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 
. What are the lengths of the sides of the rectangle in terms of 
.
The real problem here is to choose the best answer, which we can probably all agree is sides of length 
and 
.
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 
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.