Over half term some good friends visited. I had an interesting chat with Dan, who is in his fifties and gained a first in maths through the OU a few years ago. He’s just done a PGCE as a maths teacher and has been trained to build understanding through plenty of problem solving tasks.
The discussion made me reflect on the stark difference between the way I’ve taught maths to my own children at home, with the lion’s share of time spent learning to fluency, and the focus in schools on exercises to build understanding. After all, I reflected, the progress of my children has stunned even me. How is it they missed out on SO much work on understanding while accelerating far ahead of their peers?
It isn’t that I don’t appreciate that children need some degree of understanding of what they are doing. I remember when I discovered that the reason my friend’s daughter was struggling with maths at the end of Year 1 was because she had failed to grasp that crucial notion of ‘one more’. Her teacher had advised that she needed to learn her number bonds (and indeed she did) but while she did not grasp this basic notion the bonds were gibberish to her. What we call ‘understanding’ does matter (more thoughts here).
I’ve realised the reason I’ve never had to invest significant time in exercises to build understanding. It is because when my children are given a new sort of problem they can already calculate the separate parts of that problem automatically. All their working memory is focused on the only novel element of a procedure and so it is very quickly understood. Understanding is just not a biggy. Identify the knowledge necessary to calculate the component parts of a problem and get fluency in those and generally activities for understanding become a (crucial but) small part the maths diet.
The degree of focus on fluency that my children were given is highly unusual. I have huge piles of exercise books full of years of repeated calculations continued a year, two years, after they were first learned. My children learnt all possible addition and subtraction facts between one and twenty until they were known so well that recall was like remembering your own name. I did the same with multiplication and division facts. There were hours and hours and hours and hours of quite low level recall work.
Generally the the focus in schools is the opposite and this creates a vicious cycle. Children are taught more complex problems when they are not fluent in the constituent parts of the problem. Therefore they struggle to complete calculations because their working memory breaks down. The diagnosis is made that children don’t ‘understand’ the problem posed. The cure is yet more work focused on allowing children to understand how the problem should be solved and why. The children may remember this explanation (briefly) but it is too complex to be remembered long term as too many of the constituent elements of the problem are themselves not secure. When the children inevitably forget the explanation what is the diagnosis? – a failure of understanding. Gradually building ‘understanding’ eats more and more lesson time. Gurus of the maths world deride learning to fluency as ‘rote’ but perversely the more time is spent on understanding instead of fluency, the harder it is for children to understand new learning. By comparison my children seem to have a ‘gift that keeps on giving’. Their acceleration isn’t just in the level of maths proficiency they have reached it is in the capacity they have to learn new maths so much more easily.
I’ve not got everything right but I’ve learned so much from teaching my own children including that the same general principle is true of understanding maths and understanding history. If understanding is a struggle it is because necessary prior knowledge is not in place or secure.
Go back – as far as you can get away with.
Diagnose those knowledge gaps.
Teach and secure fluency.
You’ll find understanding is no longer the same challenge.