Guest Post: Ed Barbeau – Exercises that Schmeck

Ed Barbeau took up my recent offer for readers to write something for the blog, and has contributed the following, typically excellent, article. 

 

EXERCISES THAT SCHMECK

Whatever your approach to mathematical pedagogy in the schools, it appears that the pupils have to work through a lot of sludge. At the end of a section in a textbook, they might be faced with a large number of similar low level exercises that are essentially rote and restricted in their application. Continue reading “Guest Post: Ed Barbeau – Exercises that Schmeck” →

Is a Percentage a Number?

This post was requested by frequent commenter, Terry Mills, famous for his divisibility test for 53. Simply put, Terry asks the question in the title:

Is a percentage a number?

To give Terry’s question some context, we’ll include a few exercises from the Year 7 Cambridge text. To be clear, we haven’t chosen Cambridge because it is a very bad textbook; that is by the by. Cambridge‘s is just a good example of pretty standard treatment. If and when the conversation gets going, commenters can introduce better sources. Continue reading “Is a Percentage a Number?” →

Maths Anxiety Is Not a Thing, and Is Really, Really Complicated

This could be a WitCH or PoSWW but I’m not expecting anyone to bother. And I am sure as hell not gonna bother. I’ve paid my dues (here, here, here, here and here), and I’m not paying further.

Last week, the journal Nature proto-published a scientific report, a study with the pithy title,

Mathematics performance predicts cognitive and affective math anxiety through mutual mediation pathways from adolescence onward with potential working memory moderations Continue reading “Maths Anxiety Is Not a Thing, and Is Really, Really Complicated” →

Arithmetic, CAS and the Dangers of Obsession

Over the summer, my family and I took a holiday, driving around Tasmania. It was great, of course, albeit the ferry over was a bad mistake.1 Our first stop was Stanley, a gorgeous, windswept town, featuring a mini Ayers Rock, and penguins that could be viewed unobscured by an MCG-sized stadium or a crazed horde of fellow tourists. Also in Stanley, as it happened, I found a book. Continue reading “Arithmetic, CAS and the Dangers of Obsession” →

The Return of A, B and C

Clearly, readers got weary of the TNDOT exams (and I still have some tidying up to do). There is also still a bit to say about the Two Ladies Problem. So, I’m reluctant to keep going. But, Red Five has thrown the problem below at me.

The problem comes from an 1858 Cambridge exam, for “students who are not more than 18 years of age” and “who are not members of the university”, which presumably amounted to an entrance exam for senior school students. This particular exam was on “Arithmetic, Algebra, Trigonometry and Land-surveying”. Go figure.

Here is the problem.

A, B and C rent a field for £180. A puts in 12 horses for 5 months, and 45 sheep for 3 months. B puts in 15 oxen for 6 months, and 54 sheep for 2 months. C puts in 6 horses and 47 oxen for 3 months. Now 4 horses and 3 sheep together eat as much as 5 oxen and 1 horse; also two oxen eat as much as 7 sheep; how much of the rent should A, B and C pay respectively?

Ed Barbeau’s “Two Ladies” Problem

I’m a bit flat out, and working on another post, but here’s a quick one. It is a lovely problem told to me by Ed Barbeau, who credits it to Andrei Toom:

Two ladies, A and B, live in separate towns. At the same time one day, each sets out on foot for the town of the other, walking at her own constant rate. They pass at noon. A reaches her destination at 4 pm, while B reaches her destination at 9 pm. At what time did they set out?

UPDATE (29/5)

This seems an easy problem to get tangled, although it’s not too bad if you set up the diagram as above. Then, the ratios of times are the same for each part of the trip (because the ratios of speeds are the same). That leads to the equation

T/9 = 4/T

So, T2 = 36, and thus T = 6 (and also T = –6, if we permit time travel).

I just wanted to point out an alternative, very natural approach that one of my students thought up. (He framed it as a guess-and-check thing, but it is natural to frame it as a proper algebraic solution.) What the student did was consider the fractions of the trips, for each lady, in terms of T. There are various ways to do this, and here is one way.

Consider Lady A on the the first part of her trip. She covers T/(T + 4) of the trip. Similarly, Lady B covers T/(T + 9) of the trip. But together these fractions add up to the whole trip. So,

T/(T + 4) + T/(T + 9) = 1

Then, solve away: multiply both sides by (T + 4)(T + 9), expand and cancel, and we end up with T2 = 36, as before.

Of course, this is a lot messier. (The mess is lessened by adding the second halves of the trips.) Nonetheless, this seems a conceptually simpler approach, even if it is more difficult to think of.

ACARA vs. IPA

A few weeks ago, Colleen Harkin, “research fellow” and Education Thingo at the Institute of Public Affairs, did a quick LinkedIn post, comparing the “3500 pages” of the Australian Mathematics Curriculum to the “about 80 pages” of the Singaporean Mathematics Curriculum. Harkin included a “compare the pair” photo:

It is not clear that anybody paid much attention to Harkin’s post, but a week ago the Sky Clowns decided to amplify the thing. That seemed to be enough to spur The Age to check out the IPA’s slightly implausible claim. A couple days ago, The Age published a brief article, detailing Harkin’s claim and then slapping it down:

Astounding. Shocking. Also untrue, according to the body that writes the curriculum. The Australian Curriculum, Assessment and Reporting Authority says the foundation-to-year-10 [ugh] maths curriculum is only 273 pages long, including optional resources.

The core content – the achievement standards and content descriptions – comes in at a whopping 33 pages.

Who are you backing? Godzilla or Mothra?

Harkin’s claim is, of course, utter nonsense and The Age + ACARA were right to slap it down. But let’s take this step by step.

To begin, it is not clear that “only” is the most appropriate qualifier for a 273-page curriculum. It is also not clear where ACARA got that “273”. The Age article includes no links, perhaps because there were no links for ACARA to provide. ACARA’s downloads page links to a number of mathematics curriculum files but none of the files is seemingly the whole box and dice. None of the documents is huge, but presumably the IPA went scavenging through sample materials and the Curriculum’s twenty-five other dimensions and so on, which is how they got to thousands of pages. Whatever. Let’s get to the basic bones.

ACARA links to a document, Mathematics: Scope and sequence F–10 (W,I).  This document contains just the achievement standards and content descriptions, and so is presumably the “core content” to which The Age refers. The document is 39 pages rather than 33, but whatever. Precision arithmetic has never been ACARA’s strong suit. Whichever number, should we describe that 39/33 as “whopping”, or “only”? It depends how we measure.

It is natural to compare the Australian curriculum, as Harkin does, to the Singaporean curriculum, which everyone seems to agree is sane. The comparison is a little difficult, however, because Singaporean secondary schooling has various pathways. It is easier and it suffices to compare Australia’s F-6 curriculum to Singapore’s 1-6 curriculum.*

The “core content” of Australia’s F-6 curriculum runs to 24 (double column) pages. Adding in the elaborations (W,I), which then includes the Indigenous stuff et al, but not the various Tolstoy-length sermons, takes the F-6 curriculum to 76 pages. For the sake of comparison, the Singapore 1-6 curriculum is 49 pages. This includes two pathways for Primary 5 and Primary 6, the regular curriculum and the simpler, “Foundation” curriculum. Ignoring the Foundation path, the Singaporean primary curriculum is then 43 pages.

Now, how do we compare? First, the Singaporean curriculum contains a number of non-“syllabus” sections: an introduction (6 pages); a conceptual overview of mathematics and doing mathematics, on themes and “big ideas” (10 pages); pedagogy (5 pages); and assessment (6 pages). The actual Singapore syllabus, the itemisation of content, comprises 16 pages.

So, comparing bare bones to bare bones, Singapore 1-6 is two-thirds the pages of Australia F-6. This ratio, although significant, does not begin to convey the cleanness of Singapore’s curriculum and the bloatedness of Australia’s, even just for the content. We’ll get to that, but first, let’s compare flesh to flesh.

Singapore’s curriculum document is clean, with just a few clearly distinguished and well-presented sections. Australia’s curriculum, which is not even a document, is a nightmare. The website is an unnavigable labyrinth of God Knows What intermingled with Who Cares. It is a disaster, and everybody knows it is a disaster except for ACARA and, seemingly, a couple of Age reporters.

Harkin scored an idiotic own goal with her absurd claim about the thousands of pages of the Australian mathematics curriculum, and she scored another with claims that this length was primarily due to the Indigenous stuff and “social justice” and the like. It is all nonsense. But it is not nonsense that the Australian curriculum is a bloated, unworkable mess. Harkin comes out of this looking pretty silly, but not nearly as silly as the swallow-anything Age reporters.

Harkin was fundamentally right and The Age was fundamentally wrong, but they were both fundamentally off the main point. Yes, that the Australian curriculum as a delivered website is useless is a very big problem. Obviously. It is not nearly as big a problem, however, as the fact that substance of the Australian curriculum, the content, is appalling.

Let’s consider just one example: addition and subtraction at Year 2.** The Australian curriculum has no clearly labelled subsection, but the following are the directly relevant content descriptors (which appear on different pages):

add and subtract one- and two-digit numbers, representing problems using number sentences and solve using part-part-whole reasoning and a variety of calculation strategies (AC9M2N04)

recall and demonstrate proficiency with addition facts to 20; extend and apply facts to develop related subtraction facts  (AC9M2A02)

If you glance at it sideways, you can almost see what is being asked for, but then you look at it straight on and you realise you have no idea. What does “representing problems” mean, or “a variety of calculation strategies”? How does “[be able to] recall” differ from “demonstrate proficiency with”? How does one “extend and apply” which “facts”? At this stage, the confused and desperate primary teacher delves into the elaborations, hoping to find some clarification, and they are never heard from again.

How does Singapore do it? They have a clearly labelled subsection, Addition and Subtraction, with two descriptors:

2.1 addition and subtraction algorithms (up to 3 digits)

2.2 mental calculation involving addition and subtraction of a 3-digit number and ones/tens/hundreds

That’s it. That’s all the Singapore guys write and that’s all they need to write. In brief, the Year 2 kid should be able to add 345 + 287 on paper and be able to add 345 + 80 in their head, with corresponding abilities for subtraction.

There are three things to note here. First, the Singaporean curriculum is worded much, much more clearly than the Australian curriculum. Secondly, what the Singaporean curriculum is requiring of the kids is much, much clearer than what the Australian curriculum requires. Thirdly, and by a mile most important, the standard of the Singapore curriculum, the level it is requiring of the kids, is much more advanced than the Australian standard.***

It is great that IPA is attacking ACARA’s appalling curriculum. It is just a shame that they are so ideologically blinkered and so amateurishly manipulative, and thus so bad at it. And it is a real shame that The Age has absolutely no clue that, in attacking Mothra, they are also defending Godzilla.

 

*) Singapore has no Foundation year, but does plenty of mathematics in their “pre-school”. Singapore Primary 1 students are roughly the same age as Australian Year 1 students.

**) For more such comparisons, see this and this and this.

***) For more on the addition-subtraction thing, see this, and for the general awfulness of the Australian curriculum, see this and this. 

Union Leavity

There’s been a ton of media coverage of the Grattan Institute’s report on primary school mathematics, which was released a couple days ago. We had a quick, tangential one yesterday, and we hope to write something more substantial in a day or two (no promises). Generally, the coverage has been stenographic and sympathetic, which is, in turn, not great and good. Undoubtedly there will be others (hello, AARE), but so far we have came across just one dissenting voice. Continue reading “Union Leavity” →

WitCH 145: Get Bentleigh

As just reported in The Age (and The SMH), the Grattan Institute has a new report out, apparently concluding that a lot of primary mathematics teaching sucks. Well duh, and more power to Grattan. Taking Grattan’s lead, the Age report also gives Clarendon College a strong plug, which is predictable and maybe fine. (There is a disconcerting ring to “lesson plans that have been honed over several years”.) Similarly, the Age report gives Bentleigh West a strong plug, which is predictable and which we presume is fine.

These are the good guys, or at minimum the significantly better guys. We hope to give the Grattan report a careful read in the next day or so, and we’re keen to learn more about both CC and BW. But, if there is a WitCH to be done then there is a WitCH to be done. And so here we are.

Continue reading “WitCH 145: Get Bentleigh” →

TIMSS, Australia and the Iraqi Army

UPDATE (21/01/25)

No big deal, but I just stumbled across an article in the online magazine Teacher, a joint article by Wernert and Teacher editor, Jo Earp. Written to accompany ACER’s release of Australia’s TIMSS results, the article is similar to Wernert’s Conversation piece but is longer and sillier. Why these people see it appropriate to be cheerleading rather than reporting, God only knows.

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The 2023 Trends in International Mathematics and Science Study results came out last week, and it appears to have Rorschached people. The Australian cried doom and gloom, including CIS’s Glenn Fahey doing the right wing thing of suggesting that we might want to teach young kids some maths. The Age went with Victoria lagging behind other states before repackaging the report to a more Australian line. The Herald Sun had some STEM guy declare, to the surprise of no one, that what we need is more STEM subjects. The SMH and the ABC went with the persistent boy-girl gap, ABC radio following up with a mathematics professor declaring that what we need is more Eugenia Cheng. Yeah, maybe, although Eugenia would not be my top recommendation. Perhaps, rather, it is time to conclude that the years of such crusading hasn’t helped anyone, including the girls. Continue reading “TIMSS, Australia and the Iraqi Army” →