It has become fashionable to claim that much educational research is of little use to teachers. But the findings of a study we have recently completed for the Teacher Training Agency contain significant messages for all primary teachers, as well as for the Numeracy Task Force, charged with planning a national strategy to raise numeracy standards.
We were given a year and enough money for one researcher in order to try to identify what distinguished effective numeracy teachers from others, and how these characteristics were acquired.
Lacking trust in the ratings of inspectors from the Office for Standards in Education, we decided the only way to check teachers' effectiveness was to find out what their pupils learned over the year, using a specially designed set of numeracy tests for each year from Year 2 to Year 6.
We chose eight schools which, according to available test data and advice, appeared to be effective numeracy schools in value-added terms. They included independent schools and some from each of three local education authorities with varied populations. To provide a comparison we also selected two schools which were judged to be of average effectiveness, and one that appeared weak in terms of its numeracy teaching.
We asked heads from each of the schools to select their three most effective teachers. Then we observed their teaching and interviewed them on several occasions in an attempt to understand their views about teaching numeracy, and about pupils. A further interview assessed the teachers' knowledge and understanding of important ideas in numeracy.
For all 73 teachers of Years 2 to 6 in these schools we gathered questionnaire data on these topics.
The average gains in test score for all the 73 classes between October and April were calculated, and within each year group classes were sorted into three roughly equal groups, with high, intermediate and low gains.
The first interesting result was that the teachers selected by their heads as effective were, as a group, no more effective than all the others. And some of the teachers of the classes with the highest gains were not selected by their heads.
One school was outstanding in that 12 out of 13 teachers had high class gains. Nearly all the other schools were indistinguishable in terms of their results. For example, the "weak" school, in an inner-city area, did indeed have low test scores but had class gains similar to those in other schools.
The overall effect of having almost all teachers in one school with high gains was that their pupils entered Year 2 with below average numeracy test scores compared to the rest of the sample, but at the end of Year 6 their pupils were outscoring all other schools, including all three independents.
What distinguished the teachers with high gains, here and in other schools, were not the forms of their practices - some of these highly effective teachers used mainly whole-class organisation, while others favoured mostly group work. A highly effective Australian teacher used only individual silent work, in an inner-city school where he considered it the only way to cope with a difficult Year 6 class. Equally, while the most effective school had just introduced setting and used a well-known mathematics scheme, this form of organisation and scheme was also used by much less successful schools.
Nor did age and experience or mathematical qualifications distinguish highly effective teachers. In fact, the 13 teachers who had passed mathematics at A-level had smaller average gains than the other 60. Similarly, in the interviews, the most effective teachers did not necessarily seem to know extra mathematics or have superior skills .
While some of the mathematically well-qualified lacked confidence and expressed negative attitudes, the most effective teachers seemed enthusiastic about and able to explain connections between mathematical ideas, reporting how in their teaching they related these ideas to pictorial representations such as grids and number lines.
Also distinguishing the highly effective teachers, who we dubbed "connectionists", was a set of common coherent beliefs about the nature of numeracy, the way children learn it, and effective ways of teaching it.
They encouraged children to select the strategy most appropriate to a problem, especially mental strategies (for example solving 3,000 - 1,948 by mentally adding 52 to get to 2,000 and then another 1,000, rather than resorting to a calculator or standard written algorithm).
They were constantly assessing and recording, formally - in written and mental tests often given at random times or before a topic was introduced - and informally by asking pupils to explain their methods. They were concerned about pupils' attitudes, and tried to challenge all children at their own level, believing almost all pupils could become numerate.
For example a typical connectionist lesson with a Year 6 class started with children working in groups using a grid with columns labelled fractions, decimals, ratios and percentages. On each row one representation was filled in (for instance 35 per cent) and the objective was to fill in all the other equivalent representations on each row. A class discussion would follow, during which groups had to explain their methods to the rest of the class, and methods were compared for effectiveness. Finally came a discussion of the contexts in which it was appropriate to use one form of representation rather than another.
In contrast, teachers with the lowest gains seemed to be mainly subject-centred ("transmission" teachers) or child-centred ("discovery" teachers). Transmission teachers emphasised particular methods, which children were expected to imitate, but without linking mathematical ideas, and without taking account of pupils' own understandings.
Discovery teachers, mostly teaching younger groups, encouraged practical activity without a clear view of where it was leading, and protected pupils from mathematical challenge so they were not forced to examine the possibility that they could improve the efficiency of their methods.
So how did the highly effective connectionist teachers acquire the coherent beliefs and linked practices?
Attendance at sustained professional development courses in a variety of institutions (such as Grants for Education Support and Training-funded 20-day courses) was strongly associated with high effectiveness, and teachers reported the courses were "fantastic and fun" and that changes in their thinking had occurred because they were forced to examine their beliefs and practices. Sometimes this was in reaction to hearing the results of research in relation to different forms of imagery or to the variety of pupils' meanings and understandings. In the case of mental calculation, exploring methods used by their pupils and by other teachers on the course had clearly been formative experiences.
In the highly effective school, a few teachers had gained equivalent experiences without leaving the school. This was achieved through key staff who were enthusiastic and knowledgeable about numeracy and shared a common view. Each had attended and helped to run sustained in-service training courses.
Resources were allocated to enable the deputy and the mathematics co-ordinator to work in classrooms alongside other teachers, which rarely occurred in the other schools. The frequent year group planning meetings included evaluation and discussion of detailed teaching activities. New teachers were successfully socialised into the school's mathematical ethos: "Last year I had a new teacher and I was able to work closely with her. When she first came she was a sums person but she has responded really well" was one comment.
Another connectionist teacher had benefited from more informal experience, through observing his own children learning, and through spending time at weekends exchanging ideas about maths and teaching with a mathematician brother-in-law.
Time to discuss, reflect, try out and evaluate, together with access to knowledgeable support, seemed factors common to all the teachers whose classes had high numeracy gains.
Two highly effective experienced teachers in another school were unusual in managing to develop their numeracy teaching through mutual support independent of unsympathetic colleagues. "My parallel teacher and I talk a lot. Being dissatisfied with the way we felt things were going, we both started changing...we plan a lot together although we don't do every lesson the same," said one. Their access to ideas was through trying to keep up with recent trends spotted in "the good old Times Ed every week, especially the maths supplements...if I see a really good book ... there's a recent one come out on research in mathematics teaching ..."
What better testimony?
Effective Teachers of Numeracy in Primary Schools by Mike Askew, Margaret Brown, Valerie Rhodes, Dylan Wiliam and David Johnson is available free from the school of education, King's College, London. Tel: 0171 872 3139