The Disciples of Highgate

Of late, I have been having some very productive conversations with Jen Brewin, the author of the last post on my blog. She recently mentioned a nice analogy that I have decided is worth sharing, especially because it goes some way to explaining my scheme of work, and some of Jen’s blog post comments.

We both taught at Highgate School and although we only overlapped by one year, we both experienced the same ethos. In Jen’s words:

…as a teacher in this department, I was being given a great responsibility: to join in the telling of a story. This story was one that all of us were in the process of revealing to our pupils, and it was one that was revealed to them over the course of their five (or seven) years at the school.

Everyone who has taught maths at Highgate (or Westminster, where it arguably began… that’s for another time) knows the story to some extent. The extent is limited because it took many years to learn the story, and not everyone stayed for long enough to pick up the whole tale. This is partly because of the way in which it was passed on: meetings between new and experienced members of the department.

There is a written scheme, but it sadly doesn’t reflect the true brilliance of some of the ideas and leaves a lot to the imagination. So I was talking – complaining – to Jen about how it will take me so long to write up the details in order to fully explain the current spreadsheet list of topics with brief descriptions.  Jen said that she feels similarly, and introduced an analogy which I rather enjoyed: a long time ago there were oral tales of a great story, but they weren’t written down until years later by the disciples. As a devout Atheist, I’m not sure why this appeals to me: perhaps because it rather aggrandises the impact of my life’s work! Anyway, as I am aptly named Luke, here is a sneak preview of a small part of my gospel:

This particular section of the scheme of work exemplifies really well another point from Jen’s blog:

The discovery that a highly selective private school would pay so much close attention to the sequencing of teaching was something of a surprise to me.

You may have been surprised that calculators were not introduced in either of the first two occasions on which we teach trigonometry. Perhaps you found it unusual to only introduce tangent first time around; what about sine and cos?

Why these unusual features? This scheme carefully takes into account the limits of children’s working memory and doesn’t try to introduce too many new ideas at once. The gaps between these topics allow time for consolidation and retrieval practice of each new element, before the next one is introduced.

Jen and I discussed how the really funny thing is that it is Highgate and Westminster that are breaking down the content into these small chunks. These are two of the most academically selective, high-achieving schools in the country. If anywhere could get away with introducing many new ideas at once, these schools could. And yet they do not.

Secondly, there is also probably a lot more detail than you’re accustomed to in schemes: for example, the unusual approach of writing the trigonometric functions as scale factors rather than ratios. This fits in beautifully with other parts of the scheme of work: percentages, proportion, similar triangles to form a thread which link different parts of the story together.

When teachers first join the department they often find the specificity of such approaches disconcerting, especially when they are quite different from what they may have done before. But with time, they usually come to realise the beauty of the links, and appreciate the luxury of teaching pupils who have been introduced to mathematics in a familiar manner. If they still don’t like it, they can argue that the department should change: indeed, many department meetings were dedicated to discussing threads from the scheme of work. There are definitely similarities between this approach and that of the departments of Danni Quinn and Greg Ashman.

So, I’ll be writing down the rest of my gospel over the next year or so, as I build my own department around it in Geneva, whilst Jen gradually introduces her version of its ideas to her school in York and there are others out there who know it too. But one big question remains: If Jen and I are some of the disciples, who is Jesus?

Guest Post: The madness of writing schemes of work (or how I fell in love with curriculum design)

This is a guest post by a friend and former colleague, Jen Brewin, who is currently Head of Mathematics at a comprehensive school in York. I whole-heartedly agree with it.

On 16th October, Mark McCourt posted the following tweets.

Various educators joined in his lament.

Based on the numerous responses to Mark’s question there seems to be a general sense of frustration about this in our profession. Why do people keep asking us to redo this? There’s barely enough time to plan and teach and mark, so why does any school bother wasting time just shuffling things around? After all, surely whether you teach plans and elevations in year 8 or year 9 doesn’t really make a difference does it? And aren’t all schemes basically the same anyway: a glorified re-ordering of the GCSE specification with perhaps a few “must, could, should” thrown in here and there?

Five years ago I think I might have agreed with these frustrations. My experiences in departments up to that point had been that schemes of learning served little purpose other than to make sure all the prescribed content was covered by the time the exams hit. Within that, teachers were largely left responsible for their own medium-term planning, and since my students generally didn’t seem to have any coherent sense what they had learned in the preceding years anyway, what the SoW said they had covered wasn’t really all that helpful in any case.

And in the end it didn’t really matter: year 11 was characterised by a mad dash back through all of the material. Gaps needed to be addressed here, there and everywhere, and since different pupils remembered different things and with only finite lesson time left, it often came down to whether or not they were able to make good use of their revision guides. In hindsight, I had perhaps dismissed the importance of a scheme of work because, really, I’d never actually experienced a real one.

Then in 2012 I joined a department that showed me just what a scheme of work really can be. The discovery that a highly selective private school would pay so much close attention to the sequencing of teaching was something of a surprise to me. I had naively assumed that in an independent school, teachers (some extremely well-qualified mathematicians) would be left to their own devices much more than in the state school setting I was used to. And given the aptitude of its intake, I’d imagined that there would be much less concern about pupils missing out on bits of content here and there – after all they would probably figure it out for themselves.

What I realised very quickly however was quite the opposite: that as a teacher in this department, I was being given a great responsibility: to join in the telling of a story. This story was one that all of us were in the process of revealing to our pupils, and it was one that was revealed to them over the course of their five (or seven) years at the school.

It was a complex novel, with multiple, overlapping themes and carefully constructed characters. The introduction of each strand of the story had been pondered over, and the groundwork for each new concept had been established carefully in earlier chapters, well in advance of its exposition. For every student to be able to follow this story it was essential that each teacher knew exactly what part of the story they were responsible for passing on. The narrative was communicated in great detail, and students knew it.

This story was not static but as a department we rewrote it, collaboratively, over the years. We enjoyed debates about whether this idea would really fit better here, and whether this theme might be better understood with this other introduction. It was a joy and a privilege to play a small role in the telling and the rewriting of a great story.

This might sound all very flowery, and I suppose it is, but I cannot think of a better analogy for a scheme of learning. Unfortunately, the “story” that is so often passed on to pupils is far from coherent. Learners hear the same sections over and over again, yet they are expected to recall something vaguely mentioned many years earlier in order to make sense of a new theme. Some parts are told without the necessary back-story for them to make sense of it, and certain vital sections might just have been missed out entirely. It’s hard to blame someone for not following the story if the storyteller is essentially a rambling old uncle.

So when I say I have fallen in love with curriculum design, what I am talking about is the careful construction of a coherent narrative: one which respects the limitations of learners’ working memories; one which accounts for the inevitability of forgetting; one which establishes high expectations of the learner’s ability to think logically and make connections for themselves without leaving to chance whether they have all the necessary skills and background knowledge to do so.

When teachers say “stop re-inventing the wheel” I want to point at this:

This is apparently what we got the first time we invented a wheel, around 3500BC. It’s a damn good thing it was “re-invented”, and that it continues to be re-invented because there are always better wheels to be made and different things which need wheels. You can’t put that block of stone on a Ferrari or a skateboard or a chair. Luckily there are people who like re-inventing wheels, they are really good at it, and as technology develops they are doing it all the time and in all sorts of contexts.

Cognitive research has come on leaps and bounds in recent years. Educators and psychologists alike have a much better understanding of what we can do to help our students to remember and understand what we want them to learn. If this knowledge isn’t embedded in the structure of our schemes of work – if it doesn’t define (or even inform) the narrative we tell them – then our schemes of work are not fit for purpose.

Ask yourselves these questions about your scheme:

  • Does it cover the entire time the learner is going to be at the school? (I see no value in a division between key stages, for example)
  • Does it ensure the teachers know exactly what content learners should meet at what stage?
  • Does it ensure a consistent approach to vocabulary, models and representations so that pupils have a coherent experience when moving between teachers?
  • Is the sequencing of content logical (mathematically, pedagogically sound)?
  • Are ideas revisited and developed at intervals that minimise the need to re-teach?
  • Is enough time devoted to establishing the most fundamental skills and models which underpin successful teaching of the content?
  • Are review, revision and assessment built into the scheme at intervals that encourage and enable learners to develop fluency and embed knowledge and skills in their long-term memory?
  • Does the scheme demonstrate an expectation that pupils will have learned what they have previously been taught?

The point Mark made of course (that it is a time-consuming and therefore expensive exercise) remains valid. It takes great mathematical expertise and many years of iteration to go from nothing to a successful scheme of learning which really does satisfy the criteria above. But now that I am running a department, I consider this to be one of the single most important things I can establish in my role, and I am excited and daunted by the challenge. However, I know it can be done because I have seen it, and I’m not starting from scratch.

Most fundamentally I disagree with the claim that we should stop talking about what we teach and instead talk about how we teach. What we teach and how we teach cannot be separated. The starting point for successful teaching has to be the understanding of where students are now and where we are trying to get to. It’s imperative not only that schools have the best scheme of work they can, but that teachers engage with it critically. A colleague of mine at the independent school said “I get so frustrated by how schemes just degrade over time”. I think she’s right, they do, but I know there are teachers like me who find the challenge of reimagining, tweaking and improving a scheme one of the most interesting and satisfying aspects of teaching.

If you have read my posts about leaving Highgate, or my own scheme of work, you will understand why I feel the same was as Jen. She has summarised my thoughts much more clearly than I ever could.