Polynomialy Perverse

What, with its stupid curriculastupid texts and really monumentally stupid exams, it’s difficult to imagine a wealthy Western country with worse mathematics education than Australia. Which is why God gave us New Zealand.

Earlier this year we wrote about the first question on New Zealand’s 2016 Level 1 algebra exam:

A rectangle has an area of  \bf x^2+5x-36. What are the lengths of the sides of the rectangle in terms of  \bf x.

Obviously, the expectation was for the students to declare the side lengths to be the linear factors x – 4 and x + 9, and just as obviously this is mathematical crap. (Just to hammer the point, set x = 5, giving an area of 14, and think about what the side lengths “must” be.)

One might hope that, having inflicted this mathematical garbage on a nation of students, the New Zealand Qualifications Authority would have been gently slapped around by a mathematician or two, and that the error would not be repeated. One might hope this, but, in these idiot times, it would be very foolish to expect it.

A few weeks ago, New Zealand maths education was in the news (again). There was lots of whining about “disastrous” exams, with “impossible” questions, culminating in a pompous petition, and ministerial strutting and general hand-wringing. Most of the complaints, however, appear to be pretty trivial; sure, the exams were clunky in certain ways, but nothing that we could find was overly awful, and nothing that warranted the subsequent calls for blood.

What makes this recent whining so funny is the comparison with the deafening silence in September. That’s when the 2017 Level 1 Algebra Exams appeared, containing the exact same rectangle crap as in 2016 (Question 3(a)(i) and Question 2(a)(i)). And, as in 2016, there is no evidence that anyone in New Zealand had the slightest concern.

People like to make fun of all the sheep in New Zealand, but there’s many more sheep there than anyone suspects.

The Treachery of Images

Harry scowled at a picture of a French girl in a bikini. Fred nudged Harry, man-to-man. “Like that, Harry?” he asked.

“Like what?”

“The girl there.”

“That’s not a girl. That’s a piece of paper.”

“Looks like a girl to me.” Fred Rosewater leered.

“Then you’re easily fooled,” said Harry. It’s done with ink on a piece of paper. That girl isn’t lying there on the counter. She’s thousands of miles away, doesn’t even know we’re alive. If this was a real girl, all I’d have to do for a living would be to stay at home and cut out pictures of big fish.”

                       Kurt Vonnegut, God Bless you, Mr. Rosewater


It is fundamental to be able to distinguish appearance from reality. That it is very easy to confuse the two is famously illustrated by Magritte’s The Treachery of Images (La Trahison des Images):

The danger of such confusion is all the greater in mathematics. Mathematical images, graphs and the like, have intuitive appeal, but these images are mere illustrations of deep and easily muddied ideas. The danger of focussing upon the image, with the ideas relegated to the shadows, is a fundamental reason why the current emphasis on calculators and graphical software is so misguided and so insidious.

Which brings us, once again, to Mathematical Methods. Question 5 on Section Two of the second 2015 Methods exam is concerned with the function V:[0,5]\rightarrow\Bbb R, where

\phantom{\quad}  V(t) = de^{\frac{t}3} + (10-d)e^{\frac{-2t}3}\,.

Here, d \in (0,10) is a constant, with d=2 initially; students are asked to find the minimum (which occurs at t = \log_e8), and to graph V. All this is par for the course: a reasonable calculus problem thoroughly trivialised by CAS calculators. Predictably, things get worse.

In part (c)(i) of the problem students are asked to find “the set of possible values of d” for which the minimum of V occurs at t=0. (Part (c)(ii) similarly, and thus boringly and pointlessly, asks for which d the minimum occurs at t=5). Arguably, the set of possible values of d is (0,10), which of course is not what was intended; the qualification “possible” is just annoying verbiage, in which the examiners excel.

So, on to considering what the students were expected to have done for (c)(ii), a 2-mark question, equating to three minutes. The Examiners’ Report pointedly remarks that “[a]dequate working must be shown for questions worth more than one mark.” What, then, constituted “adequate working” for 5(c)(i)? The Examiners’ solution consists of first setting V'(0)=0 and solving to give d=20/3, and then … well, nothing. Without further comment, the examiners magically conclude that the answer to (c)(i) is 20/3 \leqslant d< 10.

Only in the Carrollian world of Methods could the examiners’ doodles be regarded as a summary of or a signpost to any adequate solution. In truth, the examiners have offered no more than a mathematical invocation, barely relevant to the question at hand: why should V having a stationary point at t=0 for d=20/3 have any any bearing on V for other values of d? The reader is invited to attempt a proper and substantially complete solution, and to measure how long it takes. Best of luck completing it within three minutes, and feel free to indicate how you went in the comments.

It is evident that the vast majority of students couldn’t make heads or tails of the question, which says more for them than the examiners. Apparently about half the students solved V'(0)=0 and included d = 20/3 in some form in their answer, earning them one mark. Very few students got further; 4% of students received full marks on the question (and similarly on (c)(ii)).

What did the examiners actually hope for? It is pretty clear that what students were expected to do, and the most that students could conceivably do in the allotted time, was: solve V'(0)=0 (i.e. press SOLVE on the machine); then, look at the graphs (on the machine) for two or three values of d; then, simply presume that the graphs of V for all d are sufficiently predictable to “conclude” that 20/3 is the largest value of d for which the (unique) turning point of V lies in [0,5]. If it is not immediately obvious that any such approach is mathematical nonsense, the reader is invited to answer (c)(i) for the function W:[0,5]\rightarrow\Bbb R where W(t) = (6-d)t^2 + (d-2)t.

Once upon a time, Victorian Year 12 students were taught mathematics, were taught to prove things. Now, they’re taught to push buttons and to gaze admiringly at pictures of big fish.

Factoring in the Stupidity

It is very brave to claim that one has found the stupidest maths exam question of all time. And the claim is probably never going to be true: there will always be some poor education system, in rural Peru or wherever, doing something dumber than anything ever done before. For mainstream exams in wealthy Western countries, however, New Zealand has come up with something truly exceptional.

Last year, New Zealand students at Year 11 sat one of two algebra exams administered by the New Zealand Qualifications Authority. The very first question on the second exam reads:

A rectangle has an area of  \bf x^2+5x-36. What are the lengths of the sides of the rectangle in terms of  \bf x.

The real problem here is to choose the best answer, which we can probably all agree is sides of length \pi and (x^2+5x-36)/\pi.

OK, clearly what was intended was for students to factorise the quadratic and to declare the factors as the sidelengths of the rectangle. Which is mathematical lunacy. It is simply wrong.

Indeed, the question would arguably still have been wrong, and would definitely still have been awful, even if it had been declared that x has a unit of length: who wants students to be thinking that the area of a rectangle uniquely determines its sidelengths? But, even that tiny sliver of sense was missing.

So, what did students do with this question? (An equivalent question, 3(a)(i), appeared on the first exam.) We’re guessing that, seeing no alternative, the majority did exactly what was intended and factorised the quadratic. So, no harm done? Hah! It is incredible that such a question could make it onto a national exam, but it gets worse.

The two algebra exams were widely and strongly criticised, by students and teachers and the media. People complained that the exams were too difficult and too different in style from what students and teachers had been led to expect. Both types of criticism may well have been valid. For all of the public criticism of the exams, however, we could find no evidence of the above question or its Exam 1 companion being flagged. Plenty of complaining about hard questions, plenty of complaining about unexpected questions, but not a word about straight out mathematical crap.

So, not only do questions devoid of mathematical sense appear on a nationwide exam. It then appears that the entire nation of students is being left to accept that this is what mathematics is: meaningless autopilot calculation. Well done, New Zealand. You’ve made the education authorities in rural Peru feel very much better about themselves.

Accentuate the Negative

Each year about a million Australian school students are required to sit the Government’s NAPLAN tests. Produced by ACARA, the same outfit responsible for the stunning Australian Curriculum, these tests are expensive, annoying and pointless. In particular it is ridiculous for students to sit a numeracy test, rather than a test on arithmetic or more broadly on mathematics. It guarantees that the general focus will be wrong and that specific weirdnesses will abound. The 2017 NAPLAN tests, conducted last week, have not disappointed. Today, however, we have other concerns.

Wading into NAPLAN’s numeracy quagmire, one can often find a nugget or two of glowing wrongness. Here is a question from the 2017 Year 9 test:

In this inequality is a whole number.

\color{blue} \dfrac7{n} \boldsymbol{<} \dfrac57

What is the smallest possible value for n to make this inequality true?

The wording is appalling, classic NAPLAN. They could have simply asked:

What is the smallest whole number n for which \color{red} \dfrac7{n} \boldsymbol{<} \dfrac57\, ?

But of course the convoluted wording is the least of our concerns. The fundamental problem is that the use of the expression “whole number” is disastrous.

Mathematicians would avoid the expression “whole number”, but if pressed would most likely consider it a synonym for “integer”, as is done in the Australian Curriculum (scroll down) and some dictionaries. With this interpretation, where the negative integers are included, the above NAPLAN question obviously has no solution. Sometimes, including in, um, the Australian Curriculum (scroll down), “whole number” is used to refer to only the nonnegative integers or, rarely, to only the positive integers. With either of these interpretations the NAPLAN question is pretty nice, with a solution n = 10. But it remains the case that, at best, the expression “whole number” is irretrievably ambiguous and the NAPLAN question is fatally flawed.

Pointing out an error in a NAPLAN test is like pointing out one of Donald Trump’s lies: you feel you must, but doing so inevitably distracts from the overall climate of nonsense and nastiness. Still, one can hope that ACARA will be called on this, will publicly admit that they stuffed up, and will consider employing a competent mathematician to vet future questions. Unfortunately, ACARA is just about as inviting of criticism and as open to admitting error as Donald Trump.