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# Discovering the ABC of number;Briefing;Research Focus

Teachers' lack of knowledge in mathematics is often said to contribute to relatively poor performance in the subject by their pupils.

The new curriculum for initial teacher training addresses this problem by prescribing the "knowledge and understanding" that primary trainees need in order to underpin effective teaching of the subject.

Since September 1998, teacher-trainers have also been required to audit and improve students' subject knowledge.

At the Institute of Education, University of London, we piloted the new curriculum with more than 150 postgraduate certificate in education primary trainees a year ahead of time, and audited their subject knowledge in mid-January, after the main content areas had been covered in lectures and workshops.

The audit took the form of a one-and-a-half-hour written assessment which provided guidance for further study. One of our aims is to identify what students find difficult.

Some 95 per cent of those tested could order a set of decimal numbers from least to greatest, or solve a problem by algebraic reasoning, whereas test items involving scale factors and percentage increase, generalisation, deductive reasoning and argument met with between 40 and 60 per cent success.

Analysis of responses to the more difficult items is uncovering errors and misconceptions related to gaps in trainees' subject knowledge. For example, students were asked to verify and to generalise from the three examples: 3+4+5=3x4

8+9+10=3x9

29+30+31=3x30

Five per cent of students appeared to focus on the first example to the exclusion of the other two, and incorrectly generalised "n + (n+1) + (n+2) = n x (n+1)".

We share the Teacher Training Agency's concern about prospective primary teachers who find it so difficult to perceive unity of form (let alone of meaning) in the three equations, but question the effectiveness of "guided self-study" in the face of such cognitive obstacles.

We are also investigating possible links between subject knowledge and students' performance on teaching practice.

Students were rated A, B or C for their subject-knowledge audit ("A" being perfect or near-perfect) and 1, 2 or 3 for practical teaching ("1" being strong or very strong). In this classification an "A3" student, for example, is weak in the classroom despite strong formal subject knowledge.

Data from the first teaching practice did not support the view that teaching performance is linked to formal subject knowledge, apparently confirming research at King's College, London, on effective teachers of numeracy. Indeed, our study produced one bizarre finding. Judging by the second teaching practice, the match between trainees' formal knowledge of English and their observed ability to teach number was, if anything, better than that found for formal knowledge of mathematics.

However, we also found that there was a positive association between mathematics subject knowledge and competence in teaching number.

Students with high audit scores were much more likely to do well in school, and much less likely to do badly. The converse was true for those students with low scores.

This finding does not, in fact, contradict the King's College study, which found that the possession of higher mathematics qualifications (as opposed to current knowledge or training in mathematics) did not appear to improve teachers' effectiveness.

Our next task is to investigate this association between subject knowledge and classroom performance. It is possible that: * Secure subject knowledge really does underpin and enhance teaching in the primary years. However, as there were five "A3" students, a high level of subject knowledge is not sufficient to ensure even a capable level of competence in teaching mathematics.

* A poor result on the audit could have demoralised and de-motivated students, contributing to their difficulties in teaching.

* Fundamental factors such as commitment and motivation (some would add intelligence) may underpin success in all areas - academic and professional - of a PGCE course.

Tim Rowland, Sarah Martyn, Patti Barber and Caroline Heal Further information: Dr Tim Rowland, Institute of Education, 20 Bedford Way, London WC1H 0AL, e-mail: t.rowland@ioe.ac.uk

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