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.

UPDATE (09/05/21)

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.

We decided ACARA’s psychedelic circle, which we discussed briefly here, was worthy of its own post. So, apart from the potential to trigger an epileptic fit, what is wrong with ACARA’s wheel?

ACARA labels their wheel as depicting

[the] relationship between the six strands and three core concept organisers

The six strands make up the inner wheel, and are explained here (pages 4-5). The three “core concept organisers” are the three colours on the wheel rim, with the categories and subcategories as pictured below, and which are explained here (pages 5-8).

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.