The Knowledge Quartet Story (Part 1) Tim Rowland
O
nce in a while, a questionnaire of some kind asks me to state my occupation. For most of the last decade, my response has been “Unsuccessfully retired”. That happy way of life is largely due to research carried out 20 years ago, initially with friends/colleagues here in Cambridge and others elsewhere in the UK. That research was located in the field of ‘mathematics education’, but I hope that many colleagues with an interest in teaching and learning, of any kind and in any phase, will find this account of interest. In 1997, I was working at the London Institute of Education (now a department of UCL). The Blair government was elected in May that year, with "Education, education, education" the headline mantra. Five years earlier, in a government-commissioned report, ‘three wise men’ had called for improvement of the knowledge base of mathematics teachers, and Ofsted had identified teachers’ poor subject knowledge as a contributory factor in low standards of mathematics attainment in primary schools (Ofsted, 1994). An Education Department Circular (DfEE, 1997) then set out what it considered to be the “knowledge and understanding of mathematics that trainees need in order to underpin effective teaching of mathematics at primary level”, and in September 1998, “audit and remediation” of students’ subject knowledge became statutory. The only saving grace, back in 1998 at least, was that the means of assessment (so-called ‘auditing’) of pre-service primary teachers’ knowledge was not prescribed. Drawing on a body of well-established research that had identified common ‘elementary’ mathematics errors and misconceptions, my London colleagues and I devised a 16-item instrument to assess what our students knew about the mathematics relevant to their intended careers in primary schools. We also put in place arrangements to support and assist them where ‘gaps’ appeared to have been identified (Rowland et al., 1999). Analysis of the trainees’ responses to the audit enabled us to identify aspects of mathematics, such as generalisation, that seemed to be problematic for many of our students. In a follow-up study, we found a positive association between their mathematics subject knowledge – as assessed by the audit instrument – and their competence in teaching number, as assessed by their placement supervisors. In Autumn 1999 I was back in Cambridge. In Durham (and later, in York), Maria Goulding and her colleagues were conducting their own audit-driven research into trainees’ mathematics subject knowledge. I was now enjoying working in primary mathematics with Peter Huckstep and Anne Thwaites, then Fay Turner and Jane
Warwick, and together – supported by University Research Development funding – we instigated collaborative research with our colleagues in London and York, with the acronym ‘SKIMA’ (Subject Knowledge In MAthematics). When we began, each university had its own PGCE mathematics audit procedures, but our collaboration resulted in a convergence of support and audit practices in the three institutions, with a common audit instrument. A report of our findings when we implemented the audit in our three universities (Goulding, Rowland and Barber, 2002) continues to be cited in publications on mathematics teacher knowledge. By now, we ‘knew’ that secure mathematical knowledge, as assessed by our mathematics audit, is associated with greater competence in both the planning and the ‘delivery’ of elementary mathematics teaching. The next question needing to be answered was, of course – why is it so? In what ways is primary trainees’ teaching of mathematics – planning, reflection and classroom interaction – informed by their own knowledge and understanding of mathematics? In what ways can it be seen to ‘play out’, to make a difference, as they work with their students in the classroom? As to what we did to investigate this question – I shall do my very best to ‘keep it brief’! From one cohort of 149 Cambridge primary PGCE students, 12 trainees were identified to represent, as a group, the gender balance in the course as a whole, their upper/lower primary specialism, and their performance on the mathematics audit. The usual ethical practices concerning consent and awareness of the purpose of their involvement were observed, with respect both to the trainees and the participating schools. Two mathematics lessons taught by each of the trainees during their final 8-week school placement were observed and videotaped. They were asked to provide a copy of their planning for the observed lesson. As soon as possible after the lesson (usually the same day) the observer/researcher wrote a brief Descriptive Synopsis of what happened in the lesson, in order to contextualise subsequent discussion of any events within it. As for the analysis of these lesson regarding the ‘manifestation’ of these trainees’ mathematics knowledge as they taught, since there was no relevant analytical theoretical framework for us to draw on back in 2002, we took a ‘bottom-up’ approach, undertaking so-called ‘grounded’ analysis of the data, following the lead of US medical sociologists Barney Glaser and Anselm Strauss some 30 years earlier. At the outset, we did not know what kind of ‘theory’ might emerge from our close scrutiny of the lesson videotapes. rsma newsletter september 2021 page 13
