## PISA Triangles

Tony Gardiner did some hunting and found a document containing PISA’s released mathematics items for 2022. In a comment on our PISA non-post, Tony made some remarks about a triple of triangle questions contained in the document. The questions, each with the same triangle picture prompt, are below. The questions are followed by PISA’s discussion. The detailed marking rubric can also be found in the items document (pp 9-14). Note that PISA is a test for 15-year olds; as with all such tests, the questions are intended to range from easy to difficult.

## Two PISA Crap

Below are two “units” (scenarios) used in the PISA 2012 testing of mathematics. The units appeared in this collection of test questions and sample questions, and appear to be the most recent questions publicly available. Our intention is for the units to be read in conjunction with this post, and see also here, but of course readers are free to comment here as well. The two units below are, in our estimation, the most difficult or conceptually involved of the PISA 2012 units publicly available; most questions in most other units are significantly more straight-forward.

# REVOLVING DOORS

## The Slanted Tower of PISA

Here’s an interesting tidbit: PISA‘s mathematics testing doesn’t test mathematics. Weird, huh? Who knew?

Well, we kinda knew. Trustworthy colleagues had suggested to us that PISA was slanted, but finding out the extent of that slant, like lying-on-the-ground slant, was genuinely surprising. (We’re clearly just too optimistic about the world of education.) Not that we had any excuse for being surprised; there were clues of mathematical crime in plain sight, and it was easy enough to locate the bodies.

The first clues are on PISA’s summary page on “Mathematics Performance“. The title is already a concern; qualifications and elaborations of “mathematics” usually indicate some kind of dilution, and “performance” sounds like a pretty weird elaboration. Perhaps “mathematics performance” might be dismissed as an eccentricity, but what follows cannot be so dismissed. Here is PISA’s summary of the meaning of “mathematical performance”:

Mathematical performance, for PISA, measures the mathematical literacy of a 15 year-old to formulate, employ and interpret mathematics in a variety of contexts to describe, predict and explain phenomena, recognising the role that mathematics plays in the world. The mean score is the measure. A mathematically literate student recognises the role that mathematics plays in the world in order to make well-founded judgments and decisions needed by constructive, engaged and reflective citizens.

The alarms are set off by “mathematical literacy”, a pompous expression that promises more than, while signalling we’ll be getting much less than, straight mathematics. All doubt is then ended with the phrase “the role that mathematics plays in the world”, which is so fundamental that it is repeated verbatim.

What this sums to, of course, is numeracy, the noxious weed that inevitably chokes everything whenever there’s an opportunity to discuss the teaching of mathematics. What this promises is, akin to NAPLAN, PISA’s test of “mathematical performance” will centre on shallow and contrived scenarios, presented with triple the required words, and demanding little more than simple arithmetic. Before investigating PISA’s profound new world, however, there’s another aspect of PISA that really could do with a whack.

We have been told that the worldly mathematics that PISA tests is needed by “constructive, engaged and reflective citizens”. Well, there’s nothing like irrelevant and garishly manipulative salesmanship to undermine what you’re selling. The puffing up of PISA’s “world” mathematics has no place in what should be a clear and dispassionate description of the nature of the testing. Moreover, even on its own terms, the puffery is silly. The whole point of mathematics is that it is abstract and transferrable, that the formulas and techniques illustrated with one setting can be applied in countless others. Whatever the benefits of PISA’s real world mathematics for constructive, engaged and reflective citizens, there will be the exact same benefits for destructive, disengaged psychopaths. PISA imagines Florence Nightingale calculating drip rates? We imagine a CIA torturer calculating drip rates.

PISA’s flamboyent self-promotion seems part and parcel of its reporting. Insights and Inpretations, PISA’s summary of the 2018 test results, comes served with many flavours of Kool-Aid. It includes endless fussing about “the digital world” which, we’re told, “is becoming a sizeable part of the real world”. Reading has changed, since it is apparently “no longer mainly about extracting information”. And teaching has changed, because there’s “the race with technology”. The document wallows in the growth mindset swamp, and on and on. But not to fear, because PISA, marvellous PISA, is on top of it, and has “evolved to better capture these demands”. More accurately, PISA has evolved to better market itself clothed in modern educational fetishism.

Now, to the promised crimes. The PISA test is administered to 15 year old students (typically Year 9 or, more often, Year 10 in Australia). What mathematics, then, does PISA consider worth asking these fifteen year olds? PISA’s tests questions page directs to a document containing questions from the PISA 2012 test, as well as sample questions and questions from earlier PISAs; these appear to be the most recent questions made publicly available, and are presumably representative of PISA 2018. In total, the document provides eleven scenarios or “units” from the PISA 2012 test, comprising twenty-six questions.

To illustrate what is offered in those twenty-six questions from PISA 2012, we have posted two of the units here, and a third unit here. It is also not difficult, however, to indicate the general nature of the questions. First, as evidenced by the posted units, and the reason for posting them elsewhere, the questions are long and boring; the main challenge of these units is to suppress the gag reflex long enough to digest them. As for the mathematical content, as we flagged, there is very little; indeed, there is less mathematics than there appears, since students are permitted to use a calculator. Predictably, every unit is a “context” scenario, without a single straight mathematics question. Then, for about half of the twenty-six questions, we would categorise the mathematics required to be somewhere between easy and trivial, involving a very simple arithmetic step (with calculator) or simple geometric idea, or less. About a quarter of the questions are computationally longer, involving a number of arithmetic steps (with calculator), but contain no greater conceptual depth. The remaining questions are in some sense more conceptual, though that “more” should be thought of as “not much more”. None of the questions could be considered deep, or remotely interesting. Shallowness aside, the breadth of mathematics covered is remarkably small. These are fifteen year old students being tested, but no geometry is required beyond the area of a rectangle, Pythagoras’s theorem and very simple fractions of a circle; there is no trigonometry or similarity; there is no probability; there are no primes or powers or factorisation; there are no explicit functions, and the only implicit functional behaviour is linear.

Worst of all, PISA’s testing of algebra is evidently close to non-existent. There is just one unit, comprising two questions, requiring any algebra whatsoever. That unit concerns a nurse (possibly a CIA torturer) calculating drip rates. Minus the tedious framing and the pointless illustration, the scenario boils down to consideration of the formula

D = dv/(60n) .

(The meaning of the variables and the formula needn’t concern us here, although we’ll note that it takes a special type of clown to employ an upper case D and a lower case d in the same formula.)

There are two questions on this equation, the first asking for the change in D if n is doubled. (There is some WitCHlike idiocy in the suggested grading for the question, but we’ll leave that as a puzzle for the reader.) For the second question (labelled “Question 3” for God knows what reason), students are given specific, simple values of D, d and n, and they are required to calculate v (with a calculator). That’s it. That is the sum total of the algebra on the twenty-six questions, and that is disgraceful.

Algebra is everything in mathematics. Algebra is how we name the quantity we’re after, setting the stage for its capture. Algebra is how we signify pattern, allowing us to hunt for deeper pattern. Algebra is how we indicate the relationship between quantities. Algebra is how Descartes captured geometry, and how Newton and Leibniz captured calculus.

It is not difficult to guess why PISA sidelines algebra, since it is standard, particularly from numeracy fanatics, to stereotype algebra as abstract, as something only within mathematics. But of course, even from PISA’s blinkered numeracy perspective, this is nonsense. You want to think about mathematics in the world? Then the discovery and the analysis of patterns, and the analysis of relationships, of functions is the heart of it. And what makes the heart beat is algebra.

Does PISA offer anything of value? Well, yeah, a little. It is a non-trivial and worthwhile skill to be able to extract intrinsically simple mathematics from a busy and wordy scenario. But it’s not that important, and it’s hardly the profound “higher order” thinking that some claim PISA offers. It is a shrivelled pea of an offering, which completely ignores vast fields of mathematics and mathematical thought.

PISA’s disregard of algebra is ridiculous and shameful, the final stake in PISA’s thoroughly nailed coffin. It demonstrates that PISA isn’t “higher” or “real”, it is just other, and it is an other we would all be much better off without.

## A PISA With Almost the Lot

So, that’s just about everyone, right?

## A Quick Message for Holden and Piccoli

A few days ago the Sydney Morning Herald published yet another opinion piece on Australia’s terrific PISA results. The piece was by Richard Holden, a professor of economics at UNSW, and Adrian Piccoli, formerly a state Minster for Education and now director of the Gonski Institute at UNSW. Holden’s and Piccoli’s piece was titled

‘Back to basics’ is not our education cure – it’s where we’ve gone wrong

Oh, really? And what’s the evidence for that? The piece begins,

A “back to basics” response to the latest PISA results is wrong and ignores the other data Australia has spent more than 10 years obsessing about – NAPLAN. The National Assessment Program – Literacy and Numeracy is all about going back to basics ...

The piece goes on, arguing that the years of emphasis on NAPLAN demonstrate that Australia has concentrated upon and is doing fine with “the basics”, and at the expense of the “broader, higher-order skills tested by PISA”.

So, here’s our message:

Dear Professors Holden and Piccoli, if you are so ignorant as to believe NAPLAN and numeracy is about “the basics”, and if you can exhibit no awareness that the Australian Curriculum has continued the trashing of “the basics”, and if you are so stuck in the higher-order clouds to be unaware of the lack of and critical need for properly solid lower-order foundations, and if you can write an entire piece on PISA without a single use of the words “arithmetic” and “mathematics” then please, please just shut the hell up and go away.

## A PISA Crap

The PISA results were released on Tuesday, and Australians having been losing their minds over them. Which is admirably consistent: the country has worked so hard at losing minds over the last 20+ years, it seems entirely reasonable to keep on going.

We’ve never paid much attention to PISA. We’ve always had the sense that the tests were tainted in a NAPLANesque manner, and in any case we can’t imagine the results would ever indicate anything about Australian maths education that isn’t already blindingly obvious. As Bob Dylan (almost) sang, you don’t need a weatherman to know which way the wind is blowing.

And so it is with PISA 2018. Australia’s mathematical decline is undeniable, astonishing and entirely predictable. Indeed, for the NAPLANesque reasons suggested above, the decline in mathematics standards is probably significantly greater than is suggested by PISA. Greg Ashman raises the issue in this post.

So, how did this happen, and what are we to do? Unsurprisingly, there has been no reluctance from our glorious educational leaders to proffer warnings and solutions. AMSI, of course, is worrying their bone, whining for about the thirtieth time about unqualified teachers. The Lord of ACER thinks that Australia is focusing too much on “the basics”, at the expense of “deep understandings”. If only the dear Lord’s understanding was a little deeper.

Others suggest we should “focus systematically on student and teacher wellbeing“, whatever that means. Or, we should reduce teachers’ “audit anxiety“. Or, the problem is “teachers [tend] to focus on content rather than student learning“. Or, the problem is a “behaviour crisis“. Or, we should have “increased scrutiny of university education degrees” and “support [students’] schooling at home”. And, we could introduce “master teachers”. But apparently “more testing is not the answer“. In any case, “The time for talk is over“, according to a speech by Minister Tehan.

Some of these suggestions are, of course, simply ludicrous. Others, and others we haven’t mentioned, have at least a kernel of truth, and a couple we can strongly endorse.

No institution we can see, however, no person we have read, seems ready to face up to the systemic corruption, to see the PISA results in the light of the fundamental perversion of mathematics education in Australia. Not a word we could see questioning the role of calculators and the fetishisation of their progeny. Not a note of doubt about the effect of computers. Not a single suggestion that STEM may not be an antidote but, rather, a poison. Barely a word on the “inquiry” swampland that most primary schools have become. And, barely a word on the loss of discipline, on the valuable and essential meanings of that word. What possible hope is there, then, for meaningful change?

We await PISA 2021 with unbated breath.