As I walked to school with my grandchildren recently, we talked about how some people keep a pedometer to count the number of steps they walk in a day. Ruby, seven, told me that she'd recently completed a 1,500 metre charity run for children, and wondered how many steps that was. "Well," I said, "I guess your steps are about half a metre. So, how many steps would that be?"
She wasn't sure, but George, nine, didn't hesitate. "Twice that - 3,000,"
It was all there, wasn't it? In the words of the discussion paper Using and Applying Mathematics, recently sent to local authorities by the national numeracy strategy to support the revised framework: "The nature of mathematics is such that it provides an uncluttered and consistent 'world'
within which to work and think. Problem solving involves moving between the 'real world' and the 'mathematical world'."
George's effortless double move, from the real world of steps and charity walks into the mathematical world of straight calculation, and then back again, was a simple illustration of what's meant by "using and applying" - and also a neat tribute to good maths teaching at Hearsall primary, Coventry.
The existence of the discussion paper, and the inclusion in the revised framework of "Use and Apply Mathematics" as one of seven identified strands, reflects some concern within the strategy's leadership about the way this aspect has been taught up until now.
Part of the problem is that, faced with the challenge of getting through the curriculum, some teachers have found it easier to stay within the mathematical world of straight calculations, rather than straying into the apparent complexities of problem solving.
Instead, it's sometimes been seen as an extra activity - perhaps for the end of term. Tim Coulson, who leads the numeracy strategy, says: "In the attempt to inject momentum and pace into maths teaching, we've slightly overdone it. Children need space to digest, to understand. And teachers need to give time in the lesson for that."
It's not that using and applying hasn't always been there - in the 1999 strategy, and the national curriculum and long before that. The old style verbal "problem" - "Four men with four wheelbarrows" - has actually passed into folklore.
Today's good teachers, while they don't ignore that kind of problem involving numbers, want also to cover the kind of logical challenge described to me by Debbie Weible, maths leader and assistant head at Oldway primary, Torbay. "Part of children being able to solve problems is being able to follow a line of enquiry," she says, "So, for example, I might present them with a group of people who want to play each other at tennis.
Each needs to play all the others, but they're not all available all the time, one is only free on Saturdays, another can only play on Tuesday and Thursday afternoons and so on - so the children have to work out a schedule for them." Activities like this, she points out, are ready made for the application of ICT.
Work of this kind has an effect on the style of teaching. This is a long way from the world of "Turn to page 18, do numbers one to 35, work quietly, put up your hand when you've finished."
The aim - emphasised in the strategy's discussion paper - is for children to be able to use various mathematical routines and steps they've learned, putting them together into what the paper calls, "linked chains of calculations, decisions, reasoning and communication". And, it goes on:
"This requires practice and takes time to learn." So, in the problem-solving classroom, it's apparent that there'll be quite a buzz. "It means talking," says Tim Coulson, "Talking to the teacher, to your partner, to your group - explaining, asking questions."
Teachers echo this. Debbie Weible says it gets pupils to talk about their maths. And at High Lane primary, Stockport, key stage 1 teacher Rachel Hancock spends time teasing out explanations from her children. "I don't want them to be afraid to get it wrong, I suggest trying another way."
A key part of classroom discussion in maths involves staying with wrong answers instead of moving straight on to a correction. It's in the exploration of wrong answers, where the misconceptions lie, that a deeper understanding is forged.
There's a strong sense both within the strategy and from the best primary schools that it's time to think creatively about the numeracy framework, shaking off some of the sense of "must push on" urgency. Rachel Hancock, who was one of the teachers consulted about the framework changes, feels this way, "We wanted to make sure that using and applying was embedded.
There is good practice around, and I wanted to help those who lack confidence so that they can add their own flair."
Pre-order the renewed frameworks at publications.teachernet.gov.uk
In the draft revised framework, Use and Apply Mathematics has five themes :
* Solve Problems.
* Represent - analyse, record, do, check, confirm.
* Enquire - plan, decide, organise, interpret, reason, justify.
* Communicate - describe, create, apply, explore, predict, hypothesise, test.
* Communicate - explain solutions, choices, decisions, reasoning.
The purpose of the themes, which clearly link and overlap, is to support progression. Teachers can check a child's progress against each of them.
The discussion paper has a table showing progression within each of the five themes, from Foundation through Year 6 to the transitional Year 67 stage.
The examples are quite detailed, and can provide a focus for discussing progress with children. At Year 1, under Solve Problems we have: "solve problems involving counting, adding, subtracting, doubling or halving in the context of numbers, measures or money;recognise the value of coins."
Rachel Hancock of High Lane primary, Stockport, uses a solving strategy called Lucky with her KS1 children (it'll surely work for all ages) , which can be applied to written problems. It goes like this.
L Look carefully.
U Underline the important information.
C Choose a method. (Might be a calculation, but could easily be visually manipulating a set of shapes.)
K Keep on checking.
Y You did it. You're a maths champ.