Rachel Rayner is a Primary Mathematics Adviser for Herts for Learning
Yesterday was a bit of a surreal day. Charlie Harber and I were filmed talking about mental mathematics. A day of feeling hugely embarrassed by presenting our thoughts to a camera; I’m sorry to say, I don’t think I did too well. To fit in all we wanted to say in 5 minutes was somewhat of a challenge to say the least. So this blog is an attempt to put that right …I’ll let you be the judge of how I get on!
Why the focus on mental mathematics?
Our work in research projects around this area has led us to see the gaps between those children entering school having had rich experiences of maths at home and those who have had very little. Increasingly rarely, do pupils seem to play with dice, cards, board games or dominoes for example. Neither do they spend so much time singing counting rhymes, using their very own counting environment, their fingers. Try asking your reception or Year one classes to show you 7 on their fingers – now spot the pupils who don’t have 5 as a benchmark to find 7.
Also we have found and written about in our newsletter the group we refer to as ‘fast counters’ who have never really had to develop fact recall because they constantly divert around it by counting. Nothing wrong with counting, but when they enter LKS2 and when learning formal written addition they still have to count to find 3 + 4, this is obviously going to impact on how well they learn the formal written procedure. These pupils are arguably doing harder maths. Once the number ranges increase, so the need to calculate rather than count impacts.
Finally, let’s face it the statements around mental maths are rather vague compared to the more specific written method statements. This has led some schools to focus on the written calculation progression and teaching procedures over developing adaptive mental fluency. Pupils with little mental fluency are more likely to trust a procedure unthinkingly than see it as a range of possible strategies. It is heart breaking to see some pupils trying to solve quite simple calculations tracing out formal written methods or clicking through the jumps on number lines– controversial but I believe the number line is a procedural approach that has little impact on developing fact recall or a range of strategy.
See this apparently confident Y2 child’s method I observed who was unable to use any fact recall and never referred to the numbers at all except to use them in a counting order. For the pupil these procedures done unthinkingly have multiple error points and why should one answer be more valid than any other?
Return the attention to the number
We need to refocus our pupils to attend to the numbers involved, to think what can these numbers do for me? Consider the question…
21 – 16
Believe it or not I have seen pupils attempting to count back sixteen ones using fingers. Of course this is much harder maths than the pupil who understands the concept of equal difference i.e. adjusts the numbers to 20 – 15 knowing that the difference will remain the same but that this is an easier calculation.
Figure 1 Extract from HfL Mental Fluency Progression
Consider how much easier £18.00 – £7.62 if we use this method to subtract 1p from each side even if we do see the layout in columns.
The error points become reduced. So mental fluency impacts on better written fluency.
What about the calculation 72 – 57? Most pupils do not see the number holistically as a result of the teaching of procedures. So they think 2 – 7 can’t do that so I have to exchange one of the tens from the 7 and so on.
The pupil has a good sense of number will be aware of the fact that many numbers live within 72 and 57. They will also acknowledge that 57 is quite near to 60. That might lead them to partition 72 into 60 and 12.
Pupils readily use their number bonds to ten to subtract 57 from 60 which is much easier, and will recombine the resultant 3 and the 12 to find the difference.
If we return to the formal written method, of course we can then point out to pupils that instead of exchanging we are really decomposing the 72 into 60 and 12. This makes it easier to subtract the 7 from the 12 and the 50 from the 60. Again we are then recombining to find the difference. From mental strategy we can really understand written methods.
But both of those examples require pupils to be comfortable playing with number, being able to decompose and recombine, adjust and compensate and rebalance.
How do we begin to encourage playfulness?
Well firstly, get your pupils playing with dice and cards etc. the pupils that recognises dice pattern 5 has the advantage of seeing 4 dots and 1 dot, 3 dots and 2 dots, 2 dots and 2 dots and 1 dot. That helps when later they want to add 5 to 7 for example.
Next as you can see from our blog Take one resource: the humble tens frame we work with our pupils exploring numbers to ten, decomposing and recombining, exploring one more one less, looking for patterns and considering how close to ten.
Then we explore and discuss multiple strategies for one calculation.
Which part of 17 is it easier to take the 9 from? How many will be left then?
- How much time do you give your children to discuss, explore and evaluate strategies?
- Is there more you could do to nurture children’s own innovations on mental strategy?
- Are your pupils able to interpret representations of shared strategies and acknowledge other people’s strategies?
- Do you and your children play enough with numbers?