New recommendations for the teaching of school mathematics were in the news last week. The proposal was that teaching should involve students in “important and engaging tasks”, in particular enabling them to understand “the importance of mathematics to real-world issues”. And who can possibly argue against such common sense?
The news reports follow the release of a report from the Australian Council for Educational Research. The author of the report is Peter Sullivan, Professor of Mathematics Education at Monash University and a lead writer of the Australian Mathematics Curriculum.
In his report, Professor Sullivan contrasts two possible goals for the teaching of mathematics. One goal is the practical mathematics that students will require “for general employment and functioning in society”- what is often referred to as numeracy. The second goal is the specialised mathematical training required for some tertiary studies and technical professions.
Professor Sullivan cites research that about 40% of students will be professional users of mathematics, and fewer than 0.5% of university graduates specialise in mathematics. He goes on to note that a full 100% of students will require practical mathematics. This is the basis of the foundational claim of Professor Sullivan’s report: “It is clear the appropriate priority in the compulsory years should be mathematics of the practical perspective”.
As an illustration of the application of mathematics to real world issues, Professor Sullivan has raised the question of asylum seekers. He has argued that one cannot properly understand the current debate without “understanding the mathematics of the situation, like the proportion of the Australian Population to asylum seekers”.
Yes, all students require training in practical mathematics: this is true by the definition of “practical mathematics”. However, the scope of such mathematics is being vastly overstated. In fact, the mathematics that most of us require in our everyday lives is a little arithmetic and very little else. Professor Sullivan’s suggestions otherwise are contradicted by the preponderance of arithmetic-based applications in his report.
The narrowness of practical mathematics also points to a logical gap in Professor Sullivan’s argument. The fact that 100% of students require practical mathematics gives absolutely no indication of how much time should be devoted to its teaching, or what might be sacrificed to make that time.
Now, to the asylum seeker debate, though “debate” is too respectful a word. We are as appalled as anyone by the level of ignorance and manufactured fear. But can mathematics improve matters? Well, we can do the arithmetic:
However, to suggest that any such equation will somehow counter the demonisation of asylum seekers is extraordinarily wishful thinking. And again, the mathematics involved is nothing more than simple arithmetic.
Clearly, the general standards of mathematical and scientific literacy in Australia are depressingly low. Stronger arithmetic skills would improve the public discourse. However, we do not believe that the main barrier to quality debate is an inability to do fractions: rather, it is the overvaluing of base opinion at the expense of informed argument and truth.
If we’re not convinced by Professor’s Sullivan’s campaign for practical mathematics, are we then advocating the teaching primarily of specialist mathematics? Definitely not.
First of all, let’s be clear that what is currently being taught in schools is specialist mathematics in only a very weak sense. The current curricula and textbooks are so focussed upon formulas and information, and so devoid of logic and meaning, it is all of little value to anyone. And, as Professor Sullivan correctly argues, it is a crashing bore. However, prescribing more practical mathematics, as has been done ad nauseam in the Australian Curriculum, will only make matters worse.
In considering the purposes of teaching school mathematics, Professor Sullivan has invoked a false dichotomy. We believe the primary purpose for teaching mathematics is neither the practical nor the specialist; the fundamental value of school mathematics is the teaching of the power of reason and the importance of truth.
Students must learn to think. To that end, the purity and abstraction of mathematics is not a handicap, it is a strength. The precision of the arguments, the certainty and the clarity of proven truths: these are the perfect models of logical thinking.
Can such abstract mathematics also be engaging and beautiful? Of course! It is what we attempt to demonstrate in almost every column (all except ones such as this, devoted to battling windmills). True, we might have been able to write 100+ columns on practical mathematics, but it would have been extremely painful. And Lord knows who would have wanted to read them.
Professor Sullivan has claimed that the practical mathematics of everyday lives is just as interesting as abstract mathematics. We very much doubt it. What we do not doubt is that practical mathematics is less elegant, less beautiful, and that it is far inferior for the practising of clear logical thought.
Professor Sullivan’s disregard for abstract mathematics also raises an even greater problem with his report: whatever the value of his recommendations, they are currently pointless.
The central, elephant-sized problem with the teaching of school mathematics is teacher training. Neither mathematics departments nor education faculties are sufficiently concerned with teaching what mathematics really is or how to think mathematically. Just as in schools, there is a dull and debilitating emphasis upon mathematics as information rather than ideas, as a way of thinking. The power of mathematical reasoning and the beauty that this reasoning can unveil are very distant afterthoughts.
Teachers cannot possibly engage their students in mathematics unless the teachers themselves have learned to engage with it. Many have done so and, little credit to their university training, perform a very fine job of motivating their students. Many do not, and until someone confronts the elephant, little will change.
So, there is a bright side to the release of Professor Sullivan’s report. We may disagree with his arguments and proposals, and believe his trivialisation of abstract mathematics is silly, but we are comforted by the knowledge that the implementation of his proposals can’t make things much worse.
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.
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