The paradox of Australian mathematics education

Burkard Polster and Marty Ross

The Age, 23 June 2014

There is a famous and fascinating paradox of movie-making: the French are funny, sex is funny and comedy is funny, but somehow French sex comedies are never funny. Which brings us to the topic of Australian maths education.

It's a depressing topic. It seems that each week brings a new report on Australia's mathematical doldrums. Most recently it's been the "death spiral" of year 12 advanced mathematics: about a 20% drop in student numbers over the previous decade. The decline has been even greater amongst girls, which has come off an already absurdly low base. Mathematical Barbie may still be puzzling away but she's pretty much on her own.

It's not just a problem of higher year classes. There's hardly a happy buzz over the mathematics offered at any level. Nor is there any happy buzz over what students are learning or, rather, are failing to learn. A careful study of the (almost secret) NAPLAN results reveals that students' arithmetic skills are depressingly weak. International test results are no more encouraging.

But why are things so bad? How can Australian maths education fail so dismally to be educational?

The Australian Curriculum can definitely take its share of the blame. One would hope for a strong emphasis upon core mathematics and fundamental skills; just as learning a musical instrument requires practising of scales, learning mathematics requires a strong grounding in the basics. Unfortunately, the Australian Curriculum contains the exact opposite emphasis. And any primary school teacher who mimics the Curriculum's denigration of arithmetic skills, who fails to emphasise the fundamental role of times tables and mechanical methods, is condemning their students to life in a mathematical ghetto.

Not that honing skills is all there is to a mathematics education. Though the skills are essential, mathematics must also be presented as a human endeavour, as the search for beautiful, provable truths. Again, the Australian Curriculum misses the point entirely; it exhibits no concern for beauty or history or the human element of mathematics. Instead there are bitsy nothing-topics, aimless investigations, silly pseudo-applications and an endless stream of tedious statistics.

The senior years of this ridiculous curriculum are still to be implemented in Victoria, with a draft plan having been recently released for review. Some minor improvements have been proposed for Specialist Maths and Mathematical Methods but both subjects will retain their fundamental flaws, and some grave new flaws will be introduced.

In particular, the statistics that is likely to be incorporated into the VCE curriculum is truly appalling. It is pointless for VCE students, it's incredibly boring and it will be a horror to teach. Many students may be fleeing advanced mathematics now, but just watch for the flood of refugees in a couple years.

(Readers who wish to express their views on the proposed VCE curriculum can submit comments to the Victorian Curriculum and Assessment Authority. But be advised: the VCAA apparatchiks are well-practised in listening to no one.)

OK, so the curriculum doesn't make things easy, however what really matters is what happens in the classroom. Which brings us to the teachers.

Of course no education article is complete without a good round of teacher-bashing. However your Maths Masters will forego the opportunity. We may have some criticisms but we also know that most teachers are very dedicated, that they are trying as hard as they can under very trying conditions. And, teachers are already the whipping boys and girls of every deluded government and every pompous education "expert".

Moreover the general standard of mathematics teaching is not the fundamental problem. Consider, for example, the recent news report that 40% of year 7 to year 10 maths classes are conducted by a teacher not qualified to teach mathematics. That is a huge problem, and definitive proof of policy failure, but the report makes no mention of a related problem that is just as serious.

In fact, there is as much an issue with qualified teachers as unqualified teachers. Hints of this problem can be found in an extensive review of Australian mathematics conducted by the Australian Mathematical Sciences Institute, and in their accompanying policy document. Unfortunately, the hints are small and well hidden.

The fundamental question is, what does it mean for a mathematics teacher to be qualified?

Of course the answer depends upon the mathematics in question. Teaching a calculus course requires much more background than teaching year 3 arithmetic. So, some teachers should clearly be expected to have a major in mathematics while for others such a major may be substantially irrelevant. Currently, primary school teachers are "qualified" if they have an education degree, which will typically include a couple of subjects on teaching mathematics, and that's it.

Such subject requirements are simply insufficient to guarantee that a teacher is qualified in any real sense. To teach mathematics well, one must know more than the mathematical topic at hand, the specific techniques to be taught; one must also know about the mathematics, why the topic is the way that it is. That involves consideration of the fundamental nature of mathematics and mathematical thought, including a proper appreciation of mathematics' long and difficult history.

Take fractions. They may appear simple but the arithmetic of fractions is tricky, and tricky to explain. And more than that, it is tricky to even know what a fraction is. It is very difficult to understand that 1/3 does not mean the process of 1 divided by 3, but rather the number that results from that process. It is very difficult to realise that 2/6 and 1/3 are equal, that they are the exact same thing, and that the weasel word "equivalent", which is almost always employed, has no place here.

It took thousands of years to sort out the nature of fractions, and only afterwards did the arithmetic rules emerge. These rules cannot be properly understood without some appreciation of what fractions really are, and it is an appreciation every mathematics teacher should have. The same is true for almost every concept in school mathematics.

There are few subjects that a potential teacher can take that will reflect and encourage this deeper mathematical sense. True, some Australian mathematics departments offer some excellent, thought-provoking subjects. However many subjects in many mathematics departments are turn-the-handle formulaic silliness, of little benefit to anyone and absolutely no benefit to a potential teacher. As a consequence, it is very easy to obtain a mathematics major while gaining no sense of what mathematics is about, what it means to think mathematically.

And that's the good news. Mathematics departments may have their problems but, when it comes to mathematics, the education faculties are a disaster.

Education faculties do not teach mathematics. They exhibit no concern that the vast majority of their students lack the mathematical background to teach well. More often than not, the lecturers themselves have a similarly poor mathematical background.

What takes place instead of the investigation of mathematics? Education students learn silly calculator tricks; they wallow in discussion of "numeracy", that absurd, impoverished substitute for mathematics; they are taught prissy rules of mathematical presentation, none of which anyone obeys outside of a classroom; and, they discuss the teaching of mathematics with other people who have no proper sense of what mathematics is.

Do people realise how absurd this all is? Yes, to some extent. Most teachers know they've somehow been cheated; they wish they knew more mathematics, and knew it more deeply. The (pre-Abbott) Federal government allocated $12 million for projects designed to improve the training of maths and science teachers. And, the AMSI policy document contains the excellent recommendation that the training of primary teachers include two subjects on mathematics (as opposed to mathematics teaching), to be taught "in conjunction with" the mathematics department. All this is good.

But we also feel that people don't really get it, or at least won't admit it. For example, while pointing the finger at education faculties, AMSI makes no mention of the silly cookbook subjects offered by mathematics departments, and how useless they are for prospective teachers. And AMSI's finger-pointing is much too gentle; the mathematics subjects for teachers should not be taught "in conjunction with" mathematics departments, they should be taught "by the mathematics departments, with the education faculties nowhere in sight". (AMSI might also have mentioned that plenty of mathematics departments are so wrapped up in presenting cookbook maths, they would make a complete hash of teaching mathematics to prospective teachers.) Similarly, all the Federal Government's projects involve collaboration between education staff and mathematics staff; every project actually concerned with teaching mathematics would be instantly and dramatically improved by the exclusion of the education staff.

The simple truth is, the major stumbling block for mathematics education in Australia is that teachers, qualified or not, don't learn enough mathematics and they don't learn it well enough. Discussion of anything else is pointless until that problem is resolved. And the second simple truth is that education faculties are so divorced from the study of mathematics, so lacking in understanding and appreciation of mathematics, they can play no meaningful role in fixing the problem except by getting out of the way.

It's all a hilarious state of affairs. Just about as funny as a French sex comedy.

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 helping Barbie smash calculators and iPads with a hammer.