## On the banks of denial: the Higher Maths “crocodile” question

“Come hither, Little One,” said the Crocodile, “for I am the Crocodile,” and he wept crocodile-tears to show it was quite true.

Rudyard Kipling, Just So Stories: The Elephant’s Child.

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.

The time taken, T, measured in tenths of a second, is given by

$T(x) = 5\sqrt{36+x^2} + 4(20-x)$

(a) (i) Calculate the time taken if the crocodile does not travel on land.
[1 mark]

(ii) Calculate the time taken if the crocodile swims the shortest distance possible.
[1 mark]

(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.
[8 marks]

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.

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### 35 Responses to On the banks of denial: the Higher Maths “crocodile” question

1. John Birch says:

Could I maybe throw in one observation that I had when seeing this question.

The answer in the video deduces the river is 6m wide – right angled triangle, etc.

But it isn’t. The question clearly states that this is a river, not a lake. Water in a river moves and the question also says that the zebra is upstream, so the water is moving against the crocodile (this must matter as why else would the question say that the zebra is “upstream”?).

So we need to know the speed of flow of the river – and we are not given that. So you cannot answer part a(ii).

• I think that’s a fair point — it’s another peril of the way that this mathematical problem has been “situated”, so a student who starts thinking about the context is likely to be more confused than a student who goes into it on autopilot.

• andrea says:

No, I don’t see that as a problem… The speed is embedded in the function that was given, so if there is any stream velocity to be taken into account, that’s already been done for you.

• diddle40 says:

I have never seen a crocodile stalk prey like a zebra ON LAND!!!! Does the examiner realise how fast the crocodile would have to move to catch a zebra ON LAND!

If the scene painted by the examiner is impossible, what is the point of the question?

• Ryan Lanham says:

Not at all. It is a principle of least effort question–very common in Newtonian Physics. Is it easy? It isn’t OBVIOUS. And it is designed to be tricky… which is frustrating… in part because it is linguistically biased. It isn’t, in other words, just a maths problem. It is a reading problem. The obvious thing to do in any such problem is substitute 0 and substitute the given maximum (20) for x and get a number. The 8 point equation is… harder, but only requires a derivative.

A properly trained student could deal with it. It isn’t crushing. That said, it isn’t “fair.” Too much relies on logic/application/reading rather than calculation. Those traits are not “maths.” Maths is, at its core, procedural. This is an IQ question except it cannot be answered without calculus. Messy. Not a good question. Impressively challenging, but solving it proves nothing except calmness under pressure as the maths are trivial when set up. “Setting it up” is all about reading… not maths.

• I’d disagree that “maths is, at its core, procedural” — my view is that maths is at its core about reasoning rather than applying procedures mechanically. (If you want to be hyper-reductionist you could ask whether I’m being unnecessarily mystical by suggesting that reasoning isn’t mechanical; at the level of everyday thought, though, there is a distinction between mechanical and logical thought.) I think there’s considerable educational value in setting problems that genuinely combine “reading” with procedural “maths” so that they demand some systematic reasoning, but questions like this where the “reading” part seems to get in the way of the “maths” aren’t really fair.

2. Update 2015/09/12: to nobody’s particular surprise, the coverage of today’s Scottish Parliament evidence session has focussed on whether the question was “too challenging”, not on whether it made sense. Crocodile tears all round.

• diddle40 says:

Well, the answer could quite reasonably be, “impossible, crocodiles don’t move fast enough to outrun a zebra”.

Since the examiner used living creatures to create the question, there is no reason not to point out the obvious.

3. andrea says:

After reading this (and other posts) I am still not clear as to what the problem of this question was…
For me the unclear bit is the beginning of (b): “Between these two extremes” because to me it refers to the two values we have just calculated. Using the word “extremes” it seems to suggest that one is a min and one a max, but then we are asked to find a min again. That had me think for a while.

As I don’t know what students are supposed to know at this level and I haven’t done anything like this for over 25 years 😦 😦 (but I did manage, although I forgot that the time was in tenth of a sec ) what is the problem the derivative or the enunciation of the problem? At the end of the day the question boils down to working out a min point of a function. Shouldn’t they be expected to know how to do it?
Is the enunciation of the problem to be considered so obscure?

• andrea says:

I think it should have said “between these two extremes of x”, otherwise I read it as “between these two extremes (of time)”, but that coud be me getting old 😀

• I think the problem is precisely in the “boiling down” of the problem. The core may be a perfectly tractable mathematical problem, and the derivative isn’t hard to calculate, but the task has been badly described (your point about “extremes” is another good example of poor phrasing) and also embedded within a context that tends to distract and confuse. As I noted in my post, someone who is confident (or who tends not to worry about detail) can step over the distractions and automatically correct the errors in the description, but a candidate who isn’t confident or who worries about details may be thrown. Obscure: no; clumsy: yes; and under time pressure a little clumsiness can create a lot of problems!

• andrea says:

Thanks for that.
http://www.bbc.co.uk/news/uk-scotland-33778968
let me to believe that the question was “too difficult” as it is the only one that was mentioned. I should really look at the rest of the paper as well 😀 Did you have a look at it? Was it “that” difficult (although I have nothing to compare it with).

4. I think the crocodile question probably did cause the most trouble, though I don’t find the word “difficult” helpful because it conflates difficulty due to a high academic standard with difficulty due to badly constructed questions. (To mount one of my hobby-horses, raising standards does not equate to raising difficulty — you can always make an exam more difficult by reducing the time allowed for it, or for that matter by requiring students to sit the paper wearing full deep-sea diving gear.) Looking at the paper myself, and speaking to school teachers and other colleagues, I think several of the other questions were academically challenging, though not beyond the bounds of the syllabus or of precedent…

5. denismollison says:

Seems a typical maths question where all you need is the formula, with a “real life” background sketched in. Students used to that kind of question should realise that the questions boil down to:
(a)(i) Set x=20 in formula, giving T = 5 x sqrt(436) = 104.4 approx (calculator needed).
(a)(ii) Set x=0 in formula, giving T = 5 x 36 + 80 = 110.
(b) Differentiate the given formula and set the derivative equal to zero: answer is x=8, giving T = 5×10+4×12 = 98.
The hard part (b) took my rusty brain less than 2 minutes; writing it out neatly might have made that 3, so I would expect a confident good student to do the whole question in less than 5 minutes.

As to the context, yes it is clumsy and ignores possible real life complications such as whether the river is straight and of uniform width, and whether there is a current that should be taken into account, but again this is par for the course at this level of maths teaching; hopefully these aspects are discussed in classroom teaching, but you can’t include an essay-length discussion of simplifying assumptions in every question.

• I think the word “confident” is important; I agree that a confident good student probably wouldn’t be flummoxed even under exam conditions, but unfortunately a lot of good students aren’t confident (and vice versa). Although one can’t provide a detailed description of modelling assumptions in a question, one should at least ensure that the information given is consistent with the model: including extraneous information (upstream/downstream) while omitting necessary information (lines in the diagram at right angles) still seems to me to be a breach of contract. (I’ve set plenty of optimisation questions in my time both in class and in exams, and I’m fairly confident that it’s possible to do a better job than the examiners did on this occasion…)

Yep I calculated 8 as the answer to B with a 20-year rust layer on my calculus.

6. diddle40 says:

Here is a better scenario. A crocodile sees a zebra on the opposite bank of a river, downstream. It starts thinking…….that is going to be my next meal. How can i most easily get it? That would be swimming up to the point opposite to where it is grazing. Hmmm, then I still have to get out of the water, waddle fast up to it, and then launch the attack. No, that won’t work. It is much faster than I, it will bound away. The only way for this to work, is for the zebra to already been dead.

And if that is the case, there is no need for me to hurry. It’s carcass will still be there tomorrow. I think i’ll take a little rest before I make the effort.

• This may be a little unfair to the intellectual powers of crocodiles (even excluding those of the Kipling variety). At least one mathematician has argued that his dog Elvis was capable of solving optimisation problems (“Do Dogs Know Calculus?”, The College Mathematics Journal 34(3): 178-182, 2003). But this might be straying from the subject…

7. diddle40 says:

More to the point, it reveals the intellectual power of scottish examiners! They posed the question in ‘real life’ terms, which in this case could not even be translated into hypothetical.

The crocodile, on the other hand, does have some ‘intellectual powers’, in that it hunts prey as part of its own existence. No crocodile hunts in the way that question is posed. It is not mind-boggling difficult to do a little research when posing the question on how crocodiles actually hunt.

8. Steve Jones says:

OK – there’s a little ambiguity, but the assumptions you need to make are fairly obvious and in my days you were encouraged to state them explicitly. It would have been more important if those being tested were actually asked to derive what is actually a very simple equation given the width of the river and speed of swimming/walking. As it is they’ve been handed the formula on a plate and the first two questions are just substituting numbers into a formula using some fairly simple assumptions (banks are parallel, river is a fixed width etc.

In contrast, the calculus is much harder. However, even here I just wonder if the examiners aren’t just expecting a trial-and-error approach by plugging in numbers or taking a graphical approach.

If calculus is required (are they asked to show working), then agreed it’s much more tricky – and might even qualify as an actual maths question – but if all that is required is to produce a number, a few minutes work will reveal the answer (that’s assuming they are allowed a calculator).

In all, surely not a difficult question for those being tested to university entrance standards.

• I’ve not seen the marking scheme, but I’m pretty sure from the Higher syllabus that the examiners are expecting calculus to be used in optimisation questions, rather than trial-and-error or graphical approximations; I’m also fairly sure that the working does matter, though I suspect it’s not always marked very rigorously.

I agree that if the candidates had been given a (clearly stated!) problem and asked to derive the formula before optimising then this would have been a much better question — and one that would have been “challenging” in the right way. This would have been more substantial in terms of time and working, though, which may be why it wasn’t set in this manner given the other constraints on the paper… though I do wonder whether it was how the question was stated in some earlier draft.

9. A further update, for anyone who’s curious: the report quoted by today’s BBC News article can be found here, while the marking instructions are here. I’ve not had a chance to look at them in detail yet but intend to follow this up in a later post…

10. Seb HH says:

Looks like a pretty typical school exam question (from my memory); a taught method disguised under some confusing and irrelevant context (“gotcha” as it is described in the article).

Reading about this initially I was thinking a confusing scenario was actually a good thing to prepare students for University, especially those going into engineering. The jump from school maths to university level math is a big one (in my experience) and I think it would be good if exam boards were better preparing students for university. But yet again this question is teaching student to disregard the written text, generally the opposite to what is required in a real life problem (that the question is pretending to be). “Gotcha” has it’s place in life as people will always lie, distract and manipulate but it’s overdone in school exams and I’m sure nobody from the exam board or in parliamentary discussions is arguing the true life “gotcha” case!

I hope the budding engineers that took this paper didn’t panic but managed to read through the problem, disregard the woolly stuff and apply the simple methodology required to answer it, without wasting too much time. Questions were like this 10 years ago and it looks like nothing has changed.

• But yet again this question is teaching student to disregard the written text, generally the opposite to what is required in a real life problem (that the question is pretending to be).

I’d agree entirely — this is far too often the flaw in “situated” problems, “word problems” and the like. Years ago I taught a student who’d been advised by his Higher tutor to ignore the words when reading maths questions, and instead just to look at the equations and identify a few key words to indicate the template he should apply to the question. Sadly, this had got him through Higher; inevitably he was struggling when he got to university and was trying to tackle more (genuinely) challenging problems in engineering mathematics…

11. Ben Glancy says:

Unless you were familiar with drawing graphs, finding the peaks and troughs etc you could not make sense of why 1.i how can you catch a zebra standing on the land (an unspecified distance away from the river) without getting onto land yourself. and 1ii swim the shortest distance to where? The zebra? The other side of the river? Where??
I do think it’s slightly confusing question, but not if you have drilled through a few. You would know what the question is covering and know pretty much in advance what information you need to work it out.

They could have just put some more points on like C and D and E and asked for the distance between them.

• If I’d set a question like this (there but for the grace etc.) and couldn’t extend it substantially to include marks for setting up the problem, I think I’d have scaffolded it along these lines. For example, providing co-ordinates for the zebra, the crocodile and the point P immediately removes quite a lot of the ambiguity introduced by the diagram.

As for drilling through a few, I have a nasty suspicion that this would make some students very good at solving problems involving crocodiles or zebras, or even optimisation problems with essentially this geometry, but transferring this ability to optimisation in general is considerably tougher!

12. vedanthnair says:

I really don’t know what was so hard about this question. I did my A-level maths exams last year (the equivalent of the Scottish highers that English 17/18 year olds do), and that question was a relatively easy question compared to some of the calculus questions I have done in past papers. Are maths exams in Scotland easier than English ones , or is there less of an emphasis on calculus?

• The A-level isn’t the equivalent of the Higher — the Higher is generally taken at the end of S5 (equivalent to Year 12 in England) rather than at the end of Year 13 like the A-level. The closest equivalent to the A-level under the Scottish system is the Advanced Higher, typically taken at the end of S6. (The confusion comes from the fact that Scottish students have traditionally been able to proceed to a four-year university degree after twelve years at school, as against a three-year degree after thirteen years in England… although a lot of these traditional patterns are now changing!)

13. Ernie says:

Calculate the time taken if the crocodile does not travel on land.

Well, the zebra is ON land so if the croc does not travel on land the answer is….. the croc never gets to the zebra.

Also, the speed it can run is not given. Is it a fast runner (11mph top speed) or is it a slao runner?
Also the flow of the river and the exact distance across is not given. Both of these variable would definitely effect the answer.

• In fairness to the setters, I think that if the zebra is right on the bank then the crocodile can grab it without fully emerging from the water and “travelling on land”. The crocodile’s speeds on land and water, and the width of the river, would be necessary information if one were trying to derive the formula for the travel time, but in the question as asked they aren’t required.

• diddle40 says:

My thoughts too. The zebra is even shown standing in the grass. Not at the waters edge which is the only place the crocodile would ever be able to catch it. The figures given and all the calculations would never equal the diagram, no matter whether the student got the calculation right or not!

They were asked to calculate the crocs journey to the prey, the answer is – it won’t happen. 😀

• The zebra is even shown standing in the grass.

Well, agreed — the diagram that appeared in the paper is an object lesson in why one shouldn’t use silly wee bits of clip-art in a technical schematic!