A warm Sunday morning in May. I’m up (too) early and heading off to my local football ground. Its cup-final day for the local youth leagues and I shall be officiating in a couple of games, first as an assistant referee, then I shall swap with the man in the middle who will be my “Lino” as I referee the game.

I arrive early, park up and, as I step out of my car, another vehicle pulls up and a young man, also clad in black, hops out – my fellow referee. There is a glimmer of recognition as we shake hands and I begin to quiz him about his background so I can work out where I have encountered him before. It turns out he’s on the books of a local League 2 side and I’ve ref-ed some of their U16 trials games and will have seen him on the pitch there.

We chat some more as we wait for kick off. He’s currently in Year 11 at a local school, is quietly confident about his forthcoming GCSEs and is even more excited to tell me that he has been signed on as a “Sixth Form Scholar” at the aforementioned League 2 club. No mean achievement, and he is right to be proud.

I quiz him further (I’m a nosey old git!) He’ll be going to the sixth form college in the town of his club – he admits he probably wouldn’t have chosen that college if it weren’t for his football commitments, but it is a condition of signing on as a “scholar.”

He also tells me that none of the current “lower sixth scholars” at the club have been signed on for the “upper sixth.” They have all been “let go” – they will, of course, continue their studies at the local college but ties with the football club have been severed.

Another school, another Year 11, another Football Club (but also in League Two.) I found myself chatting to this student, asking what he would be doing next year. Again, it was with some pride that he told me he would be joining …… FC as one of their football scholars.

We chatted away, I was interested to see what this meant in practice. Like the young ref mentioned above, this lad will spend several days each week training with the club, and several days each week at the local college. Asking him what he would study he told me that all football scholars at the club followed the same academic course.

And that set a few alarm bells ringing. These boys are mortgaging their future on the possibility, and, as we will look at below, a slim possibility, of “making it” as a professional footballer. The football scholars will have different skills and strengths on the pitch, and it will be the same in the classroom. To shoe-horn them all onto the same academic course, to deny them the choice of studying the subjects that they want to study is doing them a disservice. But with the lure of of a glimmer of a pro-contract in two years time, heads are easily turned.

If they make it, does it really matter that they’ve missed out on A Level Maths, or English or Geography or whatever? No, it doesn’t. But most won’t make the grade.

The Independent recently published an article promoting a book: “No Hunger in Paradise: The Players. The Journey. The Dream” by Michael Calvin, who documents the sometimes seedy story from schoolboy to professional footballer, a story littered with shattered dreams, shady agents and broken promises. In it, he offers the staggering stat that only 180 out of the 1.5 million boys who play organised youth football in England will go on to play in the Premier League. That’s a success rate of 0.012%

I thought I’d do some of my own sums, and they back up the authors claims. I’ve made many generalisations, but I think I’m in the right ball park. Here are my calculations.

20 Teams in the Premier League each with, say, a playing staff of 30, so that’s 600 players playing in the Premier League.

Assume a career length of 15 years, so each year one fifteenth of the players will retire. To replace these we need ^{1}/_{15} x 600 new players each season, or 40 new footballers each season, 40 youngsters to take the place of those who have reached the end of their playing days.

There are 48 counties in England, so its not enough to simply be the best player in your county, you’ve got to be better than that. And that’s before we even factor in overseas players.

Another ballpark figure, let us say that half the players in the Premier League are English, half not. So we probably only need 20 new English lads each year to join the elite ranks of Premier League footballers. To succeed, statistically, you’ve got to be – at least – the best player in your county and the neighbouring county.

But of course, you don’t have to play in the Premier League to enjoy a successful, and financially rewarding, career in football. Those plying their trade in the Championship, League One and League Two will enjoy a good living and lifestyle. And as you drift further down the football pyramid their is still scope to make a significant additional income from playing the game. (Alas, my playing days were so far down in the roots of “grass roots” football, I was paying to play, a long way off being paid to play!)

I don’t want to deny anyone their dreams and I wish those two young men I mentioned above the very best of luck – it would cheer me enormously to cheer them on a professional football pitch – but I can’t help but worry that, because of choices that are being made for them by their clubs, doors are being closed to them, when education should be about opening as many doors as possible.

]]>Lawro – Mark Lawrenson – former player, football pundit and expert took a bit of a battering today, with a website castigating his weekly predictions of upcoming football fixtures:

## The table based on Mark Lawrenson’s predictions is actually ridiculous

Reading the article, it goes on to rubbish Mr Lawrenson’s weekley predictions for Premier League fixtures. Joe.co.uk has turned his predictions into a league table and the Give Me Sport article goes on to say

we’ve discovered how the Premier League table would look going into the final game week if all of Lawrenson’s predictions had come true.

To say it’s ridiculous would be an understatement.

Firstly, his former club Liverpool sit top on 89 points having not lost a single game. Not biased at all…

Also, Lawrenson clearly still rates Leicester with them sitting seventh with 71 points! After the Foxes, there’s then a bizarre 22-point gap to Everton in eighth.

Despite Sunderland being rock-bottom with 24 points in real-life, Lawrenson’s predictions has them in the dizzy heights of 14th on 34 points.

Later on the article describes Lawrenson’s predictions as

embarrassing

Harsh words, so do the numbers back up such criticism?

In a word, no.

Using Spearman’s Rank Correlation Coefficient we actually discover that Mark Lawrenson has done rather well.

Spearman’s Rank is a great tool to compare how closely two different variables compare. It is perfect for a case like this where we want to compare a predicted league rank with the actual league rank.

It is based on the difference between the square of the two different rankings.

For example, Lawrenson predictions would have Bournemouth in 13th place, in reality they are in 10th place, so the difference is 3, the square of the difference is 9 (we square the values so we don’t have to worry about negative numbers). This is the formula used to calculate Spearman’s Rank Correlation Coefficient:

where d is the difference and n is the number of pairs of data (in this case 20 as there are 20 teams we’ve compared.)

The value will always be between -1 and 1, where 1 tells us there is perfect agreement in ranking.

So how did he do?

Pretty well – he ended up with a coefficient of 0.890 (to 3 decimal places) which suggests that there is a strong positive correlation between his predictions and the actual league positions. In other words, the boy’s done good.

So ignore the sensationalist headline of the Give Me Sport piece and let the numbers do the talking. Mark Lawrenson has been proven, mathematically, to be a pretty good pundit and predictor.

]]>Regular followers of my blog know that I like my sport and some of you are aware that I am a qualified football referee and regularly ply my (alternate) trade as the “man in black.”

With my referee’s hat on, I am an active member of a forum that discusses the finer points of the offside law and whether or not any particular offence as seen on Match of the Day warranted a red or yellow card. All part of the learning process, everyday is a school day, and all that.

Today someone posted a problem that was a little left field for the average referee discussion – what is the area of the “D” on the edge of the penalty area.

What a great question!

Lets fill you in with a few dimensions. From the goal line to the edge of the penalty area is 18 yards. The penalty spot is 12 yards from the goal line. When a penalty is being taken all players (other than the penalty taker and the goal keeper) must be both outside the penalty area and at least 10 yards from the penalty spot. Hence the “D” on the edge of the penalty area – it’s arc marks the points that are 10 yards from the penalty spot.

Its not a simple question, but one that should be within the grasp of of a “good” GCSE student. It’s a problem involves trigonometry and Pythagoras, sectors and segments; the area of a triangle and fractions …

I got the answer to be 44.73 square yards (to two decimal places), do you?

If you’re not sure how to approach the problem you can see my back of a yellow card working below.

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2016 will be remembered for many things, Brexit, the election of Trump, a surfeit of celebrity deaths, but it was last week, 7th February 2017, that a true giant passed away, although you might have missed the news of his passing.

Hans Rosling was a statistician, an educator, a communicator. He used facts to explain and enlighten. Four years ago, I wrote a short blog embedding one of his videos, where he illustrated the change in family size and life expectancy over the last fifty years. He really was a remarkable man with the ability to communicate some difficult and challenging ideas with ease and clarity.

Please, take the time to watch the video above – it’ll only take ten minutes of your time and, even if you learn nothing (but you will) you will enjoy the show.

In these times of alternative facts, mis-speak and other Orwellian horrors, I hope that Hans’ legacy will be a willingness to use facts and statistics to inform and shape understanding.

Hans Rosling, 1948 – 2017. The world is a better place for your passage through it.

(A quick youTube search for Hans Rosling produces a wealth of results. If you’ve watched the video above and would like to see some more, below are a few links you may enjoy)

Where are the Syrian Refugees? Although made in 2015, so the numbers may have changed, this short video is quite sobering and shows that we, in Europe, have probably got it wrong.

200 Countries, 200 Years Hans examines how the wealth and health of 200 countries has changed since 1810

Channel 4 News Interview Interviewed on Channel 4 News. Worth a watch.

]]>Large data sets – not the most inspiring of titles, but one which we teachers of A Level maths will become increasingly familiar over the next few weeks and months.

A Level maths is changing, but two plus two remains four, most of the content that is in the current A Level syllabus is in the new syllabus, to be taught from September ’17. It’s place in the syllabus may have changed – i.e. a topic that currently appears in Core 3 may now find itself in AS maths and be taught in the lower sixth/first year of A Level, but there is nothing that will be too unfamiliar to today’s teacher or today’s student.

Except for large data sets.

Candidates are to be familiar with one or more specific large data sets, to use technology to explore the data set(s) and associated contexts, to interpret real data presented in summary or graphical form, and to use data to investigate questions arising in real contexts.

… and that is new.

So its time to start thinking about large data sets, what they are, how we will teach with them, how they will impact on the exams…

The boards have, helpfully, published the large data sets that they will be using, and I’ve put copies of them here:

Further down this page you can see some sample questions from the boards that relate to the large data sets.

So what is to be expected of the student in the exam?

I must add a caveat that I am crystal ball gazing, and this is just my own view and not that of any board, but it seems that students won’t have to have whizzy excel skills to manipulate the data in the exam. They will be examined on this part of the syllabus like all others: in questions on paper in an exam hall, no computerisation of the exam. They could succeed on these elements of the exam with no knowledge of how to use Excel (or any other package) to manipulate data. But, as I will begin to explore below, manipulating the data with Excel (or similar) in lessons or homework will help them develop a good knowledge of the structure and content of the data, and this is important.

Have a look at this sample question from OCR, in particular part ii):

To effectively answer this you need to know that the data set contains different regions with different geographical characteristics, and the differences between, say Unitary Authority and a Metropolitan Borough and how the provision of public transport within different areas varies. (If you don’t believe me, have read of the mark scheme below.)

I must confess, I am a little uneasy with this – are we examining mathematical skills or geographical knowledge?

After years of staff shortages meaning Geography teachers have ended up teaching maths it now seems that we maths teachers are going to have to do a bit of Geography teaching!

However, we must do the best for our students and so we must familirise them with the nature, structure and content of the data sets. To do this I will look at how I can incorporate them into teaching as many aspects of the syllabus as possible. For example, I will get students to calculate the Standard Deviation of all, or a sample of, the data set – and not just using the single Excel formula. But that’s a post (or several) for another day.

This post was hopefully a brief introduction to Large Data Sets. Download the files for yourself, have a play and a think about how you might use them. As ever, I’d be delighted to hear your suggestions. And have a look at another couple of sample questions designed to examine the students familiarity with, and ability to explore and manipulate, large data sets.

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One of the casualties of the forthcoming changes to the A-Level syllabus is the Decision Maths module(s). The “standard” A (and AS) Level will consist of Pure Maths and a combined applied maths paper, made up of content from both Mechanics and Statistics.

I’m not sure how I feel about this – on the one hand, Decision 1 contains some really interesting maths: want to know how your SatNav determines your route? Decision 1 has the answer. Want to know how Excel sorts data? D1 explores the Bubble Sort and Shuttle Sort algorithms, as well as explaining how to determine the efficiency of an algorithm. Plus I make a cameo appearance in the EdExcel D1 text book – my 15 minutes of fame!

But answering some (many) of the exam questions can become a little tedious, with the student forced to replicate the steps taken by a computer to use an algorithm to solve a problem.

But regardless of the above, I was delighted to discover a new application of the Traveling Salesman Problem that we study in the course.

The Traveling Salesman Problem (or TSP for short) is simple in principle, but soon escalates to become very calculation heavy.

A salesman needs to visit all the towns in his area, before returning home from where he started. In achieving his goal he wants to travel as short a distance as possible and the solution to the TSP is this route.

Imagine he only has 3 towns to visit – he has 3 choices for his first town, then 2 for his second, and only 1 for his third, so there are 3x2x1 = 6 possible solutions, although we can halve this as the length of the route ABC will be the same as CBA.

Add a fourth town and there are ½ of 4x3x2x1 = 12 different routes, five towns gives ½ of 5x4x3x2x1 = 60 – the numbers soon start getting very big and even a computer would begin to struggle to compute all the various different permutations to find the shortest route.

So we use an algorithm – in fact we use two. Its such a difficult problem that we don’t necessarily find the optimal solution. We use the “Nearest Neighbour” algorithm to find the upper bound – a distance we know that can contain a route, and then we use the Lower Bound algorithm to find the shortest what the optimal route *could* be. We now have a range of values between which the optimal solution must lie. If the range is small we *may* be happy to go with our upper bound route, if not we can try the tour improvement algorithm.

Rather than try to explain these algorithms, this short video offers a good insight into how they work.

“But you promised me a pint!” I hear you cry. “Whats all this got to do with the price of beer?”

I’m getting to that now.

Earlier this week, I stumbled on a superb piece of mathematics that found the shortest route that would take in all 24,727 (yes – twenty four thousand, seven hundred and twenty seven) pubs in the UK.

You can read all about it here …

… or you can just enjoy the majesty of the map that shows the route:

Cheers!

]]>2017 is, itself, a prime number and the year kicks off with a prime number date:

1 1 17, or 1117, is a prime number

and, with a little poetic licence, the next day, 2nd Jan, is also prime, as long as we write it as:

2 01 17

20117 is prime. There are other primes in January, both the 23rd and 25th (23117 and 25117) are prime and there are many, many other prime dates throughout the year.

The 3rd February, 3 2 17, is both prime in its own right (3217) but is also “made” from 3 prime numbers: 3, 2 and 17

However 2017 unfolds for you, I hope you have a great year – and keep an eye out for those prime dates.

]]>Back in my day at school, calculators in the classroom were in their infancy – I was in the last year that were not allowed calculators in their O Level maths exam, although we did use them in A Level Maths.

We thought they were pretty powerful and quite cool, although they immediately called into question why we had spent (what seemed like) weeks learning how to use logs to help with long long multiplication and division questions when a calculator could do the same problem – with no human error – in an instant.

By today’s standards they were pretty basic – about the most exciting (non-mathsy) thing we could do was type in 5318008 and turn the calculator upside down for a cheap thrill and quick giggle.

Not so today’s student, who has access to a pocket full of power. Not only have calculators moved on in the last 30 years, so has schoolboy wit and I did chuckle when I stumbled upon the image below.

The spirit of schoolboy defiance lives on. To whoever came up with this, I salute you.

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And so, with a moth invasion rather than a pitch invasion, the Euro 16 competition has come to an end, with Portugal claiming the crown of champions with an extra time winner. Not many predicted them as tournament winners, but then that is one of the beauties of the beautiful game, its unpredictability: “Favourites win less often in football than in any other sport.”

So how did the twenty four teams involved in the competition fair compared to their official FIFA rankings?

Not too well. Or more correctly, there was little or no significant correlation between FIFA ranking and final Euro 16 position.

What I did is I looked at the FIFA rankings and ordered the teams in the tournament from 1 to 24 based on this. For example, Belgium are ranked 2nd by FIFA, but they were the highest ranked team in the tournament, hence I put them at the top the FIFA list. Some other interesting listings: England were the 6th best team in the competition, Northern Ireland 15th, Wales 16th and the Republic of Ireland 21st according to their FIFA rankings. Eventual winners Portugal were ranked 4th best out of the 24 teams taking part.

I then found each team’s final place in the competition. Some were obvious: e.g Portugal 1st, France 2nd etc., for teams eliminated in the knockout stages I ranked teams beaten more heavily lower than those who lost by a goal, and even lower than those who lost but took the game to extra time. For example, looking at the losing quarter finalists, I ranked both Poland and Italy joint 5th place as they both lost in extra time, Belgium 7th as they lost by 2 goals and Iceland 8th as they lost by 3 goals etc.

I then found the Spearman Rank Correlation Coefficient to compare the FIFA ranking and the final Euro 16 position and it came outs as **0.315**

… and that’s not a lot.

It can have a value of anything from between -1 to 1 and we would be looking to get a value in excess of circa 0.65 to suggest, with confidence, that there was any meaningful correlation between the two sets of data. You can see from the graph above that we can draw a line of best fit, but the points are all someway from that line. At GCSE level maths we would draw the line by eye and would make a statement like “there is weak (or no) positive correlation.”

By the time we reach A Level maths we begin to quantify our results and Spearman Rank is one way to do it for this set of data, and we would be expecting students to make a comment on their results. (At A Level, we would use something called linear regression to draw our line of best fit.) I would expect an A Level student to conclude that there is no meaningful correlation between FIFA ranking and where a team finished in Euro 16.

So what does this mean? Are FIFA’s rankings useless?

Not necessarily. It could just be another indication of the unpredictability of football, or it could be that the rankings are useless, but more likely a combination of both, and does a tournament really tell us which is the best team? Not many pundits, players or fans would say, today, that Portugal are the best team in Europe, but they have just won the tournament. So how do we define what is the “best team”?

As ever, statistics seems to throw up more questions than it answers. But life would be boring, back pages blank and the airwaves silent if we all agreed and had nothing to discuss. And to be fair to FIFA, they will soon be publishing their new world rankings and these will include data from Euro 2016. All I can say, is don’t use those rankings to pick out the eventual winner of the next World Cup in 2018!

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Look up to see the reason why:

Mix sunshine and an art installation and it all adds up to maths!

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