Friday, January 20, 2017

Progress 8 - newspaper nonsense

On 19 January 2017 the Daily Telegraph published the following article online.  The article has since been largely rewritten.  I have commented on the original version.  The text in red is mine.

There are several things that worry me here.  There are so many errors and misunderstandings and that isn’t good.  The biggest worry is that I tend to trust the press.  My default position (and I suspect that of many other people) is to believe what they say (not just this newspaper but others too).  If there are this many misrepresentations and errors in an article about a subject I know a little about, what are the chances there are errors in other coverage too?  Can I believe anything I read … ?

Secondary school league tables 2016: Over 1,500 schools are falling behind, figures show
19 JANUARY 2017 • 12:33PM
More than 1,500 schools are falling behind, according to the Government's new progress measure, official league tables released today by the Department for Education (DfE) show.
Almost a quarter of a million pupils monitored under the Government’s new GCSE ranking system – called Progress 8 – are at schools which were given a negative rating, meaning they are performing below the national average.
Progress 8 has been created so that the total Progress 8 score for all pupils across the country is zero.  The average score per pupil is therefore also zero.  When this is transferred to school level things are not entirely straightforward.  We shouldn’t expect there to be exactly the same number of schools above and below the average, but this is roughly the case. [The discrepancy is down to the different sizes of schools, the fact that special schools tend to be smaller and that Independent schools are not included within the figures.   To give a simplified example: if you have all of the pupils in two schools, one of which has a positive Progress 8 score and the other a negative one.  Half the schools are below average.  If you then split the pupils from the school with the negative score into two different schools you could have one big school with a positive score and two small schools that have a negative score.  Then there would be twice as many schools below average as there are above average.] 
My reading of the numbers is that in 1616 schools the pupils averaged a Progress 8 score above zero, in 48 schools the pupils averaged a score of zero and in 1994 schools they average a score below zero. 
There are three issues so far in the article.  One is that it is bonkers to complain that lots of schools are “below the national average”.  The way averages work is that some are above and some are inevitably below.  The only way to avoid having anyone below average is for every single school to be exactly average. 
The second is I can’t get “1500 schools” from the figures!  1994 schools were below zero (but it would seem odd to use “more than 1500 schools” to describe 1994 schools).  Maybe they are referring to the 1598 schools that are -0.1 or below.  But this is not sensible, because the government has said Ofsted will investigate schools whose figures are lower than an arbitrarily chosen -0.5, of which there are 705.
The third issue is that it is nonsensical to make a big deal about “Almost a quarter of a million pupils” attending schools where the average Progress 8 score is negative.  The key thing is the _individual pupils_ whose P8 score is negative.  If a child gets a negative score it doesn’t matter whether they are at a school where lots of other people did the same or not.
Free Schools, which were introduced under by former Education Secretary Michael Gove in 2010, were proportionately the worst performing schools in the state sector.
In total, 84 per cent of free schools were given a negative rating for progress.  Academies performed comparatively well, with the majority (57 per cent) rated as above the national average.
If you look at the figures for every school that is a Free School then there are 13 with a positive P8 score and 71 with a negative score, which is 84.5% negative. 
There are some important nuances that have been missed here though.  First of all, this is only 84 schools out of 227 Free Schools.  The rest have not yet got to the stage where pupils are in Year 11, so only 37% of Free Schools are included in the figures.
Then let’s look at the different types of school.  Free Schools include special schools, Studio Schools and UTCs.  It doesn’t seem fair to put all of these together and then to compare them to Academies.  For example, if we look at all special schools (whether Free Schools or not) we find that 406 of them had negative scores and 2 of them didn’t.  This is for sensible and understandable reasons, which I won’t go into here.  The point is, though, that the different types of school that make up ‘Free Schools’ is not the same as the types of schools that make up ‘Academies’. 
In fact, Special schools were only a small fraction of the Free Schools, but UTCs are very different from mainstream schools and seem to have their own particular challenges.  Again, it seems unfair to compare them to academies.  I didn’t know anything about Studio Schools so I looked them up.  Their website states that the “Studio Schools curriculum moves away from traditional methods of subject delivery with the curriculum delivered principally through multidisciplinary Enterprise Projects in the school and surrounding community”.  It also previously mentioned that students “work towards GCSEs in English, Maths and dual award Science as a minimum”.  If as a school you have some students who only work towards GCSEs in English, Maths and double Science then it is unsurprising your P8 scores are low.  (I pass no judgement on Studio schools – I am only pointing out that their curriculum requirements are not closely aligned to Progress 8.)
The figures also revealed a clear north-south divide with the more than half of the ten worst local education authorities (LEAs)  in terms of student progress situated in the north-west of England.
The league tables showed that every single school in Knowsley, Merseyside, was failing, as were 90 per cent of schools in Redcar and Cleveland, North Yorkshire.  
It might well be reasonable to look at exam results in different areas of the country, but again there are other confounding issues.  For example, if it happens that there are more PP pupils in certain parts of the country then those areas’ P8 scores may well be lower. 
It isn’t right to describe schools that are below average as “failing”.
Meanwhile, the five best performing areas for pupils achieving 5 A*-C including English and maths were all in London, with Hackney, Kingston upon Thames, Kensington and Chelsea, Barnet and Westminster at the top of the league table.
Nick Gibb, the schools standard minister, said the figures “confirm the hard work of teachers and pupils across the country is leading to higher standards”.  
The implication in this article is that Nick Gibb’s quote refers to the Progress 8 figures.  Either this is wrong and Mr Gibb was actually talking about other figures, or Mr Gibb is wrong. 
If you have a measure like P8 which has zero as its average then this is, definitionally, a ‘zero-sum game’.  You have no way of telling whether standards are going up, down or staying the same.
This year, nine of the top 10 schools for GCSE results were state schools, with the country's best three state schools all situated in north London. Henrietta Barnett School, a girls' grammar school, scored 100 per cent for students obtaining both 5 A*-Cs.
Queen Elizabeth's School – a boys' grammar school – came second, with 100 per cent of students gaining 5 A*-Cs, and St Michael Catholic School, a grammar school for girls, came third. 
Independent schools performed poorly in the tables, with 62 per cent of them coming in the bottom third of schools for 5 or more grade A*- C including English and maths.
This is due to the fact that many independent school students sit IGCSEs, which are not officially recognised by the Government. 
This isn’t getting better.  The school that came first scored 100% (we are now talking about pupils gaining 5A*-C grades rather than Progress 8).  The school that came second also scored 100%.  It is difficult to see how they have been separated.
This year is the first time that the Government’s new measure for attainment in GCSEs, called Progress 8, has been used to rank schools. It measures students’ progress in eight subjects from primary school through to secondary school.
The eight “core” subjects measured by Progress 8 are English, maths, history or geography, the sciences, and a language.
Um, no they aren’t.  This confuses the EBacc subjects with Progress 8.  While English and Maths are part of P8, pupils need three of the others that are mentioned and then have any three further subjects (which can include the other English exam that many pupils take and can include other EBacc subjects).
Mr Gibb said: “As well as confirming that the number of young people taking GCSEs in core academic subjects is rising, today’s figures show the attainment gap between disadvantaged and all other pupils has now narrowed by 7 per cent since 2011.
“Under our reforms there are almost 1.8 million more young people in good or outstanding schools than in 2010, and through our new, fairer Progress 8 measure we will ensure that even more children are supported to achieve their full potential.” 

Here is the URL for the article.  As mentioned earlier, it has changed significantly since I copied and pasted the original version.  There is no official statement that the article has been rewritten (although the time given on the article is different).

Tuesday, December 27, 2016

2016 gets a bad rap!

2016 is sitting slouched in its chair like a tired and stroppy Yr 11 student.  “I know that Brexit and Trump were my fault, but you can’t blame me for David Bowie or Prince.  Or George Michael.”  And then 2016 shouts: “Everyone’s got it in for me.”

For example:

Is it fair to have a go at 2016? 

The people whose deaths have made it into the national or international media all have one thing in common: they are famous.  It does feel as if there were more deaths of famous people during 2016 than in previous years, so what is going on?  Well maybe there are just more famous people who are getting old round about now.

Some assumptions:

  • We have noted the deaths of celebrities
  • The median age nowadays is about 75
  • People become famous in their mid-twenties
Using these assumptions, many of the celebrities will have come to prominence in the mid-1960s.  Let’s compare 1966 with 1956.

How can we find how many famous people there were in each year?  One proxy we can use is the number of movies that were released.

According to the Internet Movie Database, there were 10,460 movies released during 1956.  In 1966 there were 18,053.

This is a massive increase, presumably fuelled by the rise of the teenager, greater disposable income, a rise in population.  This will have required an increase in the number of movies, and also more actors and therefore more celebrities. 

If this has happened in movies, then presumably it also happened in popular music and on TV too.  Now some of the celebrities who have died during 2016 were not active during the 1960s, but many of them were and the rise in population after the second world war and the increase of celebrity since the 1960s are likely to have affected this.

And things are just going to get worse.  The number of celebrities in the world is increasing at a massive rate.  Think of the number of reality-TV show celebrities we see and read about each year.  In 50-odd years time, when the X-factor/Bake-off/Apprentice/Big Brother/etc contestants start to get old, 2016 will look like a quieter, more innocent time.

Maybe the reason we have seen so many more celebrity deaths in 2016 is that there are more celebrities nowadays and more of those celebrities are getting older.

So let’s stop blaming 2016; it really isn’t its fault (but hope you won’t be teaching his younger brother, 2017).


Saturday, December 10, 2016

Tokyo 2020 Logo

Olympic logos are often controversial. Lots of people didn’t like the London 2012 logo, for example. I thought the Rio 2016 one was great, though.

The controversy over the Tokyo 2020 logo was originally about plagiarism. The first logo that was chosen for the Olympics was subsequently withdrawn after it was found to be rather similar to that of the Théâtre de Liège. 

This is the new logo:

I find it a little unsettling. 

I think I have now worked out why. It feels as if it ought to have reflection symmetry. I think I start looking at the top of the logo, where my brain tells me it is symmetrical (roughly). 

As you go down the image this symmetry breaks down, and that is what makes it slightly uncomfortable to look at. 

There is, however, some symmetry in the logo. Can pupils spot it? 

It has rotational symmetry order 3. The red dots show three related points. 

Even though I now know about the symmetry I still find the Olympic logo ‘swims’ in front of my eyes. 

The Paralympic logo though; that works far better for me: 

Aah - nice to have reflection symmetry!


Thursday, November 10, 2016

Predictions are not necessarily bad if they don't turn out to be true

This was in The Guardian:

 Oh dear!  This again.  I have no drum to bang for pollsters and they may well have messed things up completely, but this paragraph is fairly hopeless. 

If we have a prediction, with a probability attached, it is natural to look at the biggest probability and to make use of this.  If there is a “15% chance that Donald Trump will win” then this is deemed to be unlikely.  But unlikely isn’t impossible.  By saying that the statement was ‘wrong’ we would also be saying that if a total of 6 comes up when you roll two dice then you must have cheated!

This crops up over and over.  An example is the (in-)famous prediction by the Met Office in 2009 of a ‘barbecue summer’.  Here is how the Daily Mail website refers to it:

The April statement says sun is “odds on” and that it is “likely” to be warmer.  That it didn’t happen doesn’t make the prediction necessarily “wrong”, just that a less likely scenario prevailed this time around.  (I am not sure it makes sense to make such a high profile prediction where you think it is only 2/3 likely to happen, though.)  

To be clear, I am not saying that either of these predictions are definitely “right”, but just that a scenario that is deemed to be unlikely can occur without the probabilities being wrong.

As teachers we also see this with FFT grades.  These grades used to be presented as percentages.  They would show that, for a pupil with a similar background (eg KS2 results, etc) a certain proportion of the pupils went on to get an A*, A, B, C, D, etc grade in each GCSE subject.  

Nowadays only the median version is given, and this is shown as _the_ FFT grade for the pupil.  In the new system these are given to the nearest third of a grade. 

If a  pupil has B+ as their FFT grade what this really means is that about half of children with a similar background to that pupil will get a high grade B or above, and half will get a high grade B or below.  It might well be the case that a small fraction actually get a high grade B.  If that pupil doesn’t get a high grade B that doesn’t mean that over- or under-achieved, or that their FFT grade was ‘wrong’, just that it didn’t happen. 

[The initial article also forms part of a Quibans.  Find it here.]


Friday, November 04, 2016

Aliens vs Barca

I usually enjoy the weekly puzzle on the Weekly Newsletter from Chris Smith (find him on Twitter @aap03102 – he will happily add you to his list of subscribers).

The puzzle in the most recent edition was a corker:

I loved this because there are so many ways of doing it.  I won’t quibble over whether two ways really are ‘different’ or not and will list as many as I and various students have come up with.  (Yr 11 thoroughly enjoyed it too!)

In all diagrams CN is labelled as a.  All explanations are sketched rather than presented as full proofs.

1) Similar Triangles

Triangles SNM and CNS are similar (look at the angles).  This means that a/180 = 180/135
Rearranging gives a = 240 and the diameter is 375.

2) Pythagoras 1

In triangle SNM the hypotenuse is 225. 
Angle MSC is a right angle (angle in a semi-circle), so we have two further right-angled triangles: CNS and MSC.  We can use Pythagoras in both of them:
(a+135)2 = b2 + 2252
b2 = a2 + 1802
Eliminate b2 and solve to get a=240.

A brief excursus – 3,4,5 triangles.
The numbers that Chris chose had some meaning (this puzzle appeared in newsletter number 375 and the answer is … 375).  It is also nice that they are numbers I recognise (usually in the form of angles).  135 = 3x45 and 180 = 4x45, which means we have a 3,4,5 triangle, so the hypotenuse is 5x45 = 225.  In my handwritten notes for these solutions I made the arithmetic easier by using e = 45 and calling the sides 3e, 4e and 5e.

3) Pythag 2
Draw in a radius (labelled r):

Do Pythagoras on triangle ONS and work out r.  Then double it.

4) Pythag 3
This is really the same as version 3 but with different labels.

Use Pythagoras on the triangle ONS

5) Intersecting Chord Theorem
This is related to version 1.

The intersecting chord theorem states that CN x NM = SN x NU
This gives 135a = 180 x 180

6) Area of a triangle

Looking at the big triangle, the area is ½ (a+135) x 180
If we treat CS as the base then it is also ½ b x 225.
Equate these and then use Pythagoras on CNS (a^2 + 180^2 = b^2) as a second equation and solve for a.

7) More similar triangles – using cos.

OT is the perpendicular bisector of SM.
We know SM = 225, so TM is half of that, 112.5.  Cos(M) = 135/225 (from triangle SNM).  But in triangle OTM we have cos(M) = 112.5/r

8) Trig 1

Work out the size of angle α (inverse tan of 180/135).
Angle OSM = α (isosceles triangle)
Angle NSM = 90 – α
Angle OSN = α – (90 – α)
So angle OSN = 2α – 90.
Use this angle, trigonometry and the side SN to work out either ON or OS.

9) Trig 2

We know what angle α is (using trig in triangle SNM) and we know that SM is 225 (Pythag).  We also know that angle CSM is a right angle (angle in a semi-circle).  Use trig in that triangle to get the diameter.

10) Trig 3
We could also find the length of CS using trig (as in version 9) and can then use Pythagoras to find CM.

11) Similar Triangles again

Triangle SNM is similar to triangle CSM. 
225/135 = diameter / 225.

How many different bits of maths did we use?
Many of these methods overlap, but it is rather nice that there are so many different aspects of maths that are involved.  It was also good (with Yr 11) to be able to focus on methods and explanation, rather than on finding the answer.

I think this can involve:
  • Circle theorems and other angle rules
  • Pythagoras
  • Trigonometry
  • Area of a triangle
  • Similar triangles

Many thanks to Chris for a great problem!

Saturday, July 30, 2016

Danny Baker's Sausage Sandwich Game

On Radio 5Live on a Saturday morning the Danny Baker Show features the Sausage Sandwich Game.  There are two contestants and a sporting celebrity involved.  Three questions are asked (unless there is a tie in which case additional questions are used to break the deadlock) and are the sorts of things that only the celebrity is likely to know.

The third question ("the question that gives the game its name") asks whether the celebrity would prefer to have on their sausage sandwich "red sauce, brown sauce, or no sauce at all".  Disappointingly there are no stats available for this, although Danny and Lynsey (his co-host) were incredibly surprised a few weeks ago when 'no sauce at all' was the winner.

This week (30 July 2016) Audley Harrison (Olympic boxing gold-medallist) was the guest.  The second question asked how many letters there were in the number of the house he lived in at the age of 10 or 12.  Danny then gave an example (so house number eight would have 5 letters, and house number thirteen would have 8 letters).  The contestants had to choose from: 3-6 letters, 7-9 letters, or more than that.

So: what do you choose?
My thoughts are below.


Here are my thoughts:

It's complicated!

The first twelve numbers all fall into the first group (3-6 letters).
But then thereafter there are very few others that have only 3-6 letters.  Multiples of 10 up to and including ninety (but not seventy) are the only others that fit.  This means that after a strong start (12 out of 12) this turns into 13 out of 20, 14 out of 30 and 19 out of 100.
There won't be any after that ("one hundred" has 10 letters and even if you get as far as 10^100 then I I think you would say it as "a googol", which is 7 letters).

The 'teens' are then all 7-9 letters.  After that, for the 2-digit numbers up to 100 (but not in the seventies), the numbers ending in one, two and six are in 7-9, while the rest are 10-or-more.

Forty-four is the point at which 7-9 overtakes 3-6 letters.
10-or-more overtakes 3-6 at fifty-seven.
The lead changes between 7-9 and 10-or-more a couple of times, but from ninety-four onwards it is all 10-or-more.  (Everything after that is 10-or-more.)

So what do we choose?   Almost every road will have at least 12 houses in it.  But many roads have more than that.  Not many have over 100 though.

Knowing a bit about the background to the person might help.  Danny must have been fairly confident that Audley didn't live in a house with a name, and knowing that Audley grew up in London will presumably help to tell us that there are more likely to be higher numbers in the streets, but realistically not higher than a couple of hundred.  (In the USA they seem to skip numbers with abandon, so house numbers of 2505 are not uncommon - we don't do that in England.)

FWIW Audley grew up in house twenty-seven, which gave 11 letters.

Sunday, May 08, 2016

Singapore Mental Math (7) – Finding remainders when dividing by …

[Part of the mental math series…]  Here are the strategies for Week 17, 18 & 19 in Grade 6.

Good things:
  • I usually see these as divisibility rules, and while it is obvious that the divisibility rules will give the remainder (perhaps with some tweaking), this isn’t commonly explored (in my classroom at any rate), so I found these interesting.
  • There is some mental maths to do.
  • There is proper use of vocabulary.

Bad things:
  • These are incredibly restricted and over-specify the conditions. 
  • There is a missed opportunity to look at why they work.
  • Some of them are expressed very badly.

Let’s look at each in turn:

Dividing by 5
I hope many pupils would spot that there is a far easier way to deal with the Note!  Rather than dividing the last two digits by 5, why not just instruct pupils to look at the final digit.  In fact, pupils might come up with an easier way to do this.  Let’s ask them to explore how we can tell what the remainder is when we divide by 5.  They could try a few out for themselves, could have an idea and might then link it to the rule for divisibility by 5.

If an integers ends in 0 or 5 then it is divisible by 5.  If it ends in 1 or 6 then the remainder is 1, if it ends in 2 then …

Dividing by 8
This only helps if you have a number with 4 digits or more. 

Why does it work?  1000 is divisible by 8, so any 1000s will automatically be divisible by 8.  Given that we only want the remainder we could also get rid of anything that is clearly a multiple of 8.  To find the remainder when 4169 is divided by 8 we can ignore the thousands, leaving 169.  160 is divisible by 8, so we can ignore that, leaving 9.  The remainder is therefore 1.

(It is a little frustrating that in all of the questions on these three strategies none of the numbers are actually divisible by the divisor.  It would be nice to have a remainder of 0 once in a while.)

Dividing by 9
Many pupils will recognise the algorithm for determining whether a number is a multiple of 9. 
This is phrased rather oddly in that it is only used here for 4-digit numbers (all of the questions are of that form), when it works for any integer.  It also doesn’t explain what happens if the single digit is 9.

Can pupils explain why this algorithm doesn’t produce zero (unless you start with the number 0)?

Can they explain why this algorithm works?  Here is an algebraic way to show it.

If a number looks like “abcd” then the algebra is actually 1000a + 100b + 10c + d.
This can be rewritten as 999a + a + 99b + b + 9c + c + d
Rearrange this to give (999a + 99b + 9c) + (a + b + c + d).

The first bracket is clearly divisible by 9, so we only need to worry about the second one, which is the sum of the digits.