How to keep application at the heart of learning
The application of maths is an issue of central importance. We want pupils to be able to apply what they learn to a wide range of contexts, in and out of school, and in all aspects of their lives. There is no point in pupils being able to perform feats of mental arithmetic in the classroom if they are unable to carry out simple calculations in the home or workplace or community.
There is a lot of evidence that many pupils do indeed have difficulty applying the maths they learn in school to problems they might encounter outside of school. One national survey found that 80 per cent of 12-year-olds could correctly divide 225 by 15. But only 40 per cent could solve the problem: if a gardener has 225 bulbs to place equally in 15 flower beds, how many would be in each bed? Most of the failing pupils did not know which mathematical operation to use.
In the National Numeracy Strategy, the importance of application is highlighted by the definition of numeracy: "Numeracy ... requires ... an inclination and ability to solve number problems in a variety of contexts." The Framework for Teaching Maths, however, gives more attention to calculation and computation than it does to application. There is a real danger that application will become an optional "add-on" to numeracy lessons rather than a fundamental part of the curriculum.
In our book Numeracy and Beyond: Applying Mathematics in the Primary School, we describe how a group of primary teachers used a variety of different approaches to help the children with application.
One of the main approaches used by the teachers was to introduce what they termed "authentic activities" into the classroom. By this they meant activities that had some kind of real-life context, either inside or outside the school. For example, Alice, a Year 1 teacher, set up a car boot sale in her classroom, using unwanted toys and real money, so that the children might apply their emerging knowledge of addition and subtraction. The children had to buy and sell goods as if they were at a real car boot sale and they also had to develop methods for keeping track of their money. Afterwards, Alice said the children had found the activity exciting, and that "it had made their use of maths in general much more important".
A slightly different approach was taken by Kathy, a Year 5 teacher. She set some of her pupils an "authntic" problem in which they had to calculate how much shelving was required to store some science resources in the classroom. The children had first to estimate how many shelving bays could fit into the available space and then to calculate how much shelving was required to make up these bays. The children's answers were used as the basis for ordering the actual shelving. Kathy thought the authenticity of the problem lay in it being a genuine exercise "that needs to be answered and where the teacher hasn't got the answer neatly stored away somewhere".
Some of the other teachers used approaches that emphasised application as a process of "making connections" between maths and the real world. Barbara did this by asking her Year 1-2 children to generate pieces of maths, such as 20-10=10, and to illustrate them with stories. She felt it was important that children could generate for themselves meaningful contexts in which maths might be used. One child, Rachel, responded in a somewhat bloodthirsty vein with: "There were 20 rabbits and 10 got shot and that left 10."
Another teacher, Laurie, used a similar approach, in which he read a story to Year 5-6 children and asked them to identify the maths involved. One part of the story ran: "Last year his income had tripled. If it doubled again this year, could he afford that new farm Annie Appleton was selling?" Laurie followed this up by presenting the pupils with a sequence of symbols, such as decimal, per cent, fractions, and asking them to write their own continuation of the story, introducing mathematical ideas corresponding to these symbols. One group continued the story's purchasing theme with: "If I put a deposit on it for .5 (decimal) of it, that's 50% (per cent) of it, which is 12 (fraction), I'll still be able to get it."
There is more than one way of helping pupils apply maths. What matters above all is that teachers are aware of the difficulties children have with application, that they give it the same priority as they do to other aspects of maths, and seek out classroom activities that help children make links between classroom maths and the world outside school.
MARTIN HUGHES, CHARLES DESFORGES and CHRISTINE MITCHELL
Martin Hughes is at the University of Bristol; Charles Desforges and Christine Mitchell are at the University of Exeter. Numeracy and Beyond (with Clive Carre), Open University Press, May 2000, pound;13.99