(Page 1 of 8) - Schisms, breaks, and islands - seeking bridges over troubled water authored by Toerner, Guenter. and Hoechsmann, Klaus.

SCHISMS, BREAKS, AND ISLANDS - SEEKING BRIDGES OVER TROUBLED WATERS: A SUBJECTIVE

VIEW FROM GERMANY

Günter Törner; Klaus Hoechsmann

University of Duisburg-Essen (Germany); Pacific Institute of Mathematical Sciences, Vancouver (Canada)

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The far-flung community of professionals dedicated to the practice or teaching of mathematics is cleft by at least three
kinds of divisions: (a) the schism between mathematics and math education, (b) the barrier between school and research
mathematics, (c) the rifts between mathematical subdisciplines. While latter are in some sense inherent and could at
most be made less jagged, we shall argue that the former two could in principle be bridged,

albeit slowly and

gradually. The small space allowed will, however, often force us to be sketchy and use metaphors to
compress meaning.

1. Cultures, Schisms, Breaks

1.1 Two Cultures

The "gulf of mutual incomprehension" separating the exact sciences and the humanities has neither
narrowed nor been bridged in the half century since C.P. Snow deplored it in his famous lecture
(Snow, 1959) on the Two Cultures (cf. the assessment by Davis, P., 1990). The same is true for the
present situation in Germany. A recent bestseller (Schwanitz, 1999) dismissed mathematical culture
in a brief sketch on the last pages, and thereby - fortunately - provoked another bestseller (Fischer,
2001) which deals with the broad cultural influence of mathematics and the exact sciences. We do
not intend to add to the ink already spilled on this topic, except to recall that the fruitful metaphor of
different cultures has been frequently applied to the various forms of mathematics instruction,
starting at the latest with the work of Lerman (1990), and has demonstrated its explanatory value
(cf. Cobb, 1991; or Cobb & Bauersfeld, 1994; Yackel & Cobb, 1996; and many others). Hence we
shall allow ourselves to look through this filter at the makers, users and communicators of
mathematics.

However, we shall also remember that an overly facile handling of this tool can lead to

embarrassing misjudgments, which are particularly visible in the old dichotomy of "pure" versus
"applied". They come from of three sources: distortion by contraction and caricature, ever-changing
subject-boundaries (e.g., knot theory is now "applied"), and the classical dynamics of human
behavior (tribalism and empire building justified by "objective" necessity).

For the sake of completeness one should not forget the distinction between "academic" and

"industrial" mathematics, which shows similar features. With these caveats in mind, we shall
approach mathematic and math education.

Since Snow’s time, the main change in the landscape surrounding his gulf is that more material

has accumulated on both sides: for instance, mathematics education on the side opposite a greatly
enlarged mathematics. Since, in the short run, no bridge is likely to pop out of nowhere to connect
the two, any hope for future communication requires that each of them understand, if the not the
substance, then at least the form of the other. Both also need to know how far they are apart.

Surely, any understanding of ’math education’ depends on some notion of what is ’mathematics’.

This apparent tautology is expressed by the geometer René Thom (1972), as follows: In fact,
whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a
philosophy of mathematics. Constructivists might object to the implicit hierarchy and asymmetry of
A resting on B, but we are more interested in the claim that a philosophy of mathematics - that is, an
idea of its nature - is required.

Where, then, do we find such a philosophy? Courant (1941) tries to answer the question 'What

is mathematics?' in a famous book (updated in 1996 by Ian Stewart) with just that title - and Hersh
(1997) has a very different answer in a book with the same title extended by the word 'really'.
Tomorrow there might be yet another answer, in stark contrast to the belief, held by not a few math
educators, that mathematics is monolithic and, in its core, eternally unchangeable. In reality, every
generation of mathematicians must be reminded to look after the coherence of their science, which
is in constant danger. In the words of Sir Michael Atiyah (1978): we must continually strive ... to