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How Much Math Is Too Much?

Andrew Hacker has a bombshell opinion piece in last Sunday's NY Times, arguing that teaching algebra to all students is a wasted effort.

Students are routinely told that math is a gateway subject—you have to take advanced math to get into a good college—and Hacker suggests that this is precisely the problem:

In the interest of maintaining rigor, we're actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources.

Hacker clarifies that he is not in favor of vocational tracking, and he believes all students can learn algebra, but forcing them to do so is a waste of resources given how little they will actually use it. He concludes:

Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions. Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven't found a compelling answer.

The reaction to Hacker's op-ed has been swift and passionate. RiShawn Biddle of Dropout Nation sees this as a problem of teaching quality:

The fact that Hacker doesn't consider that the nation's education crisis -- and the low quality of math instruction and curricula -- is the underlying reason why so many kids are failing algebra makes his piece an entirely laughable polemic.

Hacker does in fact argue that we need better math instruction, but what Biddle and others seem to miss is his central thesis: That algebra simply is not very useful to the vast majority of us. When something is not useful, too hard, expensive to teach, and mandatory, we have a problem.

I have to agree with Hacker on this point. I played the "math as gateway" game, meeting my university's "rigorous" math requirements for physical science majors. Yet I haven't actually used anything beyond Algebra I in any context, educational or work-related, in the years since.

We all can agree that a good education should teach you to work hard and have a broad appreciation for many subjects, from Shakespeare to political science to quadratic equations. How far do we push this, though, when it comes to learning that has no practical application for the vast majority of us?

It's worth remembering that there are many areas of valuable societal knowledge that you and I were not expected to master in high school. Did your high school chemistry teacher expect you to be able to determine multiple processes to synthesize complex organic compounds? Didn't think so. Advanced math is the outlier, and I would suggest that it's a poor choice. Algebra I is perfectly adequate for a well-educated society, and those who need more math for their college major or career can always take more.

Our curriculum is a mile wide and an inch deep, and you don't increase rigor by increasing breadth, but depth. If we required Algebra I of high school students, but allowed them to work up to it in more depth over time and pass it as seniors, I suspect our success rate would be much higher.

But Biddle goes on to make a great point:

one of the reasons why so many kids are innumerate is because they are also illiterate. The very skills involved in reading (including understanding abstract concepts) are also involved in algebra and other complex mathematics. Those students who are struggling in reading almost always struggle with math (as well as science), especially as they move into sixth grade, when math lessons transition from computation to word problems that require thinking through abstractions. And reading (along with strong background knowledge gained from history and other subjects) is critical is to building the muscle needed to handle math. In short, contrary to Hacker's conceit, the "brainpower" of these young minds (including the 150 who drop out every hour into poverty and prison) were wasted by American public education's inability to provide high-quality instruction, curricula, and cultures of genius long before they encountered an algebraic equation.

I agree heartily, but Hacker is actually writing about a different issue: advanced high school and college math, not the K-8 foundation that is the backbone of an educated society. Biddle goes too far in arguing the merits of such math in the workplace:

The advent of the Internet and the emergence of quantifiable data is making math skills critical to many white-collar professions; marketers and public relations staffers, for example, have to understand the arcane aspects of statistics in order to analyze data on ad campaigns, while reporters and editorialists need stronger math skills as well. The need for strong math skills has become even more-critical in high-paying blue-collar professions. Welders, for example, need to understand trigonometric equations and think through abstractions (along with master computer programming language such as C and Fortran) in order to ensure that products are shaped properly and fit together upon assembly. Elevator installers and repairmen, who are paid an average of $67,000 annually, must understand electrical and mechanical engineering (which comes from mastering high-level math) for their own jobs. link

Nope, nope, and nope. None of those jobs require advanced trigonometry (welders probably learned all they needed to know about angles in 4th grade), systems of multiple linear equations, arcane statistics, or calculus. (And Fortran? Really?)

Back to Hacker's op-ed:

Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic. But a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above.

I find this estimate to be exceptionally high. I live in Seattle and know numerous professionals in all manner of technical fields, yet I can think of only one person, an aeronautical engineer with a Master's degree, who uses advanced math to actually do his job on a regular basis. For the rest of us, advanced math is like obscure Shakespeare - good to have studied, and certainly something of a badge of honor among the well-educated, but hardly a critical skill for career or citizenship. You should recognize Hamlet references, but not every quote from Henry VI, Part 2.

student math paper, photo by attercop311 on Flickr

What about the benefits for people in jobs that don't actually require advanced math? Biddle writes:

Meanwhile in arguing that "potential poets and philosophers" (and just about everyone else not taking on the sciences) shouldn't have to master algebra, Hacker ignores the reality that strong math knowledge (along with literacy and science understanding) is critical to understanding the abstract concepts at the very heart of civilization and society. An adult with a working student with a working understanding of algebra will also be able to understand why the Laffer Curve matters in discussions about tax cuts, and be able to weigh the benefits and consequences of the Affordable Care Act on government, the private sector, and civil society. It also allows for the social mobility that has helped America bend the arc of economic and social history toward progress. A bus driver who understands calculus and trigonometry can move up socially, converse with executives, serve as a leader in his community, and even help his kids make their way into the middle class.
Again, Biddle vastly overstates the level of mathematical understanding necessary for attaining these various benefits. I looked up "Laffer Curve" on Wikipedia and understood the concept after looking the diagram for about 5 seconds, no algebra required. A bus driver who tries to chat up executives about calculus will find that they've long since lost any knowledge of calculus that they might once have had.

Jose Vilson also takes a shot at Hacker's piece. He asks:

if we taught the humanities (presumably English, arts, and yes, political science) in the same way you're suggesting for math, how would that look like?

If you're OK with English and language arts taught this way, then let's focus 100% fully on non-fiction texts ... like "How To Operate Your VCR" and "The Intricacies of Setting Up Your Passport." Useful, and often complicated, these texts would surely be of worth post-college, especially for kids who never get out there. They'd have no need for Jon Steinbeck, William Shakespeare, or Julia Alvarez; their texts aren't very relevant to what students actually encounter on a daily basis, so we'd leave it alone in the hopes that they don't have to think abstractly.

I would argue that we actually do teach other subjects to a much more appropriate depth. Unfortunately, advanced math is not a subject we teach for "literacy" or appreciation (which Hacker recommends as an alternative); we teach it to mastery. It's not enough to learn that there are such things as polynomials and that they are useful in certain situations; to get into college, you have to master polynomials and solve all kinds of problems that you will never encounter in any other context in life. Under our current approach, there's no benefit to "appreciating" most advanced math; rewards come only to those who master it, something we rarely require at such a high level in other subjects. For example, an English or social studies teacher may assign an essay in which students must develop and defend an argument, but this is largely a process-oriented task; there is not a sharp delineation between correct and incorrect the way there is in mathematics, and even if your essay isn't very strong, you can learn a lot from the process and get better over time. If you end up never writing an essay again after you graduate, at least you'll be able to use whatever level of knowledge and skill you developed to be a more informed and responsible citizen. Advanced math offers no such benefits for those who try but fail.

And yet, most Americans don't balance their checkbooks, can't make a spreadsheet, and have no idea how loan amortization works. This tells me not that we need to up the rigor of our already-rigorous math courses, which don't address such mundane-but-useful concepts, but that we're teaching the wrong concepts.

metal beam with numbers, photo by cogdogblog on Flickr

Hacker says as much, arguing that we should "familiarize students with the kinds of numbers that describe and delineate our personal and public lives." In response, USF professor Sherman Dorn writes:

[Hacker] proposes "citizen statistics," to which I (and many others) respond, how you can teach such a course without algebra? "Effect size" and "meta-analysis" should both be key concepts in such a course, but you don't get them without understanding what a standard deviation is.
I lean strongly in favor of requiring algebra, but I am aware of the difficulty some students have with it.
...while I am an historian, I continue to use my algebra skills in many ways.

Dorn seems focused on statistics, but like many critics of Hacker's piece, he vastly overestimates how much algebra is involved in statistics. Statistics is a challenging subject, but it relies very little on what's taught in other math courses after algebra I. If you can understand the statistical concepts, you can do statistics with fairly basic arithmetic. If you need more advanced stats, you probably had plenty of other advanced math courses to prepare you for them.

I am curious what algebra professor Dorn relies on in his day-to-day work. I suspect it is well within the range of what Hacker would say students should master by the end of high school.

Finally, let's turn to Daniel Willingham's comments. Willingham focuses mainly on the relevance issue, and Hacker's assertion that advanced math is not applicable to most of everyday life. He says that the transfer of any kind of academic knowledge into real-world situations is challenging, and is a matter of how we teach:

The best bet for knowledge that can apply to new situations is an abstract understanding--seeing that apparently different problems have a similar underlying structure. And the best bet for students to gain this abstract understanding is to teach it explicitly.
But the explicit teaching of abstractions is not enough. You also need practice in putting the abstractions into concrete situations.

Fair enough, but we do tons of that already. Visit any 4th grade math lesson and you'll see plenty of real-world word problems. But look through a trigonometry or calculus textbook, and if there are word problems, you'll find just what you'd expect: you virtually never need to solve this type of problem in the real world unless you're an engineer.

And again back to the workforce argument:

Economists have shown that cognitive skills--especially math and science--are robust predictors of individual income, of a country's economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008).

Very well, but teaching no math and teaching ridiculous amounts of math aren't our only two options. Just because something is a requirement for entry doesn't mean it's related to success in that endeavor.

It may be instructive at this point to take a look at the table of contents of a popular Algebra I textbook (PDF). While I would not agree with Hacker that these concepts need not be mastered by the end of high school, our current curriculum typically requires 9th graders to take algebra I, followed by several more math courses (which tend to get either easier or harder, but not to help students who fail algebra I pass it later). If instead we set these concepts as the target for 12th graders and bumped the rest to college, we could teach far more that is actually useful in everyday life, and we could help far more students master algebra I.

And here we encounter the real reason we require so much advanced math: because some kids can't do it. Colleges and universities intentionally screen students out on the basis of their math abilities—even when such abilities are wholly irrelevant to their degree programs—because few other tools can eliminate so many students from consideration.

We should teach less advanced math, teach it better, and eliminate the absurd math requirements that block access to higher education for millions of students. We'll end up with fewer dropouts, a better-educated citizenry, and greater mastery of the real math we need in our daily lives.

Photo credits:
Alan Levine

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