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Primary Teacher Trainees’ Mathematics Subject knowledge and Classroom Performance

Tim Rowland, Sarah Martyn, Patti Barber, Caroline Heal

Institute of Education, University of London

Paper presented at the British Educational Research Association Annual Conference, University of Sussex at Brighton, September 2-5 1999

 

UK government agencies have recently set in train requirements to ‘crack down’ on primary [elementary] school teacher trainees whose own knowledge of mathematics is weak. The responsibility to identify and support (or fail) them currently rests with training providers (mainly university schools of education). We describe one approach to this process and some findings concerning what trainees find difficult and how their knowledge is related to their teaching competence. We flesh out these findings with a case study of a mathematically-strong trainee whose path to qualification was less than smooth.

Recent changes in the curriculum for Initial Teacher Training incorporate a stronger focus on trainees’ subject knowledge (DfEE, 1997). Some evidence would seem to support this shift of emphasis. In the US, Kennedy’s research (1991) suggested that teachers’ mathematical understanding is frequently limited, whilst in the UK Alexander et al. (1992) called for improvement of the knowledge base of teachers in order to improve the teaching of mathematics. Inspection evidence identifies teachers’ lack of subject knowledge and confidence in mathematics as being a contributory factor in low standards of mathematics attainment of pupils (Ofsted, 1994).

Circular 10/97 (DfEE, 1997)(1) sets out what is considered to be the "knowledge and understanding of mathematics that trainees need in order to underpin effective teaching of mathematics at primary level". Since September 1998, audit and remediation of students’ subject knowledge has been statutory.

All providers of ITT must audit trainees’ knowledge and understanding of the mathematics contained in the National Curriculum programmes of study for mathematics at KS1 and KS2, and that specified in paragraph 13 of this document. Where gaps in trainees’ subject knowledge are identified, providers of ITT must make arrangements to ensure that trainees gain that knowledge during the course …(DfEE, 1997, p. 27)

At the Institute of Education, we piloted the new ITT curriculum one year ahead of time. In this paper, we describe our approach to the audit of the mathematics subject knowledge of a cohort of 154 trainees following a one-year primary PGCE course. We present some findings related to a number of different aspects of our research.

APPROACH AND TIMING

The structure of the primary PGCE at the Institute of Education is perhaps unusual, in that the methods course for each core curriculum subject is taught in three intensive 5- or 6-day blocks, one in each term. The blocks for mathematics are timetabled first in each term, so that by the middle of January, with fully six months of the course remaining, the main content areas - number concepts and operations, data handling, mathematical processes, shape and space, measures, algebra, probability - have been ‘covered’ in lectures and workshops. This therefore seemed to us to be the optimum moment for an audit of subject knowledge, giving the trainees maximum opportunity and professional motivation to recall those topics they had forgotten (for lack of use) since they did mathematics at school.

A 11/2 hour written assessment consisting of 16 test items in mathematics was therefore administered at this point of the course. Trainees had been given notice of the ‘test’ and a revision syllabus some six weeks earlier. Their response to each question included a self-assessment of their ability to complete it successfully. The scripts were marked and the response to each question coded either: secure, possibly secure, not secure. The middle category was created in recognition of the difficulty (resolved later) of making confident inferences from some of the written responses. Corresponding scores of 2, 1 and 0 respectively were recorded for each question and each student. In mid-February, an individual audit feedback sheet was returned to each student, with guidance (where appropriate) for further study.

39 students (about 25% of the cohort) who had been found to be secure in 15 or more of the 16 topics audited were invited to become mathematics peer tutors. Following training for this task, they conducted one-to-one peer tutoring sessions with all other students (on average, three per peer tutor) in April, writing a feedback sheet on each of their tutees.

TRAINEES’ MATHEMATICAL THINKING

One dimension of our research is to identify what mathematics (within the remit of Circular 10/97) primary trainees find difficult, and the nature of their errors and misconceptions in these areas. Facilities in the four ‘easiest’ and ‘hardest’ of the 16 items audited were as follows:

HIGHEST FACILITY

 

LOWEST FACILITY

% secure1

Mean score2

TOPIC  

% secure

Mean score

TOPIC

94

1.92

Ordering decimals  

63

1.46

Generalisation

94

1.92

Inverse operations  

60

1.40

Pythagoras, area

90

1.86

Divide 4-digit number by 2-digit  

43

1.17

Reasoning, argument

89

1.85

Order fractions  

40

0.95

Scale factors, percentage increase

1 The written response gives a high level of assurance of the knowledge being audited

2 A secure response scores 2, which is therefore the maximum possible for the mean.

It is important to try to interpret some of these findings within the context of the course and the audit. For example, some standard errors and misconceptions associated with ordering decimals are well-documented (Mason and Ruddock, 1986). For this very reason, these common difficulties are brought to the attention of trainees in the taught course, as an important detail of the professional knowledge they need to acquire. This may account for the subsequent low incidence of such mathematical errors in these trainees. For other topics, such as generalisation, the taught course appeared to have been less influential, or the topics themselves inherently more demanding. Others, such as Pythagoras’ theorem, were not given detailed attention in the course.

Ongoing analysis of responses to the more difficult items is uncovering errors which illustrate a continuum of ‘gaps’ in trainees’ subject knowledge in particular areas. By way of illustration, the 25 insecure responses to the following question (on generalisation) were scrutinised.

Check that 3+4+5=3x4 8+9+10=3x9 29+30+31=3x30

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

Responses were separated first according to whether or not they achieved an adequate prose statement of a generalisation of the kind (irrespective of linguistic elegance) "the sum of three consecutive whole numbers is equal to three times the middle number".

Those that did not (17 in all) were assigned to four categories in terms of progress towards such a statement, in the following notional hierarchy:

1.Blank. No response recorded by the student. [3]

2.Irrelevant/Incoherent: The response has no bearing on (or fails to communicate meaningful engagement with) the question as stated. [3]

3.Checks only: response limited to checking the arithmetic in the three examples given in the question. [4]

4.False generalisation: a prose generalisation is stated which does not encompass the three examples. [7]

The responses of the remaining eight students who did produce a comprehensive generalisation were assigned to two categories according to their ability to communicate it in prose:

5.Poorly communicated (prose): naively-expressed or rambling, but otherwise accurate. [4]

6.Well-communicated (prose). [4]

The same eight responses were assigned to two further, distinct categories which reflected their subsequent attempts to express the same generality in algebraic notation:

7.No attempt (algebra) [4]

8.Partial attempt (algebra) Failing to express fully the relationships between the variables in the algebraic statement. [4]

The following examples from three of the above categories serve to illustrate some of the more ‘interesting’ gaps identified in the audit.

5. Generalisation poorly communicated:

"Three ascending numbers may be equal, in sum, to 2 numbers that are multiplied together. The middle number of the sequence and the over all numbers are multiplied to give the same answer as those added together."

Pupils’ lack of fluency in the mathematics register is commonplace in both primary and secondary schools. The Standards for Qualified Teacher Status (DfEE, 1997) require trainees to "be taught the importance of ensuring that pupils progress from using informal mathematical vocabulary, to using precise and correct mathematical vocabulary, notation and symbols". Whilst the trainee wrote the generalisation above under duress, s/he will need to formulate similar sentences spontaneously in the classroom.

8. Partial attempt (algebra).

These were all of the kind "a+b+c=3b". Peer tutoring revealed whether the student was unable to incorporate the consecutive nature of a, b and c in the notation, or whether they simply thought it unnecessary. It was noticeable that most of those (other, ‘secure’) students who did express the consecutive relationship algebraically – especially those who wrote "(b-1)+b+(b+1)=3b" – were aware that this not only captured the generalisation, but also effectively explained it to them.

We are unable to isolate, from the audit data, those students who could simply express "one more" and "one less" than b (or similar) since the test item required a great deal more than that. All we can say is that 63% completed the whole question securely, and would not have been judged ‘secure’ unless they had concluded with a+(a+1)+(a+2)=3(a+1) or a variant such as that given in the previous paragraph. Other students also demonstrated this ability (one example follows immediately below) although the generalisation was not apparent to them. We note the finding of the Assessment of Performance Unit (Foxman et al, 1987) that slightly fewer than half of 15-year-olds would be expected to express relationships between consecutive (and other close) integers algebraically.

4. False Generalisation.

This category was perhaps the most interesting in terms of cognitive ‘gaps’. The response of one student most clearly represented the seven trainees, all of whom made the same error:

"Three consecutive numbers added together equals the product of the first two numbers. n + (n+1) + (n+2) = n x (n+1)."

These responses appear to focus on the first example (3+4+5=3x4) to the exclusion of the other two, or to indicate inability to see the second two examples as counter-examples to the proposed generalisation. The ‘3’ after the ‘=‘ is not perceived as the one constant term common to the three equations. It is perhaps not difficult to share the concern of the TTA about prospective primary school teachers who, for example, find it so difficult to perceive and communicate unity of form (let alone of meaning) in the three equations. At the same time, we would question the adequacy of "guided self-study" (DfEE, 1997, p.27) in the face of such cognitive obstacles.

SUBJECT KNOWLEDGE AND CLASSROOM PERFORMANCE

Research at King’s College into effective teachers of numeracy suggests that what matters is

… not formal qualifications or the amount of formal subject knowledge, but the nature of the knowledge about the subject that teachers have. (Askew et al, 1997, p. 93)

With this in mind, another strand of our enquiry investigates whether a significant link between subject knowledge, as measured by the audit, and students’ performances on teaching practice can be identified. Given the King’s findings, it might be expected that trainees’ teaching practice performance would be independent of the outcome of the audit of ‘formal subject knowledge.

There are similarities between our study and that of Bennett and Carré (1993), but there are also significant differences in scale and in methodology. The Exeter Study (ibid.) compared the teaching performances of a sample of primary PGCE students who had opted to specialise in mathematics within their course with a sample who had opted for different specialisms. The mathematics subject-matter knowledge of all the 59 students in the PGCE cohort was assessed against a framework of ‘substantive’ and ‘syntactic’ components. When classroom teaching performance was assessed, it transpired 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, 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, but this was not the case for mathematics specialists vis a vis mathematics.

The study that we now report offered the opportunity to compare mathematics teaching performances of a much bigger sample of students with wide variation in audited mathematics subject knowledge.

Method

Students were assigned to a 2-way classification:

I: Audit scores (maximum score being 32).

Category A, B or C corresponding respectively to audit score above 30 (i.e. perfect or near-perfect), between 30 and 24, below 24 (of whom almost all were insecure on 3 or more items)

II: Teaching practice performance

Students were categorised as 1 (very strong/strong), 2 (capable) or 3 (weak) on the basis of a grade profile given on their first (spring term) or second (summer term) teaching practice for planning, teaching and assessment.

These data were entered into a 3 by 3 contingency table (one for each practice), together with expected frequencies based on the null hypothesis that audit performance and teaching performance are independent. A chi-square test can be applied to these grouped audit and teaching practice data. Suitably low values of c 2 support the null hypothesis.

Spring Term

It should be noted that the grade-data for the first practice did not relate solely to the teaching of mathematics, since the only practical teaching assessment grades recorded covered all aspects of their teaching in school. Preliminary findings on the outcomes of this practice for each student, related to their mathematics audit performances (Rowland et al., 1998), did not support the view that teaching performance is linked to formal subject knowledge.

Summer Term

For the final school placement, specific assessments of the students’ teaching of number were made, and used with the original audit scores to compile a 3 by 3 contingency table similar to that for the first teaching practice. These data are shown below, together with expected frequencies (in brackets) based on the null hypothesis that audit performance and teaching performance are independent.

Perhaps unsurprisingly, tutors’ (moderated) assessments of practical teaching for the final teaching practice were less tentative than those for the first. Whereas, on the Spring practice, two-thirds of the students had been assigned to the middle (‘capable’) category, this reduced to a little more than one third in the Summer, when tutors were more prepared to make assessments at the extremes of the competence scale. It must be borne in mind, of course, that the first teaching assessment had been of a general kind, the second assessment used for the analysis which follows focused narrowly on the teaching of number. Five students from the cohort in the original analysis had withdrawn from the course, leaving a population of 149 students.

TEACHING PRACTICE PERFORMANCE

SUBJECT KNOWLEDGE AUDIT

1 (strong)

2 (capable)

3 (weak)

A (high)

20 (12.7)

12 (13.5)

5 (9.6)

B (middle)

28 (24.1)

28 (25.5)

14 (18.2)

C (low)

5 (14.5)

16 (15.3)

21 (10.9)

The same chi-square test was applied to these data to test the null hypothesis that initial audit performance and final performance in the teaching of number are independent. This time it turns out that c 2 = 24, with probability virtually zero (8 x 10-5), and the hypothesis of independence does not stand up to the data. There appears to be an association between mathematics subject knowledge (as assessed by the audit) and competence in teaching number.

Some striking aspects of the contingency table – notably the ‘extreme’ cells A1, A3, C1, C3 - intuitively support this conclusion. In particular, students with high audit scores seem much more likely to do well in school (as over half of them did) than the cohort as a whole (A1), and much less likely to do badly (A3). The converse is the case for students with low audit scores (cells C3 and C1), no fewer than half of whom were assessed as ‘weak’ in school.

These intuitions can be quantified. For example, the actual number of C3 students is double that ‘expected’. Is then a low audit score, in particular, a significant predictor of weak teaching performance? It is, in the following sense: on the hypothesis that the distribution of the 42 students with low audit scores across the three teaching grades is the same as that for the whole population, a binomial model B(42, 40/149) gives the probability of 21 or more category C3 trainees to be 0.0012. This is far too small to be attributable to chance ( i.e. it is highly significant) and supports the alternative view that students with a low mathematics audit score are more likely to be poor teachers of numeracy.

The polar opposite case – that of A1 students – is less striking but significant nonetheless. A similar binomial analysis (on a similar null hypothesis) shows the probability of as many as 20 of the 37 high-scoring students being strong teachers of number to be only 1.6%. The evidence suggest that students with a high mathematics audit score are more likely to be strong teachers of numeracy.

The nature of the relationship between the levels of subject knowledge and those of teaching performance can be analysed further, to yield a more refined understanding of that relationship, following a procedure proposed by Goodman (1964). The 3x3 contingency table gives rise to nine 2x2 matrices of cells (A,B/1,3 being one such matrix, for example), from each of which a statistic (denoted z2) can be computed (see Goodman, 1964 for details), the significance of which is ascertained by comparison with critical c 2 values for the original contingency table – in this case 9.5 for df=4 and p<0.05. In fact, only two of the nine 2x2 matrices give rise to significant z2 values, namely B,C/1,3 and A,C/1,3 with z2 = 12.76 and 16 respectively. (The next most significant, A,C/1,2, fails to attain even 10% significance). From this, it is possible to infer (not merely intuit) the source of rejection of the original null hypothesis, as follows. Students obtaining high (or even middle) scores on the audit are more likely to be assessed as strong numeracy teachers than those with low scores; students with low audit scores are more likely than other students to be assessed as weak numeracy teachers. Put simply and bluntly, there is a risk (statistical, at least) which is uniquely associated with trainees with low audit scores. This finding does not, in fact, contradict the King’s study, which found that the possession of ‘higher’ mathematics qualifications (as opposed to current knowledge or professional training in mathematics) did not in itself appear to improve teachers’ effectiveness.

Incidentally, Goodman’s analysis bears out what we might surmise from inspecting the middle column of the original contingency table: audit score is not a useful predictor of a ‘so-so’ – capable but not strong - numeracy teacher

Discussion

Our next task is to try to gain some insight into the nature of the association established above. For the moment, we briefly speculate on what might be going on. We raise and consider some of the possibilities that come to mind.

The most obvious is that secure subject knowledge, as assessed by the audit, really does underpin and enhance teaching in the primary years; that, all things being equal, it is better for the teacher to be knowledgeable about mathematics per se, than to be ignorant. A high proportion of students scoring poorly on the audit do indeed fall within the weak teaching practice category. This could indicate that these students’ knowledge and understanding of mathematics did not allow them to plan effectively, teach numeracy effectively or monitor pupils’ achievements and misconceptions with accuracy, in order to plan well-matched and challenging work. This is an attractive argument, and may well be sustainable, but it is not the only possibility. It is also no surprise to find – as the five ‘A3’ students give evidence - that a high level of mathematics subject knowledge is not in itself sufficient to ensure a strong or even a capable level of competence in teaching it.

Given that students with weak subject knowledge tend to fare poorly on the final teaching placement, it might be thought attractive to use the audit as a filter at the point of selection for the course. The wisdom and validity of such a conclusion are questionable, given that we audited after the content had been encountered (though not all in the same detail) by trainees in lectures and seminars during the initial third of the course. It is impossible to guess how the same students might have performed without that opportunity to recall what they had forgotten, and to realise its pedagogical significance.

There is also the possibility that those knowing themselves to be weaker in understanding of number may have been differentially disadvantaged through the anxiety or lack of confidence caused by both the test and the teaching practice situations. The audit may have been self-fulfilling in terms of students’ self-belief and performance. The effect of a poor result on the audit and the subsequent requirement to improve in this area, through self-study and peer tutoring sessions, could have had a considerable de-motivating and demoralising effect on these students, contributing to their difficulties in teaching in this area. The pressure to perform in an area where confidence is already not strong might have had the effect of lowering students’ performance.

It is only responsible, of course, to observe that the statistical relationship between subject knowledge and teaching competence (in number) need not be causal. The sheer complexity of the task of teaching and the wide range of contexts in which it is carried out and assessed, point to the danger of restricting attention to just two of the many variables in play.

An additional piece of information gleaned from this study also points to caution in assuming a causal link. The same students’ subject knowledge was also audited in English, and a similar analysis to that above was applied to uncover evidence of a link between knowledge of English and competence in teaching number. It is salutary, if bizarre, to note that the statistical link was marginally stronger (c 2 = 28.7) than that between knowledge of mathematics and number teaching competence. A Goodman-type analysis identifies the same crucial link between weak/strong performances in school and low/not-low audit scores.

Such evidence leads us to speculate about possible fundamental personal and interpersonal factors underpinning success (or lack of it) in all areas – academic and professional - of a PGCE course. Intangible factors such as ‘commitment’ and ‘motivation’ may have contributed to success in both the audit and during the final teaching practice. Perhaps there is a cycle in which commitment and motivation are, in turn, fuelled by success. A combination of good teaching practice experiences and success on the audit could have increased both, whilst, conversely, poor teaching practice experiences and weak results on the audit might be expected to have the opposite effect.

TRAINEES WITH SECURE MATHEMATICS SUBJECT KNOWLEDGE

A third dimension of our research concerns the histories, attitudes and professional trajectories of trainees who score highly against our audit of their ‘formal’ mathematics subject knowledge. In this paper, we give a profile of one such student, whom we shall call Frances.

Sixteen students out of 154 were secure on all of the 16 audit items. 11 of these have A level mathematics (although another 16 with A level are represented throughout the ‘top’ two-thirds). In the self-audit, all but one of the 16 were confident about their own ability to teach the mathematics subject content of the audit to someone else. Here, we focus on the exception – Frances, an Early Years specialist.

Frances

The portrait of Frances which follows is based on tutors’ reports, a coursework essay and a semi-structured interview with her. A mature student, Frances has O level grade 2 in mathematics, three A levels at grade A, a 2.1 Cambridge degree and a doctorate in anthropology. She is academically well qualified but not particularly so in mathematics.

A tension between her pedagogical knowledge and beliefs and the practices she encountered in school was apparent in her first teaching practice with a Key Stage 1 class. Whereas the class teacher taught mainly by using worksheets, this was not what the student wanted for her own teaching.

In a coursework essay, Frances reflected:

In my own teaching practice there was a problem in laying the basis for progression. In the scheme of work agreed, I planned to teach the concept of ‘difference’, then show how it can be represented by the minus sign. However, when I team taught with the class teacher at the end of the practice, the teacher said she didn’t like to teach subtraction as difference but only as ‘take away’ because it was "confusing at this age".

Grossman et al. (1989) believe the task of transforming disciplinary knowledge into content suitable for students is one of the central skills of teaching. Frances argues:

However, if teachers’ understanding is deeper and the learning objectives are related to how the subtraction sign fits into the whole "form of discourse" (Aubrey, 1994a), it is relevant to introduce the idea that a minus sign can mean different things to quite small children, especially once they are familiar with more than one subtraction structure.

I was consciously using my knowledge both about mathematics, its nature and its concepts and my knowledge and experience about how young children learn. Presenting young children with the concept of mathematical difference without linking it to their existing practical experience would have been akin to presenting them with a foreign language with no scope for translation.

This demonstrates Frances’ own depth of understanding, and how a student’s thinking may be more theoretically sophisticated than that of the class teacher with whom they are training.

Frances was the only one of the high-scoring group who was not confident about her ability to peer-tutor the subject matter of the audit, even though her ‘test’ result did indicate that her subject knowledge was highly secure. It is easier to make judgements about a lesson and how to extend children’s understanding if the teacher is confident of their subject matter (Pollard et al., 1997). This factor underpinned a second tension for Frances, between what she believed to be best for pupils and her reluctance to implement it. Elsewhere in her writing, she recognises her lack of confidence

To develop powers of reasoning, children need opportunities for mathematical investigation. In not capitalising on such opportunities, I failed to present mathematics as an investigative problem solving subject. This was an example of how use of subject knowledge can be affected by confidence and experience. At an intellectual level I appreciated the importance of mathematical investigation, but did not have the confidence to pursue it.

Later, she reflected that, to teach a topic, it is imperative to understand it in some depth – she needs "to understand the knowledge properly to teach it". She supposed that the audit tested ‘what she could do’, but that if she had to teach (as a peer tutor) arithmetic sequences, for example (DfEE, 1997, paragraph 13), she would need to study it "from first principles"(2). Teaching, she feels, is more about breaking knowledge down into tasks that will enable children to "get there", whereas her learning is more to do with "internalising the processes". This is supported by Aubrey (1994a, p. xx) who acknowledges that "pedagogical knowledge goes beyond subject matter knowledge to the examination of subject knowledge for teaching".

In the essay, Frances is eloquent and reflective about her practice. She is aware of why she had problems; she needs the confidence to pursue her own learning agenda and work on her own ideas. She illustrates why even a high level of subject knowledge and understanding of pedagogy are not enough: her problem is a lack of confidence in herself. All these circumstances contributed to a difficult spring teaching practice for Frances, and to formative assessment as ‘weak’ by her tutor. Brophy (1991) has noted that pedagogical subject knowledge is influenced by teachers’ beliefs, and with associated values and attitudes towards what is involved in teaching (in this case) mathematics. Mathematics had been Frances’ weakest subject at school. Frances attributes this in part to poor teaching. The PGCE course had awakened a sense that mathematics was, nevertheless, an interesting academic discipline. Frances’ academic record suggests a person with unusual intellectual capability. Perhaps her success in the mathematics audit came as a surprise to her. Her personal subject knowledge clearly failed to ensure success in the classroom in the spring placement.

Frances’ summer practice could hardly have been more different. Her tutor reported on the quality of her medium term planning, where her subject knowledge in mathematics was a real strength. Having enjoyed success, Frances reflected (in the interview) on her earlier less successful practice, and attributed her difficulties to the enormity of planning across the curriculum, which resulted in her not being "on top" of the pedagogic subject knowledge she needed in mathematics. She resorted to planning isolated activities on a daily basis with little regard for the children’s understanding. She did not spend enough time giving them tasks to assess what they were able to, do and the class teacher was not clear about her own assessments. Frances became confused, and tried to plan activities to cover a range of objectives in a topic on measurement before finding out what the children already knew. As a result of a lack of a firm diagnostic foundation, the classroom management and control fell apart. Frances responded by resorting to time-filling activities for the children, and planning on a day-to-day basis. She was unable to see where to go next, as she had not clarified her understanding of the mathematics that she was trying to teach. Interestingly, Aubrey (1994b) makes very similar comments about the nature of pedagogical subject knowledge.

Pedagogical subject knowledge includes knowing what knowledge, concepts and strategies children bring to learning, their misconceptions as well as their understandings, and the stages through which they pass towards mastery of topics within a subject area. (p. 106)

Before her final teaching practice, Frances found out what she was to teach, and went back to "first principles" to clarify her own understanding of the relevant subject knowledge, the stages in children’s learning and the significant achievements which would mark their progress. Her work focused largely on counting. Frances was familiar with the literature (for example, Thompson, 1997) and she was able to see how the different strands connected. The placement was in a nursery setting, and she found herself more in sympathy with the approach to learning that she encountered there, with more time to observe children and to assess what they knew. The "smaller steps in the learning" (as she put it) enabled Frances to think about less content, but in greater depth. She did not feel the children were on a learning treadmill, as she had when working with the Key Stage 1 class.

In his report, Frances’ tutor comments that she is reflective about her own practice, both in discussion with colleagues and tutors and in her written evaluation. She makes honest and constructively self-critical comments about her teaching, management and organisation and uses these insights to inform subsequent planning successfully. She adapted and amended schemes of work as she progressed through the practice, in the light of children’s achievements and her own understanding of effective approaches in particular situations. This sense of ownership and pedagogic clear-headedness underpinned progression and continuity in the children’s learning. She was very successful; having come ‘close to the brink’ mid-term in the course, she completed it with the highest achievable grades in both academic elements and in practical teaching.

Frances demonstrates that successful teaching of mathematics is not guaranteed by subject knowledge alone - indeed that success in the written audit may mask significant gaps in ‘relational’ understanding (Skemp, 1976). Despite an unpromising start in the classroom, Frances’ effectiveness eventually resulted from the integration of a complex matrix of knowledge and personal qualities, including:

  1. ability to perform mathematically at the level assessed by the audit;
  2. willingness and the intellectual capability to "go back to first principles" regarding the mathematical content (however seemingly elementary) that she was about to teach;
  3. diagnosis of what children already know, analysis of what they are intended to learn, and confidence to adapt others people’s material and to devise her own approaches accordingly;
  4. constructive application of skills of reflection and self-criticism;
  5. positive attitudes and increasingly confident beliefs about herself, teaching, children and mathematics.

Circular 10/97 (DfEE, 1997) recognises the importance of such qualities. For Frances, those to do with self-belief, ownership and intelligent professionalism made the difference between potential failure and notable success.

THE PEER TUTORING PROCESS

We conclude with some preliminary observations regarding the peer tutoring arrangements, which have been the focus of our fourth research initiative. 32 students who scored highly in the subject knowledge audit agreed to act as peer tutors to other students, following a briefing session. These met with their tutees one-to-one, following a prearranged schedule of appointments, and reviewed the troublesome items in their tutee’s subject knowledge audit paper. Out of 32 peer tutors, 18 agreed to audio-tape record the sessions. Of these, 15 persuaded at least one of their tutees to be tape-recorded. As a result we have several hours of recorded material that we are in the process of transcribing for analysis.

The peer-tutorials seemed highly-charged events for the participants. No doubt the negotiated audio taping contributed to this, but it seemed as if this extra dimension was insignificant compared with the self-consciousness engendered by the occasion itself. Most noticeable was the degree of controlled anxiety amongst the peer tutors. They felt their professional responsibilities keenly, notwithstanding their professed confidence; discussion with them on the day suggested that they had a strong sense that both their subject knowledge and their pedagogic skills were on the line. The material from the transcribed tape-recordings is very rich. The preliminary analysis is suggesting several themes that could be pursued.

For example, the material could be explored in terms of the range of discussion taking place around particular questions in the audit. The hierarchy of difficulty of the questions is such that much of the taped material focuses on a fairly narrow range of the more difficult topics (see the earlier table). Clearly some work on the data related to each question will be rewarding. By contrast there is a much smaller amount of material relating to the easier questions in the audit, and these are interesting cases because they tend to involve, as peer tutees, those students whose subject knowledge is apparently most seriously deficient as judged by our audit.

As teacher educators engaged on a day-to-day basis with making professional judgements about the effectiveness of the teaching of novice teachers, the most seductive aspects of a preliminary review of the material relates to the sense that some of the peer tutors are more effective teachers than others. This ‘sense’ requires a great deal of disciplined unpacking, but is informed, for example, by the contrast between the determination of some peer tutors to ‘explain’ and the willingness of others to listen. Follow-up work which invited peer tutors and/or tutees to evaluate the teaching and learning (with or without access to the taped record) would be an interesting project.

In addition, issues of tutor/tutee gender relations, of patterns in turn-taking and the exploration of many language issues are all potentially available. One of the most interesting features of this initiative is its particular context as an example of peer tutoring in the context of learning to teach. There is a long-standing interest in peer tutoring (Topping, 1996) but little work has been done on its potential within teacher education.

The relationship between subject knowledge and pedagogic knowledge is not a simple one (Shulman, 1987, McNamara, 1991), and through their participation in peer tutoring, students could be encouraged to confront and explore it. If this approach to addressing students’ needs for enhanced subject knowledge proves fruitful, we could consider extending it through sensitive use of the taped material as part of the learning experience of the course.

We need to form a view about whether these peer tutoring methods have the potential to support the remediation of deficiencies in subject knowledge and to develop teaching skills, and if they do, how to refine and develop them.

REFERENCES

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Notes

1. Since superseded by Circular 4/98 (DfEE, 1998). Reference to 10/97 has been retained in this paper, since (a) that was the current régime at the time of our fieldwork, and (b) changes incorporated in 4/94 were minimal or cosmetic e.g. ‘underpin’ is replaced by ‘support’.

2. For example, one audit question asked for the 20th term of the sequence 2, 12, 22, … We had envisaged students would arrive at 192 by reasoning "from first principles" that there must be 19 gaps of 10 after the initial 2. It is apparent, however, that many of them, having been alerted by DfEE (1997) to the likelihood of being tested on "arithmetic sequences", rote-memorised a formula of the type a+(n-1)d in order to answer "those sorts of questions".

This document was added to the Education-line database on 26 April 2000