Subject Matter Knowledge in Experienced and Novice Teachers of Mathematics.

Stephanie Prestage and Pat Perks

School of Education, University of Birmingham

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

This paper investigates subject-matter knowledge in pre-service teachers and subject leaders in mathematics. Research shows that teachers develop a variety of pedagogical content knowledge to translate personal subject matter knowledge into classroom events. Whilst learners need only to know that they have gained knowledge (learner-knowledge), teachers need to hold multiple and fluid conceptions of that knowledge to know how progress might be made through that knowledge (teacher-knowledge). Existing research makes assumptions that teachers have full access to the requisite subject matter knowledge. Our work reveals that much of this knowledge is held as learner-knowledge and lacks explicit analysis for teacher-knowledge with surprisingly little difference between novice and experienced teachers.

Introduction

Teaching requires a myriad of knowledge and skills, knowledge about pupils, systems and structures, knowledge about styles of teaching and learning, knowledge about management, resources and assessment as well as knowledge about the subject. Existing research in the area of teachers’ knowledge, offers definitions of professional knowledge as well as explanations for the different forms of knowledge that a teacher holds, (Shulman 1986, Wilson et al. 1987, Brown and McIntyre 1993, Cooper and McIntyre 1996, Desforges and McNamara 1979, Tamir 1988, Ernest 1989, Marks 1990, Aubrey 1997). Subject matter knowledge in all its complexity is but one part of this analysis. Research on pedagogical content knowledge explores knowledge about explanations, tasks and activities, about styles of teaching and learning but does not include explicit detail of how such subject knowledge is held in an intellectual way by teachers other than is demonstrated by the activities chosen or the explanations given. The results of this study raise questions with respect to the adequacy of teachers’ understanding of subject knowledge and which have implications for the development of professional practice.

In this paper we take subject knowledge to be knowledge about the subject matter in mathematics, knowledge about its structure, the body of concepts, facts, skills and definitions as well as methods of justification and proof and offer some results on the way in which teachers might hold this knowledge. Beliefs about teaching lead to us to agree with Buchmann (1984) that teachers need a rich and deep understanding of their subject in order to respond to all aspects of pupils’ needs.

Content knowledge of this kind encourages the mobility of teacher conceptions and yields knowledge in the form of multiple and fluid conceptions (ibid. p.46)

A distinction is made between two types of knowledge on the basis of the data collected. Whilst learners of mathematics need only be aware that they have gained knowledge to pass examinations (learner-knowledge), teachers need to hold multiple and fluid conceptions of that knowledge to know how progress might be made through that knowledge (teacher-knowledge). Existing research makes assumptions that teachers have full access to subject matter knowledge and that this is transformed by the activities developed for teaching. We would argue that for both experienced and novice teachers much of this subject matter knowledge remains as learner-knowledge and is not transformed into teacher-knowledge.

There is however, little evidence to suggest that the development of project teachers’ subject matter through teaching occurred. The capacity to transform personal understanding, thus, depends on what teachers bring to the classroom. Whilst knowledge of learning and teaching and classrooms increases with experience, knowledge of subject content does not… Aubrey 1999 pp159-160

This paper explores the subject matter knowledge of mathematics teachers, an area acknowledged to be problematic and one which ‘has provoked more controversy than study’ (Grossman et al 1989).

While one can infer from studies of teacher thinking that teachers have knowledge of their students, of their curriculum, of the learning process that is used to make decisions, it remains unclear what teachers know about their subject matter … Wilson et al., 1987 p.108

Whilst Shulman and others have categorised the different components of subject knowledge and discussed its transformation through classroom events, our research data is used here to investigate the ways in which teachers’ subject knowledge is held and transformed. Whilst we acknowledge that subject matter knowledge may transform into pedagogical content knowledge, it is not always the case that it moves beyond learner-knowledge.

Methods and analysis

The nature of the research has been based within an interpretive research paradigm (Bassey, 1995). The theory that emerged was grounded in the data and existing research was used as a springboard for ideas, for data collection and for explanation. We have been studying a ‘singularity’(ibid. p.7) using data from a group of experienced teachers and a group of pre-service teachers to investigate the nature of subject knowledge. The data reported here is a sub-set of a wider set of data exploring aspects of professional knowledge, based on case studies which included interviews with primary, secondary (over two years, Prestage, 1999) and pre-service teachers (over one year, Perks, 1997). The experienced teachers were subject leaders nominated for their expertise by advisors to join a National Curriculum project (Prestage, 1996). The pre-service teachers were following a one year postgraduate course in education, with the majority having a first degree in mathematics. Subject knowledge was explored through questions about planning for teaching. The teachers were asked what they wanted their pupils to come to know about in mathematics. They were asked to describe the teaching sequences chosen to mirror learning aims for the pupils and to explain the influences upon these teaching sequences. The pre-service teachers were observed in lessons and their lesson plans, evaluations and debriefs of the observed lessons used in the analysis.

The literature on subject knowledge provides a context and a springboard for the work but did not offer a way forward with the analysis of the data from the experienced teacher (Prestage, 1999). This analysis needed to consider the wider debate on the professional development of teachers and in particular to Eraut’s (1994) comment on the twofold problem for professional education (see below). The model described arises mainly as a consequence of the interviews that took place with the expert teachers and from attempts to describe the different ways that teachers talk about subject matter and to re-analyse the data from the research on lesson planning with the pre-service teachers (Perks, 1997). Eraut’s quote is crucial to this analysis:

First, certain systems of thought or paradigms dominate a profession's thinking in a way that they are passed unquestioned from one generation to the next …… The second problem is the converse of the first. To make practical use of concepts and ideas other than those embedded in well-established professional traditions requires intellectual effort and an encouraging work-context. The meaning of a new idea has to be rediscovered in the practical situation, and the implication for action thought through. p.49

Eraut presents the idea that teachers acquire knowledge through professional traditions and in order to accommodate new ideas will need to work on them in the classroom (practical wisdom) and then think through the implication of these actions (deliberate reflection). Whilst the quote from Eraut encompasses the wide range of knowledge gained by teachers, we use these ideas to classify the subject knowledge of teachers. Three phases - professional traditions, practical wisdom, deliberate reflection were defined to classify teachers’ responses to questions about subject knowledge.

1 Professional Traditions: This phase represents teachers accounting for their subject knowledge through reference mainly to the unquestioning acquisition of subject knowledge from the professional traditions of their own learning, through use departmental schemes, texts and government policies, of doing things because ‘that is the way they are always done’.

2 Practical Wisdom: This phase represents teachers accounting for their subject knowledge through rediscovering subject knowledge in the practical situation i.e. in the classroom as a consequence of observing and working with pupils. There is varied evidence in the research to suggest that subject knowledge may or may not develop as a consequence of teaching. This classification is used when teachers talk about altering aspects of their subject knowledge as a consequence of classroom outcomes.

3 Deliberate Reflection: This phase represents an explicit intellectual re-working by teachers of their subject knowledge beyond that gained from classroom practice. This classification acknowledges that more than practice might be used to develop subject knowledge and infers a deliberate ‘standing back’ from the classroom situation in order to re-think aspects of subject knowledge to plan for teaching.

The use of the word phases indicates the possibility of movement over time as subject knowledge develops and transforms. For example if a teacher were asked about division of fractions:

a professional tradition response might be based within the learner-knowledge i.e. to teach the algorithm ‘change the sign to multiply and turn the second fraction upside down’;

a practical wisdom response might be to work in the classroom with a lot of activities that would help to demonstrate the algorithm;

a deliberate reflection response might be a an explicit integration of the mathematics behind the algorithm (reciprocal, inverse, generalisation) and be able to give the ‘why’ of the classroom activities chosen.

Few teachers’ responses to questions about subject matter were classified within the third phase. Brief descriptions of the three categories are given below whilst some implications for professional practice and development are offered in the conclusion.

Learner-knowledge and the professional traditions

Graduate mathematician interviewees to our secondary pre-service course naturally hold subject knowledge as learner-knowledge. When asked to calculate the division of one fraction by another, to differentiate a function, to solve a set of equations, all respond correctly. When asked why the answers are correct they do not know. They can do mathematics but they do not necessarily hold ‘multiple and fluid conceptions’. The view of teaching for these pre-service teachers is to replicate their learner-knowledge for others to learn. But this is also a view held by some experienced teachers. The three quotes given are all from heads of department:

A lot of what I teach is based on inertia of what has always been taught. We always taught about angle so angle is an important thing and so we'll teach it again ... based on unstated perceptions... previous experience ... texts .

So having drawn and measured all your angles then it would be angle facts/lines, parallel lines, quadrilaterals, polygons, Pythagoras and trigonometry. We would not do them immediately after each other … we tend to leave a gap.

The order I did it in was angles on a straight line, angles at a point, then vertically opposite angles and then parallel line theorems, angles in polygons, interior and exterior and then onto trig

High on the list of influences referred to when justifying decisions about the curriculum are text books and other departmental resources and their own experience of learning mathematics knowledge as well as ideas related to teaching practices, (Baturo & Nason, 1996; Ball 1988,1990). This is a strong influence for the pre-service teachers, Sara was asked why she had chosen the objective "using a calculator" when the lesson had had as its major focus the choosing of the correct operations for calculation, her response was "That's what the textbook said." Neil offered an objective "revision of examination questions", with a lesson which began with interesting examples of simultaneous equations which were easily solved by trial and improve, as offered by pupils, but his lesson focused on an algorithmic solution which was justified in the debrief because "that is how you have to do it in the exam." Justifications for experienced teachers also remain in existing traditions and in the new legislative curriculum.

I don't think it matters its just the order we do them in …I suppose the order I do things in is the order that I was taught, the order I have always done them I don't think it would have occurred to me [to change the order

it is perhaps based on two influences, how I was taught and the other… using a particular published scheme which has been the case in the past and again that sort of hierarchy is laid out. .... I'm teaching nets tomorrow. Why? It was put into the scheme of work by my predecessor

The content and the ideas for activities have broadly come from a selection of ideas in the teacher’s book, from the maths scheme that we use. Because at the moment we don't have a maths policy as such, we don't have any maths guidelines and as a stop gap our maths is organised around Cambridge maths.

The power of the National Curriculum to prove ’wrong’ a curriculum that has been in use for some years is startling:

In my experience I have found that the SMP [secondary] books tend to jump around a lot rather than follow the progression throughout the National Curriculum levels in shape.. and [in particular] … as for rotational symmetry that comes much later on [in the scheme]. This is an area that I am intending to write a scheme of work around to follow the levels of National Curriculum more closely.

The overwhelming characteristic the interviews was the powerful influence of schooling from which the teachers had each developed assumptions about the nature of mathematics so when the experienced teachers were asked about generalisation the insecurity is evident. This aspect of the curriculum had not been explicit in their experience in school. One secondary head of department said that she did not know about it and did not teach it and that her department did not really do enough of it. Exploring the 'it' in each of these statements produced no further explanations. Another gave a similar response to generalisation saying ‘we were never taught generalisation’. Progression was offered in terms of getting better at spotting patterns but no mathematical explanation was given for what ‘getting better’ meant nor how a teacher might help a pupil get better at generalisation other than through experience. The power of the new imposed tradition is startling here. The teachers have no explicit learner-knowledge for generalisation. Justifications for classroom decisions were based on ‘coursework activities in year 10 and 11’. They cannot see the connections to their own learning experiences. For example, they know ‘the sum of the angles of a triangle is equal to the angle on a straight line’ but they do not know this as a generalisation. The label ‘generalisation’ was not attached to this work and was unnecessary for answering questions about angles in triangles. It is possible that learner-knowledge is only accessible in its labelled form.

The role of practical wisdom

There comes a moment for many teachers when they realise that giving their learner-knowledge directly to pupils does not work.

if you present a problem to the class and there is a need for them to know something about a particular shape and the area of it, and therefore they would set the agenda … things are encompassed in a problem and the pupils are setting the agenda.

The pre-service teachers develop such practical wisdom during their teaching experiences as shown in their evaluation of lessons, Phil wrote about one lesson on area of triangles:

Needed to spend longer than I thought going through triangles… even with further explanation, most of the pupils were still using slope and height to calculate area.… I tried explaining splitting triangle in two, drop a perpendicular…, still had a few problems… although by turning the triangle round to get long side on base (horizontal) seemed to be getting there. Needs reinforcing again.

Many of the experienced teachers talked in the interviews about the consequences of diverse classroom interactions, of altering the teaching decisions in order to respond pupils’ needs. For one primary mathematics co-ordinator, the scheme was dominant and the extra security was maintained by the fact that: “ the scheme matches the National Curriculum” though in another part of the interview he said that the use of texts was tempered by his classroom experience: “What I have actually discovered is that it is not really like that … the timing is really important and the sequence is important.”

Classroom experiences engender pedagogical content knowledge based on perceived learner needs and new explanations and contexts are found to develop learner-knowledge. The learner-knowledge is not questioned, the teacher may still offer the rule ‘‘change the divide sign to a multiply and turn the second fraction upside down” but previous lessons may have included “how many quarters in one whole, how many in two etc.” or perhaps a starting point such as ÷=÷.