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Topic: Research in Maths. Education: Letter to the Editor
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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
Research in Maths. Education: Letter to the Editor
Posted: Jun 14, 1998 11:12 PM
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FOCUS: The Newsletter of the Mathematical Association of America, Vol.18,
May/June 1998, p.8


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)

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