Most of us are intrigued by statistics, at least when they’re presented in a friendly form. That is why USA Today’s front page regales its readers with vignettes like these:
Where American pet dogs sleep: 42% on owners’ beds; 33% on bedroom floors; 22% in another room; 3% outside.
Principal causes of debt: house plus car, 25%; credit cards, 21%; college loans, 12%; medical costs, 12%.
Favorite condiments: ketchup, 47%; mustard, 22%; mayonnaise, 20%; salsa and others, 11%.
Average life spans of paper money: $1 bills, 18 months; $5 bills, two years; $10 bills, three years.
Less frivolously, these statistics appeared in an issue of the New York Times’s Sunday Review:
17.2% (health care share of 2012 gross domestic product)
Over 44,000 (articles published on obesity in 2013) $9.39 and $7.25 (real value of minimum wage in 1968 and 2013)
Up to 27,000 (ski jobs lost to lower snowfall, 1999–2010)
For his wise and witty bestseller, John Allen Paulos used the terms numeracy and innumeracy. He chose these plays on literacy to describe a comparable set of skills he felt too many people lack. The Mathematical Association of America prefers quantitative literacy, which has a more academic ring but comes to much the same thing. Harvard’s faculty, never wishing to be outdone, says it wants all its graduates to be adept at quantitative reasoning, hinting at a higher mental plane. I’ll be examining this aim and claim later on.
In fact, people can be quite nimble with numbers, at least when they involve subjects close to home or that otherwise attract their interest. Sports fans analyze players’ “stats,” parse the odds, take part in betting pools, and draw up budgets for fantasy teams. Others fill in Sudoku grids or factor coupons for the family budget. In my numeracy classes, I’ve seen students with varied aptitudes become agile with statistics. Nor should this be surprising. Numbers are a language, and all of us learn its rudiments early on. Still, if figures are used for counting, we need to be clear about what’s being tallied. You can’t analyze economic inequality meaningfully unless you know how wealth and income are measured and which sources are reliable. I’ll be showing how this can be done.
HOW INNUMERATE ARE AMERICANS?
A 2013 New York Times editorial sought to alarm its readers by stating that “only 18 percent of American adults can calculate how much a carpet will cost if they know the size of the room and the per square yard price of the carpet.” If this is so, and assuming we’re dealing only with straight dimensions, it’s truly disgraceful that 82 percent couldn’t compute the cost correctly. In our era, most adults have completed high school, with at least some geometry and algebra. Yet we’re being told they can’t even apply the arithmetic they learned in elementary grades.
But before joining the lament, let’s look at the question that spurred the accusation. It’s from the most recent U.S. Department of Education survey of adult literacy, which also covers quantitative skills. The centerpiece of the problem is an advertisement, reproduced here along with the question as it was worded.
It soon emerges that what’s being tested is as much literacy as numeracy, as real-life situations often are. In this case, it starts with a verbal feint. You must stop to wonder whether you should reduce the announced price of $9.49 by 41 percent, which would bring it down to $5.60 per square yard.
It’s best that you don’t. The 41 percent “taken off” has already been factored in, since its earlier price was $15.99. Evidently, at least some people made this misstep, contributing to the 82 percent error rate. The next ploy is wilier. Note that the dimensions for the floor area you want covered are given in linear feet. Yet the carpet’s price is given per square yards. So it’s only a first step to say the area of the carpet measuring nine by twelve feet works out to 108 square feet. You then must have the insight to transform those 108 square feet into 12 square yards, which also requires recalling how many square feet there are in a square yard. (The correct answer is 12 yards X $9.49/yard = $113.88.)
The high error rate on the carpet question should lead us to ask why many adults got the answer wrong. In fact, what it reveals is that any effort to improve quantitative reasoning must stress careful reading as well as instilling computational skills. At the same time, the question we’ve examined calls for only elementary-grade arithmetic, not any application of higher mathematics.
In the very same test, 52 percent of the sampled adults were able to compute a discount on a heating oil delivery, and 69 percent correctly interpreted a graph showing income disparities. But these were straightforward questions, with no gimmicks, tricks, or unit conversions. Even so, success rates of 52 percent and 69 percent warrant two cheers at best and certainly give rise to concerns.
My contention is that time and effort that might have been devoted to nurturing numerical agility has been given over to asymptotes, rational exponents, and other esoteric topics that only those who choose to major in mathematics in college will ever encounter again. The carpet problem is evidence that the logic ostensibly learned during years of geometry and algebra didn’t help these adults to distinguish square feet from square yards. This failure calls to mind a remark by Deborah Hughes-Hallett, a professor of mathematics at the University of Arizona: “Advanced training in mathematics,” she reminds us, “does not necessarily ensure high levels of quantitative literacy.”
EQUATIONS UNDER THE HOOD
For this chapter, I will be focusing on two sorts of statistics. I’ll call one kind public statistics and the other academic statistics. The first refers to numbers that impinge on our lives, either personally or in our social experience, like those at the opening of this chapter, which told us student loans account for 21 percent of personal debt, or that healthcare was more than 17 percent of GDP in 2012. To understand and use public statistics requires only sixth-grade arithmetic and a skeptical eye for suspicious sources and ideological bias.
I’ll be using the term academic statistics to describe a scholastic discipline increasingly taught in high schools and colleges, and continuing in postgraduate seminars and academic research. Academic statistics is a domain of teachers, professors, and scholars, who are devoted to their methodologies and have made their pursuit a respected career. Most had their initial training in mathematics, which they regard as integral, indeed indispensable, to what they study and teach. Whether for reasons of scholarship or status, the overwhelming majority of those pursuing academic statistics have no sympathy for modes of analysis that simply need arithmetic.
The following—which I’ve slightly abridged—was written by Edward Frenkel, a mathematics professor at the University of California’s Berkeley campus, shortly after an article of mine appeared. (For the record, I supported retaining algebra, but proposed that alternatives also be offered.)
In a recent op-ed, Andrew Hacker suggested eliminating algebra from the school curriculum and instead teach how the Consumer Price Index is computed. What seems to be completely lost on Hacker is that the calculation of the CPI is in fact a difficult mathematical problem which requires deep knowledge of all major branches of mathematics including advanced algebra.
So indulge me for a moment while I show how I used the Consumer Price Index with my own students. I took my cue from Richard Scheaffer of the University of Florida, a former president of the American Statistical Association. “The key to statistical thinking,” he has said, “is in the context of a real problem and how data might be collected and analyzed to help solve that problem.”
Comparing CPI findings for 1994 and 2012, my students and I focused on how and where households in those years were spending their money. We looked at forty-five categories, ranging from fresh fruits and vegetables (down 36 percent) to “personal care” (up 118 percent). The goal was to deploy numbers to analyze personal and social changes. In our class discussions, a ground rule was that you had to cite figures to support your interpretations. Thus, on one side they found less spending on alcoholic beverages (down 40 percent). On another front, there was a rise in mostly sugary soft drinks (up 18 percent). Now that we knew the figures, how to explain these shifts?
As we saw, Frenkel sees the CPI in an entirely different light. What interests him are the complex equations used to create the data in the first place and to keep it up to date. Thus in his article, he printed out:
PCannual = [(IXt+m / IXt)12/m -1] *100
This equation, he explained, is used to factor in inflation rates for items ranging from frozen foods to home mortgages. In his view, one can’t begin to use the Consumer Price Index without first absorbing enough algebra to parse equations like this one.
Here’s the issue. Frenkel is talking about the mathematics used to set up the CPI and keep it updated. While I acknowledge and appreciate those efforts, my aim is to use CPI’s figures to sharpen our understanding of ourselves and our society. I cannot see what’s gained in making undergraduates master advanced equations before they can start discussing why Pepsi-Cola is replacing Budweiser. No one would say we can’t use our laptops without first studying the chips and circuits behind the screen.
IS YOUR PHONE BILL TOO HIGH?
I attended the 2013 meeting of the National Council of Teachers of Mathematics, held that year in Philadelphia. It was a large and buoyant gathering, where faculty members from high schools and colleges exchanged ideas and promoted innovations. One spirited session was on Quantitative Financial Literacy, featuring a recently created course at a suburban New York high school. It was billed as “the perfect third or fourth year course for everybody!” While pressure is on to have all students take a fourth year of mathematics, it’s become evident that at least some can’t cope with calculus. Hence this alternative, announced as having lots of “real world applications,” such as savings account interest, withholding tax rates, and depreciation on a car. This sounded great. But of course the devil is in the details. So let’s look at one of its “real world” lessons (see the illustration on the following page).
I didn’t know whether to laugh or cry. Or both, if that is possible. Can educators believe that telephone owners, youthful or otherwise, will construct equations like the ones on the next page to check their charges? At the session’s end, I asked the presenters why this and all their other “financial literacy” lessons called for at least intermediate algebra. “We had to,” one of them told me. “Only then would the course be approved as a mathematics offering.” What we’ve seen here is how academic statistics not only impresses its stamp, but thwarts using arithmetic-based methods, which would be just as effective—and a lot more applicable—for public numeracy.
ADVANCING PLACEMENT
We turn to schools with the hope that they will transform young people into knowledgeable and thoughtful adults. Along with safe driving and safe sex, statistics is being added to curriculums. In 2013, fully 169,508 pupils took advanced placement courses in the subject, nearly triple the 58,230 a decade earlier in 2003. (Statistics about statistics!) As this growth continues, it might be argued we are on course for creating a statistically sophisticated citizenry.
On first hearing, this could seem consonant with what I’ve been proposing. So I decided to look further and arranged to attend several AP statistics classes at a well-regarded high school near my college. All AP courses use a common curriculum, designed by the College Board, which retains panels of professors to specify the syllabus.
The reason for the uniformity is that a uniform examination is given in each AP subject at the end of the semester. The presumed purpose of these tests is to tell colleges which entering students are qualified to go straight on to advanced courses. Another is to allow applicants to impress admissions officers with an ambitious transcript. Here is a typical AP examination question on statistics:
Given this way of approaching statistics, perhaps it’s not surprising that, of the 169,508 students who took the 2013 test, only 58 percent obtained passing grades. And we should recall that this is an optional class, intended for students aspiring to competitive colleges. So it’s in order to ask what is ensured by installing a national program with so high a failure rate. A common answer is that setting a high bar betokens a commitment to stringent standards. Of course, there are plenty of contests that end with very few winners, something young people already know. But here we’re talking about schools, which we ask to educate their pupils. If nothing else, those who see nothing wrong with a 42 percent failure rate should let the rest of us know their reasoning. This issue will reappear when decisions are made about the mathematics scoring system in the Common Core.
A FOUNDATION FALTERS
The Carnegie Foundation for the Advancement of Teaching was alarmed. Its officials had found that virtually all of the 1,191 community colleges in the United States have a singular requirement. Incoming students must obtain specified scores on standardized mathematics tests before they can start taking courses for credit. Carnegie’s researchers discovered that fully 60 percent of would-be community college entrants didn’t meet the bar and so were assigned to remedial sections. Even worse, 80 percent either failed these noncredit classes or subsequently failed when they took a regular mathematics course. Either way, not passing algebra meant they couldn’t start college.
So Carnegie selected nineteen community colleges, in states ranging from Washington to Florida, and offered them a deal. The colleges were asked to overlook the mathematics scores of some of their entrants and exempt them from remedial sections. Instead, they would be allowed to take a class Carnegie had created, called Statways, which it described as an introductory statistics course. Funding for full-time instructors would be provided, with the classes no larger than twenty students. During two academic years, 2011 through 2013, a total of 1,817 students were enrolled in this project.
When Statways was announced in 2010, I was heartened and impressed. In a Chronicle of Higher Education article, its creators affirmed their desire to “help solve the remedial-math problem” that takes so heavy a toll. They said they had designed “a statistics pathway that will provide a challenging alternative,” showing students how “statistical reasoning” could be “an essential aspect of their everyday lives.” These words were what I had long hoped to hear: statistics for citizens. So I looked forward to Statways’ results.
Four years later, Carnegie released a report on Statways. Perhaps the most telling statistic of its own was that of the 1,817 students, only 920—almost exactly half—had passed with a grade of C or higher. The others got Ds or Fs, or dropped during the semester. True, this is somewhat better than the overall failure rates in remedial mathematics. But the Carnegie experiment was closely monitored, with small classes and professional support. What happened?
The answer wasn’t hard to find. For reasons not given, Carnegie chose to align its syllabus with a document called Guidelines for Assessment and Instruction in Statistics Education. These protocols, which fill two volumes, are promulgated by the American Statistical Association, whose leading members are university-level professors. Some of its tenets seem sensible, like “using real data of interest to students is a good way to engage them in thinking about relevant statistical concepts.” After all, “real data” is what ordinarily distinguishes statistics from formal mathematics. But it turned out that these remarks are only the tip of the Guidelines iceberg. Its main concern is to strengthen the domain of academic statistics. Here is a sampling of the subjects that Carnegie’s community college freshmen were expected to master:
In a word, Carnegie’s alternative for struggling students ended up as a standard academics statistics course, as promulgated by a panel of research professors: the mandarins of statistics. It remains to wonder why the foundation settled for so orthodox a path. Given its shielded status outside the academy, it’s dismaying that Carnegie embraced the rubrics of a self-referential discipline, rather than devising templates of its own for teaching statistics to potential dropouts. Equally disappointing was its unwillingness to urge community colleges to abandon their fatal mathematics hurdles—that would have been an interesting experiment. After all, Carnegie itself had figures showing their ruinous consequences. In the end, the foundation settled for launching what it billed as a lifeboat, although only half of its occupants survived.
HARVARD: TALK OR WALK?
Several years ago, Harvard University decided—at least on paper—that it wanted all of its students to graduate with a grounding in quantitative reasoning. I like this phrasing. It seems akin to quantitative literacy, but with a cerebral edge. So I took a look at how Harvard’s faculty implemented a promising idea.
Harvard was not the first top-tier institution to install a cross-disciplinary “big picture” field of study. Columbia University’s undergraduate college has long had a course, which all its students must take, called Contemporary Civilization. Since its syllabus cuts across subject lines, faculties from across the campus participate. On a given morning, one might encounter a professor from economics leading a discussion of Macbeth. Across the hall, a psychologist would be parsing the Punic Wars. One of the premises of the Columbia course was that its subject—civilization—could be taught by anyone accredited in the arts and sciences. In a similar vein, Amherst College had its Evolution of the Earth and Man, which was required of all sophomores. It was taught by faculty from astronomy, geology, and biology, who rotated lectures and led discussion sections. Their collaboration exposed students to a shared scientific culture.
These might seem some possible formats for a Harvard-wide course on Quantitative Reasoning. If it followed Columbia’s and Amherst’s paths, it would have a common framework and benefit from insights of various disciplines. Unfortunately, any such proposal would face a constraint. At Harvard, once professors have full status, they cannot be compelled to take on tasks they find uncongenial. This sheltering is often seen as vital to academic freedom. While a majority of the faculty formally voted for the quantitative requirement, it soon emerged that few had any desire to assist in its teaching.
There would be nothing like a single QR 201 that all undergraduates would have to take. Instead, in a recent year I sampled, a total of fifty-three courses were listed as satisfying the QR requirement. By my count, forty-four of these were existing listings, most long offered by departments. Over half were in mathematics, including courses like Honors Abstract Algebra and Multivariate Calculus, despite the fact that these fields have no palpable connection with quantitative literacy or reasoning.
Preexisting classes in computer science, engineering, and physics were also allowed. Economics, government, and sociology added the methodological courses they already had in place for their own majors. But a closer look showed these courses to be highly mathematical and geared to specialized research, far removed from the kinds of numbers most Harvard graduates would encounter in their personal and public lives. Other professors refurbished offerings they were already giving, like deductive logic and computer programming.
By my count, members of the Harvard faculty created only five new courses that seem to capture the spirit of quantitative reasoning. One, by a professor of statistics, was called “Your Chance for Happiness (or Misery)” and explored the numbers behind our becoming rich or poor, lonely or loved, frustrated or satisfied. Another, by a professor of astronomy, admittedly more specialized, dealt with “The Visual Display of Quantitative Information,” which showcased the interplay of aesthetics and accuracy. A philosophy professor created “What Are the Odds?” which dealt with issues like risk, statistical inference, and how correlation differs from causation. A team representing the fields of philosophy, linguistics, and computer science collaborated on “Making Sense: Language, Thought, and Logic.” The fifth, taught by a computer scientist, was deftly called “Bits: Information as Quantity, Resource, and Property.” While none of these five courses claimed to cover the whole quantitative scene, they reflected the personal passions of the professors, which always makes for good teaching.
Still, the fact remains that these were only five out of the fifty-three QR offerings, with the rest being conventional courses, with or without new titles. The upshot at Harvard is that trying to install a truly new vision of quantitative reasoning will be an uphill battle. The chief problem, notably at research universities, is that professors get embedded in their turfs, which is where professional careers are made and reputations are burnished. To create a course that isn’t tied to an established discipline calls for a self-confidence not all academics have. Even holders of tenured chairs worry lest they be charged with popularizing or oversimplifying, not to mention being unscholarly. Imagine the chatter at the faculty club on hearing that a colleague was assigning Darryl Huff’s cheerful classic How to Lie with Statistics.
All told it’s not surprising that of Harvard’s 1,334 full-time faculty, only thirty-eight decided to try their hand at a quantitative course. Of the ninety-four members of its mathematics department, only two chose to join the QR team. And doubtless concerned about their colleagues’ opinions, they made algebra a prerequisite for their offerings. Attitudes like these help to explain why a survey by the Crimson, the student newspaper, found the quantitative requirement to be “the Harvard humanities students’ biggest nightmare.”
THE PRICE OF PRESTIGE
As with mathematics, professors and teachers of statistics see it as their mission to maintain the purity of their discipline. Hence their penchant for keeping their subject exceedingly arcane, on a plane far from the statistics encountered by even the general run of college graduates. In their view, if statistics is to be taught, it must always be at this rarefied level.
For my part, I admire academic statistics and rely on its equations in much of my work. Thus I have drawn on Gini Ratios to compare income distributions (Norway gets .260 versus .410 for the United States). And I’ve recently used the Pearson Coefficient to compare mathematics scores of countries with their infant survival rates. (It came to -0.093, which means no correlation at all.) But I don’t compute these formulas myself. Rather, I type the numbers into a readily available website and watch the results pop out. (Warning: you had better be familiar with the data, to gauge whether what comes out seems reasonable.)
If we want more people to be versatile with what I’ve called public statistics, we won’t attain that goal if we first demand that they become expert with least-square regressions and line multiple representations. Rather, let’s examine what happens in real life. Suppose a local newspaper prints a table showing how medical expenditures vary across the states. You may find that these figures convey some interesting information. However, it’s unlikely you will try to apply the H0: μ = H versus Ha that you had to imbibe in an AP statistics course.
So why do statistics mandarins insist on so academic a syllabus, even for high school students? Here we’re back to safeguarding status and preserving purity. Were they to admit that only arithmetic is needed to become statistically adept, it could undercut their claim to scholarly standing. The same holds at high schools, which promote academic statistics for a fourth year of mathematics. At either level, the needs of students are being sacrificed to preserve the prestige of their instructors.