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.

5 Replies to “Accentuate the Negative”

  1. I hope you continue to post; it is good to have a forum to express (oh so politely, of course) concerns over the deficiency of Australian (Victorian) Mathematics education and assessment, particularly when it does with evidence and reason as well as the appropriate derision!

  2. The VCAA has nothing on the Queensland Curriculum and Assessment Authority. You want to expose absurdity?…Come to Queensland!

    1. Hi Brian. Perhaps you meant to comment on the Methods post, but no matter. I’m fully aware that the VCAA has no monopoly on absurdity. If you want to email me some info (, I’ll happily consider posting about QCAA.

  3. Thanks for the laugh Marty. NAPLAN is national, so sorry Queenslander, we’re all equally suffering in this case…

    It seems to me that there are perhaps a couple of paradoxes here (paradox of course being totally the wrong word) that those who both care about these matters and have the time to logically and carefully point out the absurdities are ignored by those responsible for creating the tasks and those who are responsible for creating the tasks have neither the time nor the knowledge to realise the absurdity of the material.

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