## The Crocodile Maths Challenge

I’m indebted to an anonymous informant (John Peacock) for drawing my attention to a BBC Scotland story about an allegedly challenging examination question that appeared on a “Higher Maths” paper. For those of you not up with the Scottish examination system, “Highers” are taken in the penultimate year at school so are presumably roughly equivalent to the AS levels taken in England and Wales.

Anyway, here is the question that is supposed to have been so difficult. For the record, it’s Paper 2, Question 8 of the SQA examination 2015.

Call me old-fashioned, but it doesn’t seem that difficult to me but I never took Scottish Highers and there have been many changes in Mathematics education since I did my O and A-levels; here’s the O-level Mathematics paper I took in 1979, for example. I wonder what my readers think? Comments through the box if you please.

Feel free to give it a go. If you get stuck here’s a worked solution!

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October 14, 2015 at 10:11 am

I earn part of my living tutoring school kids in maths. If one of my grade 12 (final year) German school kids could’nt solve this problem relatively quickly and without difficulty, I would think that I had failed to do my job properly

October 14, 2015 at 1:55 pm

As a regular reader and some-time commentator on your blog, I’ve wondered how you finance yourself. I’ve also done quite a bit of such tutoring.

October 14, 2015 at 10:18 am

I looked at this news item a few days ago. I think there are two things that threw the students with what is essentially a routine calculus question. The first thing was not realising that the formula for T already takes into account the width of the river, the different speeds in water and on land etc. In a sense, too much information is given – a “typical” maths question would be to say find the minimum value of T for x between 0 and 20 and I suspect that would have been accepted as routine. Another possible issue was the lack of scaffolding in the last part of the question, something that in England would be unthinkable. In general the Scottish papers are more demanding than the English equivalents.

The real point here is that the assessment drives what happens in the classroom and students (rightly) complain if they have not been prepared for an assessment. Whenever there is a change in the assessment, when new specifications are introduced, there is usually a fall in the average mark – a matter of some importance when it comes to consistency of grading – which recovers as both teachers and students become more familiar with the new assessments.

October 14, 2015 at 7:11 pm

Yes, I suspect the central problem was that the detailed information led some students to think they had to derive basic results, rather than realising the formula contains it all.

Another possible difficulty is that some students may have mistakenly thought

xwas a coordinate measuring the position of the crocodile, rather than some parameter describing the geometry. Calling it something else might have helped some (perhaps at the cost later of confusing some who were used to differentiating with respect toxonly, rather than any general variable).The statement the crocodile “swims the shortest distance possible” could be ambiguous – it could mean (as intended) that the crocodile swims as little as possible, or it could be interpreted as meaning the crocodile swims along the shortest possible path to the prey (diagonally across the river).

October 14, 2015 at 7:32 pm

There are two other flaws in the question. One is that the crocodile’s transition between water and land is taken to be instantaneous. The other is that the zebra is assumed to be stationary when it would presumably have the sense to run away when the crocodile splashes out of the water.

October 14, 2015 at 12:20 pm

What the candidate has to do is derive the formula for the time to reach the zebra in terms of the crocodile’s speeds on land and in water, and its distance from the bank, and then extract those three parameters by comparison with the formula given. The form of that expression – the sum of a ‘water’ term from Pythagoras’ theorem and a linear term – is presumably given so as to help the candidate derive the general form. There are then two trivial calculations and one piece of differential calculus.

I suspect that most candidates failed to get as far as the calculus calculation because they didn’t twig that they had to extract the values of those three parameters from the formula. I think that the examiners were trying to help by giving a formula containing only numbers and a single unknown parameter, x, but such hybrid formulae are in fact less helpful than an abstract formula in which all parameters are represented by symbols and values are then substituted in one by one. That would be the correct flow of logic – and therefore how the candidate should be thinking. The question is not well phrased.

October 14, 2015 at 2:00 pm

Anton, where does it ask to derive the formula?

Part (a) is plugging the correct value of x into the equation. Part (b) is the typical ‘find the turning point’ that requires taking the derivative and setting to zero.

I expect the students were confused by the lack of guidance in what to do – it requires reading the question to understand what the equation refers to, and knowing how to find the minimum of a function. It’s understandable confusion, but it only happens when students are being taught to follow algorithms rather than how do solve problems.

October 14, 2015 at 2:12 pm

It doesn’t request a derivation of the formula but how else to get the info needed to answer the questions it does ask?

I strongly agree with your 2nd para.

October 14, 2015 at 7:42 pm

The diagram, text and given equation (with terms defined in the text and diagram) contains all the information needed. Which information is missing?

October 15, 2015 at 10:13 am

Are we talking past each other? Of course the question tells you all you need in order to get the answer, but to extract form the formula the croc’s speed on land and in water, and his distance from the bank (all of which you need), you have to derive the formula with these parameters unknown and represented by symbols; then compare. The form of the formula as a pythagoras plus a linear terms gives you a hint what is going on. But see Bryn’s comment above as well as mine.

October 16, 2015 at 9:31 am

But you really don’t have to derive anything and compare parameters! It’s cute the equation is physically intuitive but it could be any old function.

All you need to do is read what the variable x means, and plug the right values into the given equation. No derivation with unknown parameters is needed.

October 16, 2015 at 10:02 am

(To be clear, bmarcj is me – for some reason it didn’t display my name…)

October 14, 2015 at 4:14 pm

I’ve got an engineering degree but I’ve not done any calculus for over 20 years but I was able to do this question quite easily.

This would have been a practice example we would have done in the first year of A level maths.

October 14, 2015 at 4:36 pm

That would be about the right level for AS, actually…

October 14, 2015 at 4:55 pm

What is AS level? I’m so long out if this I don’t know any more. Is it 1st year A levels in old money?

October 14, 2015 at 5:16 pm

…more or less

October 14, 2015 at 5:24 pm

As a bit of fun and to exercise my brain in thinking of entering for A level maths next year. I will need to do some work on some areas as I’m sure I’ve forgotten a lot for example chain rule, integration by parts and my stats are a distant memory.

Though I have your cosmology stats book in my bag and I may read it on my 18 hour flight to India tomorrow. Your book has been on my shelf for a while and I’ve not had chance to get properly stuck in. My flight might give me the rare situation of having a long time with nothing to do.

October 15, 2015 at 10:55 am

I think a big problem with this question is a lack of a clear diagram. The animals are big, and it would have been helpful to add a ‘dot’ for their location to make the mathematical model clearer. It’s also not clear that the dashed line immediately above the river is *supposed* to be horizontal.

October 16, 2015 at 9:02 am

Consider a spherical crocodile of mass M…..

October 16, 2015 at 9:28 am

Not simple enough. It has to be a spherical crocodile in a vacuum.

October 16, 2015 at 10:04 am

It doesn’t matter if the dashed line is horizontal, does it? The path the crocodile takes is irrelevant (it might reach the bank and walk in circles for all we know; all we have is an equation giving ts time to the zebra, given x).

October 16, 2015 at 10:06 am

It’s obvious from the form that the crocodile travels with constant speed in both segments of its journey..

October 16, 2015 at 1:18 pm

That’s how I’d interpret the equation if you threatened me with a crocodile. But the question, as given, doesn’t enforce that interpretation. The definition is pretty minimal.

October 16, 2015 at 2:26 am

I’m surprised at the number of commentators who said that the question was confusing, or the diagram wasn’t clear enough, etc.

I did the question to see what the fuss was about..the first part was trivial, and when I did the second part it came out quite simply and I felt my answer couldn’t be right because otherwise, why was everyone making such a big deal about it?

But no, there seems to be a general consensus that it was too hard.. it seems that everyone’s trying to find unreasonable reasons why young people can’t just extract information from a question and do basic math.

FWIW I thought it was a jolly good question.. interesting, related to real life, and not too hard.. but still testing people’s ability with basic calculus.

Gary

October 16, 2015 at 8:12 am

I do not think people are saying it is too hard – I am certainly not. There are two things. One is the question of whether students are used to answering questions like that. If the specification has changed and the teachers have not yet caught up, the students’ preparation may not have been perfect. If you are used to standard calculus questions such a context-based problem would worry you under exam conditions. The second point is that the diagram is sufficiently ambiguous, and the situation implausible, to worry good students. When I looked at the question I too worried about the crocodile getting into and out of the water.

The point is that the teaching will adjust to accommodate questions of this type and that is a good thing.

I did not know they had crocodiles and zebra in Scotland…

October 17, 2015 at 12:06 am

Several tens of millions of years ago they did. And they are thinking of re-introducing wolves…

October 16, 2015 at 9:28 am

I agree that the question was certainly not too hard, even if many students complained that it was.

Perhaps one issue here is that some of the commentators on this blog have themselves written exam questions, although mostly at university level rather than for school students. Those of us who have set exams are (or should be) very sensitive to leading students to think in certain ways. It is easy to write questions that approach a subject from a different perspective from the ones students were introduced to during teaching. Students might have been led into solving problems in certain ways. Students then might approach problems in exams in the wrong way because they have been led into taking certain paths.

Teachers must always be aware of these issues. Are they leading students to think in certain narrow ways when solving problems? Will this cause problems if the students encounter something different, particularly under exam stress?

October 16, 2015 at 9:30 am

I think the important thing is to concentrate, get into line, play each ball on its merits, and not worry about scoring runs.

October 16, 2015 at 9:58 am

Oh. I missed the part of the question about balls. Did the crocodile throw them or eat them?

October 17, 2015 at 12:06 am

Much of the question is balls.

October 17, 2015 at 12:24 am

I tried not to write that!

October 16, 2015 at 10:40 am

This is of course a simple calculus problem, but it is also a good example of an important principle in physics, namely the principle of least time. From this, one can derive Snell’s law of refraction. This applies to light, but it also explains how all classical trajectories derive from an ultimately quantum-mechanical reality.

Historically, and often in teaching, one starts with classical physics and then “quantizes” something. Often, it is more instructive to start with the proper theory and derive the classical limit. This is one of many interesting themes in the excellent book

Sleeping Beauties in Theoretical Physics* by Thanu Padmanabhan. Expect a book review forThe Observatoryin the February 2016 issue.__________________________

*The title does not refer to pictures which Telescoper has posted of people dozing off during lectures. Nor does it refer to a paper which starts to collect many citations only several years after publication.

October 16, 2015 at 10:42 am

Version with all URLs correct (previous similar comment can be deleted, as can this sentence).

This is of course a simple calculus problem, but it is also a good example of an important principle in physics, namely the principle of least time. From this, one can derive Snell’s law of refraction. This applies to light, but it also explains how all classical trajectories derive from an ultimately quantum-mechanical reality.

Historically, and often in teaching, one starts with classical physics and then “quantizes” something. Often, it is more instructive to start with the proper theory and derive the classical limit. This is one of many interesting themes in the excellent book

Sleeping Beauties in Theoretical Physics* by Thanu Padmanabhan. Expect a book review forThe Observatoryin the February 2016 issue.__________________________

*The title does not refer to pictures which Telescoper has posted of people dozing off during lectures. Nor does it refer to a paper which starts to collect many citations only several years after publication.