Eddie Woo has been annoying for a long time. Eddie knows much less than he realises and his smiling inanities, which are invariably swallowed whole, are a continual distraction from real issues and real solutions. But he’s gotten worse. Eddie Professor of Practice Woo has graduated from being a distraction and an annoyance to being an active menace.
Yesterday, Eddie and a couple of his University of Sydney maths ed colleagues had an article appear in The Conversation:
‘Why would they change maths?’ How your child’s maths education might be very different from yours
Greg Ashman has already had a good whack at the article but it warrants another, much harder whack. Greg summarises the article as “flawed”, which is true, but which fails to capture the magnitude of the article’s badness in the way that “hole” fails to capture the magnitude of the Grand Canyon. The article, cowritten by the Guru of Australian Mathematics Education, Eddie Professor of Practice Woo, is simply appalling.
The article begins with the very funny scene from Incredibles 2, with Mr. Incredible struggling with his kid’s maths homework:
The humorous disconnect underlying the scene is the basis for the article, and the authors first confirm, sort of, Mr. Incredible’s point, that “math is math”:
Pythagoras’ Theorem is as accurate today as it was when it was discovered millennia ago …
And just exactly how accurate might that be, fellas?
The critical insight, of course, isn’t that Pythagoras is as “accurate” as it ever was, but that it is as true as it ever was. It is the rock solid, eternal truth of mathematics that makes it so powerful and that must underly its teaching. There is something almost Freudian about the authors beginning with such a gaffe, suggesting that they really don’t get it. It makes one already question if there is value in reading on. There isn’t, but we will.
The humour in the Incredibles scene is that although maths hasn’t changed, maths education has:
… teachers today also teach maths very differently from when parents were at school. … The teaching and learning of maths has undergone a transformation in past 30 years [sic].
30 years past, you say? What a coincidence. Just the other day we were pondering the world of thirty years ago, when poor kids had to suffer educational horrors such as solving problems in maths competitions.
Yes, maths is taught differently now, and of course the writers are setting the stage for arguing how much better it is now. And of course they are out of their minds.
Their argument begins with a section titled,
Mental connections not procedures
Dear God. Even if they don’t quite mean it, why would they write it? Freud knows why: because they really do mean it.
The section begins:
In the past there has been a focus on teaching students procedures, such as times tables …
Um, times tables are not procedures, they’re related and naturally organised facts. But we get it: you hate the times tables. And the canary truth is there, as always: if you do not recognise that kids gotta learn their tables, up to 12, by heart, then you have you no business being anywhere near the education of these kids.
The authors also name a couple of procedures, “how to work out the circumference of a circle” and how to “solve an equation”, whatever that is supposed to mean. Let’s assume, for the sake of fantasy, that they don’t really want to kill the teaching of such procedures. What then, is the new “focus”? Well,
We now appreciate the importance of forming mental connections between concepts.
This is the strawiest straw man that ever left Strawtown. The implicit claim that maths ed in the past was not concerned with connecting concepts is simply absurd. But maybe maths ed guys are more concerned now:
For example, when students understand the connection between similar triangles and trigonometry they understand the definition of trigonometric ratios at a much deeper level.
Sure. And where is the evidence that trig is now taught and, more importantly, now learned, with any stronger connection and, more importantly, with any greater tangible benefit, than it ever was? None is provided.
The entire section is fantastic gaslighting, all with the flavour of, and with vacuous snowjob references for, the 21st century skills nonsense. There is no clear argument, much less any evidence for an argument.
The next section of the article is no section at all. It is titled,
Problem solving and reasoning
There follows no reference whatsoever to problem solving. The section merely notes that
the shift in mathematics education is reflected in key mathematical proficiencies in the Australian school curriculum
Yes, and the Australian mathematics curriculum is appalling, including that its proficiencies completely undermine the proper, foundational role of facts and procedures. It is unclear why the authors didn’t follow up on “problem solving”, but in preemptive response to the absent idiocy, see here and here and here and here and here.
The next section of the article is, unfortunately, really there. It is titled,
And how are we to do that? Well,
[Teachers] are deliberately showing students different and multiple ways to represent mathematics problems to give students the space to develop understanding. This also give [sic] them an opportunity to reason, model and engage in mathematical thinking.
No. What it does is give the students an opportunity to get lost in a jungle of half-learned methods, none of which are ever properly practised or learned in any internalised, functional manner. For example,
your child might bring home problems to solve using the area model for multiplication, which looks quite different to a traditional method. For example, we teach how 8×27 can be modelled in parts – 8×20 and 8×7.
Boiling it down, the “area model for multiplication” referred to here is the shrivelled, deformed heart of this appalling paper, and it demonstrates just how clueless are the authors. It is thus worth considering this example in some detail.
To begin, of course most methods of computing 8 x 27 will involve breaking 27 into 20 + 7. The traditional algorithm will set up the required computation in columns. By contrast, the area model will picture each component product as the area of a rectangle, and then the “areas” are summed, as helpfully and elegantly illustrated in the linked AMSI PDF:
There are a number of things to be said about this box method. First of all, the method is quite commonly (pseudo)taught now and will be unfamiliar to many parents. As such, the examples fits well with the Incredibles joke. Secondly, there are non-trivial arguments for teaching multiplication with boxes: the boxes make visual and intuitive the commutativity and distributivity of arithmetic, preparing the path to algebra. Thirdly, there is nothing remotely “inventive” about kids learning the box method, either instead of or as well as the traditional method. Fourthly, notwithstanding the pre-algebraic arguments for teaching the box method, the arguments against teaching it are much stronger: it is a hideous method (something AMSI forgot to tell us).
The fundamental thing one desires from a mathematical technique is that it works clearly and efficiently for the purposes at the time the technique is being taught, not for purposes that will emerge three years later. And yes, the box method is vaguely efficient on examples such as 8 x 27, but these are examples that are simple enough that they should be learned to be done mentally. Going on to examples such as 38 x 47 or 315 x 526 and the box method quickly becomes a nightmare. What this means is that the box method should not be taught as the fundamental algorithm; if it is to be taught at all, it should come after the traditional column method. But, fifthly, showing students the box method after the column method may have some merit, but even then it will likely also create serious problems: since most kids will be insufficiently practised in the traditional algorithm, introducing a second method is more likely to confuse than anything else.
Finally, it must be noted that the box method as an Incredibles example, of the teaching of a new and unfamiliar method, is being way oversold. Such new methods are rare. Amusingly, the example in the Incredibles is of solving linear equations, which is done exactly the same as it ever was. What has changed, dramatically, is the manner in which these methods are presented and the manner in which students are instructed that these methods be (under)practised.
The penultimate section is,
The world is changing
This section is more of the 21st century skills nonsense, together with the selling of dynamic software and the like as technological saviour. Nothing new, and nothing of sense. Then, the article closes with its final, “don’t worry” section:
How can parents help?
Parents can help by not buying the nonsense offered by Eddie Professor of Practice Woo and his fellow maths ed nitwits. Parents can help by ensuring that their kids learn traditional techniques and practise these techniques in a traditional, solid manner. That may not be perfect and it may not be sufficient, but it is where to start. Without the solid knowledge of clear facts and a solid facility with efficient techniques the kids are doomed, the same as it ever was.
As for taking lessons from the movies, that is fine. Eddie and his mates have simply chosen the wrong movie: