CONTINUING THE SEARCH
Anna Sierpinska & Jeremy Kilpatrick
In this chapter, we attempt to draw together the principal arguments made in the preceding chapters and at the ICMI Study Conference so as to provide the reader with a retrospective view of the clearly very complex issue, 'What is research in mathematics education, and what are its results?'. To that question, we are compelled to add: What does it mean to be a researcher in mathematics education? Do we have a common identity?
A theme resounding throughout the book has been that none of these questions, or the other questions arising from them, can be given a definite answer. Like research questions in mathematics education, questions about the field itself can be explored, but there is no way to resolve them once and for all in the way, for example, one might prove a theorem. Each generation must address anew what doing research in mathematics education is all about.
Even as the preceding chapters demonstrate our inability to provide definite answers, they both highlight and elaborate the progress we have made in clarifying the meanings of the questions. As the Discussion Document indicated and the Conference reaffirmed, the aim of this ICMI Study was to identify 'perspectives, goals, research problems, and ways of approaching problems' so as to allow a productive confrontation of views to occur and our mutual understanding to grow. This chapter, then, does not portray a finished and final consensus about our field. Instead, it points toward directions in which to continue the neverending search for who we are and who we want to be.
MATHEMATICS EDUCATION AS A RESEARCH FIELD
Some of the preceding chapters address directly the fundamental question: what is the nature of research in mathematics education? The authors attempt to situate the research within categories of existing knowledge or research practice. What one generally finds in the chapters, however, are mostly statements about where research in mathematics education should be situated rather than where it actually is being situated by practitioners and their colleagues from academia. In fact, there seems to be genuine difficulty in classifying research in mathematics education within existing disciplines because of its dual theoreticpragmatic character, which many authors stress.
Several authors remind us of the interdisciplinarity of research in mathematics education. Presmeg, for example, proposes that research in mathematics education should look at the complex phenomena of the teaching and learning of mathematics from many points of view simultaneously, in a balanced manner. She mentions mathematics, psychology, sociology, anthropology, history, linguistics, and philosophy. Adda argues that research in mathematics education should belong to a discipline she calls Cognitive Science (in the singular) that has not been established yet as such but that seems to be emerging as a result of cooperation between linguists, psychologists, computer scientists, neurobiologists, logicians, and mathematicians. She thus concedes that research in mathematics education is interdisciplinary but hopes that interdisciplinary cooperation will eventually lead to the emergence of a new entity that can provide a home for mathematics education research.
Interdisciplinarity does not usually mean that results and methods from certain disciplines should be or can be transposed wholesale into mathematics education research. The reason, according to Presmeg, is that research in mathematics education has 'not only the scientific goal of theorybuilding but also the pragmatic goal of the improvement of the teaching and learning of mathematics at all levels'. A similar argument is made by Wittmann, who says that 'scientific knowledge in mathematics education cannot be obtained by simply combining results from [the neighboring disciplines]; it presupposes a specific didactic approach that integrates different aspects into a new whole which takes into account its transposition into practice'.
Presmeg lists several neighboring disciplines of mathematics education without giving priority to any; she advocates a balanced view that takes a variety of perspectives into account. Other participants in the Study, however, insisted on giving a privileged status to the links of mathematics education with mathematics. For example, in his remarks at the first plenary session of the Study Conference, Guy Brousseau stated:
There exist problems in mathematics education that are irreducibly mathematical like, for example, the choice of problems, the organization of mathematical activities for didactic purposes, the analysis of understanding in mathematics, [and] ... the structuring of the mathematical discourse. ... There is no conjunction of classical disciplines to explain the functioning of this irreducibly mathematical part of teaching.... A scientific approach to [this part] is and will be essentially the work of a mathematician.
Quoting the mathematician William Thurston, Brousseau proposed a broadening of the concept of mathematical activity to encompass not only the production of definitions, conjectures, theorems, and proofs, but also 'the communication of results, the reorganization of theories and knowledge, the formulation of questions and problems, and all that "enables people to understand and think more clearly and effectively about mathematics (Thurston 1994, p. 163, emphasis in original). This broader view of mathematical activity implies, according to Brousseau, that 'the didactique of mathematics would become an integral part of mathematics'. He did not mean that all of mathematics education research is mathematics or should be considered part of mathematics. Rather, he identified three components of research in mathematics education. In addition to the 'didactique of mathematics' proper, whose task is to identify and explain the teaching and learning phenomena specific to mathematics, he distinguished between research stemming from the application of methods and concepts from other disciplines (e.g., psychology, sociology, linguistics, pedagogy, general education) not specific to mathematics, and didactic engineering, which is a design activity devoted to the elaboration of didactic means and teaching materials for mathematics, using the results of research from the two other components.
The design of teaching processes receives a more prominent role in the approach proposed by Wittmann, for whom the 'development and evaluation of substantial teaching units' belongs to the core of research in mathematics education. He compares the domain of mathematics education to music, engineering, and medicine, observing that in these fields, too, creative, practical activity has priority over theoretical deliberations, laboratory research, and critiques.
Other authors in this book are not concerned with attempting to situate mathematics education within or among the classical academic organization of disciplines as it exists in many universities. Ernest proposes to take what he calls a postmodern view that 'does not separate research and knowledge from the group of people that do this research and produce this knowledge and from the goals they attempt to reach through these'. In Ernest's view, a field of research can be defined only by the practices considered relevant by the people claiming to be researchers in this field. Hence, the definition of research in mathematics education should be given not in the form of a sentence that follows the Aristotelian pattern of genus et differentiam specificam but by 'some nonhierarchical list of practices, reflecting a multiplicity of viewpoints, theories, frameworks, methodologies and interests'. That approach would lead not to a classification in terms of academic disciplines but to a structure whose categories would be certain ‘programs’, ‘paradigms’ or ‘trends’. Ernest attempts to describe and classify mathematics education research practices along those lines.
Ernest thus offers a reflection on what mathematics education research is rather than on what it should be. Another author who focuses on the status quo rather than a vision is Mura, whose chapter delineates how Canadian researchers in mathematics education see themselves with respect to other domains of research. Her survey shows that, in Canada, 'mathematics education tends to be institutionalized in a way which is not conducive to its becoming "an integral part of mathematics" as advocated by Brousseau'.
Ellerton and Clements remind us of the political dimensions of education in general and of mathematics education in particular. They argue that because the compulsory education of children around the world includes a certain internationalized version of mathematics and because many children fail to learn it, 'mathematics education researchers should pay more attention to their role of helping to create more equitable forms of mathematics education'. The political side of mathematics education and the expectations of various groups, including parents, governments, employers, and mathematicians, seem to indicate 'that mathematics education is not, and will never become, a purely academic discipline. Mathematics education research is a social and cultural task rather than a field of scientific inquiry.
As Bishop observes, that task increasingly demands strong efforts by researchers to relate their work to the practice of teaching and learning mathematics. Teachers face pressures from all sides to make their work more 'effective', and researchers have not been able to provide them with much assistance. In Bishop's view, research in mathematics education needs to be shaped not by theoretical considerations but by the questions and problems of practitioners.
THE OBJECT OF STUDY
Mura's survey also documents a variety of views among Canadian mathematics educators on the object of study in mathematics education research. For some of her respondents, that object is the teaching of mathematics, where study can mean either observation and analysis of the phenomena of teaching or, in addition, the search for means and approaches to help students construct mathematical concepts for themselves. Other respondents focused on the learning of mathematics. For example, research in mathematics education might mean a systematic study of how students come to know and understand mathematics. For some, mathematics education is a theoretical pursuit; for others, a practical pursuit, an art, an applied discipline aiming at improving the teaching and learning of mathematics. Still others claimed that mathematics education is not a discipline in itself but an 'integrated application of mathematics, psychology, epistemology, sociology and philosophy'. Mura concludes that the object of study in mathematics education research should be defined very broadly as, for example, 'all aspects of learning and teaching of mathematics'. She adds that the objects of study of, for example, psychology, sociology, or mathematics are similarly broadly (or even vaguely) defined, yet the identity of these domains is not in question.
In general, the other authors agree with Mura's desire not to narrow the scope of the field of study. Ellerton and Clements take an even stronger position, claiming that defining that scope can be dangerously limiting. They cite the case of the 1979 book by E. G. Begle, Critical Variables in Mathematics Education, whose influence, they say, 'straitjacketed thinking within the international mathematics education research community for many years'. One could, of course, take issue with this opinion. The book, although influential, was hardly a straitjacket, as witnessed by the many critiques it elicited when it was published. Nor did it prevent the development, in the 1980s, of original theories and research methodologies that had little to do with 'testing hypotheses using statistical controls and mechanisms'. It is enough to mention the developments in French didactique or the constructivist program endorsed by many scholars in the AngloAmerican community.
It is worthwhile remembering the historical context of Begle’s book. In his time, the emerging community of mathematics education researchers had to contend with a widespread 'armchair pedagogy' and the formulation of recommendations for curricula and teaching practice by people who had never taken a careful or systematic look at actual mathematics classrooms. In her chapter, Adda reminds us that such pedagogy has not completely disappeared. The quotation from Begle in Ellerton and Clements' chapter addresses precisely Adda’s concern.
The chapters in the present book describe the object of research in mathematics education in broad terms, but the descriptions are quite varied. For example, Margolinas sees the object of study in mathematics education research as 'the production and spreading of mathematical knowledge'. For Presmeg, it is 'the complex inner and outer worlds of human beings as they engage in the learning of mathematics'. These worlds include classroom life, and Presmeg sees the development of mathematical concepts as 'situated within this social and mental complexity'. Rather than characterize the object of research in mathematics education, Ellerton and Clements prefer to list a set of problems currently important for mathematics education. The first and most important problem for them is to identify the many outdated assumptions that underlie the ways school mathematics is currently practiced.
Wittmann rejects even the concept of an object of study. He says that as a design science, mathematics education does not have an object of study; it has a field of practices that it wants to improve. This field encompasses those practices related to the teaching of mathematics, including teaching at different grade levels and teaching teachers.
RESEARCH GOALS AND DIRECTIONS
One of the questions in the Discussion Document was related to the 'aims of research' in mathematics education. But what do we mean by 'aims of research'? At least two meanings come to mind. If we focus on the outcomes of research, then we understand 'aims' as being the ultimate goals of research. If, on the other hand, we are more interested in the process of research or of scientific pursuit as such, then we think of 'aims' as being, for example, the directions or orientations of research activity, research problems, issues for research, or problématiques.
Traditionally, the goals of scientific endeavor have been divided between ,pure' and 'applied' goals. A researcher may be satisfied with understanding how things are, how they change and why, or under what conditions they change. On the other hand, he or she may want to acquire knowledge in order to take action and change the way things are. Although this distinction can be quite useful for the purposes of methodological clarification, actual research practices can, of course, seldom be classified under one rubric or the other (especially in mathematics education, where most researchers, because they are also mathematics teachers, teacher trainers, curriculum developers, etc., are part of the reality they want to study or change). Still, there is a marked difference between the two questions below:
1. What are the mechanisms of learning mathematics and the factors influencing these mechanisms? These factors may be internal (such as individual characteristics, motivation, personal goals, mathematical ability, mathematical cast of mind or lack thereof, and cultural background that matches or does not match the school culture) or external (such as components of the school setting, teaching methods, expected levels of performance, organization and presentation of mathematical topics, use of computers and calculators, the teacher's communication skills, instruction in reading mathematical symbols, and relations with other subject areas and with the home culture, ethnic culture, and everyday life).