17 October 2021

Curriculum Sequencing

Yesterday I presented a session on curriculum sequencing at #mathsconf27 in Ashford. Thank you to everyone who attended. In this post I will summarise the main points and provide a link to the slides.

Prescribed Content
I started by talking about the national curriculum. It's statutory for all local-authority-maintained schools in England to teach the Department for Education's programmes of study. In reality the vast majority of non-local-authority-maintained schools (e.g. academies) teach them too. I talked about how heavily prescribed the content is for maths. With the exception of science (which is also heavily prescribed), all other subjects have a fair amount of creative scope over the content they teach. If you're a Head of English you might choose to teach Macbeth, and if you're a Head of History you might choose to teach the Black Death. If you're a Head of Maths you have pretty much no say over what you put on your school's curriculum (in terms of the content), other than perhaps a bit of enrichment that goes beyond the national curriculum. 

But that's not to say we have no control over curriculum at all. We can control the sequencing of the curriculum and we can control our pedagogy, resources, methods and approaches. Basically we can control the implementation of the curriculum. This implementation varies widely by school.

I talked about how odd it is that we don't all follow the same sequence, and that there is no agreed 'best order' for the topics we teach. In some countries (I talked about a specific example from Shanghai) a great deal of research is done on getting the order right, down to the minutia of 'is it more effective to introduce area first, or perimeter first?'. Yet in England our decisions about curriculum sequencing (which are typically made in isolation by each Head of Maths in over three thousand different secondary schools) are normally not research-informed. I also talked about the massively different approaches to curriculum sequencing seen in the United States, where students spend an entire year on algebra and an entire year on geometry. To us this seems unusual, and to them our approach seems unusual. I shared some images from Ben Orlin's very funny post about this. And the question I asked is, 'Has anyone actually researched which of these approaches is more effective? Which approach optimises student experience and progress?'. 

I talked about why Heads of Maths and teachers should be interested in curriculum sequencing. It's not just because Ofsted might ask them about it! I also talked about an interesting point made by White Rose in their post Order, Order! The Importance of Sequencing - although our sequencing decisions are mainly determined by prerequisites (i.e. you can't study this topic until you've done that one), there are also considerations relating to student experience.

After this long introduction about why we should think about curriculum sequencing, the rest of my presentation was broken down into three parts: prerequisites, interweaving and common practice. 

Teachers should think about the prerequisites for every topic they teach. That goes without saying. When I start teaching Pythagoras, the first thing I should do is ask myself the question 'what maths do my students need to know to access this topic?'. This is done by Heads of Maths when creating their scheme of work, and it's also done by teachers when planning their own lesson sequences. Prerequisites have a large part to play in the order of our curriculum. For example I wouldn't put area of a circle as the first topic in Year 7, because in order to access that topic I need to first ensure that my students are fluent in rounding, calculator use and squaring. 

I reflected on the fact that students at my school do rounding in January of Year 7 but then don't actually use rounding in any topic until January of Year 8. So perhaps it makes sense to move rounding to Year 8, when it can be taught and then immediately used in topics like Pythagoras, circles and volume. 

I showed examples of topics that can be interwoven and how the order of our curriculum can create opportunities for interweaving. For example, I think it's really important that angles should follow equations in Year 7. If angles is done first, it can end up being a repeat of primary school angles. But if angles is taught second then teachers can make the most of opportunities to interweave equations and angles. This adds depth and challenge to the topic of angles, as well as giving students the opportunity to make use of their newly acquired equation solving skills.

I gave the example of how I recently taught index laws to Year 8 and then expanding brackets with the same class, so was able to make use of index laws within my lessons on expanding brackets. It's important that students understand that each bit of maths they learn takes them to the next step, and allows them to access more complex maths. We don't need to answer 'Why are we doing this?' questions from students with tenuous real-life application nonsense. "All roads lead to calculus", and they are on a journey heading in that direction. 

Common Practice
In the last part of my workshop I shared examples of schemes of work. Even though we all teach the same curriculum, there is very little consistency in curriculum sequencing across schools. And even though Heads of Maths put a lot of time into designing their schemes of work, when you look at all the different orders side-by-side, it almost looks random.  

In my recent survey of almost 800 Key Stage Three maths teachers, I found huge inconsistency in when topics are taught. Look for example at the topics with few prerequisites (shape transformations and constructions) which are spread across Year 7, 8 and 9.

I talked about the idea behind spiralling in order to build depth of knowledge. I feel like spiral curriculums are sometimes seen as the opposite of mastery curriculums, but that's not necessarily the case. The worst kind of spiral curriculum touches on topics at a surface level and returns to them every year. Things are rushed rather than mastered, meaning each year it's like starting from scratch again. A better form of spiral curriculum takes a strand of maths in its entirety (e.g. angle geometry) and each topic within that strand adds a piece to the jigsaw, building on prior knowledge and deepening understanding of the strand as a whole. Angles could be taken as five topics: 'basics', parallel lines, polygons, bearings and circle theorems. Each year we teach one of these topics, and each time we do so we not only add a new piece to the jigsaw but also build depth of understanding of the strand as a whole.

The approach taken in the new DfE/NCETM curriculum sequence seems to differ to this. This has challenged my thinking, and I would love to hear the NCETM team speak about the rationale for their proposed order. I know the team behind this are very experienced and knowledgeable and will have given it a lot of thought, so this deserves our attention.

In the last part of my presentation I shared my own five year curriculum sequence. I made it as an example for the purpose of this presentation, really just to see how difficult it was to do, and it's therefore simply based on my own thoughts and experience. The main challenge was time: we just don't have enough of it. It's frustrating. I've said it for years and I'll say it again: we simply have too much prescribed content in our national curriculum. If we are going to have any hope of teaching this content properly - actually in depth, without just skimming the surface of it all - then the content needs to be reduced. They could start by removing constructions... 🙃

I ended with the message that curriculum sequencing is fascinating, and it's something maths teachers should think about and talk about.

So that's a brief summary of my talk. There was a lot more in it, but that's the gist. I hope that people who attended found it helpful.

I spoke more about all this in my recent podcast with Craig Barton. 

You can download the slides from my workshop for use with your own department. 


Finally, I just want to say a huge thank you to La Salle for organising this conference. As much as I like the virtual conferences, they are nothing compared to the in-person ones. Being in a room full of maths teachers and having impromptu conversations about schools and teaching is just so powerful. I enjoyed all the sessions I attended, and I think Kris Boulton's workshop was one of the best I have ever been to. And given that I have been going to La Salle's conferences since 2014, that's saying something. 

I also want to say thank you to those who joined me at my drinks on Friday night. At work on Friday I was stressing that I'd booked a table for ten people but might end up sitting at it all by myself, but in the end I was joined by Chris, David, Sudeep, Nathan, Nathalie and Rachel, and then later in the evening by many more conference-goers. I ended up dancing in a club until the early hours. I can count on one hand the number of times I've been out dancing like that since I became a mum a decade ago! It was so much fun. After my first big night out in my forties, I was a bit worse for wear at the conference - but it was worth it. 

See you all at the next one.

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