not widely shared among humans, despite our forcing the basics on the young.” And as with the general population, so among philosophers.

But those who have thought carefully about mathematics (e.g., Plato and Husserl) or even contributed to its development (Descartes and Leibniz) share something that Hacking describes this way:

“[T]hey have experienced mathematics and found it passing strange. The mathematics that they have encountered has felt different from other experiences of observing, learning, discovering, or simply ‘finding out.’ This difference is partly because the gold standard for finding out in mathematics is demonstrative proof. Not, of course, any old proof, for the most boring things can be proven in the most boring ways. I mean proofs that deploy ideas in unexpected ways, proofs that can be understood, proofs that embody ideas that are pregnant with further developments…. Most people do not respond to mathematics with such experiences or feelings; they really have no idea what is moving those philosophers.”

Beyond the pleasure of proof (“Eureka!”) lies unfathomable mystery – of at least a couple of varieties. One is the problem addressed in “Is Mathematics  Discovered or Invented?” by Timothy Gowers, a professor of mathematics at Cambridge University.  Be careful how you answer that question, since the nature of reality is at stake: “If mathematics is discovered, then it would appear that there is something out there that mathematicians are discovering, which in turn would appear to lend support to a Platonist conception of mathematics….”

Or to put it another way and leave Plato out of it for a moment: If “there is something out there that mathematicians are discovering,” then just exactly where is “out there”? Answering “the universe” is dodging the question. We might naively think of arithmetic or even some parts of geometry as some kind of generalization from observed phenomena, But nobody has empirical knowledge of a seven-dimensional hypersphere. So how – or again, perhaps more pertinently, where, in what part of reality – does the hypersphere exist, such that mathematicians have access to it?

A neurobiological argument could be made that the higher mathematical concepts exist in certain cognitive modules found in the brain. (And not in everyone’s, suggests Hacking’s essay.) If so, it would make sense to say that such concepts are created. But if so, the mystery only deepens. Scientists have repeatedly found the tools for understanding the physical universe in extremely complex and exotic forms of mathematics developed by pure mathematicians who not only have no interest in finding a practical application for their work, but feel a bit sullied when one is found.

Translating math’s hieroglyphics into English prose is difficult but – as the two dozen pieces reprinted in Best Writing show – not always completely impossible. Mircea Pitici, the editor, pulls together work at various levels of complexity and from authors who pursue their subjects from a number of angles: historical or biographical narrative, philosophical speculation both professional and amateur, journalistic commentary on the state of math education and its discontents.

And the arrangement of the material is, like the selection, intelligent and even artful. Certain figures (the 19-century mathematicians Augustus de Morgan and William Hamilton) and questions (including those about math as experience and mystery) weave in and out of the volume -- making it more unified than “best of” annuals tend to be.

That said, I am dubious about there being a Best Writing ’13 given the dire implications of certain discoveries (or whatever) by Mayan numerologists. This will be the last Intellectual Affairs column for 2012, if not for all time. I’d prefer to think that, centuries ago, someone forgot to carry a digit, in which case the column will return in the new year. And if not, happy holidays even so.

 

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Review of William J. Cook, 'In Pursuit of the Traveling Salesman'

Intellectual Affairs

A theoretical physicist named Eugene Wigner once referred to “the unreasonable effectiveness of mathematics” -- a phrase that, on first hearing, sounds paradoxical. Math seems like rationality itself, at its most efficient and severe. But even someone with an extremely limited grasp of the higher realms of mathematics (your correspondent, for one) can occasionally glimpse what Wigner had in mind.  His comment expresses a mood, more than an idea. It manifests a kind of awe.

For example, in the 1920s Paul Dirac came up with an equation that permitted two possible solutions, one of which applied to the electron. The other corresponded to nothing that physicists had ever come across. Some years later, physicists discovered a subatomic particle that did: the positron. The manipulation of mathematical symbols unveiled an aspect of the physical universe that had been previously unknown (even unsuspected). To adapt a line from the Insane Clown Posse’s foul-mouthed appreciation of the wonders of the universe, “This [stuff] will blow your mother[loving] mind.”

True, that. I’ve even felt it when trying to imagine the moment when Descartes first understood that algebra and geometry could be fused into something more powerful than either was separately. (Cartesian grids – how do they work?) But there’s a flipside to Wigner’s phrase that’s no less mind-boggling to contemplate: the existence of “simple” problems that resist solution, driving one generation of mathematicians after another to extremes of creativity. Fermat’s last theorem (formulated 1637, solved 1993) is the most famous example. The four-color map conundrum (formulated 1852, solved 1976) proved misleadingly uncomplicated.

And then there's the great, bewildering problem surveyed in William J. Cook’s In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (Princeton University Press). The challenge has been around in one form or another since 1934. It looks so straightforward that it’s hard to believe no one has cracked it – or will, probably, any time soon.

Here’s the scenario: A traveling salesman has to visit a certain number of cities on a business trip and wants to get back home as efficiently as possible. He wants the shortest possible route that will spare him from going through the same city more than once. If he leaves home with just two stops, no planning is necessary: his route is a triangle. With three stops, it’s still something that he might work out just by looking at the map.

But his company, pinched by the economy, needs to “do more with less,” like that ever works. It keeps adding stops to the list. By the time he has five or six calls to make, planning an itinerary has gotten complicated. Suppose he's leaving home (A) to visit five cities (B, C, D, E, F). He starts out with five possibilities for his first destination, which means four for his second stop. And so on -- one fewer, each time. That means the total number of possible routes is 5 x 4 x 3 x 2 x 1 = 120. Our salesman may be relieved to discover that it's really only half of that, since traveling in the sequence ACDBFEA covers exactly the same distance as doing it the other way around, as AEFBDCA.

Still, picking the shortest of sixty possible routes is a hassle. Let the number of cities grow to 7, and it's up to 2,520. Which is crazy. The salesman needs a way to find the shortest trip, come what may -- even if the home office doubled the number of stops. There must be an an app.

Except, there isn't. I don’t mean with respect to available software, but at the level of a method that could solve the traveling salesman’s dilemma no matter how many cities are involved. A computer can tackle problems on a case-by-case basis, using brute force to calculate the distances covered by every possible route, then selecting the shortest. But that’s a far cry from having an elegant, powerful formula valid for any given number of cities. And even the most unrelenting brute-force attack on the traveling salesman problem (TSP) might not be enough. Finding the shortest way around a 33-city route would require calculating the distances covered by an unimaginably vast number of possible tours. I’m not up to typing out the figure in question, but it runs to 36 digits. And that's for a two-digit tour.

Cook explains what would happen if we tried to compile and compare every possible sequence for a 33-city route using the $133 million dollar IBM Roadrunner Cluster at the Department of Energy, which “topped the 2009 ranking of the 500 world’s fastest supercomputers,” The Roadrunner can do 1,457 trillion arithmetic operations per second. Finding the shortest route would take about 28 trillion years – “an uncomfortable amount of time," Cook notes, "given that the universe is estimated to be only 14 billion years old.”

That does not mean any given problem is insoluble, even with an extraordinarily high number of cities. In 1954, a group of mathematicians in California solved a 49-city problem by hand in a few weeks, using linear programming -- which seems appropriate, since linear programming (LP) was developed to help with business decisions on how best to use available resources. In Pursuit of the Traveling Salesman devotes a chapter to the history of LP and the development of a multipurpose tool called the simplex algorithm. (Cook’s treatment of LP is introductory, rather than technical, though it’s not exactly for the faint of heart.)

Other tour-finding algorithms find clusters of short routes, then link them as neatly as possible. If having absolutely the shortest path isn’t the top priority, various methods can generate an itinerary that might be close enough for practical use. And practical applications do exist, even with fewer traveling salesmen now than in Willy Loman’s day. TSP applies to problems that come up in mapping the genome, designing circuit boards and microchips, and making the best use of fragile telescopes in old observatories.

But no all-purpose, reliable, let-N-equal-whatever algorithm exists. It's the sort of cosmic untidiness that some people can’t bear. Cook quotes a couple of mathematicians who say that TSP is not a problem so much as an addiction. Combining brute-force supercomputer processing with an array of tour-finding concepts and shortcuts means that it’s possible to handle really enormous problems involving hundreds of thousands of cities. But that also makes TSP a boundary-pushing test of computational power and accuracy. Finding the shortest route is an optimization problem, but so, in a way, is figuring out how to solve it using the tools at hand.

Since 2000, the Clay Mathematics Institute has offered a prize of a million dollars for the definitive solution to a problem of which TSP is the exemplary case. (Details here.) “Study of the salesman is a rite of passage in many university programs,” Cook writes, “and short descriptions have even worked their way into recent texts for middle-school students.” But he also suggests that the smart money would not bet on anyone ever finding the ultimate TSP algorithm, though you’re more than welcome to try.

Another possibility, of course, is that the right kind of mathematics just hasn’t been discovered yet – but one day it will, proving its “unreasonable effectiveness” by solving TSP as well as problems we can’t even imagine at this point. As for our hypothetical traveler, he’d probably feel envy at one of the Cook’s endnotes. The author “purchased 50 years of annual diaries written by a salesman” via eBay, he writes, “only to learn that his tours consisted of trips through five or six cities around Syracuse, New York.” He didn't need an algorithm, and got to stay home on weekends.

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2011 Fall Western Section Meeting

Date: 
Sat, 10/22/2011 to Sun, 10/23/2011

Location

Salt Lake City , Utah
United States

2011 Fall Central Section Meeting

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Lincoln , Nebraska
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