GROUP 7: RESEARCH PARADIGMS AND METHODOLOGIES AND THEIR RELATIONSHIP TO QUESTIONS IN MATHEMATICS EDUCATION

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1 GROUP 7: RESEARCH PARADIGMS AND METHODOLOGIES AND THEIR RELATIONSHIP TO QUESTIONS IN MATHEMATICS EDUCATION MILAN HEJNY (CZ) JUAN DIAZ GODINO (E) HERMAN MAIER (D) CHRISTINE SHIU (GB)

2 European Research in Mathematics Education I.II: Group RESEARCH PARADIGMS AND METHODOLOGIES AND THEIR RELATIONSHIP TO QUESTIONS IN MATHEMATICAL EDUCATION Milan Hejny 1, Christine Shiu 2, Juan Diaz Godino 3, Herman Maier 4 1 Faculty of Education, Charles University, Prague, Czech Republic milan.hejny@pedf.cuni.cz 2 The Open University, UK c.m.shiu@open.ac.uk 3 University of Granada, Spain jgodino@goliat.ugr.es 4 Universitaet Regensburg, Germany hermann.maier@mathematik.uni-regensburg.de SUMMARY AND DISCUSSION (Elaborated by Christine Shiu) The seven papers which comprised the starting point for our discussion and are published above seemed to fall naturally into three types: two dealt with global theories of mathematics education (Rouchier, Godino & Batanero), three were empirical studies which addressed specific questions in mathematics education (Bagni, D Amore & Maier, Stein) and the remaining two presented accounts of elaborated methodologies which have been devised and developed for specific purposes in research into mathematics education (Marí, Stehlíková). It was agreed in the group that we would allow one session of minutes in length (known as an A-type session) for each paper to be discussed intensively and that such sessions would be interspersed with longer ones (90 minutes, known as B-type sessions) in which general issues emerging from the particular cases would be examined. In A-type sessions the author-presenter gave a brief synthesis of the paper and added any complementary information which was deemed appropriate. The chair then moved the group from questions of clarification into the identification of the issues arising.

3 European Research in Mathematics Education I.II: Group The pattern of the sessions was as follows: Day 1 Session A1 André Rouchier, chaired by Klaus Hasemann Session A2 Juan Godino, chaired by Leo Rogers Session A3 Giorgio Bagni, chaired by Martin Stein Session B1 general discussion, chaired by Hermann Maier Day 2 Session A4 Hermann Maier, chaired by Jeremy Kilpatrick Session A5 Martin Stein, chaired by André Rouchier Session A6 José Marí, chaired by Nada Stehlíková Session B2 general discussion, chaired by Juan Godino Day 3 Session A7 Nada Stehlíková, chaired by Gunnar Gjone Session B3 general discussion, chaired by Christine Shiu The resultant general discussion involved us in a consideration of the nature, roles and functions of theories in mathematics education and how these related to specific questions in mathematics education, and how both theories and specific questions shaped and were shaped by the methods used to investigate the questions. What emerged was a triad of theory, research questions and methods, which when developed and used in a particular context, could viewed as a research paradigm. It was noticeable that specific research paradigms seemed often to be particular to the cultural or national context in which they were developed. It was soon apparent that interpretations of the meanings of many words which we were using differed among the members of the group. These differences too might be particular to their cultural or national context. Elucidation of differences framed the discussion as they were explored through a series of questions. The questions themselves were in their turn scrutinised and refined as the sessions progressed, and the following summarises the progress and scope of the discussion.

4 European Research in Mathematics Education I.II: Group Theories in and of Mathematics Education 1.1 What is a Theory and How Do Theories Arise? We took as a working definition that a theory is a framework of concepts which show how things work. Theories in and of mathematics education can be developed within mathematics education but often, quite properly borrow from and particularise theories developed in other disciplines such as psychology, sociology, anthropology, semiotics etc. Theories can therefore be imported from other fields of knowledge (and used unchanged or adapted to a certain degree) or be the results of previous research in mathematics education or inventions on the basis of an implicit and holistic pre-understanding of the field of research. 1.2 What Is the Role and Function of Theory in the Research Process? Theory is not imperialism but simplification. Theory functions in the research process as a means of reducing and controlling the variables that have to be taken into account when the researcher is studying, for example, a didactical process. Mathematics education is a domain in which there are a large number of salient variables. Researchers patterns of interaction with the reality studied necessarily involve reducing and controlling some of those variables, in order to allow them to attend to those they investigate. Theory is thus one means (the most powerful) we have of making an a priori analysis of the variables inherent in a situation, and hence of allowing us to make rational and defensible choices about what variables to control. For example, consider the work of the researcher who is creating a classroom situation with the help of didactical engineering. The a priori analysis is the study of all possible events (from a cognitive and mathematical point of view) in the development of that situation according to the specific mathematics problems and to the other components of the didactical setting. It corresponds to the task analysis of psychologists. This analysis then aids the interpretation of the actual events, so helping our understanding of the process under study.

5 European Research in Mathematics Education I.II: Group Theory is also a means of generating questions and problems that are practical, operational, open to empirical study. The same a priori analysis of the situation often identifies important phenomena susceptible to investigation. It shapes the view of the reality to be investigated and opens a particular perspective on this reality, hence raising and precising questions and equipping the researcher with a language to formulate these questions. Finally it guides the data collection and the data analysis and is thus a determinant of the research methods to be used. Sometimes it is appropriate to start with an empirical exploration of the research field without an explicit theoretical model. This happens when the researcher wants to remain open for a deep and adequate understanding of the reality. Reflection on the analysed data can result in a description or an explanatory theory. Theory generated this way is known as grounded theory, comprising a posteriori accounts emerging from observed data. This notion seems to be consistent with the working definition given above. However it is essential that the process is completed, and that researchers claims to be developing grounded theory are substantiated in their reports with specific accounts of the emergent theory being provided. Some more specific functions of theory might actually demand the production of different kinds of theory. Four possibilities are: descriptive theories which give an account of what has happened, describe what is the case; explanatory theories which seek to explain why or how something happened; predictive theories which predict what will happen in given conditions; action theories which guide action by identifying what can be done in given conditions. Identifying such possibilities raises a further question.

6 European Research in Mathematics Education I.II: Group Is It Feasible and Useful to Distinguish Different Kinds of Theory? Perhaps partial theories might better be designated models reserving the word theory for the larger frameworks of concepts which offer more global accounts of how things work. It may be noted that the field of mathematics education can be broken down into different kinds of elements and that different elements would demand different global theories. They would also lead to different research questions and different methods of investigation. In the two examples in this section of the proceedings the elements are didactical situations (Rouchier) and meanings of mathematical objects (Godino). 1.4 How Can Theories in Mathematics Education Be Evaluated? Mathematics education is not mathematics, nor is it a science. Its theories cannot be proved by an A implies B chain of logical reasoning. Nor can we look for the Popperian notion of falsification by a crucial experiment. Rather we must look for self-consistent narratives in terms of identified elements within mathematics education which are sufficient and are illuminating in accounting for the observed phenomena. The criteria by which theories in mathematics education are judged are adequacy and usefulness. Evaluation is therefore essentially pragmatic. Published theories can also be subjected to external evaluation through data generated independently of the theory. Can possessors of data apply a theory so as to see their observations in a new way? Can generators of theory interpret external data in terms of existing formulations? For example, what light can be thrown on Bagni s teaching experiment by the theory of didactical situations or by the theory of meanings of mathematical objects? The focus of the paper attends more closely to the elements of the latter theory. A relationship between the introduction of a group concept by means of an historical example and an earlier investigation of the meaning of mathematical objects could be posited. In particular, the introduction of the group structure could be read as a human activity involving the solution of social-shared problem-situations leading to the creation of a symbolic language in which problem-situations and their solutions are expressed. (Godino & Batanero 1998, p. 179).

7 European Research in Mathematics Education I.II: Group By such tests theories are more likely to be modified or extended rather than verified or falsified in their entirety. They may also be extended, modified or changed through theoretical reflection. Experience suggests that major paradigm shifts occur when a theory is found inadequate. 2. Conducting Empirical Research 2.1 What Can Be Learned from Empirical Research? Empirical research can be carried out in a number of physical and social contexts. In our three examples Bagni reported a teaching experiment carried out in regular classrooms. The textual eigenproductions (TEPs) discussed by Maier were created in classrooms but the data included teachers responses to these TEPs outside a classroom setting. In Stein s study the social unit was a pair of students working on a task rather than the whole class. The choices implied by these contexts derived from the phenomenon to be researched and the particular questions to be addressed. In all cases the description of the conditions of the study is an essential tool in allowing the reader to interpret the findings. It was noted that over the history of research in mathematics education, which was seen as a twentieth century phenomenon, there has been a marked shift from quantitative to qualitative methods and approaches. This probably reflects a view that it is the exceptional case which challenges our perceptions of what is the case, and which causes us to seek new interpretations and explanations. We therefore need phenomenological accounts thick descriptions of individual cases. On the other hand, a possible danger inherent in the exclusive pursuit of qualitative data is that we may fail to establish the typical to which our special case is the exception. We may need a broader picture, possibly established through quantitative approaches, to anchor our new interpretations.

8 European Research in Mathematics Education I.II: Group What Is the Relationship Between the Researcher and the Researched? Another issue in empirical research is the effect of the observer on the observed, especially but not exclusively, when the observer is a participant in the situation. In the above discussion of theory it was noted that studying classroom processes becomes possible when we are able to reduce the numbers of variables or to control some of them. It follows that where an external researcher is studying the interactions in a particular classroom the construction of the teaching to be done must be shared by the teacher and the researcher. The technology of building situations (to research) is supported by previous knowledge in the field (including in particular theoretical knowledge). This has been described as ingenierie didactique or didactical engineering. Action research is research carried out by practitioners on their own practice with a view to changing and improving that practice. Two of the papers (D Amore & Maier, Stehlíková) described projects which involved teachers taking on aspects of the role of researchers through a sharing of the methods of the projects. Discussion of this led to a further question. 2.3 How Can Research Best Be Shared with Teachers? There is a responsibility to report on research results in a format which is accessible to teachers. However it is often more fruitful to share the methods of research usually qualitative methods with teachers so that they can construct knowledge particular to their own situation. 2.4 Developing Methodologies What is a methodology? As with theory there is a question about how global, how all-embracing a methodology must be in order to be called a methodology. In both of the papers above (Marí, Stehlíková) the methodology is detailed and elaborated. In both cases this elaboration performs an integrating function in the first a particular topic from the school

9 European Research in Mathematics Education I.II: Group mathematics curriculum is chosen and a very thorough survey of existing research of that topic is synthesised to give a basis for the study. In the second the project examines a range of aspects of learning mathematics and seeks to integrate the work of many researchers including teacher-researchers to produce the findings. It was agreed that major projects need all-embracing methodologies. However much valuable research is carried on a smaller scale and it was pointed out that without such small scale studies there would be no existing research to survey. What is important is that in reporting research the methods used and the reasons for the choice of methods are reported as clearly as possible to allow others to follow up and build on that work. 3. Conclusions The conference structure and themes worked well for Group 7 so we relate our conclusions to the themes of the conference. 3.1 Communication All participants found great value in sharing accounts of research practice. We agreed that an important aim (possibly the central aim) of our research is the improvement of mathematics teaching. It follows that an important part of communication is communicating with teachers of mathematics. What we communicate may be research results, but it may be research methods, perhaps helping the reflective practitioner to become a practitioner-researcher. Whoever the audience of our research reports, in order to maximise communication we need to be as clear as possible about the researchers underpinning theoretical position, and about both the conditions and the methods of research. One difficulty with communication with colleagues from different research traditions lies in undeclared assumptions. Ambiguities and misunderstandings may also arise from the different meanings and connotations of words which we use in

10 European Research in Mathematics Education I.II: Group common. 3.2 Cooperation The spirit of cooperation was strong in the group. Like all groups we started from the premise that papers would not be presented, but would have been read by all. This proved to be a valid premise. Nevertheless we found that our A-type sessions allowed a useful focus on individual papers from which authors received feedback and ideas for improving final versions of papers and participants gained deeper insight into what they had read. 3.3 Collaboration As we worked there was a growing awareness of similarities and differences among our perspectives. It seemed that research paradigms are in some ways particular to the country in which they arise. Our activities were essentially the first stage of a group project to improve communication by clarifying commonly used terms such as: paradigm, theory, didactics, mathematics. 4. References Godino J.D. & Batanero C. (1998) Clarifying the meaning of mathematical objects as a priority area for research in mathematics education, in Sierpinska A. & Kilpatrick J. (1998) Mathematics Education as a research domain: a search for identity, Kluwer, pages