Building Conceptual Understanding and Number Sense
I’ve just completed the excellent open access ‘How to Learn Math’ course led by Jo Boaler (author of ‘The Elephant in the Classroom’). There were lots of ideas to think about on the course and lots that I’ll want to revisit and mull over in the coming days and weeks. For a more detailed overview of the course, Pam O’Brien has written a helpful summary.
One of the big ideas is that children learn maths best by exploring real problems with a context rather than learning and practising routines. This, I’ll admit is a challenge. I really warm to the idea of exploring maths in context and feel that embedding maths in cross-curricular work is something we’re not at all good at as teachers. I also love the idea of children exploring problems and challenges for their own sake rather than as opportunities to use and apply their mathematical skills. However, there is a more ‘old school’ part of me that feels that it’s important for children to acquire fluency in calculation skills and fairly rapid recall of number bonds and tables facts. Boaler points out that these skills are not what real maths is about and I agree with her there, and I’m sure there are examples of real mathematicians who struggled with basic arithmetic. However, I do feel that having these facts and skills at their fingertips is for most mathematicians, part of their ‘toolkit’ of mental resources. That’s not to say that other attributes aren’t even more important – confidence, perseverance and curiosity all spring immediately to mind. In the real world of education too, both the current and the new curriculum require these skills and I have no doubt that children will continue to be tested on them for the foreseeable future, so as teachers we need to think about how we can help children acquire them.
What I’m not saying here, is that these facts and skills should be taught in isolation. Boaler is keen that children build up a conceptual understanding of number and mathematical skills and I would absolutely agree with her on this. There is little point in children acquiring rapid recall of number bonds and tables if they have little or no idea of what it means to add or to multiply. Before learning any number facts, children need to be building up a good ‘number sense’, a feel for numbers and how they can be manipulated. For a child without this understanding, ‘3 + 2 = 5’ is a fact to be learned in isolation. Whereas for a children with good number sense, they may visualise a group of 3 combining with a group of 2 to make a group of 5; they may use their knowledge of doubles facts to work out that the total will be one less than 6 or one more than 4; or they may be able to use this fact to derive lots of other related facts: 5 – 2 = 3 or 30 + 20 = 50.
Building up number sense needs to be an ongoing process throughout primary education, and probably well beyond. We can help children to do this in lots of ways but one of the key ones is by giving children lots of opportunity to explore the way numbers work using concrete apparatus and helpful representations before plunging into the more abstract world of numbers in isolation. This might be by using dedicated mathematical equipment: counters, Dienes, Numicon, multilink, ten frames etc. Or it might be by using the opportunities that occur every day: counting the steps as we walk upstairs, working out how many cakes will be left if we each eat one, combining our pennies to see what we can buy at the sweet shop etc. This brings us back to exploring real problems with a context. It would be great if this was happening naturally at home for all our children long before they ever got to school, and for many I’m sure it is. Creating opportunities in school for this to happen is something that I suspect EYFS teachers are already very aware of, but perhaps it’s something for the rest of us to be challenged by as we plan our classroom environments and cross-curricular work.