More than adding up
Tim, aged nine, was working on what to add to a collection of numbers to make each one up to 500. Twenty minutes into the lesson, Tim had written 222 + 278 = 500 in his exercise book and was busy making 56 up to 500. Pressed to explain or show his working Tim reluctantly brought out a piece of paper from under his desk. The paper was covered in a long string of ungrouped tally marks. Amazingly, Tim had managed to count on correctly from 222 to 500.
Tim seems typical of lower-attaining pupils who rely mainly on counting strategies to calculate. In contrast, more numerate pupils have a range of alternative strategies to draw on, based on: "knowing by heart" - recall of some number facts (for example 5 + 5 = 10), and "figuring out" - deriving or deducing other number facts on the basis of the known facts (for example, 5 + 6 must be one more than 5 + 5). We know that pupils able to recall and deduce number facts make more progress because each approach supports the other.
In 1992, the School of Education, King's College, University of London, in collaboration with Tower Hamlets and Wandsworth local education authorities, set up a "numeracy recovery" initiative, to explore strategies for helping pupils like Tim develop competence and confidence in numeracy. This was based on ideas in the successful reading recovery programme.
Central to this work is the question: what do we mean by numeracy? Traditional definitions concentrate on the acquisition of arithmetical "facts". However, we suggest that being numerate involves being able to process, communicate, and interpret numerical information in a variety of contexts.
From the first phase of the numeracy recovery work we found evidence of: o greater competency in numeracy targeted pupils, through developing a wider range of mental and other non-routine strategies o increased pupil confidence in tackling unfamiliar problems.
However, we also identified a number of questions: o How can we systematically monitor improvements in pupil competence and confidence?
o Can the teaching approaches be easily and cost effectively implemented on a broader scale?
o Are the teaching approaches suitable for raising attainment of pupils at all levels and can they be developed for whole-class teaching?
On this basis, we have begun a second phase to the project, trialling and evaluating a feasible national numeracy recovery programme aimed at improving standards of numeracy in key stage 2. Attainment standards of targeted Year 3 children will be compared with those of a control group to explore the extent to which the programme helps raise standards.
We are also developing teaching approaches that draw attention to: the centrality of mental strategies in becoming numerate; the difficulty that many pupils have in "abstracting" mathematics from teaching; the links between problem solving and developing number concepts; and the importance of careful assessment of what pupils can do.
Training for teachers in the project includes: awareness of pupils' framework of skills and understanding through diagnostic assessment; examining the implications of the diagnosis for teaching and developing teaching activities that integrate facts, skills, understanding and problem-solving skills.
oTeaching materials from the first phase of the project, A feel for number, are available from BEAM, Barnsbury Complex, Offord Road, London N1 1QH Mike Askew is a lecturer in maths education, King's College, London