Sunday, May 31, 2020

The worst way to give 2020 GCSE grades … apart from all the others


I have read several comments over the past week or so suggesting that the grades this summer will be rather accurate. They won’t. And I wish people would stop pretending they will.

My prediction (as I wrote about in a previous post) of the way grades would be awarded was largely accurate and I cannot see any other sensible way to generate such grades. In that post I raised some potential problems with this system and these problems appear to have been forgotten or ignored.

What is the problem?

This year’s grading system is (rightly) not going to be used to grade schools.
But the system will be closer to being accurate on a school level than it will on an individual pupil level.
If it’s not right on a pupil level then it’s unfair.
But we need something and this is the least-worst method, so what can we do about it?

In this post I am going to set out some of the reasons why this system is likely to be inaccurate and unfair. And something that I think will make it less bad (although not everyone will agree!).

Various issues

The grades a school gets overall will reflect the grades the school would have got from those pupils had they done as well as their counterparts at the same school over the previous couple of years. 

Problems with this:

1)  This might not be accurate as far as the school’s overall grades are concerned.  If a school is on an upward trajectory, or if a department has made changes (either recently, or to KS3 a few years ago) then this year’s students might be expected to do better.  Ofqual have said they will not take this into account because: “any statistical model is likely to be unacceptably unreliable in predicting trends in performance in 2020”.

2)  Subjects with small numbers of entries in some schools are likely to have a larger variance in their grades each year. This year that will lead to a larger risk of error in the grades that are assigned.

3)  Subjects being offered for the first time in a school will be difficult to manage fairly.

4)  New schools won’t have prior data to use.

Teachers are being asked to provide a Centre Assessment Grade (the grade they think each student would have got had they taken the exam) and a rank order of the students. The exam boards will then use the grades they think the school ‘should’ get and will move the grade boundaries as necessary for each school, maintaining the order of students submitted by the school.

It is worth noting that the grades which are provided are essentially irrelevant: the rank order is the only thing that is important here.  (It might be helpful to start with grades in order to help create the rank order, but the fact remains that the grades will not be used by the exam boards.)

It is also worth noting that this process could have been set up to happen the other way around. It would be possible for the exam boards to tell each school how many of each grade are available in each subject and for the schools then to distribute them amongst the students.

The key thing is getting the students in the _right order_.

A 2015 report by Cambridge Assessment compared predicted grades for the 2014 OCR GCSE exams with the actual grades achieved. The report found that overall 44% of the grades were ‘accurate’, while 42% were ‘optimistic’ and 14% were ‘pessimistic’.  These exams in 2014 were based on the old specifications that, in many cases, had been used in schools for many years, unlike the fairly new exams we have now.
The report found that overall 13% of the results were more than one grade different from the predicted grade. 

5)  One in eight exams results were two or more grades away from the predicted grade. Given that it was impossible to be more than one grade too optimistic when considering a grade of A or A*, or more than one grade too pessimistic when considering a grade G, this suggests that grades in the middle are less easy to assign accurately.

I have long-argued that it is irrelevant to talk of predicted grades as being ‘right’ or ‘wrong’, because between the predicted grades being produced in March and the exam being taken in May/June a lot can change.  This year, schools were told not to collect any additional data after lockdown began, so centre assessed grades, while being submitted in early June, are based on data from prior to 20 March, just like the previous predicted grades were.

6)  This is a problem because it is extremely difficult to tell how much a pupil will change between their predicted grade and the actual exam. Should we try to factor that in when making our Centre Assessment Grades?  Or should we only use the mock exams and other work that we have records for?  This seems to risk disadvantaging those students who would have worked hard after 20 March.

This then is the major problem for me:

7)  While the grades awarded to the school may largely be ‘right’ (with the caveats above), we are stupendously bad as a profession at giving them to the correct students.  This leads to the ludicrous situation whereby: 
a.       The school results are ‘right’ overall – but the school won’t be judged on these.
b.       The individual results are incredibly iffy – and the students _will_ be judged on these.

That is plain unfair for the students.

Surely the opportunity to take the exams in the autumn means this is OK?

No it isn’t.  A further post on this will appear over the next couple of days.

A footnote to the results

Football fans will know that Liverpool were on the cusp of winning their first league title in several decades. They have dominated the league this year and have won 27 out of the 29 leagues games played so far. They are so far ahead of the rest that they only need two wins in the final 9 matches to guarantee they win the league, and if their nearest rivals lose matches even that won’t be required.

In the early days of lockdown a frequent topic of conversation on the radio was that it would be unfortunate were the season not to be completed and for Liverpool to have an asterisk or a footnote next to their league title for this year.

Similarly, I have seen it argued that it would be unfair on the class of 2020 were their grades to be treated differently from those of other years.  In fact, I think they _should_ be treated differently, for the benefit of those students.

We just don’t know that they will be accurate and it would be wrong to pretend they will be.  A student getting a grade 5 this summer in maths might have been someone who would have got a grade 5 in the exam, or might have got a grade 4 or a grade 6 or might have got a grade 3 or a grade 7 (or further away from grade 5). We cannot tell, yet by giving _a_ grade we will be saying we think we do know.

In an ‘ordinary’ year the grade a student gets might not be an accurate reflection of their ability in that subject.  Perhaps they misunderstand the way a question is phrased and lose marks when they do understand the subject content. Perhaps the vagaries of a mark scheme affect their results (see the bizarre marking of the KS2 ‘semi-colon’ question), or perhaps they happen to revise a topic over lunchtime that then appeared as question 1 in the afternoon paper, or perhaps the marking was done incorrectly (this certainly happens because post-exam appeals are sometimes successful).

We know this, though.  We know that an exam result is just that: a result on an exam. While we might use a GCSE grade in a subject as a proxy for how ‘good’ a student is at that subject, all we really know is the grade they got in that exam.  This year everything is different.  Different schools will interpret the guidance in different ways, the number of high grades available to schools might not be ‘fair’ (see earlier in this post) and the students might not get the grade they ‘should’.

I would love for us to be able to put confidence intervals on grades – in normal years as well as this year.  This would be too complicated to understand, however, and the middle value would inevitably be used as ‘the grade’ the student got. 

My suggestion then is to put a symbol next the grades to show they have been created in a different way and are not exam results.  Maybe ~B could refer to an A-level B grade that was arrived at by this year’s system.  ~4 could be a GCSE grade from this year.  This wouldn’t be to devalue the grades, but rather to point out that they are just less likely to be accurate.  If a student currently in Yr 11 needs a grade 7 to be allowed to take A-level maths, for example, the sixth form could look carefully at those with a grade ~6, rather than that automatically disbarring the student.  If a Yr 13 student needs three A grades to meet their university offer, the university could consider carefully those who get A,A,~B

To sum up:

This year’s system is not good for the very people it is supposed to be there for: the students.  There isn’t a realistic alternative, however, so we need to find a way to make this work to serve the needs of those students and those who would ordinarily make use of exam grades.


Saturday, May 30, 2020

Geometric Social Distancing

I have another blog on which I post what I call “Quibans” – Questions Inspired By A News Story.  Those tasks are aimed at Yr 12/13 Core Maths classes.  While this post is inspired by a news story, it isn’t really Core Maths appropriate, so I’m putting it here instead.  It might be of interest to Yr 10 and Yr 11 Higher Tier GCSE classes, or as some trig revision at the start of a Yr 12 A-level course.

The article has been edited to remove material that is irrelevant (more on this in a future post); nothing has been altered or misrepresented.

Here is the article, taken from the Daily Telegraph:
Parallelogram, hexagon or pentagon? The safest way for six people to picnic revealed 
A basic knowledge of geometry could come in handy for people who wish to meet when the coronavirus lockdown rules are relaxed next week 
29 May 2020 • 6:51pm  
Groups should picnic in a parallelogram, hexagon or pentagon and if they want to meet together while still maintaining social distancing and good hygiene, experts have said.
On Monday, lockdown restriction will ease so that six people can gather together for the first time since measures were imposed on March 23rd. 
Maths specialist Bobby Seagull who presents the BBC Two series, Monkman & Seagull's Genius Adventures, has suggested that the most efficient way to keep people the correct distance apart is to form a series of triangles, with each person sitting at an apex. When six people do this it forms a parallelogram. 
Mr Seagull said: “My solution is not perfect but means anyone could leave without upsetting others' social distancing. we think that this problem is trickier than at first sight.” 
However mathematicians at Oxford University believe that sitting in a hexagon or pentagon would be a more efficient solution, taking up less space and allowing minimum distance between people. 
Jason Lotay, Professor of Pure Mathematics at Oxford University said that putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 
Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m.  
Alternatively, if one person is placed in the middle so that the other five form a petagram then the maximum distance between any two people is just 3.80m.However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.   
“Of course, things are much better for the person in the middle, who is 2m from everyone, but worse for everyone else, who now have an average of 2.86m to anyone else,” added Prof Lotay. 
“I would probably say that efficiency is given by the total area required and minimising the maximum distance between any two people, in which case the pentagram model is better.   
“However, as I say, it means that one person gets preferential treatment, and it is worse for everyone else.”

There are so many issues with this, which would make it fun to do with a class!

First of all, the article is a bit of fluff, it’s not meant to be taken seriously and the errors are really not a big deal. It may appear as if what follows is being ridiculously pedantic (and of course it is!), but the bullet points at the end make it clear that none of this is really serious.  

Let’s look at some of the figures involved.
Which of these two configurations looks like it has the smaller area?
Which one looks as if the “average distance that one person has to another” is smaller?


To me it appears as if the pentagon has the smaller area and the smaller average distance between the people.

The article says:
However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.  
Ignoring the typo of “pentagram” (which isn’t the same as a pentagon or, as it’s written elsewhere in the article “petagram”), this suggests that my intuition is partly wrong, in that the area of the pentagon is smaller but the average distance for the pentagon is bigger.  Hmm.  Let’s check that out.

The Hexagon

Here’s the regular hexagon, with 2m along each side.  There are lots of ways to find the area.  We could use Pythagoras to find the height of one of the triangles:

The height is 
The area is therefore √3 and the area of the hexagon is 6√3
(An alternative would be to use the area formula   which isn’t as bad an idea as it looks, because this is  

 (we know the value of sin 60 degrees) and this is clearly also the square root of 3.)

… putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 
Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m. 
6√3 = 10.39 (2dp), so we have worked out the area required and this agrees with the article.

The maximum distance between any two people is indeed 4m.  (F and C on the diagram are 4 metres apart, for example.)

How do we work out the ‘average distance’ between people? There is nothing special or different about any of the positions, so if we work out the average distance that A is from all the others, this will be the same as the average distance of B from everyone else, etc.

AB is easy – this is 2m.
AD = 4m and AF = 2m.

AE could be worked out using the cosine rule, but we could look back at the previous diagram and see that AE is the height of two of the triangles, so it is 2√3.

AC is the same as AE.

That gives the average distance of the other 5 points from A as:

This is 2.98564 …  and not 2.29 as in the article.

It is also greater than the values given in the article for the pentagon.

The Pentagon

I think the pentagon values are accurate.  (A class could be asked to work these out, using whatever method they want.)
Here, briefly, are my workings:

Area of a triangle = 
 so the area of the pentagon is (to 2dp)



Now we work out the lengths on the diagram:

Distance AF = 2.
We can work out AB using the cosine rule in the earlier diagram:


AE = AB = 2.3511…

To work out AC, I used the triangle ACF and the cosine rule. 


AD = AC = 3.8042…

This time we need to be careful when working out the average distance, because F is an average distance of 2m away from everyone else, whereas A (and B, C, D, E) are an average of

(This is a useful reminder to use the unrounded versions of the answers …)

To work out the average distance for all of the six people in this arrangement, I did


These figures agree with the article.

The Parallelogram

Bobby Seagull tentatively suggested a parallelogram seating arrangement based on equilateral triangles:



The area and the average distances aren’t included in the article, so our class could work those out!

Here is my brief working:

Area is 4 lots of the triangle we worked out earlier, so it’s 4√3

The distance AD is the height of two of the triangles, so this is the same as AE was on the hexagon: 2√3

AD is equal to CF and BE.

AB = AC = BC = BD = CD = CE = DE = DF = EF = 2

That just leaves AF to work out. I used the cosine rule to get √28 = 2√7

AF is the furthest distance in this configuration, which is 5.29m (2dp).

The average distance is, I think, 2.78m (2dp) which is better than the hexagon but worse than the pentagon.

Real-life implications
  • There are clearly a few mathematical simplifications going on here, but there are some other issues too.
  • People are being treated as points (I think that is reasonable in that the ‘danger area’ is presumably the mouth/nose which, when looking vertically downwards, are not particularly large.)
  • How good are people at estimating 2m? I would guess that no-one is exactly accurate, so much of this is moot anyway.
  • Given that the measurements won’t be perfect, it is a bit odd to give answers to 2dp, which suggests that they are accurate to the nearest cm !
  • A practical drawback with the parallelogram is that C and D are the only people who can see everyone else (B and F can’t see each other because D is in the way).
  • Bobby points out that the parallelogram is good because anyone can leave of their own volition. This is true for the hexagon as well. In the pentagon arrangement, for the one in the middle to be able to escape, a minimum of two other people (adjacent to each other) have to move.
  • Finally, the article suggests that the person in the middle of the pentagon has an unfair advantage because they are closer to everyone else.  It will be an uncomfortable conversation for them, though, because they will need to twist their head like an owl to be able to look at everyone!

Postscript 1:
The Daily Mail has an article that starts in a very similar way. Not sure whether they both got it from the same source or whether someone has been copying their homework!  They include the error in the calculation too.

Postscript 2:
I have written above that the article is clearly a bit of fun.  The comments under both the Telegraph and the Mail articles from members of the public suggest that not many of their readers realised this …


Wednesday, May 27, 2020

Artful Maths - review


The Artful Maths activity book by Clarissa Grandi (published by Tarquin, 2020)
Review by Mark Dawes

If you like your art to be full of maths then this is an ideal collection to use with a class, with a maths club, with your own children, or for fun yourself!

One of the occupational hazards of being a maths teacher is having to try not to react when people tell you “I just can’t do maths”.  I gather that art teachers have similar frustrations when people tell them “I can’t draw”.  On the face of it then, a book called “Artful Maths” might put off rather a lot of people! 
It could have gone this way, where only those who adore both art and maths are going to be interested:

But instead it was like this, where even a non-artist like me was engaged and interested and family members who like neither art nor maths were also intrigued:




The art created here is geometrical, so a ruler and pair of compasses are required for some of the tasks, and then the opportunities for creativity about.

The charm (and genius) of the resource is that it can be used in so many different ways, such as:
·         having an emphasis on the art, on colour, on producing something beautiful
·         as a way to learn how to carry out geometrical constructions and to practise them
·         focusing on the mathematics that underpins the constructions
·         extending the tasks and emphasising deeper mathematical ideas

One of the very exciting things about this is that those who start with one of these rationales then stay for others of them too!


There are four linked resources:
An activity book, which provides clear instructions to follow (with diagrams) and pages to write on.
A teacher book, which includes curriculum links, prior knowledge required, and a suggested way of structuring the tasks with a group.
Two different types of downloadable resources (printables to go with the activity book pages and teaching resources that link to the teacher book), to save writing in the activity book and to enable the tasks to be presented to a class.

Artful Maths consists of six main activities, each of which includes explanation of the art involved and description and questions related to the mathematics behind or within the art.  The first activity is based around curves of pursuit and begins with a square but then moves into other regular shapes and even into 3 dimensions. There are opportunities for creating beautiful diagrams, involving some creativity, and there are explicit geometrical reasoning tasks that can be used alongside the drawings.

The three sets of resources complement each other well, with some questions appearing in the activity book, while others are in the teacher book.  This serves to give enough support in their activity book, without cluttering it with huge numbers of questions, but provides those additional ideas for exploration that the teacher (or parent) could introduce as they wish.

My favourite chapter is the one that shows some very clever and intriguing ways to create mazes and labyrinths: here again there are opportunities for decision-making and creativity within an initial framework. What is particularly exciting is the dual use of ‘creativity’.  I suspect that many children (and adults) will associate creativity with art, whereas these tasks also offer opportunities to be mathematically creative. The extensive extension activities in each chapter offer many ways of doing this.

Because there is a clear focus on drawing and on creating the diagrams, the mathematics that is used can be exposed or can be left hidden, depending on the interest and the mathematical readiness of those who use the activity book.  It is possible to discuss Pythagoras’ theorem or to use trigonometry, or to work with angles or sequences during different tasks, but it is also possible to enjoy and engage with the tasks without doing this, giving a great deal of flexibility.  As a way of introducing and using modular arithmetic, the golden ratio, Fibonacci and constructions this is an interesting and valuable resource.

During lockdown, when most children are not able to be in school, another benefit of Artful Maths is to give a way for children from KS2 to KS4 to practise using a ruler and compasses, and for them to develop their practical skills.

I like the way the mathematics, while optional, seems to be a natural part of each task.  These books and accompanying resources provide the opportunity to do some mathematical thinking while creating something beautiful. What a good combination!
(Disclosure: I was sent a review copy, but had already bought my own!)