Professional Judgement

As a teacher, I have been asked to make predictions as to how my pupils will do in GCSE and A-level exams more times than I can remember. At my previous school, we did this three times a year for A-level students (which made up 80% of my teaching).

I questioned the value of these predictions, especially after reading in Thinking Fast and Slow, about the illusion of expertise: the example given was of stockbrokers who consistently thought that they could out-perform algorithms in making good predictions. The data did not support them.

I had a database of several hundred A-level students from my school so I decided to calculate how accurate our predictions were and compare this to my super-hi-tech algorithm for predicting A2 performance: AS grade + 8 UMS points.

I then calculated the mean squared error in all of these predictions and you can see these numbers in the top right of the spreadsheet.

My super-hi-tech algorithm produced an error of 0.42. (note that I could have added anywhere between 6 and 11 UMS points and this doesn’t change much).

In January, the team of expert teachers (I’m not joking here: my colleagues were very experienced and effective teachers) produced an error of 0.64, in March they’d reduced this to 0.45 but it wasn’t until April, about a month before the exams that the experts finally beat the algorithm, with an error or 0.35.

This suggests that there was absolutely no point in making the earlier predictions. To be honest, I’m not sure what use the April predictions were either but at least they were slightly more accurate than the simplest model I could think of. Moreover, I think it shows how bad teachers are at judging students and why we shouldn’t use teacher assessment in reports, or school data generally. This point is also made well in Daisy Christodoulou’s blog: Tests are inhuman, and that is what’s so good about them.

How to give all UK teachers a 35% pay rise.

Warning, this blog involves lots of numbers. Don’t worry, I had a historian proof read it and he understood ūüôā

For the purposes of these calculations, I’m going to ignore inflation and talk in 2018 pounds. The teachers’ pension automatically takes into account inflation, so that makes this a reasonable thing to do.

The average teacher earns, according to the government, ¬£37,400. Each year under the current ‘career average’ scheme, my average teacher, Sarah will earn a pension of 1/57th of her salary: ¬£656 per year.

How much would this pension cost if Sarah wasn’t a teacher? At the age of 68, a pension pot of ¬£100,000 will buy an annuity, which grows with inflation as the teachers’ pension does, of ¬£3,600 per year (see note 1). Therefore, it would cost ¬£18,200 to pay for Sarah’s pension of ¬£656 p.a.

Where does this ¬£18,200 come from? As a member of the mysterious teachers’ pension, Sarah contributes 9.7% of her income, ¬£3,600. This means that each year the government contributes an additional ¬£14,600¬†(39% of her salary: see note 2) that she never sees and may not even know exists.

My sister is a lawyer and her employer contributes 3% of her salary to her pension pot. Let’s say the government adopted this approach: it takes the ¬£14,600 it currently contributes to Sarah’s pension, pays her ¬£13,100 extra, sending her salary to a healthy ¬£50,500, a 35% increase. It contributes to remaining ¬£1,500 (3% of her new income: see note 3) to her pension.

I’m using Sarah as an example but it doesn’t matter if you think she’s not representative, because we could equivalently increase all teachers’ salaries by this 35%. I’ll say it again, 35%!¬†Starting salaries for teachers shoot up to ¬£31k (¬£39k in inner London) and suddenly look a lot more competitive. On the other hand, teachers’ pensions are now terrible, along with the pensions of lawyers, accountants and most other professions. But who goes into a job because of the pension?

Could this change help solve the recruitment problem?

ps. I make no comment as to whether or not I think this is a good idea. My wise father-in-law pointed out that it’s a very Conservative suggestion: let people choose how to spend, or save, their money.


(1) I interpolated based on the figures given. This is actually a conservative estimate because I used the figures for a single pension: in fact, the teachers’ pension also pays 37.5% to a partner after the teachers’ death, so would be worth more than what I calculated. This all assumes that a private pension pot grows at the same rate at the teacher’s pension (CPI + 1.6%). I suspect that some pension schemes may do better than this, but they will be much more variable and may also go down significantly, for example in 2009.

(2) Officially, Sarah’s school contributes 16.5% (in the case of state schools, this is just the government shuffling numbers around on a page) but the¬†¬£14,600 is actually 39% of Sarah’s salary of ¬£37,400. I understand that the government doesn’t actually ‘save’ this money as teachers’ pensions are unfunded, but it does have to pay it eventually. In the short term, this policy would cost the government quite a lot of money, but in the long term, it wouldn’t make a difference.

(3) My wife thinks this paragraph is confusing, because she is eagle-eyed and noted that 3%+35% does not equal 39%. Why doesn’t it add up? These percentages are of different numbers (3% of the new salary, 35% of the old salary), a pretty classic tricky idea with percentages.

Absolute vs Relative

A recent episode of Radio 4’s “More or Less”, addressed the issue of Progress 8, which is obviously interesting to me as a teacher. However, it was the discussion about poverty in the UK which most caught my attention.

The Trades Union Congress (TUC) recently hit the headlines by pointing to statistics which showed that the number of children from working households who are in poverty has significantly increased in the last ten years. They claim that the main drivers of this have been cuts to in-work benefits and restrictions on public-sector pay. The government’s response: It doesn’t recognise the TUC analysis; there are one million fewer people living in absolute poverty.

There a few extra details in the programme, but the gist is that both claims are correct. Relative poverty is increasing, but absolute poverty is decreasing. So the question really about which we value as a society? In the UK, my feeling is that the focus should be on relative poverty (although what I have written next has made me question this slightly!). Indeed, I’m surprised that there are many people at all living in absolute poverty: I know I live in a social bubble, but I suspect that the government figures are not based on the international definition as set by the World Bank.

Returning to education, I feel that a similar debate that has been ‘raging’ on twitter for a few months now (perhaps even longer), boils down to the same issue.

Is Ofsted biased against schools in more deprived areas? Clearly, many people on twitter are convinced by Stephen Tierney’s recent blog post¬† on the topic and regular references to this graph:

It shows that schools with a high proportion of White British children receiving Free School Meals are judged, on average, much worse than schools who have fewer children in this group. The immediate conclusion is that Ofted is biased against these schools. Surely the proportions should be the same for all types of schools? No.

Why not? Because Ofsted’s standards are absolute, not relative. As Jason Bradbury and Sean Harford explain, the evidence shows that when looking at schools with the same Progress 8 measures, inspectors actually give more generous judgements to these ‘most deprived’ schools.

This¬†thorough treatment of the issue points out that there are many reasons why it’s difficult for schools in ‘deprived’ areas to attain the same absolute standards as schools in more affluent areas. However, this doesn’t mean that we should instead use relative judgements: that would be to accept that it’s ok for children growing up in disadvantaged areas to go to schools with lower standards.

What the analysis does show, however, is that it’s much harder to run a good or outstanding school in underprivileged areas. As a result, perhaps management and staff in these schools should be rewarded / treated with leniency to a greater extent than those in prosperous areas? Similarly, should these schools be funded more generously?

Overall, this has got me thinking about whether we need to get better at teaching the key idea of ‘Absolute vs Relative’ in maths classrooms. Up until now, I haven’t taught it explicitly… another one to add to my scheme of work, perhaps.

(Disclaimer: although I am rather convinced by Ofsted’s blog, I don’t think it proves beyond all doubt that there is no bias: judgements clearly account for progress 8 weaknesses, but to what extent?)

Leaving Highgate

I spent my first seven years in teaching as part of an amazing department at Highgate School. For most of that time, my HoD was Dan Abramson and Robert Wilne (past Head of Secondary at NCTM) was Assistant Head. Robert had revolutionised the department a few years earlier with some radical changes: he introduced a scheme of work with great ideas, loads of links between topics and very specific approaches to teaching. Dan continued this tradition, bringing in many ideas from AfL and relentless energy. In the words of one of my colleagues Peter Davison, Dan could be prime minister if he wanted to.

When you teach maths at Highgate, you feel like part of something special. The whole team, now over 20 teachers, is strongly encouraged to use the same approach to teaching.¬†I remember that Craig Barton sounded pretty shocked when Greg Ashman described a very similar style of department; I suspect that it’s pretty rare and perhaps not for everyone.

The advantages: It makes for an incredibly coherent experience for the students. When you take on a new class, you know what models and vocabulary they’ve seen before and how they’ve been taught to think about every topic. You can be sure that the highest attaining will not have been pushed through more material than the scheme dictates and you can reliably call upon the standard models to helped lower-attainers.

Outside of the classroom, the number of conversations about approaches to teaching was probably 20 times what I got at the (academically very similar) school in which I subsequently taught. I think that this is a great advantage of shared teaching methodology: if you wanted to change how you taught, you had to convince the rest of the department that your ideas should go in the scheme. Many teachers stayed in the office to work until 6 or 7pm; it helped that many were young and family-free but I like to think that it was partly because the sharing of ideas made it an inspirational place to work.

If I loved it so much then why did I leave, you ask. Mostly because my wife and I wanted to buy a house, which is not very achievable for a teacher in London.

What followed was two fairly depressing years. My new colleagues were passionate, highly knowledgeable and interested in the success of their students. But there was no scheme of work beyond the chapter list from the textbooks and discussions about how to teach maths were few and far between.

Then I discovered Twitter.

I remember very distinctively one mid-winter run, I was listening to Jo Morgan on Craig Barton’s podcast, talking about how she found twitter and all the great ideas out there. I’d just been through a similar process and it made me feel quite emotional to be part of a community again: I think I almost cried.

Despite my improved mood, I still childishly felt a bit sorry for the people on twitter because I suspected that the reason they were online was because they lacked departments like Highgate. Over time, I’ve come round the view that twitter actually has some advantages over Highgate: I can draw from a much wider range of experiences and ideas, and it has exposed me to many more ideas from the world of educational research. There are negatives too: there are still times on Twitter when I ask for ideas or opinions and don’t get any. In the Highgate maths office it was harder for people to ignore me!

I can’t actually remember if I discovered edu-twitter or Craig’s podcast first, but if I hadn’t found either of these, then I think it’s quite possible that I would have left teaching. Two years later and I’ve just started a job in which I’m the only maths teacher in the school. That doesn’t phase me because I’m safe in the knowledge that I have my online community of teachers.

Three Weeks In

I started teaching at a new school 3 weeks ago.¬†Overall, I’m absolutely loving it. My job is more varied than anything I’ve done before and for the first time in my life, I actually look forward to going into work on a Monday.

I’ve been writing the timetable. It has been really interesting to learn how our part time staff prefer to work and try to balance this with providing a balanced week of lessons, alongside management discussions about what proportions staff should work. I’m also excited to lead outdoor education: my climbers seem to be really enjoying our weekly indoor club and I can’t wait to get them out into the mountains proper. It has been great to be involved in discussions about the curriculum: how many periods per week should we give to each subject is not a question I’ve ever considered before.

More mixed has been my work as assessment lead. Inspired by Tom Sherrington, I’ve started with the theme of feedback as actions, sharing some of my own attempts to put this into practice, but I have yet to garner much response from my colleagues. Similarly, initiating the process of collecting pupil data is taking some time.

Despite the fact that I have very few lessons and much more experience in this area, probably the hardest part of my job has been teaching maths! Small class sizes mean that it’s been possible to personalise my teaching more than ever before, and I’m enjoying the opportunity to implement some of the ideas I’ve read about during my nine month break. However, a few issues are challenging me.

1. I’ve never taught mixed-attainment classes before and I’m struggling to find a lot of concrete advice on how to best deal with it.. Do I split the class by task or try to keep them all together? Let the children choose their own tasks or assign them myself? Do I provide extra explicit instruction to some? Should this be within, or in addition to lessons?

This flow chart from @mathsmrgordon has provided some inspiration.

2 . How much to use technology? All my pupils now have a Macbook and iPad. This gives great opportunities, I’ve got them using Desmos, Geogebra and Quizlet, but am I going too far? It’s hard to tell when it’s genuinely educational and when it’s just more fun than pen-on-paper maths. And on that note…

3. I’m following in the footsteps of a teacher who sounds like he was much more fun than me! I’ve heard that he was a great teacher, very inspirational and played lots of games. My insistence on copying down worked examples and setting of written practice and extension tasks sounds pretty boring in comparison. To their credit, the pupils have generally been working very well, but I get a sense that we haven’t fully bonded yet.

Any advice? I’d love to hear it.


Freddy: “Mr Pearce, we found your blog.”

Me: “Yes?”

Freddy: “It’s all about teaching.”

Me: “What did you expect?”

Freddy: “Some mathematical nuggets?”

Me: “I’m a teacher, not an entertainer.”

OK, so I’m still quite happy with my last line, just because I like the ring of any line of the form “I’m not a …”, which I stole from a friend Sam Bartlett who employs it comedically better than I ever could. But I was being rash – a teacher is, to some extent, an entertainer.

I clarified by saying that I think there is rather a glut of mathematical nuggets out there and so I’m looking to provide something different. ¬†However, I do like nuggets, so¬†I’ll allow myself one (and only one) blog post to share some of my favourites:

The Movie Maths Quiz


Don’t work hard…


The Venn Diagram of Bollocks


George Ford on Countdown

Report Cards for Mathematicians


The Three-Switches Problem


An interesting problem

Causes of Death


New Year’s Resolutions

Many of these ideas come from a survey of my pupils at the end of last term. So this starts with a brief summary of that rather long post.

  1. Ensure that I’m providing enough guidance and support when asking pupils to investigate unfamiliar problems, by creating more guided resources, and preparing a back up text-book option for certain pupils.
  2. Refer more frequently to the skills from Thinking Mathematically to encourage pupils to know what to do when stuck on a problem and other strategies from Helen to help them develop a growth mindset.
  3. Make use of mini-tests: Mathsbot looks like it will be a good source for these in KS3+4, I need to source something similar for KS5.
  4. Set one or two summative homeworks per class per term, in addition to supporting pupils in choosing their own questions by using Google sheets to share questions which pupils have found difficult and track their progress in re-attempting them.
  5. Take opportunities for whole-class interactivity, particularly with year 10, making use of Dan Meyer’s 3 act tasks.

And others which don’t come from the survey.

  1. Ensure I always make clear the Headache before providing the Aspirin.
  2. Write (type where possible) board notes more clearly and slowly; learning Spanish and trying furiously to copy quickly-disappearing notes down from a board has taught me this!
  3. Continue trialling Inquiry Maths lessons, in particular bringing them to sixth form as well as younger pupils.
  4. Use shared Google doc with each pupil to track their general progress and targets, alongside my target setting form.
  5. Build a website to share my approach to providing summative and formative feedback, both directly to pupils and in written reports.