Professional mathematics teacher identity in the context of pre-service training

Sonja van Putten

University of Pretoria

svanputten@gk.up.ac.za

Professional mathematics teacher identity in the context of pre-service training

Abstract

In the TIMSS 2003 report, South Africa again achieved the lowest of all the scores. This may be partially attributed to the teachers of mathematics in this country. Who is the person who wants to teach mathematics? This study hypothesises that pre-service mathematics teachers have an image of themselves as professionals in the mathematics classroom, and that this image is part of what constitutes their identity as teachers. From the sample of Fourth Year mathematics education students at the University of Pretoria (UP), a maximum variation sub-sample of six students was selected for in-depth study.

Outcomes of this research may include an improved understanding of the complex nature of mathematics teaching in this country; there could also be implications for the nature and content of methodology modules at UP whose effectiveness may be called into question as insight is gained; identity influencers motivating entry into mathematics education may be identified, which, once known, could be stimulated for the purpose of diminishing the shortfall of teachers in this field.

Key Words: Mathematics teacher identity pre-service training

1.1 INTRODUCTION

In South Africa until quite recently, as in many countries all over the world, classrooms were mainly teacher-centred. However, the National Curriculum Statement introduced in 2005 entrenches a learner-centred approach to education in South Africa. Since this curriculum is relatively new, the majority of in-service teachers’ and current pre-service teachers’ own classroom experiences were teacher-centred.

Between the two poles of learner and teacher lies the mathematics student teacher who, for four years, is in a phase where s/he is no longer a learner, nor is s/he yet a teacher. At this time identity-influencing factors from “learner-hood” (which may be teacher-centred) are interacting with identity-influencing factors inherent in tertiary training, which, at the University of Pretoria, focuses on learner-centeredness. At the end of that four-year period fledgling teachers are launched into classrooms of their own, being their professional identity.

1.1.1 Aims of the Study

This study, still in progress, aims to contribute to our understanding of the nature of the identity of the mathematics teacher in the context of mathematics teacher training, set against the background of education in South Africa, as well as the factors which influence this identity during tertiary training at UP (University of Pretoria). It is understood that “identity” is a complex concept and that it is necessary to set very clear parameters within which this construct is to be defined and explored. It may be called a cache of capacity (van Zoest & Bohl, 2005) in which such constructs as knowledge, skills, beliefs, perceptions of self, to name but a few, co-exist to form what Palmer calls “the self that teaches”(2007, p. 8). The following objectives were set: the origins and characteristics as well as the elements influencing pre-service Professional Mathematics Teacher Identity (PMTI) should be explored and the PMTI-influencing aspects of tertiary training should be determined.

Since “The study of teachers and teaching deserves much more attention than it has been given, particularly in the light of growing empirical evidence that good teaching makes a huge difference to learning regardless of the socio-economic status of the learners” (Arends & Phurutse, 2009, p. 45), it is important to begin at the beginning: there is a need to investigate who the teacher of mathematics is before that career has even begun. It is possible that, consciously or not, students in mathematics teacher training have an identity which formed across their own experiences of mathematics education during their schooling and which resonated with some element within their own personalities. This study places that identity under a magnifying glass, and explores the formation and nature of such an identity, based on the hypothesis that pre-service mathematics teachers have or are in the process of constructing an image of themselves as professionals in the mathematics classroom, and that this image is part of what constitutes their identity as teachers.

1.1.2 Rationale for the study

In the TIMSS 2003 report, the score of Grade 8 learners in South Africa was the lowest of all the scores, both internationally and of the six participating countries in Africa. This study makes the assumption that this result is at least partially attributable to problems in both the supply of mathematics educators and the quality of their teaching. This is corroborated by the findings of Arends and Phurutse (2009) who state that “Teacher competency is increasingly seen as critical if all learners are to derive benefit from the schooling system. It has been found in many developing countries, learners benefit less from education due to the poor quality and quantity of instruction” (p. 1).

It is obvious that effective teaching of mathematics requires more than just sound subject knowledge. Then it should be equally obvious that efforts to improve mathematics teachers’ subject knowledge alone will not succeed in bringing about positive change in the teaching of mathematics in South Africa. Palmer (2007) speaks of the “tangles of teaching”: “Those tangles have three important sources. The first two are commonplace, but the third, and most fundamental, is rarely given its due” (p. 2). The first two he denotes as the complexities of the subject taught, and the even greater complexities of the learners being taught. The third, and the one on which this research focuses in particular, he describes as follows:

If students and subjects accounted for all the complexities of teaching, our standard ways of coping would do – keep up with our fields as best we can and learn enough techniques to stay ahead of the student psyche. But there is another reason for these complexities: we teach who we are. (p. 2)

Who then, is the pre-service mathematics teacher in the University of Pretoria (UP) context? Why did this person choose to teach? Why did this person choose to teach mathematics, specifically? How is the tertiary training s/he is undergoing adding value to the “who we are” that is going to be taught once such training is completed? These are the questions which underpin the rationale on which this research is based.

One of the outcomes of this research could be an improved understanding of the complex nature of mathematics teacher training in this country. There could also be implications for the nature and content of modules and courses whose effectiveness may be called into question as insight is gained into PMTI as manifested during tertiary training. Another outcome may be the identification of identity-influencers motivating entry into mathematics education, which, once known, could be stimulated for the purpose of diminishing the shortfall of teachers in this field.

1.1.3 Research questions

Where does PMTI come from and what does it look like? The literature (van Zoest & Bohl, 2005; Varghese et al, 2005; Boaler et al, 2000; Beijaard, 1995; Kagan, 1992) indicates that teacher identity is not a simple, unitary construct, but has both social (in-the-community) and personal (in-the-mind) roots, and that its nature is complex (Beijaard et al, 2004; Stronach et al, 2002; Zembylas, 2003; Cooper & Olson,1996). This research is therefore structured around the following questions:

1. What are the origins and characteristics of the professional mathematics teacher identity of students in their final year of pre-service training at UP?

In order to unpack this question, the following subquestions have been formulated in this regard, so as to explore what informs this identity both internally and externally:

1.1 How may the personal (intrinsic) and sociological (extrinsic) origins of this identity be described?

1.2 What are the characteristics of this identity?

2. How does tertiary training influence professional mathematics teacher identity at UP?

Such an analysis would imply that the following subquestion needs to be answered:

2.1 What are the factors in tertiary training which contribute to the development or modification of such an identity?

3. What are the implications for the training of pre-service mathematics teachers at the University of Pretoria?

1.1.4 Context

In South Africa, all learners take mathematics as a subject to the end of the ninth grade of their school career. After that, they may choose to do either mathematical literacy or mathematics to matric level. At UP, all students in the Faculty of Education who choose to train as teachers of mathematics have to have chosen mathematics as a matric (grade 12) subject. Therefore these students have each been in a mathematics class for twelve years before commencing their tertiary studies. A number of such matriculants enter the Faculty of Education at UP to train to become mathematics teachers. As students in the Department of Science, Mathematics and Technology Education, they are required to complete, amongst others, modules about mathematical content, the methodology of teaching mathematics, and teaching practicum, and after four years they are released into the professional world of teaching.

2.1 LITERATURE REVIEW

Teacher identity is complex in its constitution from both a personal and a sociological perspective, but also in terms of how these various elements behave. Some parts of this identity seem rigid and inflexible, bending with difficulty to receive change, while others are in a state of continuous change and development, depending on the stimuli to which they are subjected.

Liljedahl (2007) suggests that teacher knowledge i.e. knowledge of teaching and knowledge of teaching mathematics, is in fact part of a belief system. He espouses the notion postulated by Leatham (2006) that beliefs are “things that we just believe” and that knowledge is “what we more than believe – we know” (p. 92). Webdictionary, like Princeton University’s Wordnet, amongst others, defines belief as “any cognitive content held as true”; Plato defines knowledge as justified true belief. Bearing these definitions in mind, Liljedahl’s notion of knowledge being a subset of belief is a logical progression of thought. His theory may explain the nature of those elements in teacher identity which are characterised by continuous growth and development – these may be constructed on a network of ‘just beliefs’ which may be open to alternative persuasions, depending on the circumstances, while the elements of rigidity may be those based on ‘what we know’. Resistance is a natural reaction when what we believe to be truth is opposed: if a pre-service teacher has convictions about who he/she is as a teacher, based on their understanding of the truth about mathematics teaching and learning, and new opposing ideas are introduced (possibly even in tertiary training), it seems obvious that such ideas would not be readily accepted. Kagan (1992) confirms this:

Pre-service students enter programs of teacher education with personal beliefs about teaching images of good teachers, images of self as teacher, and memories of themselves as pupils in classrooms. These personal beliefs and images generally remain unchanged by a pre-service program and follow candidates into classroom practica and student teaching. For professional growth to occur, prior beliefs and images must be modified and reconstructed” (p. 142).

Although viewing knowledge as a social construct is a convenient way to differentiate between knowledge and beliefs, individuals (for the most part) operate based on knowledge as an individual construct. That is, their actions are guided by what they believe to be true rather than what may actually be true. Mathematics teachers (pre-service or inservice) are no different – their actions (ie teaching) are guided by what they believe to be true about mathematics and about the teaching and learning of mathematics.

For the purposes of this study, an operational definition of Professional Mathematics Teacher Identity has been formulated. No absolute definition of so elusive a concept has been agreed upon in the literature. Bearing in mind these different aspects as revealed by the literature, a workable definition may be formulated as follows:

Professional mathematics teacher identity is part of “the self that teaches” in which a complex system of knowledge and beliefs, including beliefs about mathematics and teaching, and about learners and learning, is created and continually developed through sociological experiences and personal predispositions.

To examine PMTI, it is necessary to consider both the sociological and psychological origins of the knowledge and beliefs (van Zoest and Bohl, 2005) and the dynamism (in the sense of changeability) of PMTI. In fact, according to Lasky (2005), “Human development occurs on two planes, first on the social plane and then on the psychological. In short, that which is psychological, is first social” (p. 900). Wenger (2000) indicates that professional identity is constituted not in isolation, but in the context of a “community of practice”. For the teacher, this community of practice may be seen as the school environment, colleagues teaching the same subject, and so on. For the pre-service teacher the community of practice is the tertiary institute and its staff, and fellow students, particularly those who take the same subjects.

Investigating that which prompts a learner to become a mathematics teacher, Boaler, William and Zevenbergen (2000), contend that “students who develop a sense of identity which resonates with the discourse of mathematics are more likely to continue with their studies than their peers who do not develop such a sense of identity” (p.2). Their approach therefore is sociological and they espouse the notion of identity as argued by Lave and Wenger (1991) that learning is in fact “a social practice through which we come to know who we are” (p. 2). So, the process of construction of what could become professional teacher identity originates at school.

This sociological viewpoint is echoed by Liljedahl who unequivocally states that “…the formation of teachers’ beliefs about mathematics teaching and learning come from their own experiences as a learner of mathematics” (2005, p. 2). Ball (1988) puts it like this: “Long before they enrol in their first education course or maths methods course, they have developed a web of interconnected ideas about mathematics, about teaching and learning mathematics, and about schools” (p. 40) Liljedahl (2007), speaking about student teachers and the knowledge and beliefs upon which teacher identity is founded, corroborates Ball’s point of view:

These ideas are more than just feelings or fleeting notions about mathematics and mathematics teaching. During their time as students of mathematics they first formulated, then concretized, deep seated beliefs about mathematics and what it means to learn and teach mathematics. It is these beliefs that often form the foundation on which they eventually build their own practice as teachers of mathematics.

Beijaard (1995) is in agreement: he also found that the identity of teachers was greatly influenced by their experiences of school when they were learners, and that, for secondary school teachers in particular, it was rooted primarily in their classroom experiences of the subject in which they eventually specialised.

Therefore there is a strong possibility of a circular, repetitive pattern occurring: learners who are not taught efficiently or effectively, may become teachers who do not teach efficiently or effectively. However, the “self” of pre-service teachers is not a finished product, but is subject to an ongoing process of formation (Cooper & Olson, 1996).

The literature reveals that professional identity is not a fixed, simple concept. Not only does the formation of professional identity of the mathematics teacher begin at school and continue to change, but it is in itself complex in that it is made up of different sub-identities, each developed in different circumstances and influences. Perhaps the most significant characteristic of professional identity, and one mentioned by most of the researchers, is its propensity to change and be in a state of continuous development. Boaler et al (2000) explain this phenomenon as follows: identity is “not stable, consistent or life-long, but dynamic and situated (p. 2). The process of change is nevertheless filtered through an existing belief system, so what is taught in teacher training may not outweigh what was learnt at school as a learner. Borko and Putnam (1996) explain this phenomenon as follows: “Although learning can be heavily influenced by instruction, how and what individuals learn is always shaped and filtered by their existing knowledge and beliefs. It can therefore never be completely determined by instruction” (p. 674).

At the time when large scale reforms were being imposed on the teaching of mathematics in the USA, Deborah Loewenberg Ball (1996), herself a teacher educator, examined the effects of such definitive changes on the teaching and learning of mathematics, and in particular on the changes in teacher education. She identified one of the main problems in adapting to reforms in the fact that the teachers were required to teach in ways they had never seen or experienced: in terms of identity then, they were required to adopt into their professional identities ways of teaching which did not correspond with their knowledge and beliefs about effective teaching and learning. The resistance to change, Ball found, was not based on negative aspects of the reforms, which were generally accepted to be “attractive and inspiring in many ways. Yet there are also powerful disincentives to engage with this agenda, and some of these are deeply personal and at the heart of the identity one tries to create as a good teacher” (ibid. p. 19). So, psychological factors can facilitate or block developments within PMTI.

There appears to be some contradiction in the literature: while some researchers define the nature of identity as ever-changing and in a state of flux, others describe identity (particularly of the mathematics educator) as rigid and inflexible. However, it is possible that this is a paradox rather than a contradiction, and that, while certain aspects of PMTI are immutable, others in fact are constantly being changed by a variety of influencing factors.

It is possible that, contrary to expectation, teacher education may in fact not be educating teachers as well as one might hope. Wenger (2000) states it quite simply: “Communities of practice cannot be romanticized. They are born of learning, but they can also learn not to learn” (p. 229). There are two possible situations in which little learning takes place, he postulates. In one, “competence and experience are too close” and in the other, “experience and competence are too disconnected” (p. 233). It is possible that teacher training is so dissociated from the students’ own school experiences that school and university are polarised. Wenger (2000) uses a hypothetical situation to illustrate this point – an ordinary person is in conversation with a group of serious scientists:

Sitting by that group of high-energy particle physicists, you might not learn much because the distance between your own experience and the competence you are confronting is just too great. Mostly what you are learning is that you do not belong. (p. 233).

If the student finds that what is taught in teacher training is not something which s/he can easily assimilate because of the distance between “experience and competence”, it may well result in the pre-service teacher abandoning what s/he has been taught in favour of traditional school praxis (Smagorinsky et al, 2004, p. 21). Borko and Putnam (1996) explain that “entering perspectives” (p. 679) are the most important influencing factors on pre-service teachers’ reception of professional training, because they act “as a filter that determines how experiences within the teacher education program are interpreted” (ibid, p. 679).

These “entering perspectives” are entrenched through what Borko and Putnam calculate to be more than 10 000 hours of apprenticeship in observing their teachers while they themselves were learners (p. 678). However, one or two hours per week out of possibly twenty eight weeks in the tertiary academic calendar for maybe three years make a generous estimate of pedagogical training: more than likely less than 170 hours in total during the course of a teacher training programme. Sheer numbers make it obvious that the training programme would have to outweigh the “entering perspectives” in intensity in order for the training to make a difference. The likelihood of this happening, according to Borko and Putnam, is not strong. The finding of Feiman-Nemser et al (1987) that “prospective elementary teachers do not come to teacher training feeling unprepared for teaching” may well have a wider application than just prospective elementary school teachers, and this would mean that pre-service teachers feel that they already know how to teach, giving an element of superfluity to their teacher training.

Teacher trainers operate from the framework of their own belief system within the modules they present, while trainee teachers work within the framework of their own system of beliefs and knowledge. Each of these role players (teacher trainer and trainee teacher) have their own concept of what should constitute teaching and learning, and each has an understanding of his/her own professional identity. According to Ball (1988), teacher training seldom takes serious cognisance of the beliefs and conceptions of student teachers:

Instead of taking what they already know and believe into account, teacher educators tend to view prospective teachers as simply lacking particular knowledge and skills. This lack of attention to what prospective 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. (p. 40)

Feiman-Nemser and Buchmann identified this as the “two-worlds” pitfall as early as 1985. The basis of this dichotomy, according to Smagorinsky is the university’s concerns with ideals, and the school’s concern with “their gritty application” (p. 9).

Thus a reason for the continued frailty of the effect of teacher training on pre-service teachers may lie in the complexity of the requirements of such teacher training: students must be taught to teach, and they must be taught mathematics. Adler et al describe this as a twofold task involving identity formation:

An enduring problem in mathematics teacher education is its task to build both mathematics and teaching identities. While we have learned a great deal about some of the specialty of teacher’s knowledge, we need to understand better what it means to teach both mathematics and teaching in the same program. We do not understand well enough how mathematics and teaching, as inter-related objects, come to produce and constitute each other in teacher education practice. …The field needs to understand better how mathematics and teaching combine in teachers’ development and identities. (p. 378)

The question therefore remains – how must the in-service university courses be structured or presented so that new learner centred knowledge and beliefs not only take pre-eminence over the old teacher centred knowledge and beliefs, but actually supersede them? Simply put, it appears that mathematics teacher education generally does not have the impact on teacher identity that it aims to have, other than in the increase of subject matter knowledge. A reason for this may be the general intransigence of teacher training programmes when it comes to adapting to, or even acknowledging the changing requirements of the reality of the school environment despite inquiries and reviews in this area. Adler, Ball, Krainer, Lin and Novotna (2005) found that the ‘gap’ between teacher educators and pre-service students is increasing: “a significant note about who is (re)learning to teach mathematics, is that differences are increasing between teacher educators and their ‘learners’ – i.e., prospective and practicing teachers” (p. 361).

The literature thus appears to indicate that the deep seated beliefs of teacher educators play as large a role as those of the pre-service teachers in the effect, or lack thereof, of teacher education on the nascent teacher identity. There is evidence in the literature of intransigence in the mathematics teacher education programmes when it comes to adapting to new realities in the school mathematics classroom; in the lecturers within such a programme when it comes to understanding the gap between what they perceive as good teaching and what their students define as such, based on their own experiences; and in the knowledge and belief systems with which students enter such a programme.

Liljedahl confirms that belief systems often constitute a major problem when change is required: “unfortunately, these deep seated beliefs often run counter to contemporary research on what constitutes good practice” (p. 1). He suggests two ways in which such beliefs may be changed: one advocated by Feiman-Nemser et al (1987) in which the beliefs of pre-service teachers are challenged, forcing them to make “explicit” that which was “implicitly constructed”, making them transform their beliefs from “non-evidential to evidential” (p. 2); the second way involves “being submersed in a constructivist environment” (p. 2). Liljedahl’s application of a combination of these techniques, in what he calls “mathematical discovery” and which forces students to make explicit their beliefs in confrontation with mathematical problem solving, “has shown that pre-service teachers’ experiences with mathematical discovery has a profound, and immediate, transformative effect on the beliefs regarding the nature of mathematics as well as their beliefs regarding the teaching and learning of mathematics” (p. 2). Liljedahl thus proved that radical change in the belief system within a teacher identity is possible, albeit under highly controlled circumstances. Habitual reflectivity may also facilitate the ability to reform PMTI.

Skott (2001) espouses this idea: “Teachers’ reflections on practices, then, may turn the classroom into a learning environment for teachers as well as students” (p. 4). Therefore, if pre-service mathematics teachers may be brought to reflect on their belief system, it is possible that tertiary training may become more effective.

Most teacher training programmes include practical inservice periods in which students are required to work at a school as student teachers. In theory, this provides the student with an opportunity to put into practice new knowledge and skills; it allows the lecturers to assess the student as professional teacher; and it permits the student to test the character and robustness of his/her professional teacher identity against reality. Which belief system will be dominant – the “entering perspectives”, or those imparted during teacher training? Borko and Putnam (1996) found that “prospective teachers often experience traditional forms of teaching during their field experiences” (p. 680), which would further entrench the properties of identity as formed prior to tertiary training. It is not unusual that the school’s ways take precedence over the university’s in the student teacher’s mind during teaching practicum: “the motive of the school setting will potentially override that of the university setting because of the change in role from student to teacher and change in evaluative clout from professors to mentor teacher” (p. 22).

Teacher training, according to the literature, is therefore not as effective as it purports or aims to be. Various theories are put forward: the pre-service teacher’s beliefs and knowledge gained from school experiences outweigh the precepts of teacher training simply because of the ratio of time spent in either institution; teaching practicum reinforces the pre-service teacher’s own school experiences, not what is taught in tertiary methodology classes; pre-service mathematics teachers’ subject knowledge is insufficient to allow them to break away from the old methodologies within which they feel safe. Promoting a reflective practice as part of teacher training is suggested as a way around the intransigence of methodologically unsound beliefs, but the practicalities of instilling such practice in such a way as to alter or even replace these beliefs remain unclear.

2.2 CONCEPTUAL FRAMEWORK

The literature reveals that research into professional teacher identity is gaining more and more interest since it may explain why some teachers are more reform-minded than others or even why some are simply better educators than others. It may also improve our understanding of why inservice and pre-service training does not have the expected effect. However, since PMTI has a complex nature, any theoretical framework used to study this construct needs to address this complexity. For this reason, it was deemed necessary to combine aspects of theoretical frameworks regarding investigation of teacher identity as put forward by van Zoest and Bohl; Beijaard and Leatham.

Van Zoest and Bohl

Van Zoest and Bohl (2005) offer a framework for mathematics teacher identity based on Wenger’s theory that identity development is rooted in learning within communities. Participation in such a community of practice is identity-linked. However, van Zoest and Bohl point out the failing of Wenger’s theory to fully encompass the teaching environment because of its lack of ‘concrete reference of individual cognition” (p. 332). Since Van Zoest and Bohl operate specifically in the arena of mathematics teacher education, while Wenger’s postulation was general, they have woven into the framework “cognitive notions of thinking as residing in one’s head” (p. 315). They describe their theory as follows: “We view forms of learning and knowing as lying on such a continuum, with in-the-brain on one end, social on the other, and every variation of combinations of the two stretched between them” (p. 332).

The particular applicability of Van Zoest and Bohl’s framework in the context of this study lies in the fact that it was created against the backdrop of “shifts in mathematics education [which] in the last decade or so have caused much turmoil among people involved in mathematics education in the United States” (p. 315). Just such shifts have taken place in the South African mathematics teaching environment, with a new curriculum for the FET (Further Education and Training ie grades 10 - 12) phase being introduced in the classrooms in 2006. Van Zoest and Bohl found that the exigencies of reform brought about a renewed focus on how pre-service mathematics educators may be effectively trained. Their research brought them to an analysis of what they call “cache of capacity”– “the knowledge, skills and understanding that teachers carry with them from one context to the next” (p. 315).

Beijaard et al

Douwe Beijaard, Nico Verloop and Jan Vermunt (2000) in their study of professional teacher identity, were “inspired” (p. 751) by the work of Bromme, from which they drew the idea that teachers derive their professional identity from “the ways they see themselves as subject matter experts, pedagogical [nurturing] experts, and didactical [teaching] experts” (p. 751). So Beijaard’s theoretical framework expands Van Zoest and Bohl’s notion of individual cognition, Self-in-the-mind, by refining their second domain (pedagogy) into two: pedagogy and didactics. Beijaard et al also include as part of their framework an investigation into the factors which influence the formation of teacher identity. They look at context: the ecology of the classroom and culture of the school i.e. teaching experience. For the purposes of this research, Beijaard’s framework in which pedagogy and didactics are seen as two separate domains, is useful since it allows investigation into the knowledge involved in the teaching process, as distinct from the knowledge and skills involved in the support or nurture of learners.

Leatham

In order to better study the cache of “knowledge, skills and understanding” referred to by van Zoest and Bohl, the conceptual framework proposed by Keith Leatham (2006) is particularly appropriate. He calls it the “Sensible Systems of Beliefs”. His understanding of the interplay between knowledge and beliefs follows the Platonist notion of knowledge being justified true belief – in other words that knowledge and belief are both subsets of what is believed, or held as true. He describes the distinction as follows:

Of all the things we believe, there are some things we ‘just believe’, and other things we ‘more than believe – we know’. Those things we ‘more than believe’ we refer to as knowledge and those things we ‘just believe’ we refer to as beliefs. Thus beliefs and knowledge can be profitably be viewed as complementary subsets of the set of things we believe. It is in this sense that belief is used in the sensible system framework. (p. 92)

This sensible system framework suggests that “what one believes influences what one does” (p. 92). In the context of tertiary training, this is particularly relevant since the aim of such training is to “influence” what the students eventually “do” as teachers in their own classrooms. However, such influence will only become acceptable to the students as part of their own cache of capacity if it makes sense to them: “beliefs become viable for an individual when they make sense with respect to that individual’s other beliefs” (p. 93). Leatham’s framework, while dealing with the knowledge and beliefs to which both van Zoest and Bohl and Beijaard et al refer, speaks specifically to the issue underlying the third of the research questions in this study: tertiary education needs to fit in with the student’s sensible system of beliefs in order to be relevant and to provide efficient training.

Juxtaposition of the theories of Van Zoest and Bohl, Beijaard et al and Leatham now become a new framework in which PMTI can be examined. In this framework, Professional Mathematics Teacher Identity is recognised as a system of knowledge and beliefs which can be examined in terms of origin (in-the-mind and in-the-community), nature (as a system of knowledge and beliefs) and development, in the tertiary education context.

3.1 RESEARCH DESIGN AND METHODOLOGY

The study of any phenomenon, according to van Manen (2007), “is a project of sober reflection on the lived experience of human existence” p. 12). To elucidate his understanding of phenomenology, van Manen refers to lines in a poem by Rainer Maria Rilke (1987) which van Manen believes captures the essence of phenomenology:

kind of in-seeing,

in the indescribably swift, deep, timeless moments

of this divine seeing into the heart of things.

It has an effect on the ‘in-see-er’. Says Heidegger, “even if we can't do anything with it, may not philosophy in the end do something with us, provided that we engage ourselves with it?” (p. 13). The third research question in this study looks at the implications of its findings on the tertiary training for which, in part, I am responsible. Therefore my research philosophy is phenomenological: I investigate the phenomenon of PMTI not only to “see into it”, but also to allow that investigation to affect my practice.

This study may be described as an explanatory case study, both descriptive and interpretive, in a context-specific setting (mathematics teacher training, UP), which investigates professional mathematics teacher identity of six pre-service students in their final year of training at UP. The research in this study, qualitative in nature and exploring such a deep-seated construct as PMTI, needed to be designed in such a way that depth was the focus, not breadth. A case study design was the obvious choice for this research. It is true that case study research does not yield generalisable findings, but it was not the purpose of this study to generalise. However, as Goetz and LeCompte (1984) explain, “inductive researchers hope to find a theory that explains their data” (p. 4).

At the UP, the BEd (Bachelor of Education), a four-year degree, is currently constructed in such a way that the subject methodologies constitute a year-long module which is offered in their third year of study. The academic subjects like mathematics, are taught during the first three years of study only. For three weeks at the beginning of each of the second and third years the students are sent out to schools on a short teaching practicum exercise, in which observation is their main task. During their fourth year the students undergo further academic training for the first quarter, where after they spend the second and third terms at schools doing their “internship” or teaching practicum.

The Fourth Year mathematics education students were selected to form the sample of participants in this case study. The class consisted of sixty five students, of which thirty one were studying to teach in the FET phase. Twenty five of these students consented to be available for selection as participants in the study. A general questionnaire was administered to the whole group to provide biographical details and a basis for discussion in the initial individual interviews, after which a sub-sample of six students was selected for closer study. Purposive maximum variation sampling was used to determine the participants in this study. A maximum variation sample was chosen since the aim of this study was to find out ‘what’s out there’ in terms of PMTI, not to generalise from a ‘typical’ sample. The general questionnaire was analysed and coded using SPSS. Individual interviews were held with these six students before their teaching practicum in the second term of the school year. These students were filmed teaching the grade of their choice. Then a group interview was held after the teaching practice. An interview was also conducted with the mentor teachers of these students, in which questions were asked regarding the characteristics of the PMTI they were able to observe in their student. The interviews were digitally recorded and classroom observations were video-taped. These recordings were transcribed and coded according to the information required by the research questions.

4.1 CONCLUSION

This study is still in progress, so no final conclusions can yet be discussed. However, data analysis to date (initial individual interviews have been transcribed and analysed) suggests a preliminary answer to the first of my research questions regarding the origin of PMTI: the students who participated in this study entered their teacher training programme with a variety of ideas of what it means to be a teacher of mathematics, some of which changed dramatically during the course of their studies, but, in strong contrast to what the literature says, few of the students identified their high school mathematics classroom experiences as the origin of their desire to teach mathematics. More common, as origins, were a love for the subject and family influences. Several indicated that their own PMTI upon entering teacher training was largely inchoate and undefined.

So far, with regard to the second of my questions, it seems that the greatest changes in perceptions regarding Professional Mathematics Teacher Identity took place in the minds of the students who came from formerly disadvantaged high schools. These students generally described their high school mathematics teachers as frequently absent, incompetent, or unable to communicate their knowledge in a way that their learners could understand. So these students entered the training programme with an idea of what a mathematics teacher is which was in conflict with what they were subsequently taught. Most of the students affirmed that their training equipped them to teach in a way that was potentially better than the way in which they themselves were taught.

It remains to be seen what the nature of the influence of teaching practica was, and how the students perceived the effect of these, if any, upon their PMTI. Were they caught in Feiman-Nemser and Buchmann’s “two-worlds” pitfall? Further analysis of the data should reveal how the influence of mathematics teacher training fared during the period of praxis.

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