Does it Matter Primary Teacher Trainees
British Educational Research Journal, Vol. 28, No. 5, 2002
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
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 identied
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.
ABSTRACT
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 lled 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
ISSN 0141-192 6 (print)/ISSN 1469-351 8 (online)/02/050689 –16 Ó2002 British Educationa l Research Association
DOI: 10.1080/014119202200001554 3
690
M. Goulding et al.
contribute to this aim (Alexander et al., 1992; Ofce for Standards in Education
[OFSTED], 1994). More recently, Her Majesty’s Inspectors (OFSTED, 2000), in an
evaluation of the rst 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 condent 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 (Brown et al., 1999; Green & Ollerton, 1999) have identied primary trainees’
anxiety about mathematics as a major issue. In the past, however, ITT courses have been
successful in increasing students’ condence in their ability to teach mathematics and in
shifting their attitudes and beliefs about the subject (Carré & Ernest, 1993; Brown et 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 rst, 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 ndings 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 conned 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 eld
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)
If all were well in mathematics classrooms then this recycling of practice is not an issue,
but the USA has shared our concern about mathematics teaching for some time. More
recently, the modest standing of both countries in global league tables (Harris et al.,
1997), has heightened political concern about national standards, so interrupting the
pattern is seen as an important aim. Unpicking the tangle of teacher knowledge,
however, is a complex task.
Primary Teacher Trainees’ Knowledge of Mathematics
691
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 inuential in the USA and beyond. These classications 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). His pedagogical content knowledge
(PCK) consists of:
the most powerful analogies, illustrations, examples, explanations and demonstrations—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 specic topics easy or difcult. (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’s curricular knowledge consists 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.
Subsequent writing by Shulman and his associates (Grossman et al., 1989) takes into
account student beliefs about mathematics, while Ernest (1989) identies attitudes
(interest and condence) as important. Beliefs about the nature of mathematics may be
tied up with SMK in the way in which teachers approach mathematical situations. If they
believe that it is principally a subject of rules and routines which have to be remembered,
then their own approach to unfamiliar problems will be constrained, and this may have
an impact on their teaching. We suspect that beliefs are particularly salient in the
development of syntactic knowledge, where conjecturing, nding evidence and seeking
explanations is quite different from applying rules and routines in recognisable contexts.
If teachers lack condence in their SMK, then they may avoid risky situations in the
classroom and be inhibited in responding to children’s unexpected questions. They might
also seek refuge by opting to teach younger children, where they may feel less daunted
by the demands of the mathematics curriculum, denying themselves the opportunity to
engage with material which could challenge and develop their own SMK. On the other
hand, teachers who lack condence may be more inclined to prepare their lessons
carefully and to access a range of resource material. If teachers lack interest, their
planning and preparation may be skimpy and they may convey feelings of negativity, or
at best indifference, to their pupils. We cannot subscribe to a commonplace view that
good SMK in mathematics is somehow a barrier to teaching the subject to younger
pupils and low achievers. The kind of empathy that might result from weak SMK is no
substitute at all for good mathematics PCK that enables the teacher to identify and tackle
learners’ difculties. The teacher with weak mathematics SMK is not well placed to take
into account the structure of the subject in planning teaching and learning sequences.
692
M. Goulding et al.
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 signicant
difculties with the meaning of 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 themselves.
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. distributivity 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 knowledge 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 reections of the students on these
courses demonstrate the effect of this ‘unlearning’ on their self-awareness and their
beliefs about mathematics. One student wrote:
I’m learning about mathematics in this class … math isn’t just about memorising formulas—it is knowing why a problem is done the way it is … In high
school, [it was] memorising formulas, theorems and denitions. (pp. 43–44)
Primary Teacher Trainees’ Knowledge of Mathematics
693
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 difcult, 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 rst year of schooling.
Although the sample for this study was small (four teachers in three English schools),
the qualitative data was ne-grained and gathered over the course of one year. Aubrey
identied 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 identied progress of pupils over the year and the teaching behaviours of
the four teachers.
Pupil progress (as measured by test score gains) was, however, associated with
particular teacher beliefs and the nature of knowledge held by the larger sample in the
Effective Teachers of Numeracy Project (Askew et al., 1997) at King’s College London.
Although the mathematical focus was narrower than Aubrey’s, the progress of pupils
measured by tests of numeracy before and after the study revealed that teachers who
believed in the potential of all pupils and who themselves had ‘knowledge and awareness
of conceptual connections between the areas which they taught’ (p. 3) tended to produce
the highest numeracy gains in pupils. These so-called ‘connectionist’ teachers did not
necessarily hold advanced qualications in mathematics, i.e. A level or beyond, but they
were more likely to have attended extended continuing professional development (CPD)
courses. Initial training was perceived to have little inuence on effectiveness, raising
694
M. Goulding et al.
questions about the amount of time devoted to mathematics on postgraduate certicate
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 (Carré & 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 reect an advanced
mathematics qualication. 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-à-vis mathematics.
Synthesis
Our own work in this area has been informed by the theoretical and empirical research
summarised above, applied within the specic 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 inuential 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 inuence both students’ planning and
teaching, a cognitive dimension encompassing beliefs about mathematics, and their
condence 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 ndings 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.
Institutions are free to handle all aspects of the ITT NC in their own ways, as the
‘curricula [sic] does not specify a course model or scheme of work’ (p. 5). Examples of
practices in different institutions are, however, provided by the Teacher Training Agency
(TTA, 1999), together with some implicit guidance, i.e. that the auditing process should
Primary Teacher Trainees’ Knowledge of Mathematics
695
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 identied in the Career Entry Prole which new teachers take
into their rst 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 specied in the government circulars) primary trainees
(n 5 154, n 5 201 respectively) found difcult, 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 identied 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’ difculties 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 ndings on aspects of the trainees’ SMK as identied by the audit by
a micro-analysis of difculties, errors and misconceptions;
· report the ndings 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 signicantly 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 difculties, 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:
List these decimals in order, from least to greatest: 0.5 0.67 0.372
696
M. Goulding et al.
TABLE I. Audit topics with the highest and lowest facility ratings
(a) Highest facility
%
secure 1
98 3
95
93
92
Mean
score 2
1.96
1.92
1.90
1.90
Topic
Inverse operation s
ordering decimals
Divide 4-digit number by 2-digit
Problem solving in a money
context
(b) Lowest facility
%
secure
Mean
score
52
49
33
30
1.17
1.15
0.87
0.73
Topic
Pythagoras, area
Generalisatio n
Reasoning and proof
Scale factors, percentag e
increase
1
The written response gives a high level of assuranc e of the knowledge being audited.
A secure response scores 2, which is therefore the maximum possible for the mean.
3
All cohort percentage s have been rounded to the nearest percentage .
2
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; 2.35 3 10 2 2; two hundredths; 2.19 3 10 2 1; one fth
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 substantive subject knowledge: the lack of connection in the students’ knowledge between
different forms of numerical expression, and difculties 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 sufciently strong understanding to
respond to pupils who, for instance, nd 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
difculties 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
1
question, which required students to provide contexts for (2 4 4) and (11 2 2 5), with
Primary Teacher Trainees’ Knowledge of Mathematics
697
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
31415533 4
8 1 9 1 10 5 3 3 9
29 1 30 1 31 5 3 3 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 rst case only. Others were able to express the generality
in their own words but not symbolically. Some expressions, e.g.
a 1 b 1 c 5 3b
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 2 5 3(a 1 1)
Proof (working deductively from the sum on the left)
a 1 a 1 1 1 a 1 2 5 3a 1 3
5 3(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 2 1 1 n 1 n 1 1 5 3n 2 1 1 1
5 3n This is three times the middle number.
This question, then, exposed weaknesses in recognising pattern and relationships,
identifying signicant 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.
In the insecure responses, there were difculties in giving a general diagram or a general
algebraic expression for any odd number which could then be used in a proof. These
students were unable to provide a convincing mathematical justication in any form, not
necessarily algebraic. In the classroom, they may not alert pupils to the need for general
698
M. Goulding et al.
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 tting 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 Rowland et
al., 2001, for a detailed analysis). Most interesting, perhaps, is the fact that a signicant
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-examples 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 condently respond to an unexpected student question, ‘I
don’t know, let’s nd out’. Ma (1999) perceives teachers’ curiosity and condence 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 eld 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
The Institute team conducted studies with two consecutive PGCE cohorts, investigating
the relation between trainees’ audited mathematics subject knowledge and their teaching
competence (Rowland et al., 2000, 2001). With the rst student cohort (n 5 154), the
level of each student’s subject knowledge (based on the audit) was categorised as low,
Primary Teacher Trainees’ Knowledge of Mathematics
699
medium or high, corresponding to the need for signicant remedial support, modest
support (or self-remediation), or none. Towards the end of that course, specic assessments of the students’ teaching of number were made on the second and nal 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 identied
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 (n 5 164), more extensive data from school placements enabled
comparison of mathematics subject knowledge with teaching performance (a) on both
rst 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 signicant for three of the
four analyses, the exception being the pre-active dimension of the rst placement. This
anomaly will be the subject of future statistical verication 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 identies clarity about
mathematics teaching and learning, but that this clarity may be obscured in the planning
on the rst practice, when both tutors and trainees may be more concerned with the form
rather than content of planning les. That apart, poor subject matter knowledge as
identied by the audit was associated with weaknesses in planning and teaching primary
mathematics.
The ndings do not contradict the King’s study, where no relationship was found
between higher mathematical qualications 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 identied 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 underestimated, 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 reect 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.
Although it would be neat to claim that those students with poor subject matter
knowledge were hampered in their ability to plan, teach and monitor pupils’ mathematics
in a cyclical way, there are other possibilities. Since the audit was taken before the nal
teaching practice, it may have acted as a self-fullling prophecy, conrming these
students’ belief in their lack of ability in mathematics, increasing their anxiety and
700
M. Goulding et al.
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 condence were pointedly illustrated by the only Institute trainee who
scored highly in the audit but did not claim to feel condent 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 rst 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 condence 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-lling activities and day-to-day planning because her lack of rm 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 rst principles’ and identify and build upon the signicant points in
children’s progression. Whereas her tutor’s assessment on the rst practice had identied
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 reection and self-criticism;
increasing condence and a positive attitude.
For Frances, it is possible that her own self-awareness and knowledge of background
research made the task on her rst teaching practice too daunting, i.e. that her intellectual
Primary Teacher Trainees’ Knowledge of Mathematics
701
state was outstripping her ability to perform in a less than favourable situation, resulting
in confusion and lack of condence. 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 sufciently convinced by the research being done in their own institution
and research elsewhere that work on subject matter content accompanied by reection
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 reected on their own learning in the process. They were specically 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 evaluations that there were very mixed reactions 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 rst (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 rst 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 requirements 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 shortcomings in the audit. The Durham team also found that trainees’ difculties with proof
were particularly resistant to remediation in the postgraduate initial training year
(Goulding & Suggate, 2001).
More research will give us a better insight into the processes of auditing and
remediation from the students’ perspectives, with a view to improving provision. We are
particularly interested in the role of peer teaching and the extent to which it might
702
M. Goulding et al.
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 difcult for the rst 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 condence.
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 difcult 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 sufciently convinced by our research on the students’ errors and misconceptions 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.
In conclusion, we would argue that strengthening teachers’ subject matter knowledge
is a legitimate aim for ITT and CPD. In planning and teaching, it is one of the resources
that the teacher will draw upon, whether intentionally or spontaneously in the course of
interaction with pupils. We would argue that syntactic knowledge is a crucial element of
SMK and that beliefs about the nature of mathematics are implicit in its development.
Moreover, the boundaries between SMK and PCK are not rigid, and the use of
curriculum materials will enrich the critical
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
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 identied
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.
ABSTRACT
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 lled 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
ISSN 0141-192 6 (print)/ISSN 1469-351 8 (online)/02/050689 –16 Ó2002 British Educationa l Research Association
DOI: 10.1080/014119202200001554 3
690
M. Goulding et al.
contribute to this aim (Alexander et al., 1992; Ofce for Standards in Education
[OFSTED], 1994). More recently, Her Majesty’s Inspectors (OFSTED, 2000), in an
evaluation of the rst 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 condent 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 (Brown et al., 1999; Green & Ollerton, 1999) have identied primary trainees’
anxiety about mathematics as a major issue. In the past, however, ITT courses have been
successful in increasing students’ condence in their ability to teach mathematics and in
shifting their attitudes and beliefs about the subject (Carré & Ernest, 1993; Brown et 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 rst, 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 ndings 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 conned 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 eld
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)
If all were well in mathematics classrooms then this recycling of practice is not an issue,
but the USA has shared our concern about mathematics teaching for some time. More
recently, the modest standing of both countries in global league tables (Harris et al.,
1997), has heightened political concern about national standards, so interrupting the
pattern is seen as an important aim. Unpicking the tangle of teacher knowledge,
however, is a complex task.
Primary Teacher Trainees’ Knowledge of Mathematics
691
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 inuential in the USA and beyond. These classications 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). His pedagogical content knowledge
(PCK) consists of:
the most powerful analogies, illustrations, examples, explanations and demonstrations—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 specic topics easy or difcult. (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’s curricular knowledge consists 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.
Subsequent writing by Shulman and his associates (Grossman et al., 1989) takes into
account student beliefs about mathematics, while Ernest (1989) identies attitudes
(interest and condence) as important. Beliefs about the nature of mathematics may be
tied up with SMK in the way in which teachers approach mathematical situations. If they
believe that it is principally a subject of rules and routines which have to be remembered,
then their own approach to unfamiliar problems will be constrained, and this may have
an impact on their teaching. We suspect that beliefs are particularly salient in the
development of syntactic knowledge, where conjecturing, nding evidence and seeking
explanations is quite different from applying rules and routines in recognisable contexts.
If teachers lack condence in their SMK, then they may avoid risky situations in the
classroom and be inhibited in responding to children’s unexpected questions. They might
also seek refuge by opting to teach younger children, where they may feel less daunted
by the demands of the mathematics curriculum, denying themselves the opportunity to
engage with material which could challenge and develop their own SMK. On the other
hand, teachers who lack condence may be more inclined to prepare their lessons
carefully and to access a range of resource material. If teachers lack interest, their
planning and preparation may be skimpy and they may convey feelings of negativity, or
at best indifference, to their pupils. We cannot subscribe to a commonplace view that
good SMK in mathematics is somehow a barrier to teaching the subject to younger
pupils and low achievers. The kind of empathy that might result from weak SMK is no
substitute at all for good mathematics PCK that enables the teacher to identify and tackle
learners’ difculties. The teacher with weak mathematics SMK is not well placed to take
into account the structure of the subject in planning teaching and learning sequences.
692
M. Goulding et al.
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 signicant
difculties with the meaning of 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 themselves.
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. distributivity 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 knowledge 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 reections of the students on these
courses demonstrate the effect of this ‘unlearning’ on their self-awareness and their
beliefs about mathematics. One student wrote:
I’m learning about mathematics in this class … math isn’t just about memorising formulas—it is knowing why a problem is done the way it is … In high
school, [it was] memorising formulas, theorems and denitions. (pp. 43–44)
Primary Teacher Trainees’ Knowledge of Mathematics
693
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 difcult, 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 rst year of schooling.
Although the sample for this study was small (four teachers in three English schools),
the qualitative data was ne-grained and gathered over the course of one year. Aubrey
identied 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 identied progress of pupils over the year and the teaching behaviours of
the four teachers.
Pupil progress (as measured by test score gains) was, however, associated with
particular teacher beliefs and the nature of knowledge held by the larger sample in the
Effective Teachers of Numeracy Project (Askew et al., 1997) at King’s College London.
Although the mathematical focus was narrower than Aubrey’s, the progress of pupils
measured by tests of numeracy before and after the study revealed that teachers who
believed in the potential of all pupils and who themselves had ‘knowledge and awareness
of conceptual connections between the areas which they taught’ (p. 3) tended to produce
the highest numeracy gains in pupils. These so-called ‘connectionist’ teachers did not
necessarily hold advanced qualications in mathematics, i.e. A level or beyond, but they
were more likely to have attended extended continuing professional development (CPD)
courses. Initial training was perceived to have little inuence on effectiveness, raising
694
M. Goulding et al.
questions about the amount of time devoted to mathematics on postgraduate certicate
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 (Carré & 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 reect an advanced
mathematics qualication. 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-à-vis mathematics.
Synthesis
Our own work in this area has been informed by the theoretical and empirical research
summarised above, applied within the specic 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 inuential 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 inuence both students’ planning and
teaching, a cognitive dimension encompassing beliefs about mathematics, and their
condence 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 ndings 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.
Institutions are free to handle all aspects of the ITT NC in their own ways, as the
‘curricula [sic] does not specify a course model or scheme of work’ (p. 5). Examples of
practices in different institutions are, however, provided by the Teacher Training Agency
(TTA, 1999), together with some implicit guidance, i.e. that the auditing process should
Primary Teacher Trainees’ Knowledge of Mathematics
695
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 identied in the Career Entry Prole which new teachers take
into their rst 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 specied in the government circulars) primary trainees
(n 5 154, n 5 201 respectively) found difcult, 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 identied 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’ difculties 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 ndings on aspects of the trainees’ SMK as identied by the audit by
a micro-analysis of difculties, errors and misconceptions;
· report the ndings 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 signicantly 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 difculties, 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:
List these decimals in order, from least to greatest: 0.5 0.67 0.372
696
M. Goulding et al.
TABLE I. Audit topics with the highest and lowest facility ratings
(a) Highest facility
%
secure 1
98 3
95
93
92
Mean
score 2
1.96
1.92
1.90
1.90
Topic
Inverse operation s
ordering decimals
Divide 4-digit number by 2-digit
Problem solving in a money
context
(b) Lowest facility
%
secure
Mean
score
52
49
33
30
1.17
1.15
0.87
0.73
Topic
Pythagoras, area
Generalisatio n
Reasoning and proof
Scale factors, percentag e
increase
1
The written response gives a high level of assuranc e of the knowledge being audited.
A secure response scores 2, which is therefore the maximum possible for the mean.
3
All cohort percentage s have been rounded to the nearest percentage .
2
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; 2.35 3 10 2 2; two hundredths; 2.19 3 10 2 1; one fth
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 substantive subject knowledge: the lack of connection in the students’ knowledge between
different forms of numerical expression, and difculties 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 sufciently strong understanding to
respond to pupils who, for instance, nd 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
difculties 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
1
question, which required students to provide contexts for (2 4 4) and (11 2 2 5), with
Primary Teacher Trainees’ Knowledge of Mathematics
697
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
31415533 4
8 1 9 1 10 5 3 3 9
29 1 30 1 31 5 3 3 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 rst case only. Others were able to express the generality
in their own words but not symbolically. Some expressions, e.g.
a 1 b 1 c 5 3b
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 2 5 3(a 1 1)
Proof (working deductively from the sum on the left)
a 1 a 1 1 1 a 1 2 5 3a 1 3
5 3(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 2 1 1 n 1 n 1 1 5 3n 2 1 1 1
5 3n This is three times the middle number.
This question, then, exposed weaknesses in recognising pattern and relationships,
identifying signicant 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.
In the insecure responses, there were difculties in giving a general diagram or a general
algebraic expression for any odd number which could then be used in a proof. These
students were unable to provide a convincing mathematical justication in any form, not
necessarily algebraic. In the classroom, they may not alert pupils to the need for general
698
M. Goulding et al.
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 tting 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 Rowland et
al., 2001, for a detailed analysis). Most interesting, perhaps, is the fact that a signicant
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-examples 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 condently respond to an unexpected student question, ‘I
don’t know, let’s nd out’. Ma (1999) perceives teachers’ curiosity and condence 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 eld 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
The Institute team conducted studies with two consecutive PGCE cohorts, investigating
the relation between trainees’ audited mathematics subject knowledge and their teaching
competence (Rowland et al., 2000, 2001). With the rst student cohort (n 5 154), the
level of each student’s subject knowledge (based on the audit) was categorised as low,
Primary Teacher Trainees’ Knowledge of Mathematics
699
medium or high, corresponding to the need for signicant remedial support, modest
support (or self-remediation), or none. Towards the end of that course, specic assessments of the students’ teaching of number were made on the second and nal 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 identied
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 (n 5 164), more extensive data from school placements enabled
comparison of mathematics subject knowledge with teaching performance (a) on both
rst 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 signicant for three of the
four analyses, the exception being the pre-active dimension of the rst placement. This
anomaly will be the subject of future statistical verication 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 identies clarity about
mathematics teaching and learning, but that this clarity may be obscured in the planning
on the rst practice, when both tutors and trainees may be more concerned with the form
rather than content of planning les. That apart, poor subject matter knowledge as
identied by the audit was associated with weaknesses in planning and teaching primary
mathematics.
The ndings do not contradict the King’s study, where no relationship was found
between higher mathematical qualications 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 identied 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 underestimated, 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 reect 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.
Although it would be neat to claim that those students with poor subject matter
knowledge were hampered in their ability to plan, teach and monitor pupils’ mathematics
in a cyclical way, there are other possibilities. Since the audit was taken before the nal
teaching practice, it may have acted as a self-fullling prophecy, conrming these
students’ belief in their lack of ability in mathematics, increasing their anxiety and
700
M. Goulding et al.
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 condence were pointedly illustrated by the only Institute trainee who
scored highly in the audit but did not claim to feel condent 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 rst 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 condence 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-lling activities and day-to-day planning because her lack of rm 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 rst principles’ and identify and build upon the signicant points in
children’s progression. Whereas her tutor’s assessment on the rst practice had identied
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 reection and self-criticism;
increasing condence and a positive attitude.
For Frances, it is possible that her own self-awareness and knowledge of background
research made the task on her rst teaching practice too daunting, i.e. that her intellectual
Primary Teacher Trainees’ Knowledge of Mathematics
701
state was outstripping her ability to perform in a less than favourable situation, resulting
in confusion and lack of condence. 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 sufciently convinced by the research being done in their own institution
and research elsewhere that work on subject matter content accompanied by reection
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 reected on their own learning in the process. They were specically 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 evaluations that there were very mixed reactions 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 rst (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 rst 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 requirements 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 shortcomings in the audit. The Durham team also found that trainees’ difculties with proof
were particularly resistant to remediation in the postgraduate initial training year
(Goulding & Suggate, 2001).
More research will give us a better insight into the processes of auditing and
remediation from the students’ perspectives, with a view to improving provision. We are
particularly interested in the role of peer teaching and the extent to which it might
702
M. Goulding et al.
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 difcult for the rst 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 condence.
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 difcult 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 sufciently convinced by our research on the students’ errors and misconceptions 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.
In conclusion, we would argue that strengthening teachers’ subject matter knowledge
is a legitimate aim for ITT and CPD. In planning and teaching, it is one of the resources
that the teacher will draw upon, whether intentionally or spontaneously in the course of
interaction with pupils. We would argue that syntactic knowledge is a crucial element of
SMK and that beliefs about the nature of mathematics are implicit in its development.
Moreover, the boundaries between SMK and PCK are not rigid, and the use of
curriculum materials will enrich the critical