Newsletter #143     What is Mathematics? [AWM]

This has been a singularly uncommunicative three months, but in an excellent cause. I last wrote just as I was diving into the whirlwind of helping create revised Mathematics Standards for Washington, and whirling is not a good state for sending out coherent reports. The writing is now done, except for a spot of rapid polishing after we get a whole bunch (we hope) of feedback from a multitude of teachers at a statewide conference this week. Tranquility, however, has not returned. I am now working full tilt to prevent our opposition from tossing into a bonfire not merely the Standards, but everything else we have done in the past decade. You should picture me running madly around, doing my best with a garden hose. If anyone would like to join a bucket brigade, please let me know!
Perhaps I ought to translate that last couple of sentences into something with a little more of English and a little less of Ginger in it: our Standards-writing efforts have been co-opted, and an alternate set that omits many of what to me are the most important elements has been offered to the legislature, which is being very efficiently and energetically lobbied to adopt them rather than ours. If you would like to know more, I'll be happy to fill you in.

Meanwhile, the calendar doesn't quite stop for such events, and I was reminded a couple of weeks ago that I owed the AWM my now-annual Education Column. After a moment of flat panic I realized that I had some ideas that had been rumbling around the back of my mind without my ever having the time to pull them together. So I set to work and pulled and pushed and bundled them, and the rest of this newsletter is the result.

What is Mathematics?

At a very early age, I was made aware of the discrepancy between the views of mathematics held by those inside and outside of the field by watching my father attempt to maintain his civility in responding to “You’re a mathematician – you add the bridge score!” More gradually, I became conscious of the fact that most people regard mathematics in the same light in which they regard snakes. But it was only much later, as I immersed myself ever more deeply in mathematics education, that I realized how deeply and disastrously the two are connected.

Looking back from my current vantage point, I can see as the tip of the iceberg a pair of symptomatic situations where no gracious reply was available that could prevent me from adding my tiny tidbit to people’s mistaken vision of the field: “I’ve always loved mathematics, because every question has one answer and it’s either right or wrong and you’re done” and “You’re a mathematician? You must be brilliant!” I’ve smiled and nodded at the first, figuring the “I love mathematics” was worth the misapprehension – but I could at least have said, “It’s certainly an aspect of one part of mathematics.” That brilliance, although certainly a property of some mathematicians, doesn’t pertain to all of us is even harder to convey, especially with the specter of false modesty lurking about. Nonetheless, it is an image that does us all a lot of harm. Why? Because the belief that being a mathematician requires that one be brilliant underlies a fact we are all painfully aware of: that it is socially acceptable to write off mathematics with “I never could do math.” Accompanying that phrase is the (usually) implicit follow-up, “… and I’ve done just fine without it, thank you!”

Hans Magnus Enzensberger wrote eloquently about this situation in his long essay (or short book) Drawbridge Up: Mathematics – A Cultural Anathema.[1] He writes from the vantage point of a non-mathematician observing the whole field. My own vantage point is that of a mathematician who has become deeply involved with mathematics education. With that perspective, I have been looking not so much at the overall impact of the cultural anathema as at some of its causes and consequences within the educational system and some developments that offer a glimmer of a hope of counteracting it.

Public education was established in the US in the second half of the 19th century, when the mathematical needs of society were clear and straightforward: people needed to be able to carry out the basic arithmetic operations well, dependably, and reasonably swiftly. That ability met the needs of a very high proportion of careers, because the career itself would provide the templates within which to carry out the operations. Bank clerks added long columns of numbers, store clerks multiplied prices by numbers of items and added. School mathematics was perhaps boring, but it met a real need.

Then came the 20th century. The world outside of school changed – and school didn’t. The number of things that could be accomplished by calculation alone diminished steadily. Meanwhile the number of opportunities to use mathematics creatively in a whole variety of job (or non-job) contexts kept increasing. Most of these opportunities still involved calculation, but as a tool for accomplishing something much more interesting. In school, one could still perhaps rationalize a focus on teaching pure arithmetic, but only in terms of its being a necessary tool for carrying out other activities -- and practicing hammer strokes without any nails to hit is hard to motivate. The advent of dime store calculators put the finishing touches on even that rationale. The need to carry out calculations not blindly but with an understanding of what the calculations are doing, and how they fit together, and what can be deduced by, with, and about them became absolute.

The Sputnik era produced two new developments. One was that the mathematical community recognized that school mathematics had bogged down and diverged from what was useful or interesting. From this emerged New Math, to the joy of mathematically attuned teachers and their students and the distress of the rest who, alas, considerably outnumbered them. The other was that the public at large, and administrators in particular, realized that we had some serious deficiencies in our mathematics education. From this emerged standardized tests. They seemed innocent enough – after all, the need for accountability was clear. Unfortunately, what they tested was basic skills, and when that’s what you test, that’s what teachers focus on. If the test reveals deficiencies, the focus intensifies – now instead of just practicing hammer swings students need to practice the hand-hold, then the upswing, then the downswing, and clearly they must not be distracted by a nail. This intensity of focus fueled the back-to-basics movement and left us worse off than we were before. After A Nation at Risk, in which President Reagan’s Commission on Educational Excellence documented that state, came the NCTM Standards, and in reaction the Math Wars and the No Child Left Behind Act – the ultimate monument to standardized testing. In many states, basic skills tests not only define mathematics, but determine whether a school will be put on probation or even closed. The logical conclusion for a student to reach is that mathematics consists of a dry and disconnected collection of skills that no one in their right mind could take pleasure in.

I suspect that if you asked N mathematicians to define mathematics you might get 1.5 N answers, but I can’t imagine that any of the answers would involve dry and disconnected skills. In fact, the distance between our images and those of the world at large is sufficient to produce a communication chasm. If that chasm remains unbridged, we are at a standstill.

I promised some glimmers of hope, and there are some. A small but stellar glimmer is provided by the Math Circles that Robert and Ellen Kaplan have created, and that they describe in Out of the Labyrinth: Setting Mathematics Free[2]. Math Circles take small groups of interested people – most, but not all, of them school children – and entice them as a group into engaging with some substantive mathematical topic, leaving them free to delve where the delving is good, but gently preventing them from wandering into any mathematical deserts. On a larger scale, since the early nineties the Netherlands has been teaching “Realistic Mathematics,” where children work in genuine, real world contexts, ranging from maps to business decisions. The fact that the Netherlands does very well on the international tests on which we show poorly has gone oddly unnoted in the scramble to emulate Singapore, which comes in at number one.

To me, though, the glimmer that holds out the most hope is the fact that Standards-based teaching, for all the buffeting it has taken, continues to grow and solidify and learn from its own errors. The hope from that lies in the most fundamental of the Standards’ tenets: children learn mathematics by doing mathematics – engaging with it, grappling with it, and, with guidance, arriving at their own understanding of it, which they are then able to use, build on – and enjoy. They may well then choose to go another direction, but it won’t be either because “I never could do math” or because they regard mathematics with horror. It will just be because they like something else better, and that’s fine!

[1] Drawbridge Up: Mathematics — A Cultural Anathema, by Hans Magnus Enzensberger. A K Peters, Ltd, 2001.
[2] Robert and Ellen Kaplan, Out of the Labyrinth: Setting Mathematics Free, Oxford University Press, 2006



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