I’m in the middle (well, quite near the beginning really) of a specialist maths primary PGCE at the University of Worcester. This means we get all the usual primary stuff plus some extra “special maths”. There are some differences in how our school experience works but there is a definite discrete bit of learning we have over and above the usual. Our first little block of this culminated in a session on “What every maths lesson should include” which we had to research, present and discuss. This was a really useful exercise for me at least so I thought it was worth sharing some of our discussion.

I was working with fellow trainee and fellow Martin, Martin Zoeller. We started with the university’s primary maths teaching model: “The Prime Five”.

This is:

- Talk & discussion – maths lessons should not be quiet places. The conversations around number and strategies and so on are an integral part of deep learning.
- Questioning – asking questions that encourage thinking and expose the pupil’s mental model are vital.
- Misconceptions – following on from questioning, it is important to test potential misconceptions and address them immediately.
- Concrete-Pictorial-Abstract – if a child is to understand maths rather than just know it they have to be able to model the concepts. CPA does this and absolutely should be a feature of every lesson right up to Y6 (and beyond?).
- Problem solving & enquiry – using maths to solve a problem is inherently more motivating and makes the maths far more contextually relevant.

Martin began our presentation with a game, maybe more of a magic trick. The game is that someone chooses a number from 1-64 and then identifies if that number is on a set of 6 slides. From their answers we could work their number out quite easily to gasps of awe and wonder. Well, maybe a slightly forrowed brow.

The point of this was to show the power of intrigue. In a maths lesson if, instead of presenting maths for the sake of itself, I challenge or puzzle is presented that happens to require those skills, the outcome is a far more powerful and effective lesson. An example might be in teaching addition of two-digit numbers. Instead of a worksheet with a series of calculations, maybe including boxes of cakes and the like, a puzzle can be presented. My current favourite is a task I found on the NRICH website. In this task the story of Edward Benbow – who lived in my town of Bewdley in Worcestershire – is given. Mr Benbow apparently once held the record for the longest palindrome (text or numbers that read the same forwards as well as backwards). The task goes on to explain that if a two digit number is added to itself with the digits reversed you get a palindromic number. The challenge is given to find if there are any exceptions and why or why not? What about longer numbers? In the process of working with this task pupils will work far harder, solve far more calculations and get a far deeper understanding of the underlying maths. They might also enjoy it a damned sight more! We argued that while practice is important, there should also be an element of play and these open enquiry, low barrier high ceiling tasks are fantastic for this.

Another factor in maths lessons is for them to be multi-modal. We understand the world in a lot of different ways and to understand anything at depth we need to have multiple perspectives on it. Open tasks help enormously in lookin from multiple perspectives as does talk in general. This is where CPA comes in as there is a significant mental stretch from “sharing 12 sweets equally between 4 people” to 12 /4=_. By modelling operations in a concrete fashion (counters and so on), leading to pictorial representations such as number-lines a bridge from a physical, multisensory understanding into an abstract understanding of the operation. In a similar vein, any active maths will always be memorable from sheer novelty (which isn’t a bad thing) but will lead to a strong link to the movement involved. PE-maths could well be the way forwards!

Another really key thing is to link bits of learning together. We looked at an article by Skemp which talked about relational vs instrumental understanding. This is essentially the difference between understanding why a process works and how to do it. There is value in having processes – written methods for example are fantastic shortcuts. But, if the methods aren’t understood, the knowledge is fragile and where an error or exception occurs there is little that can be done to overcome this. If mathematical concepts are understood in relation to one another there is a far greater chance of being able to puzzle something out. Those dastardly written questions in SATs and GCSE papers are good examples of this. Something I think is really valuable in terms of connecting ideas together is the process of synthesis. Many a teacher would happily say that they only really understood something having taught it. This is because in order to teach something from several angles, it has to be understood from several angles and in relation to everything previous. I suppose that leads on to the idea of linking lessons together by starting with a recap, as happens in all good phonics lessons, and by ending with an element of pre-teaching to bridge to the next lesson. We talked quite a bit about pre-teaching and how this might help. The idea really is to plant a seed in the pupils’ minds so by the time the next lesson has come around, they already have some understanding to hang ideas on. It would be a bit like how you can rack your brains all day for some fact then wake up in the middle of the night with the answer. There is plenty of research that shows that processing – or latent learning – happens so there’s even some research behind the idea. An example might be if for example you have been looking at multiples and are going on to prime numbers, you might end with a question of what is special numbers that only have one factor pair and if there’s a pattern. Hopefully they will puzzle over it but even if they don’t there is a bridge between lessons that will help connect lessons and concepts together.

We had quite a lot of talk about whether all this connected understanding is actually valuable and I had what I thought was a good example. My son – now in Y8 – has started telling me he hates maths. He had a particular homework on multiplying out of brackets in algebra which was causing a fair amount of consternation. The main issue for him was that he could follow the rules – instrumental understanding – but once things got a bit complicated he was stuck. I gave him a pictorial of two rectangles on top of one another, both the width of the outside term and each with a height of one of the inside terms. After a couple of goes he got the idea that he was looking at the area of two rectangles summed rather than one composite shape. This lightbulb moment was followed by some furious calculation and a big grin! Not only did he now enjoy these puzzles (puzzles is a good maths word!) I can guarantee that he will be a lot more likely to still be able to solve these on another day.

The last category of requirements for a lesson were actual hands-on teaching skills. In many ways, all the things I’ve described so far have been planning type things. As well as all this it’s really important to use open questions to stretch pupils but to also identify misconceptions. Misconceptions should be directly challenged and any lesson should be inclusive and include elements such as AfL. With a conceptually rich subject like maths it’s generally a good idea to keep probing away at pupils’ understanding so continual querying and the odd mini-plenary can really help to knit everything together.

That seems like an awful lot to get into a lesson, let alone every lesson but it is far more about an overall approach than a checklist. Maybe I’ve missed some things and maybe I’ve overstated some others but it feels like a reasonable approach to me. I’d be curious to hear what people would add or take away and I have really no handle on secondary – should a secondary lesson include all these things too?