# Beethoven’s Ninth

# A viral maths question

This maths question went viral on Reddit recently:

This post in r/funny was quickly cross-posted to r/facepalm with the title ‘Orchestra logic’, and to r/consulting and r/ProgrammingHumor with derogatory titles about project managers and comments about The Mythical Man-Month.

This question going viral is interesting to me for (at least) a couple of reasons:

- I make a living writing maths questions and doing other maths-education-related things, so whenever a bit of maths goes viral I’m excited.
- I’m in the middle of rehearsing Beethoven’s Ninth for a concert in September.

Unfortunately, if you hadn’t already guessed, not everyone was as excited about this question as I was.

# Why does this look like a proportions question?

When you first read it I imagine that, like me, you saw it as a question about proportion. Using the terminology of ‘Thinking, Fast and Slow’, this is a ‘System 1’ response. If you’re lucky, your ‘System 2’ then kicked in and pointed out that, in reality, the time an orchestra takes to play something isn’t dependent on its size, never mind proportional (directly or inversely) to it. It’s not a proportion question, and the answer is ‘40 minutes’.

To solve a maths question, there are roughly two steps:

- Decide what to do.
- Do that.

When the maths question is a word problem, ‘deciding what to do’ involves converting the prose to abstract mathematics.

This question tests how good you are at this. For a student who realises that the time taken shouldn’t change, the second step is easy: just write down ‘40 minutes’. But a student who wrongly decides to solve a direct or inverse proportion question not only has some calculation to do, they’ll also get an incorrect answer — either ‘20 minutes’ or ‘80 minutes’ — at the end, assuming they implement their chosen procedure correctly.

Even if you didn’t fall into the trap, I would wager that you at least noticed it. Nowhere in the picture does the word ‘proportion’ appear and yet, at least for a moment, you ‘knew’ this was a picture of a proportions question just as surely as you ‘knew’ it was a picture of a maths question; just as surely as you ‘knew’ it was a picture of some text.

It’s hard for us to describe how we come to ‘know’ what a word problem like this is about (and I’m writing ‘know’ in inverted commas because some of the things we ‘know’ turn out to be false!) As a result, it’s hard to teach the process of making mathematical sense of a word problem. We have the same problem trying to get a computer to solve a word problem; Wolfram Alpha struggles with our question. Attempts to analyse the process humans use, codify it, and program it into computers have turned out to be less successful in applications than using statistical techniques.

You can contrast this with the routine (if not easy) arithmetic and algebraic manipulation that appears in the second step; once you know what to do, there is generally a well-defined procedure for doing it. For example, once you know that two quantities are proportional and have set up the equation, it’s straightforward to solve it. We can describe these procedures with complete precision to students and computers alike. More than that, we can explain to students why those procedures work.

I think this particular question looks like a proportions question, in particular an inverse proportion question, because it fits the mould of ‘P workers taking a time T to complete a task’. I’ve seen a lot of questions like that, as I imagine you have, and through experience I’ve learnt to recognise this structure.

I would, in fact, be worried about a student who didn’t initially identify this question as a proportions question. It would suggest to me that they hadn’t developed the kind of intuition required to identify what a word problem is about, and hence would be unable to access a genuine run-of-the-mill proportions word problem as they wouldn’t know where to start.

Developing this ‘system 1’ intuition is important. This is one of the reasons to interleave content; students can’t rely on the fact that the worksheet’s title is ‘Proportion’ or the fact that they’ve been learning about proportion all week to help them ‘decide what to do’, and so develop deeper intuition. Craig Barton’s SSDD problems site takes this further. The way to develop intuition is through carefully designed experience, but I’m not convinced that you can hone your intuition to such an extent that a question like this couldn’t catch you out.

To avoid the trap in this question, we need to reflect on and question our intuition.

That’s a ‘system 2’ thing.

# Why might a student fall into the trap?

As I’ve explained above, I really hope that a student’s intuition tells them that this is a proportions question. Of course I also hope that, on reflection, they realise that their intuition was wrong. Let’s think about why they might not have this realisation.

## Lack of knowledge about classical music

There are valid concerns about cultural capital and whether knowing who Beethoven was, what a symphony is, and how an orchestra works could give one student an unfair advantage over another. However, if all the students who got the question wrong were asked point blank ‘Does an orchestra with twice as many players take half the time to play something?’ I think the vast majority of them would confidently answer ‘no’.

I don’t think cultural capital is the main issue at play here, and although I would probably prefer a context like…

Yesterday, a train with 120 passengers took 40 minutes to get from London to Reading. Today, the same train has 60 passengers. How long will it take to get to Reading?

Let P be the number of passengers and T the time the train takes.

…I don’t think that making this change would ensure that all students get an answer of 40 minutes.

## System 2 isn’t activated

Suppose that students rush to find the answer and so neglect to spend the time required to understand the problem as a whole, or to reflect on the wider consequences of their answer. Is the solution just to tell them to be more careful?

In his book ‘How Children Fail’, John Holt talks about ‘producers’ and ‘thinkers’. I’m not going to repeat all the arguments that he makes, but he makes a good case that the schooling process pushes students to be ‘producers’ rather than ‘thinkers’.

One of the reasons we get students to show their working is to try to rectify this, to emphasise the importance of the thought process. But generally the working that students are required to show is for the ‘do that’ stage; with our question it might start with the statement ‘P and T are inversely proportional’ and continue with some algebra.

We don’t get students to start every solution with an in-depth account of how they decided what to do. Nor do I think we should; trying to put these things into words adds more cognitive load, and I’m not convinced it’s germane. Getting students to show their working isn’t a good solution to encouraging reflection in the ‘deciding what to do’ phase; we have to find other methods of signalling its importance. Perhaps we can only provide opportunities for students to experience the consequences of not reflecting.

## System 2 is activated, but is doing other things

Maybe we’re being unfair to students by saying they’re just not engaging their brains. They could be engaged in checking any number of things, for example that:

- they’ve used all the information in the question.
- they’ve done a reasonable amount of work.
- the value they get is sane.

A student who successfully applies the procedure to solve an inverse proportion question will find all the items on the checklist are satisfied.

- They use all the values in the question (120 players, 40 minutes, 60 players) and make nontrivial use of the variables P and T defined in the question.
- They do a reasonable amount of work in setting up the proportions question and solving for the unknown.
- The answer they get, 80 minutes, seems like a reasonable amount of time for an orchestra to play something. Had they got an answer of 80 milliseconds or 80 hours they might have smelled a rat.

These checks are often explicitly taught. Why? Because they are useful; any of these checks failing would arouse a student’s suspicion that either they’d decided to do the wrong thing or had slipped up somewhere in the process of doing it. The fact that all three checks pass when the incorrect procedure is applied shows how carefully the question was written.

Lots has been written about the power of checklists, particularly in aviation, but these checklists are generally incredibly specific to the precise task at hand. It would be nigh on impossible to write a checklist that would, for any maths question, tell you whether you’d solved it correctly.

Of course, checklists for individual questions exist; they’re called mark schemes. You could argue that a checklist for solving a proportion question involving a number of workers and a task should contain a check that the task takes more workers less time to do. (Anyone who has been in a meeting knows that sometimes the opposite is true.) Perhaps with this checklist a student would have avoided the trap in the question — I’ll tell you why I’m not convinced of that shortly — but getting students to memorise numerous context-specific checklists (because they won’t be allowed to bring their checklists into their exams) is clearly a silly idea.

The three checks above aren’t exhaustive but they are simple, relatively unambiguous, and as applicable to a question (seemingly) about inverse proportion as they are to any other. Of course, the first two checks rely on the norms of maths questions being followed, and in the orchestra question they aren’t.

## System 2 is overruled

Or should I say it overrules itself?

This is the most interesting case as far as I’m concerned. A student realises that it doesn’t make sense that the second orchestra takes twice the time, but nevertheless gives the answer ‘80 minutes’ because that’s what the question is asking for.

One way of understanding this phenomenon is by thinking of the student as a novice. They’re uncertain that applying this procedure is the right thing to do, but they’ve had similar experiences previously where they were similarly uncertain and it turned out just fine. They’re not expecting complete certainty; if they waited for complete certainty they’d never get anything done.

After getting an answer of 80 minutes they might still have qualms, but again, that’s not unprecedented. This answer, and the procedure leading to it, passes all three checks from the previous section. The answer ‘40 minutes’, which they might be tempted by, doesn’t; finding it doesn’t require using the number of players from the question, nor the variables P and T. It doesn’t really require any work at all. 80 minutes must be the correct answer, and the fact that it doesn’t make sense is just one of the many mysteries of maths.

Another way of understanding this phenomenon is by thinking of the student as a cynic disillusioned by repeated promises of real-world applications. They know full well that the only sensible answer is 40 minutes, but this is ‘maths world’ where people buy improbable quantities of fruit, answer reasonable enquiries about their age with riddles, and ignore friction. Leave your critical thinking at the door.

Perhaps this student has objected to contexts before and was branded a pedant or ‘smart aleck’ for their troubles. Maybe their teacher thought their criticisms were just an attempt at getting out of doing the sums. Maybe their teacher was right. In any case, while the student has stopped raising objections publicly, privately their views have only been reinforced and their disillusionment has only grown.

It was this kind of student who posted the question on social media.

# The post again

The title of the post gives away the thoughts of the person posting:

The question’s author got it wrong. The time an orchestra takes to play something isn’t inversely proportional to its size.

The person who posted the question doesn’t consider the possibility that it might be a ‘trick question’. Given the number of people who upvoted the post, it appears that many people agreed with this assessment.

Why is this?

With this title, people who see this post are primed to look for a mistake in the question and, having found one, don’t feel the need to look for an alternative interpretation of the question. This has interesting parallels with the question itself, whose structure primes you to think about proportion.

Looking at the comments you’ll see that this can’t be the whole answer; there are several people who reject the idea that this question was intentionally written to make you think:

# Some context

This isn’t the first time this question has gone viral. I first saw it last year on twitter:

The author of the question, a teacher from Nottingham, turned up in the thread a couple of days later and shared the worksheet the question was lifted from:

Although the title of the worksheet is ‘Direct and Inverse Proportion’, we can infer from the warning ‘Beware there is one trick question!’ that our orchestra word problem isn’t just a badly written inverse proportion question. Furthermore, the instruction to ‘Sort these questions into Direct and Inverse proportion’ suggests that the worksheet was written with the importance of ‘deciding what to do’ in mind.

Case closed? Not quite.

Time for some insight from the same century as Beethoven’s Ninth.

# Through the looking glass

In February 1880’s Monthly Packet, published just over 50 years after the premiere of Beethoven’s Ninth, the author Lewis Carroll (who was, by day, the mathematician Charles Dodgson) wrote about a proportional reasoning problem:

The Cats and Rats Again.‘If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?’

This is a good example of a phenomenon that often occurs in working problems in double proportion; the answer looks all right at first, but, when we come to test it, we find that, owing to peculiar circumstances in the case, the solution is either impossible or else indefinite, and needing further data. The ‘peculiar circumstance’ here is that fractional cats or rats are excluded from consideration, and in consequence of this the solution is, as we shall see, indefinite.

The solution, by the ordinary rules of Double Proportion, is as follows: —

But when we come to trace the history of this sanguinary scene through all its horrid details, we find that at the end of 48 minutes 96 rats are dead, and that there remain 4 live rats and 2 minutes to kill them in: the question is, can this be done?

Now there are at least

fourdifferent ways in which the original feat, of 6 cats killing 6 rats in 6 minutes, may be achieved. For the sake of clearness let us tabulate them: —A. All 6 cats are needed to kill a rat; and this they do in one minute, the other rats standing meekly by, waiting for their turn.

B. 3 cats are needed to kill a rat; and this they do it in 2 minutes.

C. 2 cats are needed, and do it in 3 minutes.

D. Each cat kills a rat all by itself, and takes 6 minutes to do it.In cases A and B it is clear that the 12 cats (who are assumed to come quite fresh from their 48 minutes of slaughter) can finish the affair in the required time; but, in case C, it can only be done by supposing that 2 cats could kill two-thirds of a rat in 2 minutes; and in case D, by supposing that a cat could kill one-third of a rat in 2 minutes. Neither supposition is warranted by the data; nor could the fractional rats (even if endowed with equal vitality) be fairly assigned to the different cats. For my part, if I were a cat in case D, and did not find my claws in good working order, I should certainly prefer to have my on-third-rat cut off from the tail end.

In cases C and D, then, it is clear that we must provide extra cat-power. In case C

lessthan 2 extra cats would be of no use. If 2 were supplied, and if they began killing their 4 rats at the beginning of the time, they would finish them in 12 minutes, and have 36 minutes to spare, during which they might weep, like Alexander, because there were not 12 more rats to kill. In case D, one extra cat would suffice; it would kill its 4 rats in 24 minutes, and have 24 minutes to spare, during which it could have killed another 4. But in neither case could any use be made of the last 2 minutes, except to half-kill rats — a barbarity we need not take into consideration.To sum up our results. If the 6 cats kill the 6 rats by method A or B, the answer is ‘12;’ if by method C, ‘14;’ if by method D, ‘13.’

This, then, is an instance of a solution made ‘indefinite’ by the circumstances of the case. If any instance of the ‘impossible’ be desired, take the following: — ‘If a cat can kill a rat in a minute, how many would be needed to kill it in the thousandth part of a second?’ The

mathematicalanswer, of course, is ‘60,000,’ and no doubt less than this wouldnotsuffice’ but would 60,000 suffice? I doubt it very much. I fancy that at least 50,000 of the cats would never even see the rat, or have any idea of what was going on.Or take this: — ‘If a cat can kill a rat in a minute, how long would it be killing 60,000 rats?’ Ah, how long, indeed! My private opinion is, that the rats would kill the cat.

Lewis Carroll.

*Hat tip: I first read this on **James Dow Allen’s website** many years ago.*

Although he signed his article as Lewis Carroll, his philosophy as a mathematician shines through just as brightly as his sense of humour as an author.

You may think that his criticisms are just pedantry dressed up in humour and put him in the ‘smart alecks’ camp, but I’d like to know why it should be obvious to a student that one assumption (that cat-hours and rats killed are directly proportional) is valid but another (that the number of members of an orchestra and the time it takes for them to play Beethoven’s Ninth are inversely proportional) is not. Can you come up with a checklist which the cats and rats question passes but the orchestra question doesn’t?

Take a look at the following question and have a think about how you could pull it apart in the style of Lewis Carroll.

Strawberry Pickers R Us employs 15 people to pick one field of strawberries in 10 hours. How many strawberry pickers do they need to pick one field of strawberries in 3 hours?

Let T be the time to pick the strawberries and P the number of pickers.

Certainly you would need more than 15 strawberry pickers. Would they get in each other’s way, reducing everyone’s individual picking rate, or would they be faster as they pick for less time? Do they all pick at the same rate? Maybe the the original 15 pickers, with at least 10 hours experience, are more efficient than the temps they’ve got in. Did they have a lunch break during their 10 hour shift?

The more you think about it, the less proportional this scenario seems.

This question is question 3 on the worksheet from which the original question was taken. Given that we’ve decided that question 5 is the single trick question, I guess the quantities in question 3 must be inversely proportional.

# What to do?

The word problems we have looked at all contain hidden assumptions. We have two options:

- Make the assumptions explicit.
- Accept and embrace the ambiguity.

Option 1 would involve adding something like ‘each (cat | strawberry picker) works independently at the same constant rate…’ to the question. It makes the question pedant-proof and students who might have had reservations about the question without the assumptions explicitly stated will be reassured that there is one unambiguously correct answer.

There’s a danger that students will eventually learn that such boilerplate text just means that ‘these two quantities are proportional’. For high-stakes summative assessments it might be better to just state the proportionality explicitly. You won’t have a word problem any more and so you’re removing a large part of the ‘deciding what to do’, but you’re allowing more students to access the marks for implementing the procedure.

Option 2 would put the onus on the student to state their assumptions, and later perhaps to justify them or explore different sets of assumptions. This takes us from mere word problems to the rich world of mathematical modelling. Pedants can be told that a question is intentionally open-ended, without an objective unique correct answer. ‘Smart alecks’ can be rebranded as ‘mathematical modellers’ and be told to create new models of the scenario rather than just criticising old ones.

While perhaps not suitable for high-stakes examinations, mathematical modelling in the classroom also gives students ample experience of ‘deciding what to do’. The handbooks for the m3 challenge are great resources if you’d like to learn more about mathematical modelling and its benefits. To me it’s clear that, in most cases, mathematical modelling offers a better solution to our woes than trick questions do.

To make progress with mathematical modelling, students will of course need fluency in the basics, so it’s too early to throw away the abstract proportionality questions and the simple word problems which bridge the gap. Once they’ve mastered these skills, they can revisit word problems, challenge the assumption of proportionality, and explore more sophisticated models. Hopefully by doing this they’ll learn, if they hadn’t before, that mathematics is more than just the unthinking application of routine procedures.

# Conclusion

Maybe this article is an out of proportion response (pun intended) to a maths question going viral, but I don’t think so. Understanding why a student could get the answer wrong is important if we want to help them get it right; understanding why some people believe the question is wrong is important if we want to challenge the conception of maths that leads to this belief.

There isn’t an algorithm for deciding how to answer a word problem. This requires intuition which can only come with experience. Developing and refining that intuition is an important part of learning maths, but this point can be missed by students. The curse of knowledge means that we can often miss this point ourselves, as we automatically fill in all the hidden assumptions that we know must be made to answer a question.

By interleaving content you can make sure that students don’t decide what to do based solely on what they’ve learnt most recently. Similarly, by doing mathematical modelling we make sure they don’t rely on whether they have used all the information in a question, whether they have done a reasonable amount of work, or whether some other criterion more about the norms of maths questions than the nature of reality is satisfied. See also the famous ‘How old is the shepherd’ question. Using these criteria can be good exam technique, but they should be used as a check after answering a question and not as a signpost at the start.

When I come to sing in Beethoven’s 9th at the end of September I’ll be sure to time the performance and count the number of players in the orchestra. Will it take the 40 minutes the question asserts, or the oddly precise 55 minutes and 14 seconds that the top comment on the Reddit post suggests? Perhaps it will be nearer the 74 minutes that a CD was designed to store, allegedly in order to hold all of Beethoven’s Ninth. I’ll let you know.