Or maybe we should have titled the post Springtime for Shitler. Whatever. The Productivity Commission has come out with its Final Report on the National School Reform Agreement. We made a submission, and we remarked upon the Interim Report, and so we feel obligated to write something now. But our heart isn’t in it. (And since there’s little else here, readers may as well enjoy the song.)
A few months ago, I made a submission to the Inquiry by the Productivity Commission into the National School Reform Agreement. The Commission has just released its Interim Report, and has invited further submissions. I won’t be bothering. Continue reading “Sins of Commission”
An unexpected benefit of writing this blog has been that, although now persona non grata with polite academic-journalistic society, I’ve made a number of new, interesting and valuable contacts with impolite society.* One such contact is Shelley, a kindred and kindredly nutty spirit. Shelley and I have chatted and emailed over the last year and, about a month ago, Shelley requested that I make a submission to the Inquiry by the Productivity Commission into the National School Reform Agreement.
I was not clear on, and am still not clear on, why Shelley wanted this. I also had no idea what to think of either the PC or the NSRA, and neither the Inquiry’s terms of reference nor the call for submissions were close to encouraging. Nonetheless, since I pretty much always try to do what is asked of me,** I obliged.
Below, modulo a few minor edits, is the submission I wrote in the two hours I didn’t have to write it.
*) Come to think of it, both sides of that coin are benefits.
**) God knows why. Probably her.
*********************************** Continue reading “A Submission to the Productivity Commission”
In this column, ACARA will be playing the role of the Good Guy.
Now that we have your attention, we’ll confess that we were exaggerating. ACARA is, of course, always the Bad Guy. But this column also contains a Worse Guy, a bunch of grifters called Center for Curriculum Redesign. ACARA appears to be fighting them, and fighting themselves.
Last week, The Australian‘s education reporter, Rebecca Urban, wrote a column on ACARA’s current attempts to revise the Australian Curriculum (paywalled, and don’t bother, and it’s Murdoch). The article, titled Big ideas for mathematics curriculum fails the test, begins as follows:
Plans for a world-class national school curriculum to arrest Australia’s declining academic results are in disarray after a proposal to base the teaching of mathematics around “big ideas” was rejected twice.
So, apparently Australia has plans for a world-class curriculum.1 Who knew? At this stage we’d be happy with plans for a second rate curriculum, and we’d take what we got. But a curriculum based upon “big ideas”? It’s a fair bet that that’s not aiming within cooee of first or second. We’ll get to these “big ideas”, and some much worse little ideas, but first, some background.
The sources of this nonsense are two intertwined and contradictory undertakings within ACARA. The first undertaking is a review of the Australian Curriculum, which ACARA began last year, with a particular emphasis on mathematics. On ACARA’s own terms, the Review makes some sense; if nothing else, the Australian Curriculum is unarguably a tangled mess, with “capabilities” and “priorities” and “learning areas” and “strands” and “elaborations” continually dragging teachers this way and that. The consequence, independent of the Curriculum being good or bad, is that is difficult to discern what the Curriculum is, what it really cares about. As such, the current Review is looking for simplification of the Curriculum, with emphasis on “refining” and “decluttering”, and the like.
This attempt to tidy the Australian Curriculum, to give it a trim and a manicure, is natural and will probably do some good. Not a lot of good: the current Review is fundamentally too limited, even on its own terms, and so appears doomed to timidity.2 But, some good. The point, however, is the current Review is definitively not seeking a major overhaul of the Curriculum, much less a revolution. Of course we would love nothing more than a revolution, but “revolution” does not appear in the Terms of Reference.
The hilarious problem for ACARA is the second, contradictory undertaking: ACARA have hired themselves a gang of revolutionaries. In 2018, ACARA threw a bunch of money at the Center for Curriculum Redesign, for CCR “to develop an exemplar world-class mathematics curriculum”. ACARA’s “oh, by the way” announcement suggests that they weren’t keen on trumpeting this partnership, but CCR went the full brass band. Their press release proudly declared the project a “world’s first”, and included puff quotes from then ACARA CEO, Bob the Blunder, and from PISA king, Andreas Schleicher. And the method to produce this exemplar world-class, ACARA-PISA-endorsed masterpiece? CCR would be
“applying learnings from recent innovations in curriculum design and professional practice …”
And the driving idea?
“… the school curriculum needs to allow more time for deeper learning of discipline-specific content and 21st century competencies.”
This grandiose, futuristic snake oil was an idiot step too far, even for the idiot world of Australian education, and as soon as the ACARA-CCR partnership became known there was significant pushback. In an appropriately snarky report (paywalled, Murdoch), Rebecca Urban quoted ex-ACARA big shots, condemning the ACARA-CCR plan as “the latest in a long line of educational fads” and “a rather stealthy shift in approach”. Following Urban’s report, there was significant walking back, both from Bob the Blunder, and from the then federal education minister, Dan “the Forger” Tehan. But revolutionaries will do their revolutionary thing, and CCR seemingly went along their merry revolutionising way. And, here we are.
Urban notes that the proposal that ACARA has just rejected – for a second time – placed a “strong focus on developing problem-solving skills”, and she quotes from the document presented to ACARA, on the document’s “big ideas”:3
Core concepts in mathematics centre around the three organising ideas of mathematics structures approaches and mathematising [emphasis added] …Knowledge and conceptual understanding of mathematical structures and approaches enables students to mathematise situations, making sense of the world.”
Mathematising? Urban notes that this uncommon term doesn’t appear in ACARA’s literature, but is prominent in CCR’s work. She quotes the current proposal as defining mathematising as
“the process of seeing the world using mathematics by recognising, interpreting situations mathematically.”
So, all this big ideas stuff appears to amount to the standard “work like a mathematician”, problem-centred idiocy, ignoring the fact that the learning of the fundamentals of mathematics has very, very little to do with being a mathematician.4 Really, not a fresh hell, just some variation of the current, familiar hell.
So, why write on this latest version of the familiar problem-solving nonsense? Because what has reportedly been presented to ACARA may be far, far worse.
Most sane people realise that before tackling some big idea it is somewhat useful to get comfortable with relevant small ideas. In this vein, before the grand adventure of mathematising one would reasonably want kids to engage in some decent numbering and algebra-ing. You want the kids to do some mathematising nonsense? Ok, it’s dumb, but at least make sure that the kids first know some arithmetic and can handle an equation or two. And this is where the proposal just presented to ACARA seems to go from garden-variety nonsense to full-blown lunacy.
Recall that the stated, non-revolutionary goal of the current Review is to clarify and refine and declutter the Australian Curriculum. Along these lines, the proposal presented to ACARA contained a number of line-item suggestions to accompany the big ideas. Urban quotes some small beer suggestions, such as the appropriate stage to be recognising coin denominations, the ordering of the months and the like. But, along with the small beer, Urban documents some big poison, such as the following:
Christ. If students don’t have a handle on ten-ing by the end of Year 4 then something is seriously screwed. At that stage the students should be happily be zooming into the zillions, but some idiots – the same idiots hell bent on real world problem-solving – imagine tens of thousands is some special burden.
The next poison:
Here, the idiots are handed a gun on a platter, which they grab by the muzzle and then shoot themselves. There is absolutely zero need to cover probability, or statistics, in primary school. Its inclusion is exactly the kind of thoughtless and cumbersome numeracy bloat that makes the Australian Curriculum such a cow. But, if one is going to cover probability in primary school, the tangible benefit is that it provides novel and natural contexts to represent with fractions. Take away the fractions, and what is this grand “conceptual understanding” remaining? That some things happen less often often than other things? Wonderful.
One last swig of poison, strong enough to down an elephant:
On the scale of pure awfulness, this one scores an 11, maybe a 12. It is as bad as it can be, and then worse.
PISA types really have a thing about algebra. They hate it. And, this hatred of algebra demonstrates the emptiness of their grand revolutionary plans. Algebra is the fundamental mechanics of mathematical thought. Without a solid sense of and facility with algebra, all that mathematising and problem-solving is fantasy; it can amount to no more than trivial and pointless number games.
The teaching of algebra is already in an appalling, tokenistic state in Australia. It is woefully, shamefully underemphasised in lower secondary school, which is then the major source of students’ problems in middle school, and why so many students barely crawl across the finish line of senior mathematics, if they make it at all.
What is “more complex equations” supposed to mean for 7 – 10 algebra? The material gets no more complicated than quadratics, so presumably they mean quadratics, the hobgoblin of little saviours. True, this material tends to be taught pointlessly and poorly. But “complex”? Simply, no. It amounts to little more than AB = 0 implying that either A or B is 0, a simple and powerful idea that many students never solidly get. The rest is detail, not much detail, and the detail is just not that hard.
Of course, a significant reason why algebra is taught so, so badly is that it is almost universally taught and tested with “technology”, from calculators to nuclear CAS weapons, to online gaming of the kind that that asshole Tudge is promoting. And all of this is “used as a support”? That idea of “support”, just as stated, is bad enough, bringing forth images of kids limping through the material. But all this technology is much worse than a crutch; it is an opiate.
It is a minimal relief if ACARA has rejected the current proposal, but we have no real idea what is going on or what will happen next. We don’t how much much poison the proposal contained, or even who concocted it. We don’t know if the rejection of this proposal amounts to a war between CCR and a new, more enlightened ACARA, or a civil war within ACARA itself.5 We should find out soon enough, however. ACARA has promised to release a draft curriculum by the end of April, giving them a month or so to come to terms with the truly idiotic ideas that they are being presented. ACARA has a month or so to avoid becoming, yet again and still, Australia’s educational laughing stock.
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1) We really wanted to slip “Urban myth” into the title of this post, but decided it would have been unfair. Yes, “world class” required quotation marks, or something. It seems, however, that Rebecca Urban was just carelessly, or perhaps snidely, repeating a piece of ACARA puffery, which is not the focus of her report. In general, Urban tends to be less stenographic than other education (all) reporters; she is opinionated and, from what we’ve seen, she seems critical of the right things. We haven’t seen evidence that Urban knows about mathematics education, or is aware of just how awful things now are, but we also haven’t seen her repeat any of the common idiocies.
2) We hope to write on the Curriculum Review in the next week or so, give or take a Mathologer task.
3) The proposal just presented to ACARA is not publicly available, and Urban appears to have only viewed snippets of it. It is not even clear, at least to us, who are the authors of the proposal. We’re accepting that Urban’s report is accurate as far as it goes, while trying to avoid speculating on the much missing information.
4) Urban’s report includes some good and critical, but not sufficiently critical, quotes from teacher and writer, Greg Ashman.
5) David de Carvalho, ACARA’s new CEO, appears to be an intelligent and cultured man. Maybe insufficiently intelligent or cultured, or insufficiently honest, to declare the awfulness of NAPLAN and the Australian Curriculum, but a notable improvement over the past.
We’re late to this, but it’s gotta be done.
Some State education ministers, unhappy with NAPLAN, commissioned a review, which appeared a couple weeks ago. The Review considers many contentious aspects of NAPLAN, but we’ll focus upon “numeracy”, NAPLAN’s homeopathic proxy for mathematics. We’ll leave others to debate “literacy” and the writing tests, and the timing and reporting and so forth.
So, what might the Review entail for the Son-of-NAPLAN testing of mathematics? Bugger all.
Which was always going to happen. For all the endless public and pundit whining about NAPLAN, which is what prompted this latest Review, none of the criticism has been aimed at the two elephants: the Australian Curriculum, which underpins NAPLAN, is a meatless mass of gristle and fat; and “numeracy” is not mathematics, is not arithmetic, and is barely anything. The inevitable consequence is that NAPLAN amounts to the aimless testing of untestable fuzz. As Gertrude Stein would have put it, there is no there there to test.
This misdirection of the Review was locked in by the terms of reference. No mention is made of “mathematics” or “arithmetic”. The single reference in the Terms to “numeracy” is a deadpan call for “the most efficient and effective system for assessing key literacy and numeracy outcomes”, as if this were a clear and unproblematic and worthy goal. It is no surprise, therefore, that the Review gives almost no attention to arithmetic and mathematics, and the meaning(lessness) of numeracy, and indeed works actively to avoid it.
The Review includes a capsule summary of the Numeracy tests, a superficial comparison to PISA and TIMSS, and Australia’s relative performance over time on these tests (pp 34-42). There is no proper exposition, however, of the nature of the tests. There is nothing reflecting the hard fact that NAPLAN and PISA are pseudomathematical garbage. TIMSS, on the other hand, is decidedly not garbage, so what does the Review do with that? That is interesting.
In what could have been a beacon paragraph, the Review compares the Australian Curriculum with expectations on TIMSS:
“… The Australian Curriculum emphasis on knowing and applying is similar to TIMSS but the Australian Curriculum does not appear to cover some of the complexity that is described in the TIMSS framework under reasoning. It seems likely, too, that a substantial number of TIMSS mathematics items are beyond Australian Curriculum expectations for achievement, especially at the Year 4 level.”
In summary, the emphasis on “knowing and applying” mathematics in the Australian Curriculum is just like TIMSS, as long as you don’t really care how much students know, or how deeply they can apply it, or how successful you “expect” them to be at it. Yep, two peas in a pod.
What does the Review then do with this critical paragraph? Nothing. They just drone ahead. Here is the indication that their entire Review is doomed to idiot trivialities, but they can’t see it, or won’t admit it. They see the smoke, note the smoke, but it doesn’t occur to them, or they just can’t be bothered, or it wasn’t in their idiot Terms, to look for the damn gun.
Finally, what of the recommendations proposed by the Review? There are two that concern the testing of numeracy and/or mathematics. The first, Recommendation 2.2, is that authorities
“Rename the numeracy test as mathematics …”
Huh. And what would be the purpose of that? Well, supposedly it would “clarify that [the test] assesses the content and proficiency strands of the Australian Curriculum: Mathematics”. Except, of course, and as the Review itself acknowledges, the Numeracy test doesn’t do anything of the sort. And, even to the minimal extent that it does, it just points back to Elephant Number One, that the Australian Curriculum is not a properly sound basis for anything.
The isolated suggestion to rename a test is of course a distracting triviality. Alas, not all of the Review’s recommendations are so trivial. Recommendation 2.3 proposes a new test, for
“… [the] assessment of critical and creative thinking in science, technology, engineering and mathematics (STEM) …”
Ah, Yes. Let’s test whether ten-year-old Tommy is the new Einstein.
This is a monumentally stupid recommendation. Is Jenny the next Newton? Maybe. But can she manipulate numbers and expressions with sufficient speed and accuracy to hold, let alone mould, a substantial mathematical thought in her head? Just maybe you might want to test for that first? Is Carol the new Capote? Then perhaps first teach her the basics of grammar, first teach her how to construct a clear and correct sentence. Then you can think to tease out all the great works inside her. Is Fritz another Mozart? Gee, I dunno. How are his scales? And on and on.
This constant, idiot call for the teaching of and, worse, the testing of “higher order” thinking, this mindless genuflection to reasoning and creativity, is maddening. It ignores the stubborn fact that deeper thought and creativity in any discipline can only be built upon the craft, upon the basic knowledge and skills of that discipline. The Review’s call is even worse for that, since STEM isn’t a discipline, it’s just a foggy con job.
This Godzilla versus Mothra battle is never likely to end, nor likely to end well. On the one side are the numeracy nuts, who can’t see the value of skills independent of some ridiculous application. On the other side are the creativity clowns, who ludicrously denigrate “the basics”, and ludicrously paint NAPLAN as the basics they’re denigrating. Neither side exhibits any understanding of what the basics are, or their critical importance. Neither side has a clue. Which means, unless and until these two monsters somehow destroy each other, we’re all doomed.
There’s much we could write about Matthew Bach, who recently gave up teaching and deputying to become a full-time Liberal clown. But, with great restraint, we’ll keep to ourselves the colourful opinions of Bach’s former school colleagues; we’ll ignore Bach’s sophomoric sense of class and his cartoon-American cry for “freedom”; we’ll just let sit there Bach’s memory of “the sense of optimism in Maggie Thatcher’s Britain”.
Yesterday, Bach had an op-ed in the official organ of the Liberal Party (paywalled, thank God). Titled We must raise our grades on teacher quality, Bach’s piece was the predictable mix of obvious truth and poisonous nonsense, promoting the testing of “numeracy” and so forth. One line, however, stood out as a beacon of Bachism:
“But, as in any profession, a small number of teachers is not up to the mark.”
We is thinking that is very, very true.
This adventure game comes courtesy of Victoria’s Department of Education and Training. Click on the graphic, or go here. (Don’t try to do it all. You will die.) If you can be bothered, you can also complete a survey here.
A few days ago we received an email from Aaron, a primary school teacher in South Australia. Apparently motivated by some of our posts, and our recent thumping of PISA in particular, Aaron wrote on his confusion on what type of mathematics teaching was valuable and asked for our opinion. Though we are less familiar with primary teaching, of course we intend to respond to Aaron. (As readers of this blog should know by now, we’re happy to give our opinion on any topic, at any time, whether or not there has been a request to do so, and whether or not we have a clue about the topic. We’re generous that way.) It seemed to us, however, that some of the commenters on this blog may be better placed to respond, and also that any resulting discussion may be of general interest.
With Aaron’s permission, we have reprinted his email, below, and readers are invited to comment. Note that Aaron’s query is on primary school teaching, and commenters may wish to keep that in mind, but the issues are clearly broader and all relevant discussion is welcome.
Good afternoon, my name is Aaron and I am a primary teacher based in South Australia. I have both suffered at the hands of terrible maths teachers in my life and had to line manage awful maths teachers in the past. I have returned to the classroom and am now responsible for turning students who loathe maths and have big challenges with it, into stimulated, curious and adventure seeking mathematicians.
Upon commencing following your blog some time ago I have become increasingly concerned I may not know what it is students need to do in maths after all!
I am a believer that desperately seeking to make maths “contextual and relevant” is a waste, and that learning maths for the sake of advancing intellectual curiosity and a capacity to analyse and solve problems should be reason enough to do maths. I had not recognised the dumbing-down affect of renaming maths as numeracy, and its attendant repurposing of school as a job-skills training ground (similarly with STEM!) until I started reading your work. Your recent post on PISA crap highlighting how the questions were only testing low level mathematics but disguising that with lots of words was also really important in terms of helping me assess my readiness to teach. I have to admit I thought having students uncover the maths in word problems was important and have done a lot of work around that in the past.
I would like to know what practices you believe constitutes great practice for teaching in the primary classroom. I get the sense it involves not much word-problem work, but rather operating from the gradual release of responsibility (I do – we do – you do) explicit teaching model.
I would really value your thoughts around this.
Warm regards,
Aaron
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.
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.
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.