“Come hither, Little One,” said the Crocodile, “for I am the Crocodile,” and he wept crocodile-tears to show it was quite true.
The new-look Higher Maths hasn’t got off to the best of starts. The exam this May provoked an unusual degree of outrage: teachers reported some of their strongest pupils leaving the room in tears, while a social-media storm among students led rapidly to a petition against the difficulty of the exam and culminated in questions to the First Minister. At the root of much of the dismay was a question about a crocodile.
It took about a month before someone in my department got hold of a copy of the egregious paper, and up to that point most of us had felt fairly safe dismissing the protests as the usual twitter-assisted teacup storm. Then we saw this question, and we understood what the fuss was about.
The question, Q8 on Paper 2, is as follows — verbatim save for the diagram, which I’ve redrawn.
A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river.
Crocodiles travel at different speeds on land and in water.
The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, x metres upstream on the other side of the river as shown in the diagram.
(a) (i) Calculate the time taken if the crocodile does not travel on land.
(ii) Calculate the time taken if the crocodile swims the shortest distance possible.
(b) Between these two extremes there is one value of x which minimises the time taken. Find this value of x and hence calculate the minimum possible time.
To my mind, this shows all the signs of a “gotcha” question gone horribly wrong.
Let’s be clear: there’s a place in exams for hard questions, even for tricky questions, and by the reckoning of school teachers whom I trust the new Higher wasn’t short of these — for which the setters should be commended, as Higher Maths for some time has seemed to reward blind algorithmic approaches rather than actual thought. Q8 seems to have been set up to be one of these tricky questions. Optimisation is always a tough subject for students; on top of that, this question involves an unfamiliar scenario, a problem statement that must be synthesised from the text and the diagram, and a routine but non-trivial piece of calculus. So far, so tough-but-fair. Where things go wrong is in the number of minor errors and obscurities that the examiners managed to squeeze into the question as they asked it.
The worst offence against clarity is that third sentence, which defines x as the particular distance that minimises the travel time, although both (a) and (b) require it to be treated as a variable. The formula for T(x) is introduced without explanation, and though it requires only a little thought to see where it comes from, I can see why this could be off-putting. The diagram is shoddy; for the question and formula to make sense, the river must be straight — as it is not drawn — and the vertical line must be perpendicular to the river bank — as is never stated. Measuring T in tenths of a second seems unnecessary and slightly bizarre. Finally, the phrasing of part (a) left some of my colleagues confused, though others saw no problem with it; these two marks are available essentially for parsing the diagram, so it’s unfortunate that they precede the essentially algorithmic part (b). On the whole, the question gives the impression that it was originally written in a somewhat different form (perhaps lacking part (a) but including the derivation of the formula) and rather ineptly hacked into its current shape, and whoever refereed it deserves to be dragged through a thorn-bush.
One can object that most candidates wouldn’t notice these errors enough to be bothered by them. As DLBMaths’s video solution demonstrates, it’s certainly possible to work through the question, calmly and systematically, in about ten minutes, well within the twelve to thirteen minutes notionally available for ten marks. This, I suspect, is how an unexceptional candidate who had swotted up on optimisation would have tackled it. Equally, an experienced maths teacher faced with the question can easily tut at the phrasing, fill in the obvious gaps, and stride over the swamp of minor errors without too much effort. The candidates who were liable to suffer are those who fall between these endpoints: the bright but nervous students, hag-ridden by perpetual assessment, hitting every question at a run and horribly vulnerable to stumbling; and the compulsive sense-makers for whom a question must be understood completely or not at all — the budding mathematicians, in other words.
Any kind of assessment relies on an implicit contract between candidate and examiner. The examiner’s side of the contract might be phrased something like this: we may examine you about anything that’s in the curriculum; we may draw on other knowledge and basic skills that one could reasonably expect someone of your level of education to possess (though admittedly this can be a ticklish point); we may choose to examine topics in different ways from year to year or to combine topics in ways you hadn’t seen before; but we will play fair throughout. This means that we won’t ask you to display blinding flashes of inspiration under exam conditions; the tasks we set will be proportional to the time available to complete them; and the questions we ask will make sense under scrutiny. The last clause is the one that the crocodile question violates, and in a written exam it is essential. In an oral, a student can respond to a vaguely phrased question by asking for immediate clarification, and in a coursework assignment s/he has time to contact the examiners and await a reply, but in a written exam the statement of each problem must be coherent and self-contained, or the candidate has no chance.
The wider issue that this contract relates to is confidence. We’re used to the idea that students should develop confidence in themselves — the kind of confidence required, for example, to stand up and defend their ideas in public — and this has been built into the basic aims of CfE. However, this is not the only kind of confidence needed in mathematics. A student also needs to develop a confidence in the subject: the confidence that’s required to say, when faced with an unfamiliar problem, “I can’t follow a recipe I’ve seen before, but I know that if what I do is mathematically valid then I will make some progress, and I won’t have to invent or guess some new voodoo method to crack the problem”. And the student needs to have confidence in the assessment process, because if the examiners aren’t playing fair then the exam might as well consist of those “what’s the next number in the sequence?” questions that boil down to little more than “what number am I thinking of?” — and passing will be a matter of grace, not of mathematics.
The psychological problem is that when one of these forms of confidence is attacked, the rest can fall with it. Sometimes this is necessary: my defence of “gotcha” questions is based on the observation that sometimes students have more confidence in their ability or in their grasp of the assessment process than is justified by the facts, and this needs to be broken before progress can be made. But there’s a time and a place for this, and to try it in a time-limited high-stakes examination shows all the honesty of purpose of the Crocodile tempting the Elephant’s Child down the banks of that great grey-green, greasy Limpopo River.
I don’t think the setters of the new Higher deserve to be heaved into a hornet’s nest for trying to make elements of the paper more demanding. I don’t think they should be criticised, as they have been, because the tone of the new Higher couldn’t be predicted from the tone of the old past papers — and in both the “specimen” paper and the “exemplar” paper they gave fair warning that optimisation was now very much on the agenda. But they do deserve all the spanking they get for failing to proof-read one of their questions properly, and thus introducing an element of distrust into the relationship between candidates and examiners, which I suspect will take several years to eliminate.
And beyond the setters of the Higher paper there’s a greater failure, which the crocodile question merely points up. That failure is that the entire public debate around exams is still conducted in terms of whether a given exam is “too difficult” or “too easy”, not of whether the exam or the tasks it requires make sense. Until we regard this as the crucial element — until we put aside the juvenile yearly argument over upward or downward trends in pass rates and ask whether our maths courses and maths exams teach or test anything resembling mathematical thinking — then any tears we may cry on behalf of the bewildered Higher candidates will be strictly of the crocodile variety.