Down under, the Australians are going through many of the same arguments about mathematics education that we are here in the US. In this column from *The Age*, Marty Ross — who holds a PhD in mathematics from Stanford — lambastes the Australian mathematics education community in ways that might seem eerily familiar to those who follow the similar issues in America. Quote:

[H]ere is an exercise from a current Victorian year 9 maths text: a farmer has 2C cows and 3H horses. The exercise is to find the square of the sum of the farmer’s animals.

The Victorian texts are not uniformly that pointless or that bad. But not much is good. Definitions are clumsy, problems are contrived, natural connections and beautiful insights are overlooked. The texts do not reflect a mathematical culture.

It is not just the textbooks. Teachers are poorly trained; the curriculum is moribund, rife with silly, contrived applications; and everywhere there is pointless calculation. And calculators – the cane toads of education.

Is there still proof? Proof is the source of the power of mathematics, the reasoning and the understanding: it’s what holds the discipline together. But it is practically dead. The very little proof that remains is meaningless and ritualised: maths as Latin Mass.

How did it get this bad? Primarily, it results from the failure to involve mathematicians, the people for whom mathematics is their life’s blood. The simple fact is, many of those responsible for mathematics education do not know sufficient mathematics to do the job.

There are lots more “ouch” moments in the article. Ross concludes by saying:

What do I want from a national curriculum? I want a dodecahedron in every classroom, and beautiful diagrams to ponder. I want students to know why there are infinitely many prime numbers, and for them to realise no one knows about twin-primes. I want them to know what the golden mean is, and why it is irrational, and why we care. I want pattern and play and beauty. And I want the times tables.

Is teaching any of the above useful? It is exactly as useful as teaching Harry Potter and Shakespeare.

Mathematicians do mathematics because it is fun and it is beautiful. If the curriculum is not written in that spirit, and if teachers are not trained in that spirit, then we are doomed. We will have yet another generation devoted to gradgrinding students into hating mathematics.

I’ll agree on many of these points — especially why mathematicians are motivated to do mathematics, the criticism about the lack of proof in the math curriculum, and to some extent Ross’ critiques of the mathematical background of mathematics education people. But what do you think — is Ross’ assertion that fun and beauty form the proper basis for a mathematics curriculum really sound? I mean, I’d like all my students to know about the infinitude of primes too, but does that sort of thing make a reasonable organizing principle for an entire curriculum?

Certainly not every mathematics class a student has to take will present them with notions that satisfy their aesthetic leanings (note that I’m referring mostly to those who “have to”, not “choose to” take said courses). On some level, it shouldn’t have to– abstract reasoning is an important measuring stick of one’s ability to function in more demanding positions in society, so demonstrating it shouldn’t necessarily require an outside motivation.

On the same token– on the other extreme, mathematics has really been whittled down to a set of tables and charts in most schools. Many teachers don’t seem to display methods that instill mathematics in students as much as, I dare say, teach it “at” them. Crunching numbers, memorizing formulae, and the minimal amount of theory and application included seem to miss the point of math entirely. Sprinkling in concepts like the golden ratio or even bits of Gödel can go a long way to show the power of mathematics not often seen in the classroom, but even these may as well be considered bits of fairy dust without the deep foundations that surround them. Students are astounded by the power of mathematics when they realize (a) the wide range of ideas it encapsulates, to be sure, but also (b) that the tools to explore those ideas are never out of reach (just sometimes not conveyed well to them).

To continue the Latin Mass analogy– why is Latin Mass often viewed as meaningless and rote? Not because the message of the sermon has no intrinsic value, but often because it is incomprehensible (at least to the modern congregation), and little real spiritual education surrounds it if already you can’t understand it. You can spew magnificent ideas all day with no real qualification, and it’s amazing for a short time– you give people something they can latch it to, and suddenly you’ve made them smarter.

Short answer– yes, Ross has some wonderful ideas, but they need to be implemented with a sense of why.

As a former math teacher, and one who loves the study of mathematics although not particularly good at it, I must shout out the obvious. First, it is all about money. I simply make 3 times as much to be an electrical engineer as I did as a teacher, having to put up with bad management, clueless and disinterested parents who do not support nor understand intellectual pursuits of any kind (other than utilitarian efforts to get into med or law school, etc.), and also a certain hubris not unlike that portrayed by Russell Crowe in A Beautiful Mind. Yes, math education is a weed-out phase, like physics, to see who has the endurance and smarts. It leaves most people hating the subject and glad to be done with it. It is old school mentality that can still exist, and oh, yeah, lets not forget the foreign accents that many TAs out there have adding to the pain. Another reality is that you have to be naturally smart to go very far with it. Only some are able to concentrate at length or follow arguments. It will always be an uphill battle for the western mind that seeks the path of least reistance in any course of action – it is only natural.

I think all kids should be taught the infinite primes proof as soon as they are young enough to learn it. I think third graders could learn it. If you understand what a remainder is, you can be taught what a prime is. The trick is that the concept of indirect proof might be a bit hard to get across.

I understand what you mean about people becomeing “gradeground”. I do private tutoring in symbolic logic. My clients seem to want there to me some kind of magic set of rules for translating sentences in English into symbolic notation. While there are indeed some basic guidelines that can get you some initial traction, there is no exaustive set of rules for how to get it right. I try to explain that it’s really mostly a matter of imagination and reading comprehension. Read the sentence, get a clear picture in you mind what the sentence says, and then write a formula that corresponds to the picture, don’t get hung on specific words in the sentence. But they usually get hung up. They look for a harder answer when there is an easier answer in their imaginations.

One of the more difficult examples I use is the old Jack-in-the-Box slogan:

“Your food just tastes better when we don’t make it until you order it.”

Everybody can imagine what a JITB restaurant is like if this is true and what it would be like if this were false. Yet most of my clients have an extremely hard time coming up with a correct symbolization. They look harder at the paper instead of looking up into their minds.

Jack’s claim can be translated using this symbolization dictionary and basic propositional notation:

T = your food tastes better.

M = We make your food.

O = You order your food.

Any logic types want to give this a go? I’ll check back in a day or two to see your responses.

why is the latin mass considered rote( that I take means no good) answer it is the very Mass That Satan is trying to get rid of, it is the Mass Christ taught the Apostles between Easter Sunday and Ascention Thursday and standardised by pope Pius V