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What is fluency in mathematics - quick recall of facts or number sense?

Thursday, 5 February 2015
The NC Programmes of Study for primary maths pushes memorizing to a new high – 9 year olds are expected to know all the multiplication and division facts up to 12x12 with all the fallout of failure and anxiety that will bring. Interestingly the US is going the other way, putting less emphasis on rote learning of facts in the new Common Core State Standards.
Jo Boaler, Professor of Mathematics Education at Stanford University, recently updated her paper on the conflict between memorizing and number sense when teaching maths. The short paper, ‘Fluency without fear: Research evidence on the best ways to learn maths facts’, is well worth a read – it is one of those that you will find yourself, at regular intervals, saying ‘Yes… absolutely… I agree… good point… please tell the DfE…!’
Jo Boaler points out that the research out there shows unequivocally that children who have a ‘feel’ for number, use numbers flexibly, select strategies, think about numbers and generally have number sense will be high achievers in mathematics. This has little to do with quick recall of facts, however useful that may be, which in itself is such a small part of mathematics. 

The memorizing and quick recall of facts is not what ‘fluency’ is about – this is concerned with using number sense to solve problems. However, the quick recall of facts is certainly useful and can help a child become fluent with mathematics. The importance of knowing some facts shouldn’t be dismissed and we should encourage children to try to remember key facts. I think the issue is in the actual process of learning these facts.
Many children are put off maths for life because they were given timed tables tests or were asked to recall an addition fact quickly under the gaze of the whole class. I can still remember a maths teacher pointing at me, aged 11, and asking for a quick answer to a subtraction - the heat moving up through my face, feeling myself reddening and my mind a complete fog. I enjoy teaching maths, but I certainly didn’t enjoy learning it!
One of the seminars with the PGCE trainees last year included the use of cycle cards. Each had a card with a calculation on one side and an answer to a different calculation on the other so it made a chain or cycle around the class. The trainees were encouraged to answer as quickly as possible to beat 3 minutes to complete the cycle. We talked afterwards about the activity and many enjoyed the idea of the game, but once they were playing it they felt under pressure and found it stressful. Simple errors were made because of their concern about getting the answer wrong or being too slow. 

So how can we use games like this that involve speed and competition, without the anxiety and fear of failure that may result? It seems to me that the ethos and atmosphere you set as teachers in the classroom is the key. We could avoid this type of activity or we can use them within an ethos of support and a positive ‘can do’ approach to maths. Making mistakes is part of learning mathematics and every child in the class can learn maths and be involved in this type of activity with differing amounts of support. When playing this with my class each table helped each other, some children had number grids on them, some used tables squares. I tried to encourage a sense of personal achievement - self-esteem can have such an impact on confidence with number.

Another thing to consider is pitching the questions so that it matches what the child has just learnt so the practice is useful. For many quick recall questions children either know it or they don’t and so they can be a little pointless – particularly for those that know the facts. Practice is fine, but not if overdone, so think about the purpose of the speed activity.

So speed games and activities can have their place, but if we want our children to have number sense we also need to give them time. Quick recall is a tiny, tiny part of mathematics - depth of understanding is far more valuable. Fluency, reasoning and problem solving involves children in playing, investigating, questioning, arguing, discussing, justifying… the processes of mathematical thinking. If the balance of their maths experiences is tilted towards rich activities that allow children time to develop these processes then there is a good chance they will have a love of maths, not a fear of it.
This positive climate for maths in the classroom and the idea of a growth mindset is something that Jo Boaler has written widely about. Her article and poster Setting up positive norms in math class gives seven of her favourite messages and they are definitely worth considering when thinking about the teaching and learning of maths in your classroom.

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