The Wrong Way to Teach Math (Andrew Hacker)

Struggles over how best to teach math (or science, history, and English) have marked decade after decade of school reform. Mental discipline vs. learning by doing and other dichotomies have characterized the professional and lay public’s back-and-forth over teaching this subject. These public debates over how best to teach math and other subjects not only have a long history but also mirror the unending struggle over the role that schooling “should” play in a society anchored in democratic capitalism. In the past thirty years of school reform aimed at marrying public schooling to economic growth, the “should” of teaching algebra, geometry, and calculus has tilted toward mental discipline because of the apparent necessity of getting a college degree in order to land a decent paying job. Andrew Hacker’s article raises again the unending issue of connecting school subjects to the real world that students inhabit rather than what adults believe should be studied for its abstract and economic benefits.

This op-ed appeared in the New York Times, February 27, 2016. Andrew Hacker teaches political science and mathematics at Queens College and is the author of the forthcoming book “The Math Myth and Other STEM Delusions,” from which this article was adapted.

HERE’S an apparent paradox: Most Americans have taken high school mathematics, including geometry and algebra, yet a national survey found that 82 percent of adults could not compute the cost of a carpet when told its dimensions and square-yard price. The Organization for Economic Cooperation and Development recently tested adults in 24 countries on basic “numeracy” skills. Typical questions involved odometer readings and produce sell-by tags. The United States ended an embarrassing 22nd, behind Estonia and Cyprus. We should be doing better. Is more mathematics the answer?

In fact, what’s needed is a different kind of proficiency, one that is hardly taught at all. The Mathematical Association of America calls it “quantitative literacy.” I prefer the O.E.C.D.’s “numeracy,” suggesting an affinity with reading and writing.

Calculus and higher math have a place, of course, but it’s not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads. Ours has become a quantitative century, and we must master its language. Decimals and ratios are now as crucial as nouns and verbs.

It sounds simple but it’s not easy. I teach these skills in an undergraduate class I call Numeracy 101, for which the only prerequisite is middle school arithmetic. Even so, students tell me they find the assignments as demanding as rational exponents and linear inequalities.

I’m sometimes told that what I’m proposing is already being covered in statistics courses, which have growing enrollments both in high schools and colleges. In 2015, nearly 200,000 students were taking advanced placement classes in statistics, over three times the number a dozen years ago. This might suggest we are on the way to creating a statistically sophisticated citizenry.

So I sat in on several advanced placement classes, in Michigan and New York. I thought they would focus on what could be called “citizen statistics.” By this I mean coping with the numbers that suffuse our personal and public lives — like figures cited on income distribution, climate change or whether cellphones can damage your brain. What’s needed is a facility for sensing symptoms of bias, questionable samples and dubious sources of data.

My expectations were wholly misplaced. The A.P. syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors. Many students fall by the wayside. It’s not just the difficulty of the classes. They can’t see how such formulas connect with the lives they’ll be leading. Fewer than a third of those enrolled in 2015 got grades high enough to receive credit at selective colleges.

Something similar occurred when the Carnegie Foundation for the Advancement of Teaching created a statistics course for 19 community colleges in 2012. It was advertised as an alternative to remedial algebra, with its sadistic attrition rates. In Statways, as it was called, here is some of what students were asked to master: chi-square test for homogeneity in two-way tables, line multiple representation of exponential models. Even with small classes and extra suppor t, almost half of the students got D’s or F’s or dropped the class.

The Carnegie and A.P. courses were designed by research professors, who seem to take the view that statistics must be done at their level or not at all. They also know that citizen statistics is not the route to promotions. In the same vein, mathematics faculties at both high schools and colleges dismiss numeracy as dumbing down or demeaning. In fact, figuring out the real world — deciphering corporate profits or what a health plan will cost — isn’t all that easy.

So what kinds of questions do I ask my students…?

I … ask them to discern and analyze changing trends. Each January, the National Center for Health Statistics releases its hefty “Births: Final Data.” Its rates and ratios range from the ages of parents to methods of delivery. I ask students to scan these columns, looking for patterns. They found, for example, that women in Nebraska are averaging 2.2 children, while Vermont’s ratio is 1.6. Any theories?

Other tables focus on changes over time. Fertility rates for white and black women in 1989 stood at 60.5 and 84.8 per thousand, a discernible difference. By 2014, they were 59.5 and 64.5, a much smaller gap. There’s a story here about how black women are reconfiguring their lives.

Finally, we talk about how math can help us think about reorganizing the world around us in ways that make more sense. For example, there’s probably nothing more cumbersome than how we measure time: How quickly can you compute 17 percent of a week, calibrated in hours (or minutes, or seconds)? So our class undertook to decimalize time.

Imagine if we had a 10-day week, each day consisting of 10 hours each. The class debated whether to adopt a three-day weekend, or to locate an “off-day” in midweek. Since a decimal week would have 100 hours, 17 percent is a flat 17 hours — no calculator required. You have to think both numerically and creatively if you want to, say, chuck out our current health care system and model the finances of a single-payer plan.

Mathematicians often allude to “the law of the excluded middle” (a proposition must be true or false). The same phrase could be applied to a phenomenon in our own backyard. We teach arithmetic quite well in early grades, so that most people can do addition through division. We then send students straight to geometry and algebra, on a sequence ending with calculus. Some thrive throughout this progression, but too many are left behind.

The assumption that all this math will make us more numerically adept is flawed. Deborah Hughes-Hallett, a mathematician at the University of Arizona, found that “advanced training in mathematics does not necessarily ensure high levels of quantitative literacy.” Perhaps this is because in the real world, we constantly settle for estimates, whereas mathematics — see the SAT — demands that you get the answer precisely right.

Indeed, it often turns out that all those X’s and Y’s can inhibit becoming deft with everyday digits.



Filed under how teachers teach, school reform policies

8 responses to “The Wrong Way to Teach Math (Andrew Hacker)

  1. Laura H. Chapman

    I am not competent to judge the best way to teach math, but I get the legacy of the “new math” post Sputnik and the long tail of that Brunerian trust-the- experts-in-the-disciplines to frame what to teach and how. It is still present in schools.
    In graduate studies I did poorly on my first statistics course with referents on the production of nuts and bolts to “make it real.” I managed to do much better when the course content was tied to my interests, meaning differences in the performance of students, classes, and schools, test construction issues and the like.
    I had the good fortune of surviving those courses and also taking courses with one of the math/science/research mavericks of that era, Egon Guba, who was then tinkering with qualitative research. He used three of his students in the arts to test ideas, including the question of how to address testing in the arts. Two of us went on to become consultants on the first and second NAEP tests in the arts.
    In recent years I have become much more informed than I ever expected to be about the role of economists in shaping the metric-driven educational landscape and defending VAM as well as the sustained navel gazing of many in the qualitative research camp.
    In any case, the latest OECD press on math testing is making news.

    • larrycuban

      Thanks for comment on your own experiences in grad school and Egon Guba, Laura. It is precisely these international comparisons on math, science, technology that feeds policymakers and pundits in the U.S. to call for more disciplinary knowledge and continue the “wars” over how best to teach academic subjects.

  2. Karen Anne Kramer

    I taught my girls how to do Math starting at 3 years of age. With seashells and stones. One Doctor and 2 others Business Professionals. Thank God for nature and common sense !

  3. Chester Draws

    The issue with teaching “practical” maths is two-fold.

    Firstly, the minute it becomes a school subject it more or less becomes detached from reality for many students. The teacher may understand the relevance, but if it is going to be tested it just becomes another chore (and if it isn’t going to be tested it is going to go the way of all non-tested subjects).

    More importantly, real life maths is always context dependent and that context changes. We might calculate a budget for someone, and round the figures for income because budgeting shouldn’t deal in cents. Yet the next question is what a person should be paid an hour for a given salary, and woe betide any finance officer who doesn’t pay to the exact cent!

    So every situation has to be taught separately, which isn’t remotely feasible.

    Or we do what we have always done — teach context dependent stuff after school in the context it sits in. To help do that the people need various pieces of set skill — how to do areas, how to use a formula etc. You know, all the stuff that we actually do teach in school.

    There’s plenty of evidence that teaching in contexts actually inhibits generalised understanding and that it doesn’t transfer across contexts (as any physics teacher who has to reteach maths skills the students already know, but think of as “maths”).

    So lessons in fertility planning are likely to lead to a better understanding of fertility, and little else, regardless how enjoyable they are. Try an experiment — teach in one context, say fertility, and then test in another that is mathematically identical but socially different. They will struggle to use their apparent knowledge. (For the record, I have tried this. It is very disappointing.)

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s