Rachel Rayner is a Primary Maths Adviser at Herts for Learning
It’s a good question. In my experience working with schools nationally, pupils default to the written method often unthinkingly. See the lovely examples here of just that happening.
As teachers we value mental fluency and we want our pupils to have it. But are we working in the right way to engage our pupils over a sustained period of time, out of unthinkingness and into causing pupils to think deeply enough about the facts and skills they are adept at and how they might use them to form a strategy?
Was it the fact recall?
Let’s start with one part of that, the acquisition of automatic fact recall. Or as I often see the territory of snake oil merchants! These merchants claim to be able to ‘fix’ this problem of pupils not developing adequate fact recall, usually by providing schools with a bank of timed tests to use weekly – torture tests as one of my team refers to them as. And then we see invented challenges and charts where we rank our children visibly against other children. What do we use them for? Who are they impactful for? Who are they not impactful for? Why? There is no magic wand – we’ve just got to give the right things some welly!
Recently after delivering a series of 2 hour sessions on teaching times tables effectively, several teachers approached me to confirm that what I was saying was that times tables needed teaching and sequences of learning needed to be built, rather than sending the facts home to be learnt and tested a week later. Yes, yes, yes. We can’t leave the learning of facts to chance. Knowing these facts release the cognitive load and enable pupils to focus on deeper structures and procedures in mathematics, they may have felt a lot less stressed by having that fact recall. So, we need to allow pupils time to commit these facts to memory, and if memory is the footprint of something we have thought deeply and repeatedly about, then we must engage pupils in learning these facts in frequent, varied and meaningful ways. Testing offers us little insight into pupils’ actual fluency it is often only repetitious fact by fact recall. We have seen that this has and continues to cause maths anxiety within a significant part of our population of learners, and even those who ‘perform well’ display nervousness (Boaler 2012). Kling and Bay-Williams (2014) carried out a study of 8 year olds learning over a year where no aspect of timed testing or rote recall was allowed. Instead pupils played regular games and explored patterns within numbers through discussion. By the end of the project pupils displayed the ability to recall those facts appropriately and automatically (within 3 seconds) 95% of the time. This directly echoes our own work in KS1 where we work each term developing fact recall through gaming, number patterns and strategy engagement for pupils alongside KS1 teachers. This is only a 6-8 week project in which we request teachers to refrain from timed tests, yet we see increases in automaticity, accuracy and the use of efficient strategy in that short space of time. We really must write some of our fact learning games down, but let that be for another blog.
Are pupils being encouraged to avoid fact learning – unintentionally?
There is a certain strategy, long beloved in the primary classroom which, in my opinion, impacts negatively on the learning of facts – that being the number line. Now before you howl indignantly at me, consider, are your pupils still beholden to the number line in Y6? Is it the only strategy? Are your pupils having to write each stage rather than use it to inform mental routes? If so, it isn’t working. The numberline is a brilliant conceptual representation dealing with difference to those landmark numbers, but it has become to procedurally taught and long winded, too many steps often to be used efficiently. But more than that, it encourages procedural counting rather than building up facts and deploying them flexibly, my experience in the classroom and through research is that pupils over rely on it and that it can become incredibly labour intensive and even when facts are utilised, there are often too many facts to deal with. We still advocate a count up strategy for finding the difference…but only when that is the most efficient route.
Consider which facts provide most access to other facts or the hardest to learn facts.
If you were only allowed to learn only 20% of the times tables facts, which facts would make it onto your list? Why?
These could be the facts to focus on more frequently as they are often the ones that make the greatest footholds towards the learning of other facts.
Have you identified the facts that are the hardest to learn?
Again, more time spent exploring and rehearsing these is often beneficial.
And don’t forget what we think is an obvious set of facts (the x/÷ 1 and 0 facts), isn’t necessarily to our pupils! Those facts can all be eliminated once the pupils recognise the effect.
An effective way that is working for me currently is to play spot the important fact. Look at the question below.
Is the related fact 132 x 10, 11 x 12, 8 x 12 or 11 x 11? Why?
Was it about skills development?
Partitioning, rebalancing and rearranging numbers are all skills that are exciting but take time to develop and rehearsal too.
Let’s consider partitioning. It is the ability to decompose and recombine those numbers.
Consider how we might partition 72 to subtract 57. Typically pupils are conditioned to only partition numbers into tens and ones for example. But by partitioning the 72 into 60 and 12, we can take the 57 from the 60 and recombine the remaining 3 with the 12.
Two skills are modelled in this Herts for Learning video ‘Nimble with Number’ https://vimeo.com/180281520
Was it the practice? The art of simmering…
Have you considered why footballers who play in the premiership still practice frequently? Yes it’s their job and they are well paid for it. But surely they are already great. So let’s apply that analogy to rehearsal of core facts and skills. Which skills and facts need to be simmered – for a long time? Do we take the time to offer pupils the variation they require in examples? There are just a few suggestions below for this.
- Different missing number box placements
- Orientation of the fact
- Playing with the equal sign placement as well as inequality symbols < >
- Pictorial representation
- Change numbers for language
Practice commonly seen by our team is lots of the same which the pupils can click through unthinkingly, have a go at changing the practice slightly after every 3 or 4 examples so that pupils constantly have to reengage with the process of practice.
…or did we move too fast and forget to continue the basics
No, pupils have not learnt everything they need to know about numbers to ten by the end of reception or even by the end of year 1. We do need to keep going well into Year 3 when you think about the implications for other areas of mathematics. Same with adding single digit numbers to any number. We need to be rehearsing and talking about how we use our low number facts way beyond KS1. Or think about how the x2, 5 and 10 facts could be reignited and simmered in Y3 to support the learning of 3s 4s and 8s?
How might we have used each of these facts we could use to solve this question?
9= 8 + 1 468 – 10 9 + 1 = 10
Were they conceptually fluent?
There are a number of concepts that if not secure contribute to weak mental fluency. Consider those pupils who have not had sufficient time to secure the concept of conservation i.e. understanding that 500 is 500 no matter how we partition it, it will still be 500. They must also be aware that all numbers to 500 live inside 500. If they have no understanding of conservation then they will be unable to compose or decompose number effectively using the skill of partitioning, as they just don’t trust that the number remains the same number. Being able to break up and recombine number impacts greatly on flexibility and adaptively.
96 ÷ 4 I can partition the 96 into 80 (20 x 4) and 16 (4 x 4) or
96 is one group of 4 less than 100. I know there are 25 groups of 4 in 100. 96 is one group less so there must be 24 groups of 4 in 96.
Place value is another multilayered concept that is assumed learnt before it actually is. A fair number of questions relied on a good understanding of this in the 2016 KS2 Paper 1 (13 by my reckoning). I often see pupils multiplying and dividing by powers of ten for example with little understanding of the consequences of this and why it is necessary. Mental skills impact greatly on written methods of course.
50 x 70
5 tens multiplied by 7 tens
Ten multiplied by ten is 1 hundred. So we could call it 5 x 7 x 100. That is 3500. 50 x 70 is 3500.
Are they caught in a procedural trap?
Finally, some of our pupils are caught by having been only taught to mimic the teacher’s method (see the comments on the numberline above). This often falls apart as pupils have not necessarily engaged with the method, rather repeated it, got it all correct and then forgot about it. Job done. Assimilating a range of strategies and comparing and evaluating them is crucial as our own research proved. Trying hard to understand someone else’s method helps you to better understand your own and that which your teacher has taught.
If it is 3:45pm now. What time will it be in 50 minutes?
Champion your strategy
Partition, rebalance, compensate
And then there’s returning the pupils attention to the numbers rather than just churning out a procedure. What do the numbers tell us? Recently I have been using the ‘Trick and Treat’ approach with Y6 pupils, showing them calculations which lend themselves to mental strategy but that they may get tricked into using a written method for. Here is some work I did with a Y6 class this term, helping them to learn a new strategy and how it works before applying it to larger numbers. You can see in this example also how mental strategy can improve efficiency for written strategy, cutting down the margin of error.
Over to you…
Developing mental fluency is a long term and whole school effort, Year 6 is a shared responsibility, one that all year groups contribute towards. The bottom line is that pupils felt more confident with written methods than mental, no matter whether they were the most efficient methods or not. There are no real quick fixes, though a clear focus on this in Year 6 will impact positively and we have moved some of our cohorts a long way towards getting them to attend to the numbers, in reality, a whole school approach will have far greater, safer and longer lasting impact. Can we really leave it to chance or Year 6? Which of the areas above is it necessary to work on first as a school?
Rayner, R and Harber, C. (2015) Billy Counts. Primary Mathematics vol
Boaler, J. (2012) Timed tests and the development of math anxiety. Education Week. Accessed online http://www.edweek.org/ew/articles/2012/07/03/36boaler.h31.html .
Kling, G. and Bay-Williams, J. (2014). Assessing Basic Fact Fluency. Teaching Children Mathematics, 20(8), pp.488-497
https://vimeo.com/180281520 Nimble with number