Are we trying to stir up trouble? No and, of course, yes. And yes. If we were really stirring up trouble, we’d be asking for the worst Aboriginal and Torres Strait Islander elaboration. But yes, as with our previous competitions,1 the intention is to damn an aspect of the draft mathematics curriculum by making evident the faintness of the possible praise. Moreover, given that there is essentially no tradition of Aboriginal or Torres Strait Islander mathematics, something has to be said about this aspect of the curriculum. We do so.2
We’re working on a long ACARA post, which, hopefully, will be up in a day or so. In the meantime and as a bit of background for the coming post, readers may wish to have a wander through the Singaporean Primary Mathematics Syllabus.* (The syllabus begins with explanatory chapters, and the content description begins on page 34.)(Added 12/10/21 – The Secondary Syllabus 1-4 is here.)
Rebecca Urban has a report in The Australian today (Murdoch, paywalled):
Education Minister Alan Tudge says the board of the country’s schooling authority must substantially rewrite its draft national curriculum, warning he will not endorse the proposed document amid concern student outcomes would be harmed. …
In the letter, seen by The Australian, Mr Tudge urged the [ACARA] board to seriously consider recent feedback from education experts, who have flagged concerns that the proposed changes amounted to a weakening of learning standards.
Maths experts reaffirm support for curriculum changes as leading group sounds alarm
The “leading group” refers to AMSI, which indeed sounded the alarm, calling for a halt of ACARA’s review of the mathematics curriculum. The claim, however, that “maths experts” reaffirmed support for the curriculum changes is, in a word, bullshit. Obviously AMSI did not do so, but also, to a reasonable approximation, no one did.
Yesterday, a good friend and colleague, let’s call him Mr. Big, threw a book at us. By Alexandre Borovik and Tony Gardiner, the book is called The Essence of Mathematics Through Elementary Problems. The book is free to download, and it is beautiful.
There is much to say about this book. It is, unsurprisingly, a collection of problems and solutions. By “elementary”, the authors mean, in the main, in the domain of secondary school mathematics. Note that “elementary” does not equate to “easy”, although there are easy problems as well.
The problems have been chosen with great care. As the authors write, the problems are included for two reasons:
- they constitute good mathematics
- they embody in a distilled form the quintessential spirit of elementary mathematics
As indicated by the the Table of Contents, the problems in The Essence of Mathematics are also arranged very carefully, by topic and in a roughly increasing level of conceptual depth, and the book includes interesting and insightful commentary. Their twenty problems and solutions embodying 3 – 1 = 2 is a beautiful illustration.
The Essence of Mathematics also contains an incredibly important message. Here is the very first problem in the book:
1(a) Compute for yourself, and learn by heart, the times tables up to 9 × 9.
Regular readers will know exactly where we’re going with this. Chapter 1 of Essential Mathematics is titled Mental Skills, it includes simple written skills as well, and the message is obvious. As the authors write,
The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight.
In particular, on their first problem,
Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of
(i) the arithmetical behaviour of the ten digits 0–9, and
(ii) the index laws for powers of 10.
Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.
What does all this have to do with ACARA’s draft curriculum? Alas, nothing whatsoever.
The draft curriculum is the antithesis of Essence. The “problems” and “investigations” and “models” in the draft curriculum are anything but well-chosen, being typically sloppy and ill-defined, with no clear direction or purpose. The draft curriculum also displays nothing but contempt for the prior mastery of basic facts and skills required for problem-solving, or anything.
Essence is not a textbook, but the authors clearly see a large role for problem-solving in mathematics education, and, with genuine modesty, they can imagine their book as a natural supplement to a good curriculum. Such a role can mean slow and open-ended learning, or at least open-ended teaching:
Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time.
The value of such digressions and explorations, however, does not negate the primary goal of mathematics education:
Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build. …
The effect [of political pressures] has been to downgrade the more important challenges which every student should face: namely
- of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
- of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.
Building systematic schemes out of the mastery of techniques. Or, there’s the alternative:
A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy).
There is problem-solving, and there is “problem-solving”. ACARA is shovelling the latter.
Mrs. Big, AKA Mrs. Uncle Jezza, has given the draft curriculum a very good whack in the comments, below. As part of that, she has noted an excellent quotation that begins the Preface of Essential. The quotation is by Richard Courant and Herbert Robbins, and is from the Preface of their classic What is Mathematics?
“Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.”
“Young man, in mathematics you don’t understand things. You just get used to them.”
Understanding is a fine goal, but it can also be a dangerously distracting goal. ACARA’s “deep understanding” is an absurdity.
Send me a postcard, drop me a line
Stating point of view
Indicate precisely what you mean to say
Yours sincerely, wasting away
Give me your answer, fill in a form
Mine for evermore
Will you still need me, will you still heed me
I’m nineteen sixty-four
Below is a document from a foreign country, one of yesterday’s finds. We have our thoughts, and we shall update the post pretty soon. First, however, we’ll give people a chance to ponder. Think of it as a WinCH.
Courtesy of the Evil Mathologre, the PDF below now has (somewhat clunky) OCR. That means you can search for words, such as “mathematising”.
We went to a strange jungle-bookshop yesterday, hunting for copies of the mythical Fitzpatrick and Galbraith. No such luck, but we did find plenty of fascinating and forgotten items. And, below is the lovely shop assistant Jill, totalling our many purchases.
Of course we rebuked Jill for not doing the sum in her head.
We’ll write about some of these very interesting finds in the near future.
That’s pretty much it, except for the “bemused” bit. But our cheap-joke title is also asking a genuine question: what does ACARA think is the essence of “problem-solving”? How do you solve a problem like ACARA (does)?
However, Mr de Carvalho said problem solving was at the core of the curriculum in Singapore, whose students consistently topped the global education rankings, …
There is plenty more, similar bait in ACARA’s comparative study of Australia and Singapore (discussed here and here). So, let’s take a closer look at the bait, at the problem-solving “core” of Singapore’s mathematics education.
A “problem” in mathematics can mean many different things. In particular, a problem can be absolutely routine, what would normally be referred to as an “exercise”, and is there for the practice of basic skills. But not all exercises are routine. An exercise may require more care in setting up, or involve nastier numbers entailing trickier computation, or more subtle manipulations of the equation(s). It is still an “exercise”, in the sense that it is there primarily there to test and to practise specifically chosen skills, but it can be a hard exercise. It may stretch the student, but within clearly defined parameters, with the required facts and skills clearly understood.
At some point such hard exercises would more naturally be called “problems”. If they’re sufficiently difficult you might call them “hard problems”. But none of that changes the essential nature of these exercises/problems, that they’re there for the testing and practicing of clearly defined facts and skills. And, in that way, these problems presume some prior mastery of those facts and skills. The harder the problem, the greater the mastery presumed.
This is the way to understand “problem-solving” in Singapore’s mathematics education. We have more to learn, but everything we have found so far points to exactly what one would expect: in Singapore, “problem-solving” largely amounts to the serious practicing of hard, up to very hard, exercises, based upon a prior mastery of fundamental facts and skills.
It is easy to get a sense of this simply by searching for “Singapore test papers”. This is one such site, and this is a Primary 6 test paper from that site. Not all the questions are hard, but they get plenty hard. Some of that difficulty is in the material being more advanced — Primary 6 students do a decent amount on rate and ratio problems, including some algebra — but that’s not the only reason. There is plenty harder, and the reader is encouraged to hunt, but here is a quick, telling example from the Primary 6 paper:
Which of the following fractions is nearest to 2/3?
1) 3/4 2) 5/6 3) 7/9 4) 1/3
That’s a Singapore maths problem. Just a fraction comparison question, but a hard fraction comparison question. You can’t possibly do the question quickly without being light on your fraction toes.
That’s the bait, Singapore’s problem-solving. And now, the switch: what does ACARA mean by problem-solving?
It is abundantly clear that ACARA’s notion of a “problem” is not remotely like Singapore’s focussed and difficult exercises. ACARA’s “problem-solving” is of a much more open-ended and exploratory nature. It is inquiry-based learning, with the little kids being intrepid little Lewises and Clarks. This is immediately clear from De Carvalho’s conscious decision to highlight a ridiculous “why”-hunting exercise, with the kids supposedly discovering Pythagoras for themselves.
It is also abundantly clear from ACARA’s documentation. Front and centre in the draft mathematics curriculum is the diagram below. It is one of the silliest, over-egged pieces of nonsense we’ve ever seen:
This craps smells very much of CCR. Whatever its origin, notice that at the bottom of the pretty blue list of “Mathematical approaches” is “problem-solving and inquiry”. This is then explained:
Problem-solving and inquiry – skills and processes that require thinking and working mathematically to understand the situation, plan, choose an approach, formulate, apply the relevant mathematics, selecting appropriate and efficient computation strategies, consider results and communicate findings and reasoning; Problem-solving and inquiry approaches that involve thinking and working mathematically include experimenting, investigating, modelling and computational thinking.
Ugh! But let’s go on.
ACARA is explicitly linking “problem-solving” to inquiry based learning, but it is worse than that. This problem-solving is more than an approach to the curriculum, it is the curriculum. From ACARA’s What Has Changed and Why:
The content descriptions and the achievement standards in the consultation version now explicitly include the critical processes of mathematical reasoning and problem-solving from the proficiency strands. This results in a mathematics curriculum that supports deeper conceptual understanding to make mathematical learning more meaningful, applicable and transferable to students. [emphasis added]
That is, “problem-solving”, meaning inquiry-based learning, is now to be part of the content of the Australian Curriculum. De Carvalho can claim that “ACARA is not making any recommendations about pedagogical approaches”, but his claim is clearly, ridiculously false. And here is the falsehood in the flesh. Here is just one of a zillion such content descriptions, this one from Year 2 Number:
model situations (including money transactions) and solve problems involving multiplication and division, representing the situation as repeated addition, equal groups and arrays. Use a range of efficient strategies to find a solution. Explain the results in terms of the situation
This is garbage and, with absolute certainty, it is not Singapore.
What is Singapore doing while Australia is playing these idiot inquiry games? The students are learning their damn multiplication tables, so they can go on and do Singapore problems. Problems worth doing. Problems that the vast majority of Australian students haven’t a hope of being able to do.
It is a blatant and insidious lie to claim that ACARA’s problem-solving push in mathematics is even remotely like Singapore. And it is a hugely damaging lie. Inquiry-based learning is a disaster; it is already here in Australia and it is already disastrous. As we have written elsewhere, the poor kids aren’t Lewis and Clark; they’re Burke and Wills. They don’t have a chance in hell of getting anything solid, of retaining anything from these aimless treks.
Does one need a proof that inquiry-based learning is a disaster? No. It is obvious on its face, to anyone with any decent understanding of what mathematics is and how children learn. But, for anyone who needs a proof that dumb is dumb, Greg Ashman has written an excellent post on ACARA’s Singapore nonsense and the evidence for the failure of inquiry-based learning.
ACARA are bait-and-switch swindlers and swill merchants, and they should be disbanded. That’s how to solve a problem like ACARA.
Sigh. So much crap …
This one is even more brazen than Chico. It’s more like
Who are you gonna believe, me, or your own eyes and me from two minutes ago?
Rebecca Urban, The Australian‘s education reporter, is notable for her poor stenography skills. Urban has this peculiar habit of not simply buying and then repeating unchallenged a formal authority’s propaganda. Urban’s strange style was on display yesterday, with a report on the Daft Australian Curriculum: Curriculum changes to maths and science are not adding up to success (Murdoch, paywalled).
Urban reports and pushes back hard on ACARA’s problem-solving crusade, this crusade of course in no conceivable manner contradicting David de Carvalho‘s statement that “ACARA is not making any recommendations about pedagogical approaches“. Urban quotes a number of people to query De Carvalho’s nonsense, including Greg Ashman, who is always worth reading and is always too polite. Ashman has a very good and, for him, very snarky post on the Daft Curriculum, and there is probably more to come.
Pretty much everything De Carvalho is quoted as saying in Urban’s report is nonsense, But there is one particular line that rises above and beyond the Chico level of gaslighting:
However, Mr de Carvalho said problem solving was at the core of the curriculum in Singapore, whose students consistently topped the global education rankings, …
Singapore, huh? Well, David, we’ve looked at Singapore, and we’ve also looked at your looking at Singapore. So, we’re sorry, but we’ll choose to believe our own eyes, and that other you from two minutes ago.
Here is what you, ACARA, wrote about Year 6 mathematics in your Australia-Singapore comparative study (p 77):
The [Singapore syllabus] builds on the depth and fluency of Mathematics established in previous years. For example, operations with decimals are considered complete and time is given to completing mastery of the four operations with fractions without the use of calculators. … The comprehensiveness of the problem sets offers Primary 6 students a sense of mastery and confidence in applying Mathematics in useful ways. [emphasis added]
There is plenty more: the comparison of the earlier years is hilarious, as long as you appreciate black humour. Does Singapore do problem-solving? Sure, lots, at least of certain, specific types. But it is absolutely clear to anyone with eyes — or anyone who reads ACARA’s literature with sufficient thoroughness and thought — that Singapore’s problem-solving is built upon a really solid grounding in arithmetic and algebraic facts and skills, a grounding that Australia simply does not offer and is looking now to undermine further.
Urban quotes Fiona Mueller, former Director of Curriculum at ACARA, to counter De Carvalho’s Singapore nonsense. There is also, however, one person Urban quotes supporting ACARA’s Daft Curriculum, and this is well worth noting:
Australian Mathematical Sciences Institute director Tim Marchant backed the changes.
“Adjusting the curriculum to focus on problem-solving is crucial to improve their skill sets and deliver students that are able to take knowledge and apply it to solve challenges,” Professor Marchant said.
That is Professor Tim Marchant, Director of the Australian Mathematical Sciences Institute there, claiming problem-solving is the “the way to improve [students’] skill sets”.
What can one say in the face of such ignorance? Just, as usual, the Evil Mathologre is correct.