Numeracy, algebra and the cavern in between

by Burkard Polster and Marty Ross

The Age, 3 September 2012


A provocative opinion piece titled “Is algebra necessary?” recently appeared in the New York Times. Written by Andrew Hacker, emeritus professor of political science at Queens College, The City University of New York, the piece constituted a full-blown attack on the traditional school mathematics curriculum, with algebra as its central target.

Professor Hacker described the algebraic “ordeal” of the millions of “struggling” students, emphasizing their high failure rate and subsequent exclusion from tertiary studies. Hacker distinguished abstract mathematics, which he regards as pointless for most students, from “basic numerical skills” and the “quantitative reasoning we need on the job”. So, yes to long division and a loud no to perfect squares.

Professor Hacker’s article demonstrated a number of important points. First, it demonstrated that a prominent professor writing in a prominent newspaper is capable of producing very prominent nonsense. Secondly, the article was a powerful illustration of that old maxim: people in political science houses shouldn’t throw stones.

An ignorant assault on the teaching of abstract mathematics is not news, but it is astonishing to witness such an attack from a professor of political science. Professor Hacker seems oblivious to the fact that the shortsighted glorification of employment skills has been a major weapon in delegitimising and defunding the arts and humanities.

Professor Hacker acknowledges that mathematics is more than a collection of workplace tools. He wishes students had a “greater understanding of where numbers come from, and what they actually convey”. Unfortunately, Professor Hacker seems to be unaware that algebra is essential to any such understanding.

The essence of algebra is giving a name, X or whatever, to a quantity or thing that we don’t know. This simple step can cast a brilliant and illuminating light on foggy mathematical fields. In our columns we endeavour to keep technical details to a minimum, but nonetheless we have resorted to algebra time and time and time again to untangle a tricky problem.

Let’s recall a notorious example: infinite decimals. They’re puzzling, annoying things and they’re very difficult to avoid. Infinite decimals appear when considering circles and π, and triangles and Pythagoras, and even seemingly simple fractions.

Now, a question, indeed our favourite question: is the infinite decimal 0.999… equal to 1? This question causes no end of consternation. The overwhelming majority of students, and more than a few teachers, suspect that the two numbers are not equal. But let’s do some algebra.

Since we’re not sure what 0.999… is, we’ll call it X. Multiplying by 10, we move the decimal point and that means 9.999… must equal 10X. Now place the first equation underneath the second, and subtract:

So 9 = 9X, and dividing by 9 we see that X = 1. QED, as we like to say.

Who gets to see this beautiful proof? Regular readers of this column do, but it is exceedingly rare to see such an argument published outside of the nerdy, education corners of a newspaper. Your maths masters have a genuine appreciation of the arts, but this appreciation is not always reciprocated by newspaper editors.

It is true, many people would not follow or even attempt to follow the above argument, and of those who do many will remain unconvinced. All such responses are understandable. To fully comprehend infinite decimals is difficult, and it requires much more algebra.

Nonetheless, the above proof is simple and elegant, and it should be known and celebrated. It should be presented to every school student, just as one is presented a poem. A student may not be aware of its full beauty but they will know something is there. They may think about it.

Much more than a formulaic topic, algebra is essential for clear mathematical thought. The magic of algebra is that it provides a solid identity to mathematical objects that initially may be vague or poorly understood. It is a model of careful reasoning, applicable far beyond mathematics.

True, there is much in the teaching of algebra, and mathematics generally, that could be greatly improved. Plenty of school mathematics is ritualised and aimless. Indeed, Professor Hacker’s beloved long division, a pointless anachronism, is a perfect example.

Professor Hacker is to be applauded for targeting the topics and the teaching of school mathematics. It is just regrettable that his aim is so poor.

One silly American opinion piece is neither here nor there. However, Professor Hacker’s naïve, pseudo-utilitarian approach to mathematics is also endemic in Australia. Here, the rallying cry is “numeracy”.

Numeracy is a tricky concept, employed in tricky ways, but in practice it corresponds to Professor Hacker’s notion of (supposedly) practical mathematics: functional arithmetic and statistics, some token geometry and little else. Numeracy constitutes a very small subset of mathematics, but increasingly numeracy is being presented as if that’s about all there is.

Consider, for example, the NAPLAN testing scheme. Leaving aside the serious question of the value of this scheme, it is telling that mathematics is not tested, only numeracy. And needless to say, whatever NAPLAN is testing will be grossly overemphasised in most schools.

The new Australian Curriculum is similarly choked with an emphasis on numeracy. The mathematics curriculum contains little of beauty and few invitations to real mathematical reasoning. Instead, there is a mountain of mundane facts, calculator nonsense, dishwater-dull statistics and pedestrian applications, all of little genuine use and even less appeal.

Of course, there are still opportunities to present beautiful mathematics, and last week we celebrated Mathematics Week. Except, it wasn't actually Mathematics Week, it was National Literacy and Numeracy Week. And it goes on and on.

Yes, algebra, and mathematics generally, can be difficult. That is not a bad thing. School is not about learning facts. School is, or at least it should be, about learning to think. And mathematics can be so enlightening and so beautiful.

Numeracy is important, but it is not that important. And numeracy contains no poetry.


Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

Copyright 2004-∞ All rights reserved.