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. 

Prerequisites
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. 

Interweaving

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.

5 Maths Gems #148

Welcome to my 148^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Distance Time Game

Thank you to @MrChapmanMaths for sharing this Graph Game from @davidwees. Best played on a computer (not a phone), this game is absolutely brilliant for developing understanding of distance-time graphs. It's really fun.

This reminds me of an activity I did on my PGCE where we walked/ran across a room trying to track distance time graphs. That was fun too, but required specialist equipment that I haven't seen since. 

The Graph Game led me to more content from David (listed here) including this excellent Factors Game. I played this with my daughter and we were quickly deep in discussion about factors and primes, and practising division. What a great game!

2. MCQ Generator
@mrshawthorne7 has created a dynamic multiple choice question generator. Select a topic and it generates random questions that can be used to identify and diagnose misconceptions. You can choose to hide the choices initially, to encourage students to do some thinking before they see the possible answers.

3. Percentages Task

I like this task shared by @MrsEVCartwright. Students are shown the working, and have to work out what the question might have been. 

4. Factorising Task

Here's another nice task, this time from @canning_mrmaths. It's an Open Middle task on factorising quadratics.

5. Probability Spinners

I recently presented a CPD session on probability where I talked about how spinners make an excellent fuss-free teaching tool. Following on from this, the team at MathsPad have created a Probability Spinners Interactive Tool which is free to use. It has sections on finding probabilities as fractions, decimals or percentages, expectations of frequency of outcomes after a given number of spins, and the results of repeated trials, including a bar chart representation and a relative frequency table. 

Subscribers can also access a Probability Trees - Spinners Interactive Tool. MathsPad's interactive tools are always excellent - you can see their full collection here. 

It's also great to see that MathsPad's new range of curriculum booklets is expanding, with the recent addition of an Expressions booklet for Year 8.

Update
My last blog post about resource design went down well. If you missed it, you can catch up here.

I recently recorded a podcast with Craig Barton where we chatted about teaching for depth and curriculum design. You can listen here.

I've spent the last week dealing with hundreds of access requests for the resources I store on Google Drive. This is because Google did a security update which broke all my links (thanks Google!). Links have to be fixed individually unfortunately - every evening I come home from work to find dozens of emails from people trying to access resources, at which point I fix those particular links. I have a feeling this is going to continue for months until they are all fixed! Apologies if you click on a broken link on my blog at any point. I'm working on it!

Are you going to the upcoming maths conference? I'm really looking forward to an in-person conference. It will be good to catch up with everyone. It's happening on 16th October in Kent and you can get a ticket here. I'll be speaking about curriculum sequencing in Period 4. 

Here are a few other things you might have missed:

• @jemmaths published a video 'Line Managing the Mathematics Department' which is highly recommended if you're a Head of Maths.
• @andyparkinson31 has indexed five hundred Numberphile videos by curriculum area and shared this resource on TES.
• The NCETM published an excellent set of subject knowledge audits which will be incredibly helpful for ITE providers.

Last week saw the launch of the DfE's Key Stage 3 Maths Guidance, which was written by the secondary team at the NCETM. This guidance suggests an ordering of the Key Stage 3 curriculum. I was surprised to see this! It links closely with the workshop on curriculum sequencing I'm running at mathsconf in two weeks.

The guidance provides valuable material which can be used in department meetings to help teachers prepare to teach each topic on the curriculum. For each area of maths, there are sections exemplifying the key mathematical ideas. These sections feature information on the common difficulties and misconceptions and suggest questioning and other strategies for teachers to use.

My school held Open Morning yesterday. Limited tickets plus a very clever one-way self-guided tour worked really well to eliminate any crowding. I am fortunate to work with a brilliant team of mathematicians - they are pictured below (shout out to Mariam who was off sick so missed the photo!). We will have Year 11 for the first time next year and we'll be recruiting. If you want to join our team, look out for a job advert in the Spring Term. 

I'll leave you with this units meme from Reddit which I first saw shared by @MrYoungMaths. I showed my Year 9s, who looked at me with blank faces while I chuckled away...

Worksheet Upgrades

I was going to call this blog post ‘pimp my worksheet’ but I realise then unless you’re familiar with MTV programmes from the mid-noughties, you won’t get the reference and you’ll think I’m just weird.

The point I want to make in this post is a simple one:

Instead of spending ages hunting for challenging activities, you may be able to quickly and easily increase the challenge level of an existing exercise.

Take this example, created last week by my excellent colleague Sarah. She made this for her top set Year 10 class. I borrowed it. I like that it starts with scaffolding and, with a simple tweak to the question style, ends with challenge.

At both the White Rose Secondary Maths Brunch in January and the MA Conference in April, I talked about the use of ‘working backwards’ in tasks to get students thinking. I shared examples of tasks which feature elements of working backwards, such as 'fill in the gaps’ activities. Here's an example from Kyle Gillies. 

And here's an example from Andy Lutwyche.

You can read more about tasks like this in Paul Rowlandson's excellent blog post "This Way, That Way, Forwards and Backwards".

After Sarah shared her resource with my department last week, I realised that it’s pretty simple to convert a standard drill exercise into a ‘working backwards’ task that requires more thinking. It’s just a case of editing a few questions.

Here’s an example that I used yesterday in Year 10's first lesson on surds. Notice how they’re not just simplifying, they’re also ‘unsimplifying’ which I think might help build depth of knowledge.

It's a simple idea to combine scaffolding and challenge in a single task. Many resources like this already exist, but it was probably quicker for me to make my own rather than search for one. It took only a few minutes to create.

Note to new teachers: once you become a pro at using equation editor shortcuts, it's super quick and easy. Press the alt button and equals at the same time in Word or PowerPoint to insert an equation, then follow this guide from Jamie Frost. You'll get the hang of it quickly and won't have to keep referring to the guide.

My lessons often combine short 'drill' exercises (i.e. practise a procedure) and longer 'thinking' tasks (i.e. use reasoning). The former normally come from screenshotting from Corbett Maths or CIMT. The latter are typically MathsPad or Don Steward. I draw on many other sources, normally going via my own resource libraries to save time. For example, in my next surds lesson I've taken an extract from an old Solomon worksheet that I found in my surds listings.

Note to new teachers: screenshot using the snipping tool to easily insert tasks into your lessons. You don't need to write exercises from scratch when they already exist.

Incidentally, in the same Solomon worksheet I spotted these questions:

I will be using these with Year 8 in a few weeks! Their first topic of the year is index laws, and then they move onto expanding double brackets. This task neatly combines the two topics. A lovely bit of interweaving.

Anyway, I've digressed. Back to my original point:  

You can add more challenge to a straightforward exercise by tweaking it slightly. Blank out some of the questions and provide the answers instead. Getting students to work backwards makes a task less procedural and gives opportunities for reasoning ("if this is how it worked going forward, how must it work going backwards?"). Even if it's just a tiny tweak it may be a worthwhile one, to avoid mechanical repetition and make students pause and think.

 

5 Maths Gems #147

Welcome to my 147^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Dr Austin's Resources 
Amanda Austin (@draustinmaths) has been updating her website draustinmaths.com over summer. She's added a number of new resources - see her Twitter feed for the highlights.

I've started to add Amanda's resources to my resource libraries, to save teachers time when planning lessons.

2. PolyPad
Mathigon's awesome interactive tool PolyPad has a load of new features including lots of cool stuff for both 3D geometry and statistics.

Big news: Polypad now supports 3D geometry! 🎉⚽️🎁 Use one of our built-in solids, or create your own net using polygon tiles. Then click “fold” and watch the magic happen… pic.twitter.com/lMeBZbvGSo

— Mathigon (@MathigonOrg) August 2, 2021

The clocks are new too:

The latest components added to Polypad are clocks! ⏰ If you want to learn more, sign up for our upcoming #BackToSchool workshops at https://t.co/L3QuGWHc3k pic.twitter.com/2Yc3JtZT9l

— Mathigon (@MathigonOrg) August 16, 2021

I love Mathigon and have blogged about it many times. Have you seen the Almanac of Interesting Numbers?

And this awesome Sieve of of Eratosthenes? This is brilliant for teaching students about prime numbers.

The new number grids on Polypad make it really easy to find prime numbers using the Sieve of Eratosthenes. You can colour or resize them quickly, and we’ve also added similar addition and multiplication tables! https://t.co/HDEd7BNKaK pic.twitter.com/NFqKjPZSLd

— Mathigon (@MathigonOrg) August 20, 2021

Great stuff from Mathigon.

3. Ofsted Maths Research Review
If you haven't had time to read the Ofsted Maths Research Review yet, or if you've read it and want to share the key messages with your department, then you might find this useful: George Stone (@DrStoneMaths) has shared a summary of the main points here.

George's PowerPoint includes reference to the criticisms of Ofsted's review. For more on this, it's worth reading the response of the Joint ATM and MA Primary Group here which explains some of the criticisms in detail.

4. MathsDIY
Thanks to Victoria Jennifer (@vics_jennifer) for tweeting about a website I'd not heard of. On mathsdiy.com, an experienced maths teacher shares GCSE Maths and A Level Maths past paper solutions, topic booklets and resources. Questions are drawn from WJEC papers. 

5. Guided Reading
Thanks to Jenny Hill-Parker (@JennyHillParker) for sharing a set of guided reading resources here.

Guided reading is a really interesting idea - check out my Gem Awards 2021 for more on this.

Update

Did you catch my last gems post? I published it in late July. In previous years I've done a lot of blogging over the summer holidays but this year I've just not had time, so two gems posts will have to do. I did manage to get away with my family for a couple of weeks though, which was lovely. The highlight for me was visiting Woolsthorpe Manor and sitting under Isaac Newton's apple tree. It's really delightful there - I highly recommend a visit.

I also had a lovely evening at Jamie Frost's maths teacher drinks. Jamie used to host these events three times a year but lockdown put a stop to it. It was so nice to see everyone again.

 

I'm looking forward to the return of conferences over the next few weeks. I find that going to a conference right at the start of a new academic year is a great way to get excited and inspired for the year ahead. First up is #mathsconfmini2 next Friday night. I'll be speaking about probability, and also more generally about reducing cognitive load.

Then on 4th September it's the national researchED conference which, incredibly, is an actual in-person event in London. I'm really looking forward to the pre-conference night out! I'm delivering a session at the conference called 'Skimming the surface of the maths curriculum'. Here's the blurb:

In this session we will look at the maths curriculum, considering both the content and the way it is delivered. We'll analyse research findings and discuss whether the practice of 'skimming the surface' of maths is widespread in schools. Are students accelerated onto new topics too quickly? Is adequate time provided to teach topics properly? Do teachers know how to teach topics in depth? And how can we do things differently?

If you're coming to the conference, do come along to my session - I have some interesting insights to share from the Key Stage 3 survey I did recently. Unlike all my other workshops over the last eighteen months, this one won't be recorded!

If you're preparing to return to school in a couple of weeks, here are two pages you might find helpful:

• Year 7 Maths Activities
• Displays

And finally, just a reminder to those of you who subscribe to my blog posts by email: the emails will now look a bit different so please don't be alarmed! I had to swap the subscriptions over to a new provider (because Google stopped its email subscription service), and I managed to lose a lot of subscribers along the way. If you want to receive my blog posts by email, you can sign up here.

I'll leave you with this nice little problem from @BerrySlime3 which would be suitable for Higher GCSE students. Hat tip to @blatherwick_sam for sharing it.

5 Maths Gems #146

Welcome to my 146^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Pythagoras 
Teaching Pythagoras was one of the highlights of the year for me. What an awesome topic. Thank you to @giftedHKO for sharing some lovely Pythagoras tasks on her blog - I look forward to using these next year.

In case you missed it, do check out @Mr_Rowlandson's recent blog post Playing With Pythagoras and Trigonometry which features lots of clever ideas.

2. CIMT Textbooks

I've made heavy use of CIMT textbooks since my PGCE. I own physical copies but when planning lessons I usually download the CIMT resources from TES, where they are in neat bundles along with the answers. The challenge is working out which bundle to use. Though they are organised by topic, sometimes it would be helpful to be able to search through all the resources. Thankfully, @nathanday314 has come to the rescue by merging all the CIMT practice books into one searchable, fully-indexed, 2100 page PDF. Thanks Nathan!

Well, that's today's project complete...
All the CIMT practice books merged into one searchable, fully-indexed, 2100 page PDF. pic.twitter.com/ut7HCvyFI3

— Nathan Day (@nathanday314) July 4, 2021

3. Surds

After a couple of years of only teaching Key Stage 3, my school has finally grown to the point of having GCSE classes! Hurrah. We'll be starting Year 10 with surds, and I like this introduction from @JaggersMaths.

For more ideas for teaching surds, checking out my surds in depth workshop.


4. Problems

Thank you to @SarahFarrellKS2 for sharing a PDF document of her first hundred #dailymathspuzzle questions. These are designed for Key Stage 2 students but would be suitable for many Key Stage 3 students too.

I've added this link to my problem solving page which features collections of problems and puzzles.

5. Powers and Roots
I'm always a fan of @MrDraperMaths's blog posts. His latest post focuses on powers and roots and is packed full of great tasks.

Update
Term ended on Friday for me and I now have five weeks to recover (and, sadly, catch up on the vast amount of student timetabling work I didn't get time to do before the end of term). This year was unbelievably brutal. To those of you who contacted me through Twitter or email in the summer term and didn't get a reply: I'm incredibly sorry. My role got totally out of control for a while there, and I failed to keep on top of anything outside of work. I'm hopeful that things will improve next year. 

In case you missed it, do check out Jemma Sherwood's (@jemmaths) excellent post 'Consistency and Autonomy', which follows on from some ideas I wrote about in a blog post back in May.

Also have a read of Anne Watson's latest post in the Dose of Don series. This post explores Don's thinking around angle bisection. 

If you're looking to spruce up your classroom or corridors this summer, you'll find loads to choose from on my displays page.

I can't wait to get my own classroom back. I'm sure many teachers feel the same. On the first day of the holidays I took my daughters into school (always a treat for them!) and they helped me clear out all the rubbish left in there by the various nomadic teachers who used it over the course of the year. I'm much happier now it's all sorted. I'm not fussy about pretty displays but I do like a splash of colour to brighten up a dreary room, so I used Sarah Carter's maths symbols display to cover a blank board (no backing, borders or laminating so it was a five minute job - and I'm not bothered that it's a bit wonky!). I've moved the desks away from the wall to stop students ruining it.

Symbols display

After an exhausting year which has left me feeling totally defeated, spending an hour preparing my classroom has cheered me up and made me look forward to September.

Nothing special, but I can't wait to get my classroom back!

Thank you to Kyle Evans for sending me a copy of his new book Maths Tricks to Blow Your Mind. My daughters are doing a swimming crash course this week and this makes perfect reading while I'm waiting for them by the pool!

I'm off on holiday with my family on Monday, and I will do my very best to switch off completely for two weeks. Have a lovely summer everyone!