Studying mathematics for teaching inside teacher education

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1 Studying mathematics for teaching inside teacher education Jill Adler 1 and Zain Davis 2 1 School of Education, University of the Witwatersrand 2 School of Education, University of Cape Town adlerj@educ.wits.ac.za; zd@humanities.uct.ac.za In this short paper we describe the rationale and methodology, and particularly aspects of the language of description we have developed in the QUANTUM project. We have drawn on sociological theory, Shulman s seminal work on pedagogic content knowledge and the field of mathematics education to construct a framework that enables us to describe and investigate what and how mathematics and teaching, as dual objects in mathematics teacher education, come to be co-produced in and across different sites of practice. Introduction One of the central concerns of the QUANTUM Research Project is that of answering the question: what is constituted as mathematics knowledge for teaching in formalised in-service teacher education in South Africa and how it so constituted? Previous work related to the study is reported in Adler (2002). Previous and forthcoming work elaborating the project and aspects of the findings of the study are reported in Adler (2005), Adler & Davis (2003), Adler, Davis, Kazima, Parker and Webb (2005), Long (2003) Parker, Davis and Adler (2005), and Davis, Parker and Adler (2005). Our rationale for the study is three-fold, with epistemological, practical and methodological elements to it. Each of these has contributed to the methodology and design of the study. In this short paper we will elaborate the rationale and the implications for the design of the study. We will then describe our methodology and the language of description (Bernstein, 1996) that we have developed and are using to analyse the data generated. Rationale A distinguishing feature of mathematics teacher education is its dual, yet deeply interwoven objects: teaching (i.e. learning to teach) and mathematics (i.e. learning mathematics for teaching), These are two very different knowledge objects. In Bernstein s (1996) terms, while they are both horizontal knowledge structures, with multiple strands in their respective discipline or field, they have very different grammars. Mathematics has a strong grammar a mathematics text is easily recognizable as such; in contrast, teaching texts have a week grammar, drawing from various fields (such as psychology) as well as from practice. Mathematics and teaching are deeply intertwined in the practice of teaching, and so there are important questions, from an epistemological perspective, as to how these two different knowledge structures come to work together. The French school of mathematics education research, commonly referred to as Didactics, presents the argument that the consideration of mathematics in the context of didactical situations is necessarily informed by both mathematics and pedagogy. From this perspective, there isn t a general abstracted pedagogy, but rather that pedagogy and mathematics are dialectically related in educational practice 1. 1 See the July 2005 issue of Educational Studies in Mathematics. The entire issue is devoted to reporting on studies informed by the French school of mathematics education research. 160

2 Shulman s seminal work recognized this feature of teachers knowledge or knowing in action, and posited the notion of pedagogic content knowledge (PCK), as a key feature of what it is teachers know and do (1986). Our interest is what and how PCK, or what we have come to call Mathematics for Teaching (MfT) comes to be produced in various sites of mathematics teacher education. In this way we hope to contribute to the body of research on PCK that has developed in the past two decades, by providing further description of this phenomenon related to teachers knowledge. We refer to MfT in order to be more specific in our focus on Mathematics, and to signal our interest in knowing-in-action. This is the epistemological rationale for QUANTUM, which thus entails a study of knowledge production as this takes place in Teacher Education practice - and so through pedagogic discourse. Because mathematics and teaching are deeply interwoven in practice, this dual object is writ large in inservice teacher education programs where new and/or different ways of knowing and doing mathematics combine with new and/or different contexts for teaching. Such are the conditions of continuing professional development in South Africa. In practical terms, any teacher education programme, and particularly formalized in-service professional development programmes need to select aspects of mathematics and teaching into their courses. Post-apartheid South Africa has seen a proliferation of formalized mathematics teacher education programs. Debate continues as to how much mathematics and how much method or educational study should comprise such programmes, and then whether and how these programs should integrate or separate out opportunities to learn mathematics and teaching. This is the well-known subject-method tension. Programs range across this spectrum, and vary in the degrees to which opportunities for teachers to learn are focused on mathematics per se, and/or embedded in problems of (mathematics teaching) practice. That this practical issue persists was reflected in numerous presentations on national systems of mathematics teacher education at the First African Regional Congress of the International Commission on Mathematical Instruction, that took place in Johannesburg in June There was a similar lament across presentations from different countries. On the one hand, there was a view that in much of mathematics teacher education, and particularly in teacher preparation programmes, there were too many courses in mathematics per se, and not enough attention to methods. At the same time, there was a strong view that teachers in practice, that is, those qualifying from such programmes, did not know their mathematics well enough to teach it effectively. This points to MfT, and thus the importance of understanding and being able to describe this knowing in action. Here too, there emerged clear directions for how we conducted our study of MfT: We needed to study teacher education practice across sites differing in how they combined mathematics and teaching in the programme. In addition to these epistemological and practical motivations guiding our study, there were also lessons from the field of research in mathematics teacher education. In their survey of research related to mathematics teacher education, Adler, Ball, Krainer, Lin and Novotna (2004) made and evidenced a number of claims based on an analysis of research reported in the Journal of Mathematics Teacher Education since its inception in 1998, and in PME proceedings Firstly, mathematics teacher education research is dominated by case studies, with a significant proportion of the studies focused on single teachers. Secondly, most (mathematics) teacher education research is conducted by teacher educators studying the teachers with whom they are working. Commenting on the claim, Novotna noted that this phenomenon of insider and small case study research is not surprising and Krainer argued the benefits of insider participation in education research processes. Adler raised the issue of nearness and distance in such research: it is difficult, when you have an investment in who you are teaching, to take a skeptical stance towards that work. Important questions that need to be asked might be missed. An obvious way to enhance skepticism is to invite external eyes onto what we are doing. Adler suggested that another way is to develop strong and effective theoretical languages that enable us to create a distance between ourselves and what we are looking at. Hence, in the QUANTUM research project, where we are 161

3 studying teacher education practice in sites where some of us are working, we needed to work across sites (and so generate some comparative advantage) as well as construct distance and so a strong theoretical language. Table 1 below summarises the what and how of our study and captures how the rationale for the study, which in turn is informed by a study of the field, had implications for our methodology. Table 1. QUANTUM rationale and methodological implications. QUANTUM Rationale Implications for study Question: What and how is MfT produced in Mathematics Teacher education Epistemological Practical Methodological Mathematics and Teaching are different but dialectically related knowledge objects in the practice of teaching. Study of knowledge production through pedagogic discourse. Selections into teacher education come from both Mathematics and Teaching (education). Both amounts and levels of integration are contested. Study a range of sites where combinations of mathematics and teaching differ Issues of insider research, and single case studies. Study across sites, and through a strong theoretical language Methodology Not surprisingly then, in QUANTUM we have selected three contrasting sites (in terms of Mathematics and Teaching) for in-depth study and two courses within each site: a mathematics course and a mathematics education course. This selection followed an initial review of fourteen programmes across eleven South African universities 2. The three focus sites offer a continuum with respect to the integration of mathematics and teaching (content and method) within courses. The first site offers an Advanced Certificate of Education (ACE) and an Honours program where all courses integrate content and methods. These are aimed at teaching for the senior phase (Grades 7 9). The second site offers an ACE and is aimed at the Further Education phase (Grades 10 12). Courses in this program include, but separate, post secondary level mathematics courses and mathematics education courses. The third site offers a specialised Honours Degree. Here mathematics education courses run alongside mathematics courses, all of which have been designed specifically for secondary teachers, and so each with a strong eye on (and so influenced by) the other. For each of the six courses, we have video tapes of all contact session, all course materials and assessments as well as a post course interview with lecturer. Our data analysis tool (See Diagram 1 below) has been developed to systematically interrogate hours of pedagogic practice, and a large volume of qualitative data. As implied in our question, our goal is not to evaluate different programs and approaches, but rather to understand what and how mathematics and teaching (and so too, mathematics for teaching) come to be produced in these settings, in other words to examine practices inside teacher education. We hold that this will provide insight into opportunities for teacher learning. For ethical and practical reasons (space does 2 See Adler & Davis (forthcoming) for a report on phase 1 of the study and the examination of assessment tasks across all sites initially in the study. It was that phase, and the study of assessment tasks that facilitated the selection of focus sites for phase 2, and the depth study of three sites of practice referred to in this paper. 162

4 not allow their elaboration here) some members of the QUANTUM research team are working in one or other site we are studying. Thus, as discussed above, we faced the challenge of distance. To deal with this, and more generally to construct a principled and rigorous study, we have developed a set of theoretical propositions a strong external language to guide and frame the research. We have used this language, together with literature in the field, and our data archive to produce an analytic tool to probe the production of mathematics, teaching and mathematics for teaching in mathematics teacher education practice. Briefly (again space limitations prevailing), the tool emerges from our use of Bernstein s sociological theory of pedagogy. 3 This overarching theoretical orientation provides us a well developed theory of pedagogic discourse, and thus a means for studying knowledge production through pedagogic discourse (see Table 1 above) 4. According to Bernstein (1996) any pedagogy transmits evaluation rules. Evaluation condenses meaning, and transmits the criteria by which learners displays of knowledge are judged. The unit of analysis is referred to as an evaluative event, that is, a teaching-learning sequence focused on the acquisition of some or other content, which derives from our use of Bernstein s (1996) proposition stating that the whole of the pedagogic device is condensed in evaluation, as elaborated in the theory of the pedagogic device (cf. Davis et al, 2003). However, any act of evaluation has to appeal to some or other authorising ground in order to justify the selection of criteria. We call these legitimating appeals. ACTIVITY (e.g., Mathematics, Teaching) Evaluative Event (Aimed at the production of a knowledge object) M Mt Tm T (MT, mt) Legitimating appeals Mathematics Mathematics Education Everyday metaphorical Experience Curriculum Authority Diagram 1. Grounds for legitimating appeals. As is illustrated by our methodological tool in Diagram 1, in QUANTUM we are looking at evaluative events across teacher education programs, on the assumption then that these would reveal the kind of mathematical and teaching knowledge privileged in these programs. The tool we have enables us to 3 Our theoretical work has been led by Zain Davis, and is published as a QUANTUM working paper, See Davis et al (2003). 4 Our work builds on that of Ensor (2001) and her use of Bernstein s sociology of pedagogy to study recontextualising practices in teacher education and teaching in South Africa. Our focus on knowledge production is somewhat different, and thus takes a different path through and out of Bernstein s work. 163

5 identify mathematical and/or teaching notions made available and how these notions unfold over time 5. In particular, we are able to interrogate the resources mathematics teacher educators draw on (where they appeal) to make available (and then evaluate) mathematical and teaching ideas. Our data analysis has proceeded as follows: We have chunked pedagogic discourse into evaluative events, discerning these from the time a notion comes into being in class, and then is legitimated in some way in other words, there is an attempt to fix meaning. We have been able to identify the central notion in each event, and indicate whether the notion is Mathematics (M) or Teaching (T), and the extent to which there is an integration of these two notions within the event. We have marked these as Mt, Tm and so forth, indicating whether if the notion is mathematical, for example, this is presented within a teaching context, and vice versa. We were then able to identify and code what kind of legitimating appeal (fixing of meaning) was made, and particularly what kind of ground was appealed to. We found, for example, that appeals were made to teachers experiences of teaching, to principles of mathematics itself, to the field of mathematics education research as well as to the curriculum. In the latter case, we refer to appeals to a text book for example, to justify a definition of a concept (as opposed to the principles of mathematics underpinning the definition. Our initial analysis of data sources across three different teacher education sites suggests that the tool is indeed useful. While hard claims are not yet possible, the tool is showing us that in the practice of mathematics teacher education, appeals (for evaluative judgements) are distributed across mathematics, mathematics education, experience and curriculum knowledge. At this stage it appears that our exploration will contribute to strengthening the grammar (description) of mathematics for teaching, and build on work being developed by others e.g. Ball, Bass & Hill (2004). In the presentation of this paper, we will elaborate on each of these appeals, providing indicators that enable their recognition, and point to the results emerging from the study. Acknowledgements This paper is based on work supported by the National Research Foundation under Grant number Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Research Foundation. This paper forms part of a wider research project on Mathematical knowledge for Teaching, directed by Professor Jill Adler, at the University of the Witwatersrand. References Adler, J. (2002). INSET and Mathematics Teachers Conceptual Knowledge-in-Practice. In C. Malcolm & C. Lubisi, eds., Proceedings of the 10 th Annual SAARMSE Conference, Durban, Kwazulu-Natal January 22-26, pp Adler, J. (2005). Mathematics for teaching: What is it and why do we need to talk about it. Pythagoras, 62. Adler, J. & Davis, Z., (2003). An analysis of the structuring of evaluative tasks: A focus on level 6 mathematics INSET. In Jaffer, S., ed., Proceedings of the 9 th National Congress of the Association of Mathematics Education of South Africa, University of Cape Town, July Adler, J. & Davis, Z. (Forthcoming) Opening another black box: Researching mathematics for teaching in mathematics teacher education. Mimeo. 5 For the temporal analysis of evaluation, we draw on Hegel s theory of judgement, and the movement over time from immediacy of a notion, to reflection and necessity (what it is and is not and how this can be established). See Davis et al (2003). 164

6 Adler, J., Davis, Z., Kazima, M., Parker, D. & Webb, L. (2005). Working with students'mathematical productions: elaborating a key element of mathematical knowledge for teaching. In Chick, H.L. & Vincent, J. L (Eds.) Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Department of Science and Mathematics Education. Melbourne University. Australia. Volume 2, pp.1-8. Adler, J., Ball, D., Krainer, K., Lin, F.L. & Novotna, J. (2004). Research on mathematics teacher education: Mirror images of an emerging field. Plenary Presentation. ICME-10, Survey Team 3, Copenhagen, July Ball, D., Bass, H. and Hill, H. (2004). Knowing and using mathematical knowledge in teaching: Learning what matters. In Buffgler, A. and Laugksch, R. (Eds.) Proceedings for the 12 th Annual Conference of the South African Association for Research in Mathematics, Science and Technology Education. Durban. SAARMSTE. Bernstein, B. (1996). Pedagogy, Symbolic Control and Identity: Theory, Research, Critique, London: Taylor & Francis. Davis, Z., Adler, J., Parker, D. and Long, C. (2003). Elements of the language of description for the production of data. QUANTUM Research Project. Working paper #2. Johannesburg: University of the Witwatersrand. Davis, Z., Parker, D. and Adler, J (2005). Imaginary and symbolic identification in formalised in-service mathematics teacher education and the differential constitution of mathematics for teaching. Paper presented the Annual Kenton Conference. Rhodes University. October Mimeo. Ensor, P. (2001). From pre-service mathematics teacher education to beginning teaching: A study in recontextualising. Journal for Research in Mathematics Education, 32, 3, Long, C. (2003). Mathematics knowledge for teaching: how do we recognise this? In Jaffer, S. and Burgess, L. (Eds.) Proceedings of the Ninth National Congress of the Association for Mathematics Education in South Africa (AMESA) Cape Town, July Cape Town. AMESA. Parker, D., Davis, Z. and Adler, J. (2005). Mathematics for teaching and competence: Pedagogies in formalised in-service mathematics teacher education in South African Universities. In Goos, M., Kanes, C. and Brown, R. (Eds.) Proceedings of the 4th International Mathematics Education and Society Conference. Centre for Learning Research. Griffith University. Australia. Pp Shulman, L. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15, 2,

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