Mental mathematics fluency and the importance of perspective

Imagine this: you are one of two people entered into a race with a large amount of money as a prize. You both arrive at the starting line fancying your chances in a foot race over any distance. (In this imagined scenario, you are quite the athlete.) You are each presented with a bicycle that you can choose to ride if you so wish. Sadly, neither you nor your competitor has ever learned to ride a bicycle. Before the two of you have time to realise that your entire existence is merely a part of a blogger’s stretched metaphor, the starting pistol goes and you’re faced with a decision – whether to learn to ride the bike or whether to leave it behind and just run. What should you do? The correct decision depends entirely on one crucial fact: the distance over which the race is to be competed. If the race is 200 metres long, then obviously the sensible decision is to ditch the bike and run. If not, your opponent will cross the line and walk off with the cash prize before you even learn how the brakes work. But what if the race is to be competed over 500 miles? Then, clearly you will stand a far better chance if you spend a few hours – or perhaps even a day or two – learning how to ride the bike. It will slow you down at first, but the eventual gains will more than make up for it. The length of the race is the crucial factor. Lose sight of that, and you’re prone to inefficient decisions.

Now imagine a year 3 teacher. Today’s lesson is on written addition. She has explained the column method, and she’s content that – having explored the dienes and place value counters – the children understand the method. She sets the class a question – 29 + 38 – and begins to purposefully wander around the room. Almost immediately, she is struck by what she sees from the majority of the children: they are adding the digits in the ones column – 9 + 8 – using a ‘counting on’ method, starting on the 9 and counting forward 8, keeping track of this by using their fingers. The majority seem to be getting the correct answer, though the ‘addition by counting on’ they are doing is a mentally laborious way to go about it. The teacher knows that this is far from ideal. She knows that these children will find all sorts of mathematics easier in the future if they develop a strategy for adding one-digit numbers based on known facts (e.g. seeing 9 + 8 as a near double; using compensation to see 9 + 8 as equivalent to 10 + 7; or perhaps even just knowing that 9 and 8 sum to 17). So, what should she do? Should she carry on as before? Or should she begin to spend five minutes every day – spread over several weeks – developing her class’s fluency with number bonds inside 20, eventually saving her children the mental effort of an inefficient ‘counting on’ strategy? She’s painfully aware that taking this time will slow her down; doing this might be beneficial in the long term, but there’s only so much time in the year. (The year before, for example, she taught bar charts and pictograms in a single lesson towards the end of the year just to cover everything she was supposed to.) She decides, this once, to leave the majority with their ‘counting on’ strategy. After all, it might be inefficient, but the class seem to be getting the correct answers. Thanks in part to the pressures and artificial delineations of individual lessons, topics and year groups, she has lost sight of that one crucial factor: the length of the race. Her kids have set off on their 500 mile trek, and their bicycle is at the starting line.

With the day-to-day pressures of teaching, it is all too easy to forget that our kids are in this for the long haul. We must remember that we are building something. Mathematics, in particular, is a highly hierarchical subject; if a foundation is poorly built, the effect it has on the construction above is exponential. In my experience, the most confounding gaps in children’s mathematical abilities are those that are difficult to teach in the space of a few lessons. There is much in primary school mathematics that is understood with relative ease, but requires a great deal of practice before it becomes fluent; number bonds inside 20 and the related subtraction facts are a perfect example of this.[i] I challenge anyone to ‘teach’ this fluency to children over just a few lessons. Like fluency with multiplication facts (and the related division facts), these need to be seen regularly for a considerable period – though perhaps just for a few minutes each day – for them to become fluent. This is not to advocate any particular method for teaching this. Children can practise these against the clock, keeping their answers private; they can quiz each other using cards; they can play games that expose them to these facts over and over. How they see them is, in my view, less important than how often they see them. Regardless, one thing is certain: if we don’t take every opportunity to prioritise long-term efficiency over short-term convenience, then we are doing the children in our classes a major disservice.

A few things I find useful in teaching underlying mental skills:

1. ‘Little and often’ is perhaps the most time-efficient way to develop fluency with mental mathematics. Five hours spread over ten weeks (roughly five minutes per day) seems to be far more effective than the same amount of time spread over one week.

2. Take the pressure off. It’s perfectly possible to quiz children – and have children mark their own work – without there being any stress about performance or speed. In my experience, if you give children a separate strip of paper with the questions and ask them to mark their own work when you share the answers, they soon realise that the only pressure is their own desire to improve. Children almost always take pleasure in developing their fluency over time. It often becomes the part of the lesson they like most.

3. Use routines and don’t be afraid to use the same activities and resources repeatedly. Routines allow children to engage with the actual learning rather than dealing with the superficial elements of an activity.

4. Be persistent. Beyond the rarest of exceptions, all children can learn fluency in the underlying mental mathematics skills. Have faith that this will lead to considerably more effective learning in the long term.

[i] This is not to suggest that the different interpretations of basic addition and subtraction do not require a great deal of conceptual understanding; they absolutely do, ideally through a variety of physical and visual structures. Peter Mattock’s book Visible Maths is superb on this subject.

4 thoughts on “Mental mathematics fluency and the importance of perspective

    1. I think ‘counting on’ as an initial strategy is an important and necessary interpretation of addition. However, it should ideally be superseded by more efficient calculation-based strategies by year 3.

      Liked by 1 person

  1. When we forget that all subjects are long haul, we have indeed lost. In my experience as a parent, teachers appear constrained by the demands of ‘what must be covered’, which reduces the time necessary to become a proficient cyclist. From an N=2 at school and my experience of undergraduate teaching, I would say this is a problem at primary and secondary levels.


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