Does it Matter Primary Teacher Trainees

Does it Matter? Primary Teacher Trainees’

Subject Knowledge in Mathematics

MARIA GOULDING,University of York TIM ROWLAND, University of Cambridge PATTI BARBER, University of London

ABSTRACT The mathematical subject knowledge of primary teacher trainees in England and Wales now has to be audited in line with government requirements for initial teacher training. This article examines how this knowledge has been conceptualised and presents the results of research in two institutions into the audited subject knowledge of primary teacher trainees and its relationship with classroom teaching. The research identiŽed weaknesses in understanding, particularly in the syntactic elements of mathematics, and a link between insecure subject knowledge and poor planning and teaching. The dilemmas and policy issues which this focus on subject knowledge presents are discussed.

Introduction

Government control of the curriculum and assessment in schools in England and Wales since the introduction of the National Curriculum in 1989 has been mirrored by increasing prescription of the curriculum and assessment for initial teacher training (ITT). For instance, there is now an ITT national curriculum (ITT NC) for primary English, mathematics and science, with a strong emphasis on trainee subject knowledge in these core subjects. For each, the knowledge and understanding which will ‘underpin effective teaching’ in the primary phase has been prescribed (Department for Education and Employment [DfEE], Circulars 10/97 [1997], 4/98 [1998]) in one of the three sections of this curriculum. This is labelled ‘Trainees’ knowledge and understanding of mathematics’ and amounts to a list of mathematical topics, together with related examples from the programmes of study for pupils in school, although the relationship is clearer for some topics than for others. Institutions are required to audit this knowledge and, where gaps are found, to make sure these are Žlled by the end of the course.

This new emphasis is not unexpected, since there is a national drive to improve standards in literacy and numeracy, and those close to policy-makers have clearly believed for some time that strengthening primary teachers’ subject knowledge will

contribute to this aim (Alexander et al., 1992; OfŽce for Standards in Education [OFSTED], 1994). More recently, Her Majesty’s Inspectors (OFSTED, 2000), in an evaluation of the Žrst year of the National Numeracy Strategy, located weaknesses in teachers’ subject knowledge, particularly ‘teaching of progression from mental to written methods; problem solving techniques; and fractions, decimals and percentages’ (p. 6). Given these arguments and the reality of the subject-based National Curriculum in primary schools, it seems sensible, then, to include a focus on teachers’ subject knowledge at the training stage. Even if we were conŽdent about trainees’ knowledge and understanding of mathematics, we would need to recognise that the vast majority of trainees have tended to specialise in non-mathematical subjects after the age of 16, and may need to refresh and deepen their understanding of mathematics before entering the classroom. This needs to be tackled by ITT providers with some sensitivity, as recent studies (Brownet al., 1999; Green & Ollerton, 1999) have identiŽed primary trainees’ anxiety about mathematics as a major issue. In the past, however, ITT courses have been successful in increasing students’ conŽdence in their ability to teach mathematics and in shifting their attitudes and beliefs about the subject (Carre´ & Ernest, 1993; Brownet al., 1999).

Although as teacher educators we have a long-standing interest in subject knowledge, more recently we have been responding to the Government ’s requirements as set out in Circular 4/98 (DfEE, 1998) and using them as an opportunity to contribute to research in this area. This article is organised into three parts. In the Žrst, we clarify our own position on subject knowledge with reference to theoretical conceptions and empirical evidence from the UK and elsewhere. In the second, we present some Žndings from our research into the mathematical subject knowledge of primary teacher trainees and its relationship to classroom teaching. We conclude the article with a discussion of the dilemmas and policy issues which this focus on subject knowledge presents.

What is Mathematical Subject Knowledge and What Do We Know about It?

Concern with teachers’ knowledge of mathematics is not conŽned to England and Wales. In the USA, both theoretical frameworks and empirical research on teacher knowledge are well developed (Grouws, 1992). Much of this research is premised on the belief that learning is a product of the interaction between what the learner is taught and what the learner brings to the learning situation. For pre-service teachers, therefore, what they bring to training courses would seem to be crucial. One researcher in the Želd of elementary (primary) teachers’ mathematics subject knowledge has commented that:

This lack of attention to what teachers bring with them to learning to teach mathematics may help to account for why teacher education is often such a weak intervention—why teachers, in spite of courses and workshops, are most likely to teach math just as they were taught. (Ball, 1988, p. 40)

Conceptualising Subject Knowledge

Lee Shulman’s categories of teacher knowledge, and, in particular, his constructs of

subject matter knowledge, pedagogical content knowledge and curricular knowledge

have been very inuential in the USA and beyond. These classiŽcations are generic but can be applied with very little adjustment to the discipline of mathematics.

Subject matter knowledge(SMK) is the ‘amount and organization of the knowledge per se in the mind of the teacher’ (Shulman, 1986, p. 9), and is later (Shulman & Grossman, 1988) further analysed into substantive knowledge (the key facts, concepts, principles and explanatory frameworks in a discipline) and syntactic knowledge (the rules of evidence and proof within that discipline). Hispedagogical content knowledge

(PCK) consists of:

the most powerful analogies, illustrations, examples, explanations and demon-strations—in a word the ways of representing the subject which makes it comprehensible to others … [it] also includes an understanding of what makes the learning of speciŽc topics easy or difŽcult. (Shulman, 1986, p. 9)

The boundaries between SMK and PCK may well be blurred; students who have multiple representations for mathematical ideas and whose mathematical knowledge is already richly linked will be able to draw upon these both in planning and in spontaneous teaching interactions. In such cases we would argue that the students’ subject matter knowledge is ripe for exploitation and that in turn the experience of teaching will feed back into and enrich subject matter knowledge.

Shulman’scurricular knowledgeconsists of knowledge of the scope and sequence of teaching programmes and the materials used in them. Again, there could be a feedback loop, as teachers encounter new ways of thinking about and representing mathematical ideas in textbooks and other resources.

Research on Teachers’ Mathematics Subject Knowledge

Deborah Ball echoes Shulman’s constructs of substantive and syntactic knowledge in any discipline by making a distinction between knowledge of mathematics (meanings underlying procedures) and knowledge about mathematics (what makes something true or reasonable in mathematics). Investigating both elementary and secondary pre-service teachers’ understanding of division, she found (Ball, 1990a) that both had signiŽcant difŽculties with themeaningof division by fractions. Most could do the calculations, but their explanations were rule-bound, with a reliance on memorising rather than conceptual understanding. They were also at a loss as to how they could justify answers. Although they wanted to be able to give pupils’ meaningful explanations, their own limited subject knowledge would prevent them from being able to realise their aim. They believed that mathematics could be meaningful but lacked the knowledge of these meanings them-selves.

Division by fractions was also the focus of a short account by Swee Fong Ng (1998), working with trainee teachers on illustrative diagrams which gave meaning to the answers obtained by using a division algorithm. Liping Ma’s (1999) comparative study of Chinese and US teachers’ knowledge of ‘fundamental’ mathematics also investigated dividing fractions by fractions as one of four problem scenarios. Compared with the American teachers, who mostly had a limited and sometimes faulty understanding, the Chinese teachers were able to do the calculation, provide examples of the concept in context and discuss concepts underpinning the understanding of division. In other topics, they were also able to articulate salient mathematical structural properties, e.g. distribu-tivity to justify the standard procedure for ‘long multiplication’. Ma concludes that no amount of general pedagogical knowledge can make up for ignorance of particular mathematical concepts.

In her earlier study, Ball (1988) explored the distinction between what mathematics should be known and how it should be organised. The ‘what’ tends to be listed, as in Circular 4/98 (DFEE, 1998), and the ‘how’ tends to be described qualitatively by words such as ‘exibly’, or ‘in-depth’. Her term ‘connected’ (Ball, 1990b), derived from comparison of the knowledge held by expert and novice teachers, has since been used by others as a way of describing the quality of subject knowledge that teachers need (e.g. Askew et al., 1997). So in the case of division, which would be expected to appear in any list of content of mathematics subject knowledge for primary teachers, the knowl-edge displayed by Ma’s Chinese teachers could be described as connected, or in her terminology, ‘profound ’. Swee Fong Ng’s approach with prospective primary teachers could be seen as a way of developing connected subject knowledge in training.

In their training, Ball argues that not only should mathematics be revisited, but also that pre-service teachers may also need to ‘unlearn’ what they know about teaching and learning of mathematics. Her programmes at Michigan State University are designed ‘to surface and challenge’ assumptions by requiring students to engage with some unfamiliar mathematics (e.g. permutations) themselves, observe her working with a young child exploring the concept, and then take on the role of teacher in exploring the concept with someone else, not necessarily a school pupil. The reections of the students on these courses demonstrate the effect of this ‘unlearning’ on their self-awareness and their beliefs about mathematics. One student wrote:

Research on the Relationship between Subject Knowledge and Classroom Teaching

If categorising teachers’ knowledge is a complex task, then trying to explore it in action is even more difŽcult, quite apart from making judgements about the relationships between variables such as subject matter knowledge and pupil progress. Aubrey (1997) acknowledges this in her comprehensive review of the literature on competent teaching performance, largely derived from US comparisons of expert and novice teachers, but argues for ‘the central importance of disciplinary knowledge to good elementary (primary) teaching’ (p. 33).

Aubrey’s conceptualisation owes much to Shulman but she uses a superordinate category of pedagogical subject knowledge, incorporating SMK, knowledge about children’s understanding and curricular knowledge. Her study set out to investigate teachers’ pedagogical subject knowledge, the informal knowledge of pupils at school entry and their teachers’ understanding of what their own pupils know and how they think about mathematics, in order to consider the implications for the mathematics curriculum in the Žrst year of schooling.

Although the sample for this study was small (four teachers in three English schools), the qualitative data was Žne-grained and gathered over the course of one year. Aubrey identiŽed pedagogical subject knowledge through observation of classroom discourse— the representations teachers used, what they said, did or demonstrated, what the children said, did or showed, and from interviews. These observations in a ‘real’ setting contrast with both Ma and Ball’s approaches, which drew on task-focused interviews with the teachers. It is important to note these differences, although an extended discussion of measures of effectiveness is beyond the scope of this article. Aubrey’s criteria for teacher effectiveness were qualitative descriptions of teachers’ behaviours, with one of the teachers ‘[displaying] precisely those teaching behaviours associated in the effective instruction literature with higher achievement scores’ (p. 145); for example:

· _{the systematic presentation of new ideas;}

· _{making explicit links between different representations (verbal, concrete, numerical,}

pictorial).

These behaviours, claims Aubrey, were possible because of the interaction between the beliefs about learning and strength of the SMK held by this teacher, enabling her to coordinate and use both her own mathematical knowledge and that which these young pupils already had on school entry. Although Aubrey did use pre- and post-observation interviews with the children, she claims that it was not possible to make any link between the identiŽed progress of pupils over the year and the teaching behaviours of the four teachers.

questions about the amount of time devoted to mathematics on postgraduate certiŽcate in education (PGCE) courses and the relative contribution of ITT and CPD to deepening understanding.

Doubt about the effectiveness of mathematics specialists had earlier been highlighted in one part of a study of primary PGCE trainees (Carre´ & Ernest, 1993) which involved a comparison of the teaching of mathematics by the specialists, with that of the whole cohort. The ‘specialist’ trainee teachers were so designated on the basis of their choice of specialist training in the course, which did not necessarily reect an advanced mathematics qualiŽcation. When classroom teaching performance was assessed, it emerged that ‘there is virtually nothing to distinguish mathematicians and others in teaching mathematics’ (p. 161). The situation was very different in the teaching of music (Bennett & Turner-Bisset, 1993), where specialists were judged to perform at a higher level of competence than other students. This was in accordance with the researchers’ expectations (p. 164), since music specialist students had been assessed as having a high level of music subject matter knowledge relative to other students, while this was not the case for mathematics specialists vis-a`-vis mathematics.

Synthesis

Our own work in this area has been informed by the theoretical and empirical research summarised above, applied within the speciŽc context of government regulation of ITT in England. The construct of connected SMK, incorporating ideas of knowledge of (substantive) and knowledge about (syntactic) mathematics, was inuential in our design of the audit assessment items and our approach to the interpretation of the written responses. We intended to investigate further the relationship between SMK and teaching performance, hypothesising that SMK would inuence both students’ planning and teaching, a cognitive dimension encompassing beliefs about mathematics, and their conŽdence in the classroom. This link was examined quantitatively using a statistical analysis and also qualitatively using case studies. Our understanding of trainees’ substantive and syntactic mathematics SMK was enhanced by micro-analysis of the written responses to some of the more problematic test items. In this way, the study provides new information about British primary teacher trainees’ SMK and the relation between SMK and teaching. In bringing together closely related work by researchers from more than one institution, our account has the advantage of providing insights derived from alternative instruments and interpretations from several perspectives. The synthesis of our Žndings has been made possible by a year of collaborative effort, which is still ongoing.

Teacher Trainees’ Subject Knowledge and Classroom Performance within the National Curriculum for Initial Teacher Training

The subject knowledge required (by Circular 4/98 [DfEE, 1998]) to be audited in ITT courses goes beyond the primary curriculum and represents a much wider range of mathematical content and process than that covered in either the Aubrey study or that at King’s College.

be developmental, that links should be made between the subject knowledge and school classrooms, that the audit should be diagnostic but manageable and that areas of strength and weakness should be identiŽed in the Career Entry ProŽle which new teachers take into their Žrst post as a basis for induction and CPD.

We have been involved with ITT programmes in three institutions: the University of Cambridge, the University of Durham and the Institute of Education, University of London. At the time of this study, each had its own procedures for addressing the subject knowledge audit, with similarities and differences of emphasis and timing. In each, however, there was a written audit consisting of 15–20 items under somewhat ‘formal’ conditions. Support, feedback and follow-up varied, although working with peers was a common feature. At the Institute, peer teaching was, and continues to be, a planned element, organised on the basis of an early self-audit. Our ongoing collaboration has resulted in a convergence of support and audit practices in the three institutions, with a common audit instrument.

At the Institute of Education and at Durham, independent research in 1998 and 1999 explored what mathematics (as speciŽed in the government circulars) primary trainees (n₅154, n₅201 respectively) found difŽcult, and the nature of their errors and misconceptions (Rowland et al., 2000; Goulding & Suggate, 2001). In addition, the Institute team also researched:

· the relationship between subject knowledge as identiŽed by the audit and students’

performance on teaching practice; and

· the histories, attitudes and professional trajectories of those trainees scoring highly on

the audit.

Further information on students’ difŽculties with aspects of proof was obtained in the subsequent year, together with a more detailed exploration of the relationship between audit performance and teaching practice performance (Rowland et al., 2001). In the following section we will:

· summarise our Žndings on aspects of the trainees’ SMK as identiŽed by the audit by

a micro-analysis of difŽculties, errors and misconceptions;

· _{report the Žndings of our enquiries into a possible link between SMK and teaching}

performance, using a statistical analysis and a case study.

Subject Matter Knowledge

Facilities in the four ‘easiest’ and ‘hardest’ of the 16 items in the 1998–99 audit at the Institute of Education are shown in Table I. Our attempts to get beneath the raw numbers have been signiŽcantly assisted by keeping in mind the distinction between substantive and syntactic kinds of subject matter knowledge. This distinction has been described earlier in this article, and can be thought of in terms of the differences between product knowledge and process knowledge.

Substantive knowledge. In comparing the differences in the difŽculties, errors and misconceptions revealed by the audit process in the two institutions, we need to acknowledge that there are differences in the actual assessment instruments used. For instance, at the Institute, a topic with high facility (94% secure) involved ordering decimals, as assessed by the item:

TABLEI. Audit topics with the highest and lowest facility ratings

(a) Highest facility (b) Lowest facility

% Mean % Mean

secure1 _score2 _{Topic secure score Topic}

983 _{1.96 Inverse operation s 52 1.17 Pythagoras, area} 95 1.92 ordering decimals 49 1.15 Generalisatio n 93 1.90 Divide 4-digit number by 2-digit 33 0.87 Reasoning and proof 92 1.90 Problem solving in a money 30 0.73 Scale factors, percentag e

context increase

1

The written response gives a high level of assuranc e of the knowledge being audited. 2_{A secure respons e scores 2, which is therefor e the maximum possible for the mean.} 3

All cohort percentage s have been rounded to the nearest percentage .

In contrast, at Durham, only 56% were successful in ordering decimals, as revealed by the question:

Arrange in order from the largest to the smallest:

0.203; 2.35₃1022_{; two hundredths; 2.19}

3 1021; one Žfth

Note: The Durham students were allowed to use calculators, and did so to evaluate standard form expressions, so this was not the source of error.

The Institute question was encouraging in showing that students were free of the common LS ‘largest is smallest’ or DPI ‘decimal point ignored’ errors revealed from research on school pupils (Mason & Ruddock, 1986). In the same area of ordering decimals, however, the Durham question exposed weakness in two elements of substan-tive subject knowledge: the lack of connection in the students’ knowledge between different forms of numerical expression, and difŽculties with more than two places of decimals. So, even if students had the knowledge necessary to deal with common pupil misconceptions in decimals, they may not have a sufŽciently strong understanding to respond to pupils who, for instance, Žnd reference to decimals or standard form when researching, for instance, the planets, or small particles.

At Durham, the question with the lowest facility (over 80% of students making errors) involved judging an appropriate degree of accuracy in an area calculation, an aspect not investigated at the Institute.

Some children were measuring their desk to the nearest centimetre. They found it was 53 cm by 62 cm. State the possible limits to the length of their sides. Work out the area to an appropriate degree of accuracy.

Discussions post-audit revealed a healthier understanding of approximation when the problem was thought of in a practical context rather than a test item, but there were difŽculties with accepting that measurement is continuous, e.g. in knowing that there could be measurements between 53.4 and 53.5. In our view, this problem relates not only to understanding the real number line but also to having a good feel for the measurement ideas which underpin practical work in science, as well as mathematics.

Whole number division was performed correctly by most students in both audits, although a few Institute students gave 2912 ₄ 14 equal to 28, the output from a (faulty) algorithmic computation. Weaknesses in the meaning of integer subtraction and fraction division, highlighted in studies mentioned previously, were revealed by the Durham question, which required students to provide contexts for (2 4 14) and (112

23% unable to provide a context or providing an inappropriate one for both questions. These understandings of subtraction and division are crucial if pupils are to be enabled to accommodate ideas of subtraction other than ‘taking away’ and division as ‘sharing’ (Nunes & Bryant, 1996; Anghileri, 2001).

Syntactic knowledge. Notable student weaknesses were exposed by items involving generalisation in both studies. For instance, just over half the Institute sample was insecure in the item designed to address the ability to express generality in words and symbols:

Check that

31415₅334 819110₅339 29130131₅33 30 Write down a statement (in prose English) which generalises from these three examples. Express your generalisation using symbolic (algebraic) notation.

Some students did not recognise the features common to all three examples and tried to derive a generality from the Žrst case only. Others were able to express the generality in their own words but not symbolically. Some expressions, e.g.

a 1 b 1 c₅3b

captured part of the picture but omitted the crucial condition that the three numbers being summed are consecutive.

Secure responses illustrated the generality using symbols, and the reason why the relationship holds was implicit in the symbolism. To prove that this relationship must hold for all sets of three consecutive whole numbers, their general statement gave them the starting point.

General statement a 1 a 1 1 1 a 1 2₅3(a11) Proof (working deductively from the sum on the left)

a 1 a 1 1 1 a 1 2₅3a 1 3

53(a 1 1) This is the expression on the right.

Although not asked to do this in the question, some students produced reasoning similar to that above and some used their symbolism more elegantly:

n 21 1 n 1 n 1 1₅3n 21 1 1

5 3n This is three times the middle number.

This question, then, exposed weaknesses in recognising pattern and relationships, identifying signiŽcant elements and in formulating expressions to represent these relationships. This relates to syntactic subject knowledge, since reasoning and proof may involve testing and then proving or refuting conjectures derived from a small number of cases by inductive reasoning.

In the Durham item designed to assess proof (40% secure), a general statement was already given so the relationship did not have to be recognised:

Prove that if any two odd numbers are added together, the result will be an even number.

arguments rather than accepting what may seem obvious from a few examples. An Institute item on reasoning and argument was also amongst those with lowest facility (33% secure). Insights into students’ understanding of reasoning and proof were elicited by the following question

A rectangle is made by Žtting together 120 square tiles, each 1 cm2_{. For}

example, it could be 10 cm by 12 cm. State whether each of the following three statements is true or false for every such rectangle. Justify each of your claims in an appropriate way:

(a) The perimeter (in cm) of the rectangle is an even number.

(b) The perimeter (in cm) of the rectangle is a multiple of 4.

(c) The rectangle is not a square.

Only one-third of students made a secure response to the whole question and 30% either gave insecure answers to all three parts or did not attempt the question (see Rowlandet al., 2001, for a detailed analysis). Most interesting, perhaps, is the fact that a signiŽcant number of students did not seem to perceive the second statement as amenable to personal investigation on their part, which (for those who did so) uncovers counter-ex-amples to refute the statement. Some claimed, for example, that the statement must be true because 120 is a multiple of 4, or even because the rectangles have four sides. These prospective teachers evidence little or no sense of mathematics as an experimental test-bed, in which they might conŽdently respond to an unexpected student question, ‘I don’t know, let’s Žnd out’. Ma (1999) perceives teachers’ curiosity and conŽdence to investigate such propositions as an important ingredient of what she calls ‘profound understanding of fundamental mathematics’. Ma describes a ‘dilemma’ situation in which a primary pupil says that she has discovered something about rectangles: ‘The bigger the perimeter, the bigger the area’. The pupil’s evidence is comparison of a 4 3 4 rectangle with an 8 ₃ 4 rectangle. Ma reports that some teachers (in the USA) presented with this dilemma accepted the claim on the basis of this evidence. The majority said they didn’t know whether it was true, and would have to look it up in a textbook. Most of the Chinese teachers, by contrast, set about investigating different ways of increasing the perimeter, and the effect on area in each case.

In both the audits, then, weaknesses in substantive knowledge and in syntactic knowledge were revealed. At Durham, where Želd notes were taken during the taught sessions pre-audit, problems with fractions, algebraic symbolism and proof were very clear and occasionally accompanied by expressions of inadequacy, fear and even panic (Buxton, 1981). This was not unexpected, and tutors sought to reassure students that extra tutorial support would be given, with opportunities to work on and remedy any weaknesses after the audit. Students were also strongly encouraged to work with peers on the support material. Accommodating students’ anxieties did, however, lead to some instrumental teaching on the part of tutors, narrowly focused on the type of assessment item to be found on the audit, despite misgivings about this strategy.

The Relationship between Audited Subject Knowledge and Teaching Performance

medium or high, corresponding to the need for signiŽcant remedial support, modest support (or self-remediation), or none. Towards the end of that course, speciŽc assess-ments of the students’ teaching of number were made on the second and Žnal school placement (against the standards set out in Circular 4/98 [DfEE, 1998]) on a three-point scale of weak/capable/strong. These data did not support a null hypothesis that the spread of performance in the teaching of number was the same for the three categories identiŽed in the subject knowledge audit. There was an association between mathematics subject knowledge (as assessed by the audit) and competence in teaching number. Further analysis pinpointed the source of rejection of the null hypothesis: students obtaining high (or even middle) scores on the audit were more likely to be assessed as strong numeracy teachers than those with low scores; students with low audit scores were more likely than other students to be assessed as weak numeracy teachers. In effect, there is a risk which is uniquely associated with trainees with low audit scores.

For the second cohort (n₅164), more extensive data from school placements enabled comparison of mathematics subject knowledge with teaching performance (a) on both Žrst and second placements, and (b) with respect to both ‘pre-active’ (related to planning and self-evaluation) and ‘interactive’ (related to the management of the lesson in progress) aspects of mathematics teaching (Bennett & Turner-Bisset, 1993). In fact, the association between audit score and teaching performance was signiŽcant for three of the four analyses, the exception being the pre-active dimension of the Žrst placement. This anomaly will be the subject of future statistical veriŽcation with a larger sample, together with close scrutiny of a sample of trainees in school, with the aim of enhancing our understanding of ‘what’s really going on’ in all four aspects of the analysis. For the moment, we conjecture that the assessment of preactive aspects identiŽes clarity about mathematics teaching and learning, but that this clarity may be obscured in the planning on the Žrst practice, when both tutors and trainees may be more concerned with the form rather than content of planning Žles. That apart, poor subject matter knowledge as identiŽed by the audit was associated with weaknesses in planning and teaching primary mathematics.

The Žndings do not contradict the King’s study, where no relationship was found between higher mathematical qualiŽcations and teaching effectiveness. The audit was assessing students’ current knowledge of material more directly related to primary school mathematics. ‘Connectedness’, the factor associated with teaching effectiveness by Askew et al. (1997), may have been elicited better by the audit than by the school examinations taken some years earlier. Nonetheless, we need to be clear about the relationship identiŽed at the Institute between audit score and teaching performance. We are not claiming that this statistical relationship is causal, since there are many other factors involved in teaching. In particular, the importance of context cannot be underes-timated, as the case study included later will demonstrate.

If the audit is measuring connectedness in some way, then perhaps this is a feature of students’ knowledge across a range of subjects and may reect some general quality of understanding, either cognitive or dispositional, or a combination of both. Some might claim that the good teachers of number were those with strong personal motivation, or even with high general ‘intelligence’, who would tend to do well in the audit.

leading them to avoid spending time in planning mathematics sessions in favour of subjects about which they felt more positive. It is possible that the emphasis on the audit and the remediation process had a demotivating effect on these students. Evidence from the Durham study suggests that this is a real possibility for the very small number of trainees who expressed anxiety during follow-up sessions, or whose behaviour was resistant or defensive during them. However, a downward spiral due to a poor audit score does not resonate with our knowledge of most primary PGCE trainees. There is stiff competition to gain a place on these courses; most students are resilient, well motivated and goal-oriented. Indeed, the majority of those who required some remediation, including some very weak students, appeared to respond positively to the opportunity and reported in evaluations that they were pleased to address some of their weaknesses before the main teaching practice. There is no room for complacency here, however, so anxiety and its potential for depressing performance on teaching practice will be investigated in the next phase of the research.

Subject Matter Knowledge and Teaching Performance: a case study

The importance of context and the complexity of the relationship between subject matter knowledge and conŽdence were pointedly illustrated by the only Institute trainee who scored highly in the audit but did not claim to feel conŽdent about teaching the same mathematics to someone else. The portrait of Frances (Rowland et al., 2000), an early years specialist, based on tutors’ reports, a coursework essay and a semi-structured interview, reveal that successful teaching of mathematics is not guaranteed by subject knowledge alone.

In her Žrst practice, Frances experienced a disabling tension between her own pedagogical knowledge, beliefs and practices and those of the class teacher. Although her coursework essay demonstrated a sophisticated understanding of the concept of subtraction supported by knowledge of the research literature, and a desire to develop a richer understanding in the pupils than that expected by the class teacher, Frances lacked the conŽdence to put her intentions into practice. She also wanted pupils to experience powers of reasoning by engaging in investigative problem-solving activities but resorted to time-Žlling activities and day-to-day planning because her lack of Žrm diagnostic assessment caused problems of management and control.

In the second practice, however, Frances was able to coordinate her own knowledge, and that of the pupils, as described by Aubrey (1997), in a classroom setting more in tune with her own approach to learning. She was now able to clarify her own understanding ‘from Žrst principles’ and identify and build upon the signiŽcant points in children’s progression. Whereas her tutor’s assessment on the Žrst practice had identiŽed formative assessment as weak, this aspect was assessed at the highest level in the second. This success could not be attributed to any one element but it was almost certainly an integration of

· _SMK;

· a personal disposition to analyse what she was going to teach;

· the developing ability to use formative assessment;

· her use of reection and self-criticism;

· increasing conŽdence and a positive attitude.

state was outstripping her ability to perform in a less than favourable situation, resulting in confusion and lack of conŽdence. With more experience, and a different placement, she was better able to coordinate the different strands of her pedagogy. This is not to say that SMK was not a factor in her performance; rather, that it required the right conditions to be brought into play.

Dilemmas and Policy Issues

Before the auditing of subject knowledge became statutory, the course at Durham included a subject knowledge element for trainees in the Key Stage 2 (ages 7–11) phase. Tutors were sufŽciently convinced by the research being done in their own institution and research elsewhere that work on subject matter content accompanied by reection on learning would feed in to pedagogical content knowledge and curricular knowledge. Students chose their own topics to pursue in depth, investigated avenues new to them and reected on their own learning in the process. They were speciŽcally asked to consider the role of discussion and the role of the more experienced other (peer or tutor) in the process. This aspect was included in presentations made at the end of the project, although most students chose to focus on the mathematics on which they had been working. It was clear from tutors’ informal observations and students’ written evalua-tions that there were very mixed reacevalua-tions to this part of the course, with some students relishing the opportunity and others failing to make a connection between this and their work as primary teachers. We are bound to record that two consecutive OFSTED inspections were also divided—the Žrst (conducted by an HMI) suggested that the subject knowledge sessions should be extended to include the Key Stage I (ages 5–7) cohort, whereas the second (conducted by an Additional Inspector, himself a Žrst school headteacher) criticised the lack of relevance to primary teaching. Circular 4/98 (DfEE, 1998), then, gave some legitimacy to this element of the course, although radical changes had to be made to comply with the ITT NC requirements—choice was replaced with coverage, and the presentations with the audit and follow-up.

In the light of this division of opinion, the research reported here lends weight to the argument that there is a connection between SMK as assessed by the audit and the teaching of mathematics. We are persuaded that the relationship involves both cognitive and affective dimensions. One of the dilemmas, however, is whether the new require-ments are creating anxiety and dislike of mathematics or acting as a useful lever for development. We do not want students to jump through more hoops and add further layers of instrumental understanding to existing ones. This dilemma raises questions about how the requirements are implemented, the nature and function of the audit, the mathematical topics covered, and the relationship of this element to what is taught on the mathematics methods course and the training course as a whole. There is also an important policy issue as to whether SMK can be dealt with adequately during initial training or whether it should be an ongoing concern, supported during induction and CPD. Our research demonstrates that this is particularly acute with regard to syntactic aspects of subject knowledge, with two-thirds of the Institute trainees exposing short-comings in the audit. The Durham team also found that trainees’ difŽculties with proof were particularly resistant to remediation in the postgraduate initial training year (Goulding & Suggate, 2001).

replace or complement tutor support. How the audit is used—as a way of weeding out unsuitable trainees or as a way of identifying areas of weakness with a view to strengthening and deepening understanding—is a matter of concern. Although weak SMK was found to be associated with poor classroom teaching, there were exceptions to this, just as there were some students with strong subject knowledge who turned out to be poor practitioners. At a time when recruitment to traditionally popular primary courses is becoming difŽcult for the Žrst time, ITT providers must be reluctant to reject or fail students who have the desire and commitment to become primary teachers. Improving teaching and learning on training courses would seem a much better option, although weaknesses in SMK associated with poor teaching performance could be grounds for failing a student at the end of the training course when all assessment opportunities have been exhausted.

There are still debates to be had about what should be included in a subject matter knowledge ‘list’ of content. The rationale for the list provided in the ITT NC is not self-evident and although early drafts were subject to consultation (with some welcome deletions, e.g. proof by induction for primary trainees), the process of its compilation and the anonymity of its authors does not inspire conŽdence.

Our discussion of generalisation and proof opens up a bigger question. This moves beyond the coverage of content to beliefs about the nature of mathematics, whether it is rule-bound and arbitrary, or meaningful, open to investigation and approachable from different directions. We believe that students with an impoverished view of mathematics should have an opportunity to make this shift, and that this change in perspective may have a profound (but difŽcult to measure) effect on their orientation to teaching mathematics. In saying this, we have to acknowledge our own special preferences, because similar arguments could be made for enriching students’ knowledge of language, music or art; for example, through writing, making music or producing art.

We are sufŽciently convinced by our research on the students’ errors and misconcep-tions that there are some ‘big ideas’, notably in generalisation and proof, which cannot be satisfactorily addressed in the training phase, but which are important underpinnings for primary teaching. We would argue that students do need to understand that symbol manipulation is only a technical aspect of algebra and that ideas of structure, regularity and generality can be sown in the primary phase. But this will only happen if the teacher appreciates how these ideas underpin the regularity of the number system, exible methods for number operations and patterns in sequences. Mathematical proof is based on rationality and reasoning: it is about convincing mathematical argument, with roots in questions such as ‘Will that always work?’, ‘Why is that?’ and ‘Can you explain that to someone else?’ If students do not experience these enquiries themselves, how can they hope to encourage mathematical questioning in pupils? Acknowledging that such big ideas need more time and that connections with primary mathematics may not be made in the course of a one-year course has implication for CPD.

conŽdence derived from this security. It may be, therefore, that time needs to be found in training courses for speciŽc teaching sessions devoted to both substantive and syntactic elements, and that peer teaching has a role in this. Conducting and analysing the audit has highlighted the strengths and weaknesses of student teachers’ subject knowledge for the researchers; whether it has served its purpose or is still useful in encouraging student teachers to take this aspect of teacher knowledge seriously is open to debate. On balance, we would argue that the disadvantages outlined in this article are outweighed by the advantages. In other words, it is a device to ‘surface and challenge’ (Ball, 1988) trainees’ assumptions. The Chinese elementary teachers in Ma’s study have a comparatively light teaching load. Lessening contact time for our primary teachers, at least in the Žrst few years of their careers, so that they can continue to develop their understanding of big ideas in mathematics or other subjects is a possibility. Both of these recommendations depend on political and professional will, and would need economic backing.

Correspondence: Maria Goulding, Department of Educational Studies, University of York, Heslington, YO10 5DD, UK.

REFERENCES

ALEXANDER, R., ROSE, J. & WOODHEAD, C. (1992)Curriculum Organisatio n and Classroom Practice in Primary Schools(London, HMSO).

ANGHILERI, J. (2001) British research on mental and written calculatio n methods for multiplicatio n and division , in: M. ASKEW & M. BROWN (Eds) Teaching and Learning Primary Numeracy: policy, practice and effectivenes s(Southwell, Nottinghamshire , British Educational Research Association) . ASKEW, M., BROWN, M., RHODES, V., JOHNSON, D. & WILIAM, D. (1997)Effective Teacher of Numeracy

(London, King’s College).

AUBREY, C. (1997) Mathematics Teaching in the Early Years: an investigatio n into teachers ’ subject knowledge (London, Falmer Press).

BALL, D. L. (1988) Unlearnin g to teach mathematics,For the Learning of Mathematics, 8, pp. 40–48. BALL, D. L. (1990a) Prospectiv e elementary and secondar y teachers ’ understandin g of division ,Journal

for Research in Mathematics Education, 21, pp. 132–144.

BALL, D. L. (1990b) The mathematica l understanding s that prospectiv e teachers bring to teacher education, Elementary School Journal, 90, pp. 449–466.

BENNETT, N. & TURNER-BISSET, N. (1993) Case studies in learning to teach, in: N. BENNETT& C. CARRE´ (Eds)Learning to Teach, pp. 165–190 (London, Routledge) .

BROWN, T., MCNAMARA, O., JONES, L. & HANLEY, U. (1999) Primary student teachers ’ understandin g of mathematics and its teaching ,British Education Research Journal, 25, pp. 299–322.

BUXTON, L. (1981)Do You Panic about Maths?(London, Heinemann).

CARRE´_{, C. & ERNEST, P. (1993) Performance in subject-matte r knowledge in mathematics, in: N. BENNETT} & C. CARRE´ (Eds) Learning to Teach, pp. 36–50 (London, Routledge) .

DEPARTMENT FOREDUCATION ANDEMPLOYMENT(1997)Teaching, High Status, High Standards : Circular 10/97 (London, HMSO).

DEPARTMENT FOREDUCATION ANDEMPLOYMENT(1998)Teaching, High Status, High Standards : Circular 4/98(London, HMSO).

ERNEST, P. (1989) The knowledge, beliefs and attitudes of the mathematics teacher: a model,Journal of Education for Teaching, 15, pp. 13–33.

GOULDING, M. & SUGGATE, J. (2001) Opening a can of worms: investigatin g primary teachers ’ subject knowledge in mathematics,Mathematics Education Review, 13, pp. 41–54.

GROSSMAN, P., WILSON, S. & SHULMAN, L. (1989) Teachers of substance : subject matter knowledge for teaching , in: M. REYNOLDS (Ed.) Knowledge Base for the Beginning Teacher, pp. 23–36 (Oxford, Pergamon).

GROUWS, D. A. (Ed.) (1992)Handbook of Research in Mathematics Teaching and Learning(New York, Macmillan).

HARRIS, S., KEYS, W. & FERNANDES, C. (1997) Third Internationa l Mathematics and Science Study: Second National Report, Part 1(Slough, National Foundation for Educationa l Research) .

MA, L. (1999)Knowing and Teaching Elementary Mathematics : teachers ’ understandin g of fundamenta l mathematics in China and the United States(Mahwah, NJ, Lawrence Erlbaum Associates) . MASON, K. & RUDDOCK, G. (1986) Decimals(Windsor, APU/NFER-Nelson).

NG, S-F. (1998) What is a fraction divided by another fraction?Mathematics Teaching, 165, pp. 28–29. NUNES, T. & BRYANT, P. (Eds) (1996)Children Doing Mathematics(Oxford, Blackwell).

OFFICEFORSTANDARDS INEDUCATION(OFSTED) (1994)Science and Mathematics in Schools : a review

(London, HMSO).

OFFICEFORSTANDARDS INEDUCATION(OFSTED) (2000)The National Numeracy Strategy : the Žrst year: a report from HM Chief Inspecto r of Schools(London, HMSO).

ROWLAND, T., MARTYN, S., BARBER, P. & HEAL, C. (2000) Primary teacher trainees ’ mathematics subject knowledge and classroom performance , in: T. ROWLAND& C. MORGAN(Eds)Research in Mathematics Education Volume 2(London, British Society for Research into Learning Mathematics) .

ROWLAND, T., MARTYN, S., BARBER, P. & HEAL, C. (2001) Investigatin g the mathematics subject matter knowledge of pre-servic e elementary school teachers , in: M. VAN DENHEUVEL-PANHUIZEN (Ed.)

Proceeding s of the 23rd Conferenc e of the Internationa l Group for the Psychology of Mathematics Education , Volume 4, pp. 121–128 (Utrecht, Freudentha l Institute , Utrecht University) .

SHULMAN, L. (1986) Those who understand : knowledge growth in teaching ,Educational Researche r, 15, pp. 4–14.

SHULMAN, L. & GROSSMAN, P. (1988) Knowledge Growth in Teaching: a Žnal report to the Spencer Foundation (Stanford, CA, Stanford University) .