Why do we inflict a full menu of mathematicsalgebra, geometry, trigonometry, even calculuson all young Americans, regardless of their interests or aptitudes? While Andrew Hacker has been a professor of mathematics himself, and extols the glories of the subject, he also questions some widely held assumptions in this thought-provoking and practical-minded book.
Does advanced math really broaden our minds? Is mastery of azimuths and asymptotes needed for success in most jobs? Should the entire Common Core syllabus be required of every student? Hacker worries that our nation’s current frenzied emphasis on STEM is diverting attention from other pursuits and even subverting the spirit of the country. Here, he shows how mandating math for everyone prevents other talents from being developed and acts as an irrational barrier to graduation and careers. He proposes alternatives, including teaching facility with figures, quantitative reasoning, and understanding statistics.
Expanding upon the author’s viral New York Times op-ed, The Math Myth is sure to spark a heated and needed national conversationnot just about mathematics but about the kind of people and society we want to be.
“Hacker’s accessible arguments offer plenty to think about and should serve as a clarion call to students, parents, and educators who decry the one-size-fits-all approach to schooling.” Publishers Weekly, starred review
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About the Author
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The "M" in STEM
This century finds America in a struggle to preserve its pride and prestige. Almost daily, evidence accrues that the United States lacks the resources to reign over world affairs. Even minor countries feel free to display their disdain. The years 1900 through 2000 were recognized as America's century. In its first half, we led the world in manufacturing and living standards. In the second, we surpassed in education and military might. But the era ahead holds no comparable promise. Other countries are already matching us in ability, efficiency, and vigor.
For example, we are warned:
In less than fifteen years, China has moved from 14th place to second place in published research articles.
General Electric now has most of its R&D personnel outside the United States.
Only four of the top ten companies receiving United States patents last year were firms based in this country.
The United States ranks 27th among developed nations in the proportion of undergraduate degrees in science or engineering.
Hence the search for solutions to arrest incipient decline. Piled on my desk are reports from august committees and commissions bearing titles like The Gathering Storm, Before It's Too Late, and Tough Choices or Tough Times. All call for national revival and renewal. It's revealing that we hear no worries that the United States may lag in literature or the arts, or that no concern is voiced over declining enrollments in philosophy and anthropology. Rather, the focus is on the now well-known acronym STEM, symbolizing competences the coming century will ostensibly require. We must, we are counseled, devote more of our minds, careers, and resources to science, technology, engineering, and mathematics.
Even a decade ago, the Business Roundtable was urging that we "double the number of science, technology, engineering, and mathematics graduates with bachelor's degrees by 2015." We've passed that year, but awards in those fields have barely budged. More recently, a panel appointed by President Barack Obama asked for another ten-year effort, this time to add "one million additional college graduates with degrees in science, technology, engineering, and mathematics." Where the missile race tallied nuclear warheads, now the countdown is STEM diplomas.
THE SOLUTION: AZIMUTHS AND ASYMPTOTES
Indeed, mathematics is the linchpin, heralded as the key to the other three. Thus we're told that if our nation is to stay competitive, on a given morning all four million of our fifteen-year-olds will be studying azimuths and asymptotes. Then, to graduate from high school, they will face tests on radical notations and elliptical equations. All candidates for bachelor's degrees will confront similar hurdles. Mathematics, we are told, will armor our workforce in a merciless world. Its skills, we hear, are foundational to innovation and a lever in the international arena. In an age of high-tech weaponry, azimuths can turn the tide more than human battalions.
On these critical fronts, our young people are no match for their agemates across the globe, with American mathematics scores lagging behind even Estonia's and Slovenia's. A Harvard study calculates that if this shortfall persists, our gross domestic product will drop by 36 percent. The American Diploma Project reports that proficiency in algebra will be needed in 62 percent of new jobs in the decades ahead. We are further warned that there's already a shortage of graduates with STEM skills. As a result, vital work is being sent abroad, or credentialed immigrants are enlisted to fill the vacancies. If coming generations want a quality of life they feel is their due, they must be prepared to master what many find the most difficult of all disciplines.
Back in 1841, a Scotsman named Charles Mackay (it's his spelling) published a book called Extraordinary Popular Delusions and the Madness of Crowds. He showed how hoaxes, frauds, and hallucinations come in varied guises, from stalking witches to marching off to wars. In more sophisticated times, delusions must show a surface plausibility if they are to ensnare an ostensibly educated populace.
Among our era's delusions are the powers ascribed to mathematics, spurred by a desperate faith in skills abbreviated by the STEM acronym. Together, they have animated a major mythology of our time. Like all myths, they start with a modicum of truth and can be beguiling on first reading. In the chapters ahead, I will show why these beliefs, even when sincerely held, are wholly or largely wrong, lacking in factual support, and usually based on wishful logic. More consequential, these illusions and delusions are already taking a heavy toll on this country, most markedly on the humane spirit that has made America exciting and unique.
FEARED AND REVERED
The Math Myth began nearly twenty years ago, when I started making notes, conducting interviews, and collating files. For much of the time, it had an intermittent schedule, competing with other projects. This changed in 2012, when editors at the New York Times heard about what I was up to and asked me to write an opinion article on mathematics. Responses poured in, at close to record levels, which told me it was time to finish the book. So I did, and here it is.
I mention the early beginning because even with the passage of years much of the terrain remains unchanged. Interviews conducted at the outset remain fully relevant today, as are facts and figures I amassed. If anything, myths about mathematics — the central subject of this book — have become more entrenched. This is why I believe that The Math Myth is needed. This country has problems. But more mathematics isn't one of the solutions.
Other books of mine have ranged broadly, from race and wealth to corporate power and the gulf between the sexes. I've also sojourned in philosophy, writing on titans like Plato, Hegel, and Rousseau. So why now mathematics? My answer is that I've found it an absorbing example of how a society can cling to policies and practices that serve no rational purpose. They persist because they become embedded, usually bolstered by those who benefit. Nor are the issues entirely academic. Making mathematics a barrier ends up suppressing opportunities, stifling creativity, and denying society a wealth of varied talents.
Although I have taught in a department of mathematics at a respected college, strictly speaking, I'm not a mathematician, in that I have no degrees in the discipline. Still, I can admit to being a social scientist, which has always had a quantitative side. I also have a fair reputation for being agile with numbers. But this is not a mathematics book, in the sense of being a textbook or a volume on the beauties of topology. Rather, it is about mathematics, as an ideology, an industry, even a secular religion.
As a long-gone wag once put it, mathematics is both feared and revered. Feared by those who recall it as their worst academic subject, if not a class that blighted their day. Mathematics is also revered, as an inspiring achievement, all the more esteemed because so much of it is beyond our ability to grasp. Amid this aura of awe, it's easy to argue that even more of it should be taught and learned. Yet on the other side is the obdurate fact that of the millions of high school and college students subjected to mathematics, distressing proportions are slated to fail.
This book asks a seemingly ingenuous question: why do we impose so prolonged a sequence of a single discipline, with no alternatives or exemptions? Given our skeptical age, I find it curious that hardly anyone has asked.
The mathematics regimen is already well entrenched. Under current expectations, every young American will study geometry, trigonometry, plus two years of algebra, with talk of adding calculus to the menu. These requirements have consequences. Currently, one in five of our young people does not finish high school, a dismal rate compared with other developed countries. Of those who manage to graduate and decide on college, close to half will leave without a degree. At both levels failure to pass mandated mathematics courses is the chief academic reason they do not finish. (Notice the italics. There are other causes, including prison and pregnancies.)
The usual responses reflect this country's can-do spirit. Calls are heard to return to rigor and end feel-good nostrums. Our goal should be to make the entire nation mathematically adept, starting early with those of school age. Just as wars were declared on poverty and drugs, so our classrooms must be viewed as battlegrounds, armed with more qualified teachers and stringent curriculums.
Clarions like these gave birth to the Common Core State Standards, which at this writing hold sway in more than forty states. On coordinated dates, all public school students in the states will take the same or parallel tests on specified subjects. The most decisive tests — gauged by the possibility of failure — will be in mathematics, where pupils will confront questions like these:
To believe that such equations are a solution is yet another instance of self-delusion. I will propose that we can deploy our material and human capital in better ways. Our goal should be to keep our young people in high school and later in college, where they can discover and develop their talents At this point, we're telling them that they must unravel reentrant angles and irrational numbers if they want a high school diploma and a bachelor's degree. I'll be offering alternatives.
"THE GREAT BOOK OF NATURE"
In no way is this book "anti-mathematics," if such a stance is possible. I would be elated if everyone understood what mathematics is and does, along with the breadth and depth of its achievements. Sadly, few mathematicians seek to evoke this appreciation, whether in their classrooms or to wider audiences. I will argue that if mathematics is to join the liberal arts, it needs to meet the rest of the world halfway.
I'd also like everyone to appreciate how mathematics under-girds our lives. Peter Braunfeld, on the faculty at the University of Illinois, once remarked to me that "our civilization would collapse without mathematics." Its models design racing cars, tell retailers how many sweaters they need for a holiday season, conjure crowd scenes in fantasy movies, and guide hotels on rates that will fill as many rooms as possible. I'd also like everyone to know how trigonometric functions enabled the Wright brothers to keep their plane aloft for a historic minute, just as calculus now allows a 450-ton jetliner to soar nonstop from Hong Kong to New York. Courtesy of equations we never see, our lives are safer and more varied and interesting. But this isn't being taught by mathematicians, because it doesn't align with their rigid syllabus. Nor are they inclined to allow anyone else to tell this story. As a result, one of the most exciting sagas of our time remains unknown and untold.
So, yes, I honor mathematics as an awesome intellectual enterprise. I will gladly have my tax dollars go for research on Goldbach's Conjecture and parametric cycloids. Not the least of my concerns about STEM is that it casts mathematics largely as an arm of technology, in a global competition gauged to gross domestic product, military might, and electronic surveillance. I would like all liberal arts students, indeed everyone, to know what Galileo meant when he called mathematics "The Great Book of Nature." Or why Isaac Newton subtitled his Principia Mathematica a study in "Natural Philosophy." I would urge schools and colleges to draw on perspectives like these and accept them as coequal to precalculus and trigonometry.
THE IMPORTANCE OF ARITHMETIC
There are some other observations I'd like to include before this book gets under way. One is to highlight the difference between mathematics and arithmetic, if only because the terms tend to get conflated. For instance, we speak of "mathematics scores" of third-graders, when all they were tested on was plain-vanilla arithmetic. Mathematics basically begins in high school, extends from geometry through calculus, and ultimately soars to the ethereal pursuits of specialists and scholars. Arithmetic is addition, subtraction, multiplication, and division, followed by fractions, decimals, percentages, and ratios, and the statistics we encounter in our everyday lives. All of us should get arithmetic under our belts in elementary school.
At this point, the challenge is not to immerse more people in more mathematics. Rather, it is that many, if not most, young people and adults are insufficiently agile in arithmetic. Of course, they can add and subtract. But a 2013 study of adults in twenty-three countries found the United States third from the bottom in ostensibly simple tasks like using odometer readings to submit an expense report. If that's a cause for worry — and I believe it is — it needs to be remedied on its own terrain. There is ample evidence that mathematics as it is currently taught does not improve quantitative literacy or quantitative reasoning, or facility with the figures that inform and organize our lives. Even if most American adults once studied algebra, geometry, and phases of calculus, it hasn't enhanced their numerical competence.
What's needed is what I'll be calling adult arithmetic, or what John Allen Paulos has termed numeracy. Nor does this mean revisiting fourth grade. Numeracy can and should be taught at a demanding level. I'll give some examples in my closing chapter, where I'll show how arithmetic is all that's needed to interpret charts in the Wall Street Journal or graphs in The Economist, as well as public documents and corporate reports.
The editors at the New York Times chose to title my article "Is Algebra Necessary?," implying I thought it was not. (For my part, I would have phrased the question "How Much Mathematics Is Too Much?") I admit that in this book I occasionally use "algebra" as a surrogate for the full mathematics sequence. That noted, I want to affirm that basic algebra is definitely necessary for everyone. I use it every day myself. If a twenty-muffin recipe specifies thirty-five ounces of flour and we want only thirteen muffins, how much flour (x) do we need? The equation we jot down — 20 is to 13 as 35 is to x — is elementary algebra, or simply "solving for x." True, this involves only multiplication and division. Still, every teenager and adult should have this skill and understand the reasoning behind it.
The current regimen expects students to master not only x-equations, but also associative properties, squared binomials, and prime factorization. Anthony Carnevale and Donna Desrochers once remarked that there can be such a thing as "too much mathematics." Those words deserve respectful pondering.
OPTIONS AND ALTERNATIVES
Arithmetic has always been a required subject, as it certainly should. But ought mathematics be made optional? Should ninth-graders be allowed to forgo geometry because they've heard it's hard? Paul Lockhart, a virtuoso mathematics teacher at St. Ann's School in Brooklyn, thinks so. "There is no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum," he says of his chosen field. Part of me agrees with him. As a college professor, I teach mostly electives, so I'm spared from facing sullen conscripts. Still, once Lockhart's door is opened, there's the problem that students might also ask to opt out of science, literature, and history, or physical education.
Yes, I want changes to come, but not solely on the votes of fourteen-year-olds. Among its many audiences, this book is addressed to adults who preside over our schools, and parents who support them. I want to urge them to consider alternatives to the current mathematics syllabus. Growing numbers of reputable colleges now accept applications without asking for reports from the SAT or ACT, which says they feel no need to scrutinize mathematics scores. Moreover, the students who decide not to submit them have been found to end up doing just as well in their courses. In this vein, I urge innovative school systems to experiment with allowing several of their high schools to offer arts or humanities diplomas, perhaps starting in the tenth or eleventh grade, akin to options many European systems offer. Such sequences might also create courses in statistics or quantitative reasoning, in lieu of conventional mathematics.
ACROSS A SOCIAL SPECTRUM
When I propose allowing alternatives, it is not to spare young people from lessons they'd rather avoid. A common response is that varying the syllabus will result in "dumbing down" the curriculum. In fact, my proposals don't constitute a downward move at all. On the contrary, they call for smartening up. I will show that versatility with figures is as exacting as trigonometry or geometry, while becoming deft with statistics calls for as much reflective reasoning as advanced algebra.(Continues…)
Excerpted from "The Math Myth"
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Table of Contents
1 The "M" in STEM 1
2 A Harsh and Senseless Hurdle 13
3 Will Plumbers Need Polynomials? 27
4 Does Your Dermatologist Use Calculus? 47
5 Gender Gaps 63
6 Does Mathematics Enhance Our Minds? 81
7 The Mandarins 97
8 The Common Core: One Size for All 117
9 Discipline Versus Discovery 131
10 Teaching, Tracking, Testing 147
11 How Not to Treat Statistics 165
12 Numeracy 101 181