Schrödinger's Marmite

When Erwin Schrödinger devised his famous feline thought experiment, which he described as "quite ridiculous", he can't have imagined that it would give his surname a linguistic life of its own. His name popped into my head this morning as I struggled to describe a piece of maths that I both love and hate simultaneously. The pair of words Schrödinger's Marmite sprang to mind. I was very pleased with my witty originality until I Googled the term and discovered that Urban Dictionary had got there first. Of course they had, those pony proud poseurs.

Schrödinger's Marmite - you either love it and you hate it

Three weeks ago I started teaching at a new school in Wiltshire. Starting school is just as disorientating for new teachers as it is for new students. The environment is discombobulating; from one side of the maths block you can see a white chalk horse on a hill and, from the other, Europe's second-largest neolithic monument, also rumoured to be the resting place of Merlin (the wizard from Shrek 3).

Folklore provides indisputable proof that Merlin is buried yards from my classroom.

While surrounded by a thousand unfamiliar faces, an unfamiliar lexicon, an extraordinary array of architecturally spectacular but geographically specious buildings, at least I could console myself with the fact that the maths remained the same.

Or so I thought.

I thought I'd lead my Hundreds (this means year 11 - no, I don't know why either) into the first Fixed Priv (a weekend when there are no lessons or activities and all the students have to leave) by solving some simultaneous equations where one is quadratic. A straightforward factorisation starter ended with me open-mouthed in bewildered awe at a technique that some members of the set unhesitatingly used to factorise a non-monic quadratic. They couldn't tell me how it works, just that it did, so I promised I'd look into it and blog about it.

Here's how the algorithm, if we might call it that, is used. Suppose we wished to factorise

Step 1: We multiply the constant term (-7) by the coefficient of the quadratic term (6).

Step 2: We consider how we'd factorise the resulting quadratic if it was monic. That is we factorise the expression

 

Determining that this factorises to

we, instead, write this down with the original quadratic coefficient multiplying each x term.

Step 3: We see that each bracket contains a common factor (3 in the left and 2 in the right) and we divide the brackets by their respective common factors.

This, by some extraordinary miracle, is the correct factorisation of the original quadratic.

I hate it because none of the students could explain to me why on earth they were doing what they were doing. They computed the correct answer through a series of seemingly random mis-steps.

I love it because it works and there is a certain elegance to the reason why. It's a perfect example of Schrödinger's Marmite.

Let's start an explanation from the rear. Suppose we were to expand the following brackets, which will represent the factorisation we are trying to reach:

We should note that the pairs A,B and C,D must be coprime.

Expanding the brackets give us the quadratic expression we are originally challenged by:

 

In Step 1, we multiply the constant term, BD, by AC and then Step 2 sees us consider the related monic quadratic

The factors of this are easy to spot. We want to find two numbers that add to give AD+BC and multiply to give ABCD. These are handed to us on a plate:

Assigning these to brackets as in Step 2, we reach:

Noting that the left-hand bracket contains a common factor of C and the right-hand bracket contains a common factor of A we perform Step 3 and reach our fully factorised destination via a lengthy detour of various pot-holed minor roads:

Perhaps it should be called the TomTom method because, in many ways, it is not a sensibly planned route but a hastily calculated compromise of the sort required when we get stuck in traffic.

But don't even get me started on SatNavs...

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Terror Threat Levels and Self-Exciting Functions

The UK is suffering a dark few months; tourists and a policeman were killed in an attack outside parliament; a suicide bomber targeted the end of an Ariana Grande concert in Manchester; a trio of thugs murdered those enjoying Saturday night on London's vibrant South Bank. Terrorism is not new to the UK but it is only since August 2006 that the current system of threat levels has been published and regularly reviewed.

There are five possible levels as explained in this screenshot from the MI5 website:

The descriptions are not particularly helpful in my opinion. I don't think there's an objective difference between "unlikely" and "possible, but not likely" although that point is moot because the levels LOW and MODERATE have never been used.

For nearly three years before the Manchester bombing, the threat level was SEVERE. The day afterwards (23rd May) it was increased to CRITICAL. To many this may have felt like shutting the stable door after the horse had bolted. Four days later (27th May) the threat was downgraded again to SEVERE as it has since remained. The London Bridge attack occurred on 3rd June.

It seems reasonable to ask two questions:

1) Why was the threat level raised after the Manchester attack and not the two London incidents?

2) Why is the level raised after an attack when surely common sense would suggest that attacks are reasonably spaced out over time?

I suspect the answer to the first lies in the nature of the murders. The London terrorists needed no support network; only the ability to hire a vehicle and arm themselves with blades. The Manchester bomber, however, was in possession of a far more sophisticated device that required significant expertise to assemble and detonate.

The second question is more interesting from a mathematical perspective. It concerns how the frequency of terrorist attacks can be modelled in time. Analysing data has led Statisticians to notice that in the period shortly following a terrorist attack, a second attack is more likely. They are rather like earthquakes in that manner; a fault can lie resting for a decade and then a large earthquake is followed by several aftershocks. The events seem to be self-perpetuating.

Mathematicians use Hawkes Processes to understand these phenomena; they emulate self-exciting events and are named after Alan Hawkes, professor of Statistics at Swansea University.

A.G.Hawkes

He came up with the idea for a function that reacts to events; increasing in value during the period shortly after each. I shall illustrate how this works using numbers plucked from nowhere and just designed to make things look reasonably nice.

Suppose we lived in an extremely volatile country where long term data analysis suggests that there is approximately one terrorist attack in every twenty June days. Suppose we let p(t) be the probability of an attack happening on any one day under normal circumstances. This gives a graph that is rather dull in its uniformity; the threat of a terror attack is identical on each day.

Now suppose there was a terror attack on 5th June. According to past observations, this will increase the daily probability of attack for a little while. The additional threat will subside over time. We can use a decay function to simulate the effect of this event on the likelihood of subsequent events.

In the above function the number 5 in the exponent represents the day of the event, 0.4 represents the likelihood that one attack will yield another, and 0.25 is the reciprocal of the length of time we'd expect to wait until the second linked attack (which is 4 days in this case).

Suppose there was indeed a further attack after four days; perhaps one that had a 10% chance of yielding another attack. Then all we would do is add another term to take effect from June 9th.

The more events there are, the more terms you add, and the longer it takes to get back near the background threat level:

Of course, you can't really use this to predict when a terrorist attack will occur. It merely gives us a heightened understanding of the underlying randomness but it does explain why it is worth raising threat levels in response to major attacks. It might also help us to determine how to deploy resources within the emergency services.

A spatial dimension is added by the company, Predpol, to help US law enforcement agencies direct their patrol officers. They harvest crime data and produce hot-spots where crime is likely to occur. Police are sent to the areas and their presence deters potential offenders. As one police chief puts it, "Burglars and thieves work in a mathematical way, whether they know it or not". The efficacy of the system is nothing short of remarkable. In Alhambra, California, burglary frequency fell by 32% and car theft by 20% during the first year of use.

It is perhaps the randomness of terrorism - very violent events happening to ordinary civilians without warning - that is its most significant feature. I find, therefore, a crumb of comfort that there is an understood underlying shape to these events and that authorities can use this information to make decisions about our security. The London Bridge attackers were shot dead by armed police within eight minutes of the first 999 call; while little consolation to the families of those affected, it was an astonishingly fast resolution in an enormous city. We have data analysts to thank for having the right people in the right place at the right time.

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"Impossible Exam Questions" - A Premeditative Strike

It's nearly time for this year's GCSE cohort to start sitting their mathematics exams. Good luck to them.

It's also nearly time for the recently emergent lazy journalistic trend of trawling twitter for students decrying particular exam papers. Now, don't get me wrong, I have absolutely no quarrel with those who have been in a stressful exam hall venting their frustration on the internet - that seems to be the modern day equivalent sitting down with your mates and consoling each other - but I do, however, think that upgrading this to actual news is utterly ridiculous.

It all started in 2015 with Hannah's sweets:

Having been the subject of various memes, the question was plastered all over the national press; The Telegraph; The Independent; The Guardian; The Daily Mail. They all featured an angry backlash as students sought to mitigate their performance fears by shifting the blame onto the examiners.

The following year, an apology was issued after students complained about this question in a Scottish Higher Exam:

In my opinion, both questions are run-of-the mill. In fact, the latter could feasibly have been made more difficult; I wonder if there was a draft version of the paper where students had to derive the expression for T themselves. The most surprising thing about these problems is that they caused such a stir; the fact that an exam board was moved to apologise beggars belief. It seems to me entirely arbitrary that these questions were singled out and I suspect that it has everything to do with the social media profiles of those sitting the papers and nothing fundamentally to do with the questions themselves. Those who shout the loudest get noticed.

Following the outcry, Durham University's Professor Robert Coe argued that fears over social media backlash would force exam boards into writing overly predictable exam papers. If this were to be true, it would put the credibility of the qualifications at risk. Here's the rub: Exams are supposed to discriminate and they cannot be designed so that every individual swans out beaming broadly because they've aced the paper. If we want the top grade to be sufficiently exclusive so that only 5% of entrants achieve it, then there will necessary have to be questions (in maths at least) that the vast majority are unable to successfully conquer.

Perhaps some expectation management, on the part of both exam boards and schools, is key; if students know that some questions, towards the end of each paper, will be tough then they are less likely to complain. Take this question from a 2016 (the year of the crocodile) Cambridge Pre-U Mathematics exam:

Like the crocodile question, it also involves some elementary Calculus but is far heavier on the algebra and probably more intimidating. Nevertheless, it didn't appear in the press. Why? Well perhaps because students sitting this qualification are used to papers varying significantly from year to year. They might well bemoan the fact that they couldn't answer a particular question but they won't necessarily feel the degree of outrage that would compel them to accuse the exam board of culpability regarding their performance.

It is the nature of mathematics qualifications at present that not all exam questions will be accessible to all students. In fact, if there doesn't exist at least one question on a paper that stumps most candidates then the examiner will not have ensured their paper is sufficiently challenging. Students are well within their rights to bleat about it but journalists should ignore it; even if it is trending worldwide on twitter and thus it goes against their every instinct. Our qualifications system must resist being jumped into a position where the tail is wagging the dog. If there is a genuinely unfair paper then there are proper channels through which teachers can make a case. Perhaps this would become news. Otherwise, giving column inches to the tweets of disgruntled students is, at best, a waste of time and, at worst, irresponsible and unlikely to help anyone anywhere.

Alternatively, I suppose, one could rethink maths exams entirely and assess students by the quality of responses to open ended questions rather than by the difficulty level of the problems they can proficiently solve. Perhaps the following O-Level paper from 1957 is the way forward. Candidates had two and a half hours to answer five questions including the first:

Maybe I'll tweet this to anyone moaning about their maths exams this year to see if they'd have preferred an essay paper.

No, probably not. That won't help anyone either.


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Approximations for Adults

***WARNING. THIS POST CONTAINS EXAMPLES OF PUERILE HUMOUR***

It is a sorry fact that most equations cannot be solved using neat algebra. It's probably true that most of the ones you saw at school turned out nicely but this is because you were sheltered from the harsh and uncompromising elements of the mathematical universe by your kindly teacher.

If you studied maths in the sixth-form then you would also have discovered how to find the area under a curve by integrating its corresponding function. Similarly, however, most functions can't be neatly integrated; you tend just to come across the friendly ones in school. There are, however, tried and tested methods for finding approximate areas under curves.

One such method is known as the Trapezium Rule (or, in the US, the Trapezoidal Rule) and involves calculating an estimate for the area under a curve by drawing a series of straight lines between points lying on it. This Wiki-illustration demonstrates the process:

The trapezium, while a superb method due to ease of area calculation, is far from the only shape it is possible to use. In a moment of procrastination you might, therefore, imagine my piqued interest when I clicked on a weblink labelled, "uncommon methods of numerical integration". Looking forward to a juicy academic article I was first shocked and then paralysed with laughter when the computer screen returned this:

After pulling myself together and printing it off to leave in a colleague's pigeon hole, I got round to asking the all important question: Does it work?

This post aims to put some meat on the bare bone of the above conjecture.

Firstly, the diagram above is a little ambiguous regarding the shape of the approximating appendages' tips. It does, however, appear that the shafts are of equal girth. With this in mind I shall assume that the round edges are semi-circles, each rectangle has the same width, and the phallic maxima all touch the curve.

Suppose, then that we were to split a certain area into n little fellas of equal width. The x-coordinates of the left, centre, and right of the rth are as shown in the following diagram.

Using this information, we can investigate the dimensions of this shape in further detail:

The total area of this element is the sum of the rectangle and semi-circle:

This is slightly hamfisted. If we relabel the width as w and simplify we get:

And, should this method work, we claim:

Rather than proving this in general, I'll verify it in a particular case. 

We know, from simple Calculus that

If we use the approximation above we get:

In this final line, it's easy to see that as n gets very big, most of the terms get very small. The only exception is the very first term which tends to the required answer.

We can evaluate this expression for various values of n as shown in the following table:

So what can we conclude about numerical integration using this rather subversive method? As you can see from the table, the estimates begin quite small and growth is disappointingly slow. It is, however, inevitable that if what we are plugging in becomes large enough, we will get there in the end. 

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Improving the McDonalds Maths Meme

If you have an online social medium then you will doubtless have scrolled through the following meme at some point in the last few years:

I'm all for a bit of casual mathematics but this problem is rather disappointing; the deduction required to solve it is trivial. All you have to do is read down the page remembering the piece of information gathered from each line. The answer, provided that you are happy to assume that the two packets of fries symbol is worth twice the one packet of fries symbol, is 15.

Sorry. Spoiler. If you hadn't worked it out, however, I'm confident that I can compensate by tweaking the puzzle to give you a superior version.

There are three reasons the McDonalds Mathematics Meme is less than elegant:

1: There is virtually no challenge here and no consequent satisfaction when finding a solution.
2: The multiplication sign in the last line seems incongruous.
3: The relationship between the two types of fries symbol requires explanation.

I've taken the liberty of altering a few details to produce the following upgrade:

Have a go at solving it. There's an easy way and a pen-and-paper way. Feel free to comment below about how you found it and how hungry it made you.

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Diane Abbott and Numeracy Shaming

Last week, almost as soon as the starting gun been fired for the commencement of the 2017 general election campaign, the media and proletariat revelled in collective schadenfreude as they enjoyed the first major gaffe of the attempted Labour fightback. In a radio interview for LBC, Diane Abbott (the honourable member for Hackney North and Stoke Newington) claimed that a Labour government would recruit 10000 extra police officers at a total cost of £300000. Nick Ferrari immediately asks her to clarify - "What are you paying them?" - and the following few minutes, as she attempts to dig herself out of this hole, are very difficult to listen to. If you are brave enough, you can do so by clicking here.

Ferrari bemused by Abbott.

It is clear that, on this particular question, the shadow home secretary was abysmally briefed. She panicked. She tried on the spot to undertake some mental arithmetic but got lost in a sea of shifting place values. This is an occupational hazard for politicians and you can almost hear the resignation in her voice as the interview unravels.

Diane Abbott is a divisive politician (maybe it's because she's black - maybe it's because she's a woman - maybe it's both - I don't know) so the response on social media was inevitable. I wish to confront a selection of them in this post because a number of the jokes focused on numeracy. The deduction that her mathematics must be substandard seems, at first, reasonable:

25(7+5) will do it

Thank you, Mr Brexit. Very droll!

63rd April is actually 2nd June is it not?

Even Aunty Beeb waded in.

Here's a fact: Diane Abbott is incredibly intelligent. She went up to Cambridge University having studied at a state school where her History teacher discouraged her from applying lest it "give her ambitions outside her social station". I'm sure her numeracy is just fine and, in any case, this isn't really a post about an individual; an experienced politician quite capable of brushing off any mud slung in her direction. This is a post about my dismay at the speed at which Brits are willing to belittle others' apparent lack of mathematical comfort.

National Numeracy is a charity that was founded in 2012 to promote numeracy among adults and children in the UK. Their report, Attitudes Towards Maths, summarises some of the challenges it faces: "It is culturally acceptable in the UK to be negative about maths, in a way that we don’t talk about other life skills. We hear ‘I can’t do maths’ so often it doesn’t seem a strange thing to say.[...] We talk about maths as though it is a genetic gift possessed only by a rare few, and inaccessible to the general public."

If this is true then many of those enjoying the Diane Abbott memes are rank hypocrites. Would we happily admit we struggle with maths while eagerly castigating someone else's failure to reason arithmetically under pressure? Either that or we are competent with numeracy ourselves and happy to laugh at someone less capable - this is bullying, obviously. I see the latter occasionally at school; if one student makes a mental slip some others, who have thought of the correct answer, might groan or worse. I am always careful to calmly stamp out this behaviour and it is not very difficult to explain why this is necessary.

According to National Numeracy, one of the three causes for negative attitudes towards maths is anxiety: "Maths makes some people feel anxious, leading them to avoid situations where they may have to use mathematics." If we leap on people's arithmetic blunders it isn't likely to help anyone. As a teacher I am often frustrated by students' lack of willingness to attempt a hard question because they can't immediately see the path to a solution. They are quick to say, "I can't do it" before having made a single mark on the page. This indicates low confidence.

Good mathematicians are willing to make mistakes and spend time reasoning their way into cul-de-sacs. They can do this because they are not worried about reaching a solution immediately and, besides, some of the apparent dead-ends might turn out to be interesting. If you are new to a neighbourhood, you will become far more familiar with it by following your nose and wandering around than you would armed with a knowledge of your destination's postcode and a smartphone. There is an enormous difference between knowledge and information. We can only develop the former with some fearless exploration.

So, by all means, hold politicians to account. It is, after all, part of the job of an MP to deliver responses on message and convince the populace of their potential to govern. Numeracy shaming, however, is not something I can stomach. It compounds the cultural problem we have in this country: It is apparently nerdy to be good at maths and shameful to be bad at maths. Maths is shrouded in negativity whichever way you lean.

I would like to applaud Diane Abbott for having a go. She was in a tight corner and it would have been easy, as so many politicians do, to say, "Well I don't have those figures with me". Instead, she tried to compute an estimate there and then. She misjudged it spectacularly, yes, but in doing so will have improved her mathematical ability.

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Sunday Sebunzus

In my last post I described how one might make up a newspaper-style quiz. I called them "sebunzus" and some people told me they liked them. By popular demand, then, here are some more:

Now that's done, it's time to find another excuse not to write school reports.

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Sebunzu

Newspaper puzzle pages have, in recent years, witnessed an explosion in grid-based number games. Since The Times featured their first Sudoku in 2004, countless spin-offs have made their appearance as publications compete for our coffee break procrastination time.

A Puzzle page from an edition of i, a UK newspaper,

Newspapers generally outsource the curation of their puzzles. Those in the illustration above are the work of Clarity Media whose clients also include three other major British newspapers and two publishers involved in creating educational materials.

Seeing as though it's half term, I thought I'd have a go at inventing my own newspaper-style puzzle rather than tear my hair out trying to complete someone else's. All that is needed is a repeatedly alterable puzzle-meme, an invariant set of instructions, and a catchy Japanese name.

After at least five minutes scribbling, I decided my puzzle would involve presenting the reader with a square grid containing a selection of integers between 1 and 5 and asking them to divide it into polyominoes each containing a total of seven.

For example, if I were to offer the following grid

You could simply divide it up as follows

As it happens, there is more than one solution here. We could swap the colours of the two 3s. Uniqueness is important, but a greater priority is to make it more entertaining. The puzzle is very easy at present and the natural way to complicate it is to enlarge the grid.

How about this?

This is marginally more engaging, but it is still possible to construct different correct solutions. These two, for example:

A problem with easily determining whether a solution is unique lies in the great flexibility with which we may partition the grid. Evaluating the number of possibilities is, I suspect, a seriously ugly problem and I can't find any literature on it. There are 10480142147 ways of partitioning a set of 16 objects so there is our upper bound but, in terms of our square grid, some of those ways will be geometrically impossible and we would have to discount any sets greater than 7 in size because these could not be part of a solution.

I suspect, however, that the number will still be vast and grids with unique solutions will be rare. So, to complete the construction of my puzzle, it's time for a wild card.

I shall make one or more of the numbers red. The red numbers can be used more than once. In fact, let's say that the red numbers must be used in precisely the number of sets it denotes.

Take this puzzle, for example

I need to use the red 3 in three different sets. I can't pair it with the five beneath, so it must be in sets with the numbers above, to its left, and to its right. After a bit of further deduction, we land on the unique solution:

This has potential to be far more entertaining.

Now we need to check that there are a large number of potential puzzles. Each grid square contains one of five numbers and each number is either black or red. This gives us an upper bound of ten million billion grids. A proportion of these will have solutions and a smaller quantity will have unique solutions. I won't, for the reasons stated earlier, calculate how many but I would wager there are enough to fill several newspapers daily for the rest of my life (after which I wouldn't really care).

Finally, I need a convincing sounding Japanese name. If you use Google to translate "sevens" into Japanese, it returns the answer "sebunzu". This doesn't actually mean "sevens" in Japanese. It's just phonetic. But it will do.

Here then, is the world's first Sebunzu puzzle for readers to solve. Enjoy:

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