There should be no such thing as boring mathematics.
Edsger Dijkstra
In terms of the mastery pedagogy and those 5 big ideas, we all probably have our go-to strand- the one we know we can comfortably deploy in our classrooms. I know for me, in the early days, the easiest one to ‘tick off’ was always representation and structure, thinking it simple meant whipping out the manipulatives, in order to bring to life the structure of the mathematical concept being taught (how wrong I was…) However, over time, through making it my mission to invite myself into multiple maths lessons, near and far, I came to notice that when teaching to mastery, we are instinctively and even accidentally often evoking all of the 5 big ideas in turn; my thoughts have increasingly come to this conclusion: you cannot have one without the other...
However, with this in mind, if I had said to myself a few years ago I was using variation theory, I would have replied, sorry what? And I certainly would not have been able to name whether it was either conceptual or procedural.
This ramble starts with a short story, taking me back to where I first twigged there was a method to this mastery madness. It was a post-lesson study chat, with a Singaporean teacher that planted this seed. This learning conversation made me feel both incredibly uncomfortable and curious. Uncomfortable in the sense that I’d been teaching for 4 years, and thought I knew the ‘big ideas’ like the back of my hand; it was evident that I had, perhaps, ran before I could walk. Curious, because I had seen wonderful things, and it had certainly sat in a new league from other mastery lessons, which had gone before. What stood out the most was this: when this teacher talked, they eloquently wove the big ideas together in their lesson reflections. However, what stood out to me the most, was how he ping-ponged in and out of the term variation in and amongst the big ideas. It was like variation was used as his teaching and learning needle and thread, weaving together the utterly outstanding mathematical experiences he had just provided the children with. My takeaway was simple: variation had been a deliberate driving force to support children’s learning.
So variation I thought to myself, what is it? And how do I use it?
Making the most of variation theory is certainly easier than understanding it. When I started digging around for ideas I began to understand why applying variation theory had been somewhat accidental in my practice to date. It sits somewhere in the realms of studying the structure of mathematical concepts, noticing relationships, and exploring for similarities and differences. My maths lessons have typically provided space for this sort of practice resonating what Watson and Mason (2006) name good mastery practice needing to encompass:
…it is not sufficient just to ‘do’ all the products. Learners need to contemplate relationships, to consider effects of changes in one particular aspect
However, when I got to the nitty-gritty, I realised that this sort of practice stemmed from research from Gu Ling Yuan, who coins it Bianchi (or teaching with variation). There were other scholars who had similar things to say, however the central idea that sung out of variation theory was this idea of highlighting the essential features (Gu, Huang & Marton 2004)- simple!
I started to see where my practice, previously, had accidentally fallen into the realms of variation theory. I would often draw attention in my lessons to the clever conceptual changes I had made alongside the similarities I had kept the same, to the children. I was starting to see how this had helped the children to make meaningful mathematical connections. As I increasingly started to make sense of what had existed in my classroom by accident, I decided there was no time like the present to make this now deliberate!
Although I have created both time and space to really come to terms with the theory underpinning the differences between and the most importantly the use of conceptual or procedural variation and examples or non-examples, I don’t believe getting lost in theory is truly vital. I believe that all you really need to do is find out enough to help you simply turn them into powerful teaching tools that will enable your children to think deeper in their understanding and make connections within mathematical concepts…
Here are some headlines:
In terms of conceptual variation, this is probably one of my favourites, and one I found easiest to deliberately deploy. Here it is about giving time to preparing multiple perspectives and experiences of a concept or ‘object’ of learning. Sounds complicated, but it can refer to an abundance of things! From representations, such as how part-part-whole models and bars can be explored together. Looking at the sameness, but equally the differences.
However, we need to hook onto the a ‘concept’ that is being both exposed and explored, otherwise we risk it becoming a ‘pick and mix’ and variety not variation. A concept is a ‘big idea’ or set of rules or statements. It is that what we need to draw attention to. Something else I noticed whilst exploring the idea of ‘concept’ was this word- mis-concept-ion… That helped me situate this idea in an area I was more familiar with, knowing how important it is to plan for these ‘misconceptions’, but now, considering what this looks like, without the ‘mis’!
What I have noticed is this: children tend to spot what is different before they see what is the same. An excellent talking point, and one which is very inclusive of all mathematical minds.
This is where the use of non-examples is situated perfectly and is an incredibly easy way to bring this to life. To give this a quick context, if you were introducing a shape, for example, a square, it would be good practice in terms of conceptual variation to present it alongside other shapes. This then helps students identify with what it isn’t, and thus begin to distinguish what makes a ‘square’ a ‘square’, naming those all important essential features. Our teacher talent of posing questions is vital here and why I argue this type of variation theory often occurs incidentally or even accidentally within our classrooms. However, it must become intentional, as when we carefully craft out questions to evoke children’s understanding of what has stayed the same or changed they are more likely to both understand the mathematical structures, and make those all important meaningful connections.
Procedural variation is something I do believe takes a little more thought and deliberation. The internet, can be, procedural variations enemy here, as we can run the risk when researching this aspect, of bumping into pre-made resources that promise to ‘evoke’ procedural variation. I would personally steer clear from these or at least look at them critically and use them carefully.
As much as we keep complete teacher control in how our questions guide the exploratory nature of conceptual variation, we must have the same professional rigour applied to our application of procedural variation in our classrooms. It takes time, but again when deliberate, it is very powerful. However, to avoid it becoming onerous and complicated we just need to keep in our minds that procedural variation is formed by the richness of varying problems and the ability to transfer strategies, using very small steps.
And that is why it is absolutely not about procedures, or even providing a ‘variety’ of them. It is some much bigger than that. It is the ‘process’, the ‘how’ we proceed through the learning, taking us back to those all important small steps! We need our maths lessons to be built on a well reasoned, connected pathway, however we sometimes forget that the size of these steps matter alot! I suggest you go away and become familiar with the term ‘Pudian’ (Gu 2014 p. 340) and you will see why!
It focuses, again, on the concept but helps children understand mathematical relationships through making clever connections between problems. This is why I go back to the whole idea of it needing to be teacher-tailored and deliberate. Ultimately, procedural variation provides a process for the deep understanding of mathematical concepts stage by stage, through varying problems in a rich manner.
So what could this look like? I find ‘varying’ the problem can be something we overthink. It can, and should, begin small (go back to Pudian!) so starting with a single aspect of a problem is the perfect place to practise this in action. It can be a tiny change! For example, when thinking about addition, changing just one addend to rehearse a connection when bridging through 10 can really evoke procedural variation. What is vital, is that those steps stay small and well connected. Know what you are changing, and why! There is an example below, to get you thinking…
So I think I have rambled on enough. In short, think about relationships, similarities, differences, examples, non-examples, and most importantly explore how you can dabble with variation to make it a deliberate part of your children’s mathematical adventures.
…And from my investigation and interrogation of variation theory- here are some useful reads and resources that helped me piece together this for my practice.