Post-intervention Progress?

This idea is totally stolen from a former colleague of mine, Andrew Dales.

His question: what happens after an intervention? Specifically, he framed the question perfectly as the following graph:

So, assuming that the intervention produces a greater increase in attainment than control (which I’ve exaggerated significantly for the purposes of clarity), what happens afterwards?

Does the intervention instil in pupils some newfound ability to continue progressing at a greater rate than they would have before (route A)?

Do the pupils return to the same rate of progress as before the intervention, staying ahead of where they were before (route B)?

Or do they, post-intervention, progress at a lower rate for some time, returning to their original path (route C)?

Of course, this will depend on the intervention. As teachers, I feel that we should consider which approaches will help our pupils to follow route A (or at least B!), not just those which produce the largest improvement over the time span of the intervention.

In the same way as Daniel Kahneman introduces new terminology in Thinking, Fast and Slow, I think it could be valuable to introduce the new terminology into educational research: “Is this a Route A intervention, or will the students revert to control via Route C?”

I wonder how much research tries to answer this question? My suspicion is not a lot.

To Mark Or Not To Mark

One of the two big takeaways which I discussed in my clip for Craig Barton’s ‘slice of advice’ podcast was that you don’t have to mark pupils’ written work.

I think the first place that I heard of this radical idea was from several of the teachers at Michaela School. They promote the idea of whole-class feedback: instead of marking pupils’ work, look through it, note down common strengths and weaknesses or misconceptions and use this to plan feedback which you deliver to the whole class.

Many of the advantages of this approach are discussed by Andrew Percival from minutes 14-18 on Craig’s podcast: It’s easier to give detail and nuance verbally, it gives teachers more time to think about planning lessons, it encourages focus on ‘improving the pupil, not the work’ (a quote from Dylan Wiliam).

For me, another major advantage is not having to spend hours writing individual comments… I did so for 10 years and it was the one part of my job that I hated, even though I knew that the research showed that it was more effective than giving numerical scores. I guess I just didn’t think enough about whether there may be an alternative.

I used whole-class feedback last term together with online homework. I set this through but next term I plan to trial, particularly because I like the fact that there is the option for adaptive level of difficulty in the questions. Pupils get immediate feedback as to whether or not their answers are correct and hence the chance, which many have taken, to correct their mistakes.

Once all pupils have completed the task, I look through the answers and choose which questions I wish to discuss in class. Last term, I then presented solutions to these questions, or asked a pupil who had correctly solved the problem to do so. Next term, I think I will try to present a slightly different question (possibly just changing the numbers) because this will be less frustrating for those pupils who have already answered the question correctly. It will also give those pupils who struggled originally the chance to re-do the question to show that they have improved their understanding.

I’ve just mentioned one negative of the whole-class feedback approach: some of the feedback won’t be relevant to some pupils, so we may see this as a waste of time. An alternative would be try to give verbal feedback based in a more individual way, but this would take a large amount of time with a whole class. One advantage of individual written feedback is that producing it does not use up the limited resource of lesson time.

Another advantage of written feedback may be its longer-term nature. A pupil may well have forgotten what you said to them yesterday, but if you wrote it down they can go back and look at it again. As I developed as a teacher before I stopped writing comments, I was getting better at directing pupils to look back at my past comments when they had not taken my suggestions on board.

Overall, I feel that these two issues do not outweigh the advantages of whole-class feedback (though from my comments above, you can tell I’m a bit biased!) However, they mean that I haven’t completely ruled out written feedback.

What do you think?

Scheming, Part 2: Spaced Practice and Interweaving.

You can tell that I had 9 months off work last year because I’m spending an unnaturally large amount of my summer holidays working. Most of this time has been improving my scheme of work. In part 1, I talked about how I’d been thinking carefully about which topics were prerequisites of each other to ensure that everything was taught in a logical order.

This time, I was to discuss this:

If you’re not mathematically or computationally inclined, please don’t be put off! This bit of code simply answers the question: “When is this idea revisited in the scheme of work?” in order to help me build spaced practice into the scheme and improve my interleaving.

Here is what my scheme of work looks like:

I can see that the first topic “Fraction Division A” (in my scheme: a model for division, writing it as a fraction and learning the relevant vocabulary) is ‘revisited’ after 2, 14, 16, 29, 52 and 66 sequences of lessons. Each sequence in my scheme takes around 3 hours / 1 week of teaching, so you can think of this roughly as weeks. By ‘revisited’, I mean either that it is listed as a prerequisite or application of a future sequence.

How are these numbers useful? Well, I’m not sure but I think that in an ideal world, they would go something like 2, 6, 20… I’m basing this roughly on something Mark McCourt said in his Slice of Advice, which comes from analysing data from his complete maths platform. I did read another blog recently which suggested in some cases linear spaced practice (eg. 4, 8, 12 , 16) may be more effective in some situations (annoyingly I can’t find this blog now).

Regardless, with Fraction Division A, the gap from 2 to 14 seems a bit large so this may prompt me to move the topic which comes 14 sequences later forward in my scheme.

Other topics have fewer applications. For example Rounding A had no close follow-ons at all. In this case, I didn’t want to move Rounding B forward in the curriculum, so I looked for another topic which could be used to apply rounding A and decided I could apply that when teaching Area A. In this way, it’s promoting me to think much more about genuine interweaving (not just interleaving) of topics.

Another harder question that immediately arose was constructions A, as there is a vague link after 10 topics but the next comes after 76, by which point the students will surely have forgotten most of what they learned. In this case, I need to think more carefully about whether I really want this topic at this point in the curriculum at all, as it doesn’t have many links to the rest of the curriculum (a few people on twitter helped me with this recently, but I’m still slightly lacking links).

I’ve only just started using this formula, but I already feel that its very powerful and is allowing me to make decisions about where to place topics and how to interweave in a much more informed way than I ever had before.

Are there any other uses for these numbers that you think I’ve missed?

What are the weaknesses of this approach?

Are you impressed by my excel skills?!

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.

Draft Assessment Policy

Over the past two months, I’ve been writing the assessment policy for my new school. It was inspired very much by Making Good Progress, but also contains ideas from several policies that teachers kindly shared with me on twitter.

I have a two-hour INSET slot on assessment at the start of next term. One of my ideas is to share this policy with our teachers, ask them for their views and work on improving it as a team.
Before I do that, it would be great if anyone would help me by making any suggestions for improvement.

It will cover all subjects (as we currently only have one or two teachers in each) so I wonder if it lacks flexibility: it’s written very much from my maths teaching perspective?

Here you go:

GES Assessment Policy

Formative Assessment

Formative assessment refers to practices used by teachers which assess their pupils’ progress in order that the teacher and pupil can plan their future learning more effectively.

Teachers should use assessment:

  • to determine pupils’ prior learning
  • to check on pupils’ understanding
  • as a form of retrieval practice to improve memory
  • to correct factual and literacy errors / poor effort
  • to modify teaching and decide whether or not to move on

In order to achieve these aims, they may use the following forms of assessment.

Individual Verbal Feedback
Verbal feedback should be specific and positive (‘do this’, rather than ‘don’t do that’) and should ensure that the responsibility to improve the work remains with the pupil. Teachers should spend a brief amount of time giving feedback to a pupil; if feedback takes longer, then further instruction is needed and the teacher should assess whether a reteach of the topic/concept is required.

This has two main purposes:
Sharing knowledge and understanding around the class;
Checking on the knowledge and understanding of the class.
In the first case, teachers may accept ‘hands up’ but the majority of the time, teachers should use named questions. This enables teachers to assess whether pupils have understood the topic they are teaching, rather than just hearing from the highest-attaining pupils.
It is wise to ask a question, pause for the whole class to think, then target the question towards an individual, to encourage all pupils to engage in thinking.

Critiquing Good Work
Teachers photograph a good piece of work and project it on the board. The class discusses the strengths of the work and how it can be improved.

Teachers ask questions and pupils write their responses on mini-whiteboards which they hold up for the teacher to see.
Pupils should generally wait to show their answers at the same time, so that they are not encouraged to rush.
Complex tasks should be broken down into smaller steps for this activity.

Multiple Choice Questions
A teacher presents a multiple choice question and pupils can hold up a number of fingers to indicate their answer.
The wrong answers should ideally contain common misconceptions.
Higher-attaining pupils can be encouraged to think through what misconceptions might lead to the other answers given.

Self and Peer Assessment
If trying to address misconceptions, self-assessment is often more effective as pupils learn from there own mistakes more readily than mistakes of others.
Peer assessment can have the advantage of pupils being exposed to and learning from the ideas of others, but this may be more effectively managed through the strategy of ‘critiquing good work’.

Tests are an effective way to encourage pupils to retrieve information and test their understanding. In studies, pupils learn most from such tests if they are low-stakes: self-marked, no negative consequences for poor performance or even no-stakes: the teacher doesn’t even find out the score.
Such tests should not be assigned a percentage or grade. The focus should be on what the test tells the pupil and teacher about the next steps required for the pupils to improve.
Short, specific tasks usually provide better formative information than complex tasks, because they help to highlight the exact misconceptions of a pupil.

Teachers are encouraged to look through all homework, make brief notes and to give ‘whole class feedback’ the following lesson.
Teachers are encouraged to make notes about common errors and add them to the schemes of learning in order to address these potential pitfalls when the topic is delivered in the future.

If teachers wish to give individual written feedback on homework, it should not come in the form of a grade, but should comment on what is specifically good about the work and give one or two suggestions for improvement.
These suggestions may come in the form of follow up tasks (possibly one of the 5R’s: see appendix 1.)
Ideally, the teacher should check that these follow up tasks have been completed, but we do not want to encourage an endless cycle which burdens teachers with unmanageable workload. The follow up tasks should be more work for the pupil than for the teacher.
Teachers should encourage pupils to look back on previous feedback before completing future tasks.

Summative Assessment

The main purpose of summative assessment is to give all stakeholders (pupils, teachers and parents) an idea of how pupils are performing and progressing over time.
In order for this to happen, the results of such assessment needs to be reliable and valid, and to communicate shared meaning.

A twenty minute test will not give a reliable picture of a pupils knowledge and understanding of an entire subject. homework is not a reliable indicator of performance, as the level of time, effort and assistance sought can vary significantly.
In order to be as reliable as possible, they should be long, and ideally set over several different days to allow for pupils having a ‘bad day’.

A test on the conditional tense in Spanish will probably not be a valid indicator of how well a pupil will perform in GCSE Spanish. Similarly, the quality of a long-term project will not be a valid indicator for a subject which is assessed by examination.
Tests will be valid if they sample from a large and wide ranging proportion of the expected knowledge and understanding for a pupil of this age.

Shared meaning:
A raw score (e.g. 21 / 30) or percentage (e.g. 53%) does not communicate shared meaning because there is no common basis of understanding. It is not clear to pupils or parents, and to some extent even teachers, whether 21/30 or 53% is a ‘good’ score, nor what ‘good’ even means in this context. In order to communicate shared meaning, summative assessments results should be scaled appropriately.

How do we apply these principles at GES?

Each year group takes part in an extended period of exams towards the end of the academic year.
In year 7 and 8, these tests are sat in classrooms within the normal school timetable.
In year 9 and above, these tests are set in an exam hall over the course of one week. Revision periods are allocated between exams.
Where a department demonstrates that examination is not the most reliable predictor or GCSE success, flexibility will be given as to the method of assessment used.

Each subject is tested at least twice, with the length of exam being related to the number of lessons taught in each subject and the age of the pupils involved.
Pupils in year 7 and 8 can expect at least 2 hours of tests in Maths, English and Science. Pupils in year 9 and above can expect to take at least 3 hours of tests in these subjects. The length of these tests help make them a reliable indicator of a pupil’s performance.
These tests will aim to cover as much of the material taught up to the point as possible, in order to make them a as valid assessment as possible.

Results for each subject will be provided as a standardised score, such that the average score for the year group in each subject is 100 and the standard deviation is 20. This helps us to compare pupils’ performance between subjects and from year to year. See appendix 2 for more detail on this process.

In year 9 and 10, there will also be an indication of what a score of 70, 100 and 130 might mean in terms of a ‘working towards’ GCSE grade. These will be produced using CEM data, alongside assessments from national comparative judgement assessments in English and Maths. This will help to communicate shared meaning to parents, without giving the false impression that we can accurately predict grades at this stage.
In year 11, mock exams will be sat in February and the grades will be reported to parents, alongside that term’s progress report from teachers.

As a school, we only use summative assessment once per year because, in order to be reliable and valid, the tests must take up a significant amount of potential teaching time. We also feel that formative assessment is more important for pupils’ leaning; summative tests are not easy to use formatively as they include complex tasks which require a variety of knowledge and skills, making it less clear to the teacher which of these are lacking.
As a result, teachers are discouraged from using summative assessment at other times of the year.

Reporting to Parents

Assessment of Effort

We use the following effort descriptors:

  • Listens carefully during whole-class discourse.
  • Works hard during individual tasks in class.
  • Collaborates well with peers.
  • Completes homework carefully and on time.
  • Asks questions to clarify or probe as appropriate.

The score for each criteria is on the following scale:

  • Almost always
  • Mostly
  • Sometimes
  • Rarely

Pupils self-assess their effort before teachers assess it.
Teachers meet with parents and pupils. They discuss the effort assessments and agree upon one or two targets for the pupil to work on.
Pupils create a Google document with their targets, share it with their tutor, who makes sure they know what they need to do in order to meet their targets.

Teachers write brief comments on how each pupil is working towards the targets they set in Autumn term. They should be aimed at the pupils and hence written in the second person.
They are sent to parents, pupils and tutors, who discuss them with the pupils.

Early May
Pupils self-assess and teachers separately assess pupils’ effort.
Teachers meet with pupils and parents to discuss the effort assessments and progress towards their targets. During this meeting, targets are revised if appropriate.
Pupils update their target sheet and discuss this with their tutors, particularly focussing on targets that have remained from the autumn term.

Late June
Teachers mark the end of year assessments and work with the head of assessment to convert the scores into scaled scores, which are then reported to parents.

Appendix 1


Appendix 2

Lets say Jamie scores 75% on an English test and 60% on a science test.
It appears at first that’s he’s doing better in English, but this does not take account of the difficulty of the test.
It could be that the class average in English was 80% and the average in science was 50%. Then Jamie is actually below average for English and above average for science. Pupils intuitively know this, which is why they want to ask their peers how they did after results of a test are delivered.
There is also a more subtle issue, which is that the results of different tests may be more spread out than others.

To account for differing averages and spread, we can standardise the scores in the following way:

The standardised score in every test will have an average of 100 and a ‘spread’ of 20. In the example of Jamie’s test results above, his English grade may (it would depend on the spread of results) be standardised to 93 and his Science grade may be standardised to 120.
This will allow his tutor and parents to compare these results fairly: he can’t use the classic excuse “but everyone did badly in English”.

Next year, he will receive another science grade on the same standardised measure. Let’s say this is 115. In this case, we should be careful not to assume that he has done worse this year than last / made less than average progress in science. If however, his score is 90 in science in the second year, this significant drop is probably worth investigating.

This system is not perfect:
It does not allow us to compare the performance of departments or teachers but we don’t believe that we should use test results to do this.
It doesn’t give students an idea of how they’re doing nationally. This issue it tackled in the feedback policy by relating standardised scores to GCSE grades.

Note that we are only talking about summative tests here, in which the aim is to “track pupils’ attainment and progress, to give them, their teachers and parents an idea of how they might perform in future external exams.”.
Formative tests, which form the vast majority of testing, should not be analysed in this way and pupils should be discouraged from comparing their performance to each other.

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 🙂

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.

Scheming, Part 1: Sequencing Topics and Prerequisites

I have probably spent about 40 hours in the past two weeks working on my scheme of work.

I started with the base scheme of work from my first school in Highgate. It stands out amongst schemes because it spends just 2-3 lessons on each ‘unit’ before moving on. This isn’t because it tries to pack in the whole syllabus into two years, but because each ‘unit’ consists of just one major new idea.

I really enjoyed teaching this way and was surprised when I first found out that other schools spend around a month teaching one narrow around area of maths before moving on to a different area. I wonder how on earth the students are going to remember a topic that they last studied two years ago. If you teach in small blocks, the difficulty is that pupils may not remember the prerequisite material. However, this forces you to constantly recap and provides spaced-practice by default.

So, what have I done to improve (in my opinion) on Highgate’s scheme? I have put a lot of effort into checking that the topics flow well from year to year, with each new ‘unit’ introducing a similar amount of new material. I have referred to some other schemes in the process:

I used these mostly to give me an idea of what year group pupils usually meet a topic. Jemma’s scheme in particular gave me new ideas of the key points within a topic and helped me to break down some of my topics more carefully. Where I needed more help on this, I also asked for help on twitter. This poll was the culmination of loads of great suggestions I received on teaching HCF and LCM, which led to me introducing a section on algebraic forms of these as a prerequisite to factorising, adding algebraic fractions and more.

Some of my considerations in designing the scheme were:

I don’t want to recover material that pupils have already learned at primary school, so I haven’t included the lessons on basic number and geometry that many secondary schemes do. Instead, I will check pupils’ prior understanding when introducing new topics as part of my planned mastery approach.

I want my scheme to only list each new idea once. Of course, there will be need for review and if necessary, reteaching, but in general, I really dislike schemes which have exactly the same ideas in two different places. I want to know what the pupils should have already learned (even if they haven’t fully learned it!) and what’s new.

I tried to put at least one bit of each ‘topic area’ in each year group. I feel that this is generally a good idea, as pupils have to see a topic each year and so have an opportunity to recap it. It’s also particularly important in my international context, as we have a higher turnover of pupils, so it will allow me to get new pupils up to date with each topic.

I don’t want to accelerate higher-attaining pupils and I don’t want to finish teaching before Easter in year 11, so the ideas are spread evenly throughout the 5 years.

I used specifications to mark the topics which only feature on the higher tier. With a few exceptions, I made sure these topics are in year 10 and 11 so that if I decide to enter pupils for the foundation tier, they will be able to spend more time on the other topics in these final two years.

So, the final product… You can see / download it here:

I’d really welcome any criticism to this as it’s definitely still a work in progress.

In particular, I’m working through the geometry tabs to check that I have listed all the relevant prerequisites. I’m also aware that while the topics are very interleaved, there is little genuine interweaving of them.

In Defence of Detailed Written Plans

I didn’t write a lesson plan for the first 5 years of my teaching career. It was a mistake.

For starters, my claim may be hard to believe for many teachers; how did I get away with such a lax approach? I taught in an independent school in Highgate. No lesson plans were required, observation was frequent but informal and observation grades were an alien concept until I joined the GTP in my 6th year.  The fact that this may surprise many teachers is a sign of a disconnect between the maintained and independent sectors, which manifests itself in many areas.

From my external perspective, the above conversation is funny; Sam now teaches at Highgate and Katharine has (as far as I’m aware) always worked in the state sector. Whilst, to Katharine, no observation grades may have been ‘unthinkable’ 10 years ago, this was exactly what Sam’s independent school was doing then (though it was long before he worked there).

Anyway, back to the main point: I love a debate, and this time I’m taking on two big names: @teachertoolkit and @teacherhead, who have both made a similar point in recent blogs.

Whilst these blogs make some excellent suggestions (click the extracts above to see the whole blogs), they both suggest avoiding written lesson plans. On the contrary, I think we should strongly encourage teachers to write down lesson plans.

A detailed written plan is like a path worn into the hillside.

When I come to teach a topic, what do I have at my disposal? I have years of classroom management experience, an understanding of major misconceptions that pupils have in my subject, but do I remember the finer details of when I last taught the topic two years ago? Probably not. Even if I think I do, my memories are not necessarily accurate.

A friend of mine once explained why, as a classicist, he loves paths: they represent years of accumulated knowledge and experience. As a teacher, my written plans are my paths. When I head in a direction that leads to dead-ends or rocky ground, I backtrack and find a different route. In this way, the best path becomes more worn and so easier to follow over time. As a Mountain Leader, I am capable of navigating away from paths, but when doing so I’m less likely to point out the interesting geological features or flora and fauna along the way.

To be fair to @teachertoolkit, their post suggests that there shouldn’t be an obligation to write lesson plans, which I suppose I agree with. And if you read the detail,@teacherhead says that teachers should have a list of objectives and resources, which is what my ‘lesson plans’ mostly contain. To be precise, I’m not specifically advocating individual lesson plans (though they may be useful for some teachers) but rather ‘topic plans’. However, I don’t believe that ‘lesson planning’ should feature on such lists because this suggests to teachers and managers that written plans are not helpful. For me, if you write down a plan, update it after the lesson to emphasise good bits and delete or alter what went badly, this will reduce your workload in the long-term, as you don’t have to plan afresh every year.

I learnt a lot of ‘big-picture’ ideas in my first 5 years of teaching, but many of the finer details were lost; which examples and tasks drew out the point well and which turned out to be too simple or complicated. I met so many interesting ideas, but many don’t feature in my teaching because I didn’t write them down.

I started to write detailed plans in earnest 4 years ago: a few of them I now deem good enough to warrant sharing online. However, I still look at them and ponder how much clearer the paths would be if I had been walking them for longer.

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?)

Things that are NOT TRUE

A colleague at a previous school produced this poster (it goes on to 29 points: click the picture to download the full version)

If you’ll allow me a brief aside, credit for this poster goes to Robin Bhattacharyya. Robin was the most knowledgeable person I have ever met and at the time, one of my heroes. Originally a University Challenge champion in 1995, he went back for more in 2011 Christmas special, in which he led his college to victory over a more recent hero of mine, Daisy Christodoulou. (strictly, it was Trinity vs Warwick but from memory, it may as well have been Bhattacharyya vs Christodoulou!)

Anyway, back to the maths… A recent tweet from Stuart Price reminded me of this poster.

Stuart was focussing in on one specific type of misunderstanding which features heavily in Robin’s poster: the over- or under-generalising of the distributive property.  Trying to head off these problems early, I’ve been making a specific effort to teach year 8 about distributivity this week. It has gone better than in the past, so I thought I’d share what (I think) I’ve learned.

In the past, the phrase “multiplication is distributive over subtraction” sounds clunky and just saying it, let alone understanding it, has been trouble enough for my pupils. I’ve replaced it with “multiplication distributes over subtraction“. This small change has had a surprisingly large effect. I think this is partly because it relates more closely to the usual English usage of the verb ‘to distribute’; indeed, this also allows for visualisation: the multiplication is literally being distributed over the subtraction.

Secondly, Having checked whether several operations distribute over each other, I asked my pupils to generalise which operations distribute over others. In this section, Colombe introduced a metaphor which helped her peers to remember the general rule: Multiplication and Division are the government: they distribute (resources) to the citizens, Addition and Subtraction but not to themselves. This also ties in very nicely with Order of Operations.

I also used a three-act-math approach in these lessons, introducing the topic with a classic moment from countdown (if you’re a maths teacher, you must have seen this!)

This was my hook: how on earth did he do this? Did you notice that he didn’t actually know the intermediate numbers? This gained interest, and later on in the lessons I tried to set some questions in which pupils applied the distributive property to simplify some expressions, leading up to (318 x 75 – 50) / 25, but it didn’t go well. If I had used the strategy that John Mason suggested on Craig Barton’s podcast, of imagining what I was going to say and how the lesson would proceed, I feel like I may have anticipated this. It wasn’t clear that the equivalent of 318 x 3 – 2 was the ‘final answer’ in the simplification problems I had created. In my updated lesson plan, I have removed these questions, leaving just the explanation of the countdown genius as a teacher-led section of the lesson.

Another issue that came up as part of these lessons was a good question from my pupils: If it doesn’t distribute, what does happen? My answer at the time was fluffy… something like “it depends on the situation, usually you just apply the operation to one of the numbers”. I feel like I need to work on this, perhaps I need to add another sequence of lessons to my combining operations thread.

Overall though, I feel that this is the first time I’ve been even vaguely successful when teaching the distributive property, mostly because I managed to distill it, with the help of a pupil’s metaphor, to this:

One final reason why I think it went better than in the past: I was more committed. Previously, I would always question the value of teaching such lessons, because I suspected that the pupil’s future teachers wouldn’t make reference to the distributive property. Now I know that I will be teaching these pupils for several years and as HoD, I can better integrate this topic into our scheme of work, it is worth the investment.