FOCUS: The Newsletter of the Mathematical Association of America, Vol.18, No.5, May/June 1998, p.8
LETTERS TO THE EDITOR
To the Editor:
We are grateful to Saunders MacLane (FOCUS, February 1998, p. 6) for his opening a reasoned discussion of issues related to research in mathematics education. The timing could not be better. At the Baltimore meetings there was unprecedented interest in, and thoughtful discussion of, both basic and applied issues in mathematics education research. We agree with the urging of Brown and DeVries (FOCUS, October 1997, p. 19) that mathematicians should look to the science of mathematics education.
Established mathematicians will recall earlier periods when, each in their turn, applied mathematics, statistics, and computer science were nurtured within the field of mathematics until they developed to the point of independence. Here we offer some thoughts, grounded in our experience, about what happens when we try to educate ourselves about theory and research in an area that is new to us. Interestingly, many of the issues raised today about mathematics education are highly reminiscent of comments made then about those fields.
It is clear that the traditions, vocabulary, standards of evidence, modes of critique, and nature of research differ by area within the mathematical sciences. Judgments about the quality of the work in a new area, or interpretations of complex theories in an unfamiliar domain, are difficult to make. They often lead to discussions like the current one, in which practitioners of a relatively new field (like us) respond to concerns of highly respected individuals in more established fields (like MacLane) with explanations, interpretations, and yes, even corrections.
To characterize constructivism, which is a theory of knowledge, as a philosophy that urges students to construct their own mathematics is a serious misinterpretation. As with most theoretical perspectives, various interpretations exist; treatments range from the strictly philosophical (see, for example, Von Glasersfeld, "Learning as a Constructive Activity," in C. Janvier (ed.), Problems of Representation in the Teaching and Learning of Mathematics, Hillsdale, NJ: Erlbaum, 1985, pp. 3-17) to those grounded in cognitive psychology (see, for example, pp. 2-27 of Schoenfeld, A. H. (ed.) 1987, Cognitive Science and Mathematics Education, Hillsdale, NJ: Erlbaum).
Despite the differences, one generally accepted tenet of the constructivism is that humans do not perceive reality directly, but that we "interpret" our experiences and behave consistently on the basis of those interpretations (take any optical illusion, or any student's misconception, such as the belief that (a+b) 2 = a2+ b2, as examples of this phenomenon).
Another point on which there is considerable agreement is that students construct their own understandings of mathematics--not their own mathematics. The central theoretical/empirical issue, then, is how we make sense of the phenomena we experience, including those in mathematics. The central pedagogical issue, in the words of von Glasersfeld, is how teachers work to develop a "fit" between students' developing understandings and the understandings established in the field of mathematics. Many constructivists feel that one can never know precisely what another person thinks or understands, but we can (and constructivists do) maintain that whatever this understanding may be, it must be consistent with the mathematical phenomena that everyone encounters.
MacLane offered "reviews" of three mathematics education articles. It would be one thing to try to read and perhaps devise personal abstracts or reviews of articles in a field that is not our own, but to write caricature reviews doesn't seem to be a productive venture. Nonetheless, it is indeed tempting to imagine what a tongue-in-cheek review of a major research paper about, say, computer science, written by a person who thinks mainly about algebraic topology, might look like. We defer from actually undertaking the exercise. In general, any random selection of articles in a field is unlikely to yield a sense of its breadth, richness, depth, or quality.
As mathematics educators who teach and have taught mathematics at all levels, we wish to make several points. Theory and research from mathematics education have had numerous positive influences on our teaching. Mathematics education research can be useful to mathematicians. Just as one cannot learn mathematics by deciding to read some random assortment of papers, it is unlikely that people who do not nave formal preparation in the field of mathematics education would find it fruitful to read only a few papers. In fact, one argument for providing reviews of mathematics education papers is to allow the mathematics community to see a range of type, quality, and focus of mathematics education papers, as a screening process for actually reading the papers. Reviews can help orient people to the field, and publishing well considered critical reviews of published papers is one way to improve the quality of research.
Research and theoretical presentations in mathematics education range in depth and quality--as do papers in all fields of mathematics. At its best, research in mathematics education offers better, deeper and increasingly more useful explanations of teaching and learning, and the contexts in which they happen. We each find the intellectual stimulation, as well as the benefit to our teaching, of the evolving and growing field of mathematics education to be enormously important. We hope that discussions such as this will lead to more examples in our field of research at its best, and we welcome MacLane's help in this quest.
Polarization in the discourse between mathematicians and mathematics educators is neither productive nor practical. It takes energy away from the important business to which we all are committed, that of improving mathematics teaching and learning at all levels. Secretary Riley, in his remarks during the Baltimore meeting, encouraged the mathematical sciences community to work together toward such improvement. We feel that the various mathematics education programs at the Winter Meetings in Baltimore, with their tones of sharing information, considering a variety of ideas, and showing mutual respect for those with whom there was the strongest disagreement were a huge step in the direction of improving the learning and teaching of mathematics. We urge our colleagues to stay focused on that end, to continue and expand the sorts of discussions and collaborations that mathematicians are engaged in as evidenced in Baltimore, and to structure the discussion, collaborations, and cross-disciplinary interactions likely to further continued improvement.
Ed Dubinsky, Georgia State University
Alan H. Schoenfeld, University of California, Berkeley *********************************************************** Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 62901-4610 USA Fax: (618)453-4244 Phone: (618)453-4241 (office) E-mail: JBECKER@SIU.EDU