Charlie Harber is Deputy Lead Adviser for Primary Mathematics at Herts for Learning. This blog aims to take the seemingly simple operation of addition and demonstrate how we can vary presentation in order that pupils see connections and do not fall into shallow procedural thinking.
Would you agree that these are all shades of grey? (Who hasn’t spent time deliberating between shades on a paint chart?) They are all different. So how can they be all grey?
We are generalising what is meant by GREY. Oxford online dictionary define grey as ‘Of a colour intermediate between black and white’ – so under this definition all these colours, despite being different, are grey because they share this similar property. For children to understand what grey is, what would you do? Would you just present them with a single shade? More likely, one activity you would do is present them with a range of objects in lots of different colours including shades of grey and ask them to sort the objects based on their colours. In mathematics we could refer to this as part of generalisation – the ability to see what is common amongst a range of different situations.
Variation theory has for a long time now demanded my attention, and rightly so. It is a complex beast with many different masks. Procedural variation and conceptual variation are very broad headings which encapsulate a huge range of ways that we should be varying the mathematics so that the children engage with a wide range of situations to deepen their understanding of a general principal. The aim is to avoid them thinking that…
Recently, I have found myself scribbling frantically on post it notes (other removal adhesive notes are available) different ways that a single mathematical concept can be expressed either in symbolic form or spoken language (gaining concerned looks from those around me). So I set myself a challenge – can I come with 50 different ways to express addition. But I will return to this in a moment…
To prove a simple example of what I have been exploring, just pause for a moment and consider this question from last year’s KS2 SAT test paper 1. (Interestingly it didn’t reappear in the arithmetic paper this year.)
What is the difference between these two questions?
The complexity of the calculation required is not much more challenging in Q29 that in Q25, so why were the national correct response figures so different? Do the children know how to read the calculation? Can they translate the mathematical symbols? Can they use language to help them understand what is happening? Or are they trapped in a language pit?
To put it simply, the presentation of the concept has been varied. Q25 represents how we might have ‘traditionally’ taught this concept. However Q29 is not ‘a trick by the nasty test people’ it is just as mathematical correct and the children we work with will need to have encountered this representation. This is how we can support their linkage of concepts across the maths landscape.
With my children I would be deepening and exploring children’s understanding by playing with different types of concrete representations but also written and symbolic language that I use with the children.
So here I need to think for a moment; how many different ways can we present addition?
I start with the different ways of reading the addition symbol? Different words could mean add. I have to add a proviso now reader – in my classrooms we have a shared understanding that the words you see below are not ‘trigger words’ that we underline in problems. These live and breathe different meanings depending on the context in which we present them in.
OK, I think that’s a good start…but I still have a long way to go… it might not be all about alternatives for the word add. Commonly we work with pupils to use the inverse (and here you can clearly see why using trigger words procedurally won’t work). What do pupils notice about these examples?
And I can also present different ways that addition can be explored. For me this about interrogating the parts and the wholes and why the calculation is addition, as much as it is about the calculating.
And I can’t forget we could also use pan balances, bead strings, real life objects, number balances or the Slavonic abacus to further deepen experiences… but I’m starting to feel like I am cheating…
Next I consider variation in the presentation of the question itself, I’ve often seen pupils come a copper here. Instead of a page of calculations that all look the same and lead to unthinking clicking through steps, we can design practice that allows pupils to reengage thinking.
Using words rather than digits or a combination is also a well-trodden classic in my classrooms.
I think varying the context is quite common but I’d be tempted to place examples like this within sequences of learning rather than leave it until after methods have been learned. In my experience some pupils tend to put learning in boxes without seeing the connections unless those are interwoven.
How about negative numbers? For many children they don’t appear to follow the accepted rules.
Again, starting to feel like I am cheating as we could apply addition in all the different domains – statistics, geometry, measures, algebra etc. but as earlier, pupils making connections is vital, I wouldn’t want addition in measures to be seen as a separate entity to pure addition.
There are lots of other ways that addition could be and should be explored, and not just for the younger children to ensure that they have a deep generalised understanding of addition and its reciprocal relationship to subtraction. Don’t even get me started on its relationship with multiplication….
If you have any other ways, please feel free to share
Always looking to hit the magic 50 shades of addition!