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Heinz Steinbring's conception of a dialogue between theory and practice
in mathematics education takes this perspective of "teachers as experts" into
account. He provides an introduction to the discussion on reconceptualizing
INTRODUCTION TO CHAPTER 2
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ROLF BIEHLER
the relation between theory and practice in the didactics of mathematics and
summarizes insights from projects under the heading of Systematic coop-
eration between theory and practice, in which teachers and researchers have
been trying to establish new kinds of relations: Overcoming the widespread
"teaching as telling" (the "broadcast metaphor") in the classroom is related
to overcoming the broadcast metaphor in teacher education as well. With re-
spect to teachers' knowledge, the paper is based on the assumption that a
deeper understanding of the epistemological nature of mathematical knowl-
edge as theoretical knowledge with its specific relation between objects,
symbols, and concepts is necessary if teachers are to cope adequately with
problems in the classroom. The author gives examples from the teaching
and learning of fractions. The role of diagrams for communicating and
working with theoretical knowledge is one focus. In this respect, the paper
relates to the analysis of representations for mathematical teaching, learn-
ing, and thinking by Kaput (this volume). With regard to in-service teacher
education, the important function of shared situations (in the shape of lesson
transcripts), besides theoretical knowledge, is elaborated for stimulating re-
flection and communication between researchers and teachers. Steinbring
respects teachers as experts with a lot of intuitive knowledge but tries to
transform and elaborate this knowledge by means of a dialogue.
Tom Cooney's analyses on the application of science to teaching and
teacher education are concerned more explicitly with overcoming the unsat-
isfactory practice of mathematics teaching. Complementary to Steinbring's
contribution, he discusses what kind of didactical research and didactical
theory is necessary in order to not just mirror existing practice but open up
ways for innovations. Research is necessary to broaden our understanding
of how teachers come to believe and behave as they do, where and how
their attitudes toward mathematics and its teaching are created, and how this
may be changed toward a more adaptive and reflective teacher with a "sci-
entific attitude" to his or her own teaching practice. From this point of view,
research on teachers' cognitions as well as on the efficiency of in-service
programs is reviewed. Research points to the limited view of mathematics
that teachers communicate in the classroom and the lack of that mathemati-
cal sophistication (especially in elementary teachers) that would be needed
to implement innovative mathematics teaching such as described in the
NCTM standards. However, a simple extension and broadening of the
knowledge related to mathematics in teacher education can hardly be suffi-
cient, because of the complex social situation of the teachers' work place
and longstanding habits. For Cooney, it is necessary to "create contexts in
which teachers . . . can envision teaching methods that reflect reasoning,
problem-solving, communicating mathematics, and connecting mathematics
to the real world . . . and yet feel comfortable with their role as classroom
managers." Discussing with teachers new forms of problems for assessment
that reflect the above innovative ideas are seen as an important possibility of
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REFERENCES
Dörfler, W., & McLone, R. R. (1986). Mathematics as a school subject. In B. Christiansen,
A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 49-97).
Dordrecht, Netherlands: Reidel.
Fennema, E., & Franke, M. L. (1992). Teachers' knowledge and its impact. In D. Grouws
(Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). New
York: Macmillan.
Grouws, D. (Ed.). (1992). Handbook of research on mathematics teaching and learning.
New York: Macmillan.
Hoyles, C. (1992). Mathematics teaching and mathematics teacher: A meta-case study. For