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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
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Toerner, Guenter. and Hoechsmann, Klaus. "Schisms, breaks, and islands - seeking bridges over troubled water" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117518_index.html>

Toerner, G. and Hoechsmann, K. , 2004-10-21 "Schisms, breaks, and islands - seeking bridges over troubled water" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117518_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: This article will try to identify the causes of the deficits metaphorically alluded to in its title, and to describe possible contributions to their removal, albeit slow and gradual. The small space allowed will, however, force it to be sketchy and often use metaphors to compress meaning. 1.1 Two Cultures. Any understanding of ‘math education’ depends on some basic 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. Where, then, do we find such a philosophy? Courant (1996) tries to answer the question ‘What is mathematics?’ in a famous book 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 many math educators, that mathematics is monolithic and eternally unchangeable. Instead, every generation of mathematicians must be reminded to look after the coherence of their science. In the words of Sir Michael Atiyah (1978): we must continually strive [...] to unify. Though aware of their own diversity, mathematicians by and large tend to regard math education as a rather dull but straightforward affair. Hence they see math educators as most of the latter see them: as a clique of uniformly narrow-minded and feckless academics. Assigning to mathematicians an appropriate role in the transmission of their science (cf. Bass (1997)) might, under these circumstances, do more harm than good. It is therefore more urgent than ever to slow down (perchance to stop) the drifting apart of these two communities -- for instance in meetings like this one -- by tracing the roots of this double myopia. Here are some details from Germany. 1.2 Schims. When parts of a professional community find their aims and interests increasingly diverging from the rest, they naturally tend to form separate entities. Thus, in 1890, under the leadership of Georg Cantor, the newly created German Mathematical Society (DMV) (cf. http://www.mathematik.uni-bielefeld.de/DMV/) split off from its more generally science oriented parent, the GDNÄ. The next century brought schisms within mathematics itself: statistics and computer science, for instance, are no longer seen as belonging to it, and more schisms seem to be in the offing. Mathematicians with a strong commitment to education had also founded their own professional organization, the Society for Mathematical Didactics (GDM) (cf. http://www.gdnae.de), and thereby prepared their gradual drifting away from mathematics. In this manner, simple administrative regroupings can eventually lead to the formation of new ‘disciplines’. 1.3 Breaks: For individual careers, especially of teachers, these organizational and social separations imply mind-boggling discontinuities. Felix Klein (1908) -- one of the major mathematicians of his time to worry about education -- describes them aptly: ‘The young student sees himself at the beginning of his university course confronted with problems which in no point remind him of things he was concerned with at school; of course this is why he forgets all these things rapidly and thoroughly. However, when he enters a teaching position after completion of study, he is expected to teach traditional elementary mathematics in the traditional school manner; as he can hardly relate this to his university mathematics, he will in most cases embrace traditional teaching within a short time, and the university course will remain to him only a more or less pleasant memory that has no influence on his lessons. This twofold discontinuity… ’, This description is, unfortunately still valid now, a full century later, and – alas – not only in Germany, but world-wide. 2.1 Islands. As rifts widen, islands are created. In mathematics, this general pattern is exacerbated by a coast line of daunting cliffs. Even inside the subject -- which is really an archipelago -- intelligent inhabitants of different parts are often mutually unintelligible. In lieu of bridges, a system of tight ropes, used only by a few acrobats, holds the subject together. The rank and file are busy digging for theorems in their own back-yards. They could use some of their time to whittle down the cliffs, but such work is scorned. Addressing the International Congress of Mathematicians, the poet Enzensberger (1999, p.18) uses the metaphor of a fortress surrounded by a moat, with all drawbridges up and not working. But let us stay with islands, because they may have unexplored terrain below the cliffs, where mathematicians and educators could meet. 2.2 Prejudice. What seems to separate these two communities at the deepest level is the mutual underestimation of each other's scientific work. Isn't mathematics the hardest of sciences? Instead of playing on words such as 'hard' versus 'soft' or 'easy' (cf. Berliner (2003)), why don't we just agree that it is one of the oldest and least popular? Its esoteric nature makes it difficult for outsiders to identify trashy mathematics, while anyone can easily dig up shallow and ill-written papers in the more accessible ‘humanities’. As Horup (1994, p. 277) recognizes: ‘mathematicians, however, also tend to have a handicap. The particularly important position of the logical argument in mathematics easily leads to the opinion that everything not belonging to mathematics, particularly political and moral thought and convictions, is illogical and beyond argument. Furthermore it is not uncommon that mathematicians mistake this epistemological dichotomy between demonstration and subjectivity for a social dichotomy and, [...] take their own inveterate persuasions and prejudices for objective truth’. Such prejudice is reinforced, when the others make pronouncements about a subject they admittedly don’t wish to know. All this quibbling within the ivory tower, takes its greatest toll among the most defenceless in the world outside: the preservice teachers and, through them, the students. 2.3 Bridges. But there are hopeful signs: this PME-NA conference, for example. As far as I know, every meeting of the AMS or CMS now has an education section. Moreover, there are a number of recent articles in the Notices of the AMS arguing about math education as a discipline. Schoenfeld (2000), for instance, first quotes Pollak’s saying that ‘there are no proofs in mathematics education’ and then explains the nature of the subject in analogy to natural science, while Ralston (2003) stresses: ‘Work in education or the social sciences will almost never lead to provable, ironclad results’, and endorses Berliner’s (2003) distinction of hard versus easy sciences, without as boldly proclaiming educational research to be the hardest science of all. But words must be followed by action. Like Bass (1997), Stiff (2003) calls on the mathematical community to fulfill that part of its mandate: mathematicians as educators. Indeed, prejudices on both sides will be most effectively softened by collaboration on concrete tasks. Nobody will expect mathematicians to show consummate pedagogical skill, nor demand great theorems from educators. But the public has the right to expect each side to show a little more interest in, and knowledge of, what the other one is doing. 3.1 Wake-up Calls. In the particular case of Germany, the discussion about these problems was given new impetus by TIMMS. Its sobering results were suitable neither for the usual finger-pointing between specific groups nor for the classical clichees dear to the media: incompetent teachers, uninterested students, and other-worldly professors. For the first time in a long while, joint declaration were issued by the professsional societies concerned: the German Mathematical and Didactical Societies (DMV) and (GMD), as well as the Association of Mathematics and Science Teachers (MNU). Each of them admitted the urgent need for correction and action in its domain. 3.2 Official Inertia. Unfortunately, this new coalition was not recognized as a chance for positive change by the educational administrators on both the federal and the state level. A promotional program launched by a commission formed by them completely ignored the teacher training rôle of universities. Honest burden-sharing was thereby aborted, and so was the multilateral potential in the discussion of standards, which was more recently triggered by PISA. With very few exceptions, the responsible actors, mostly teachers, teacher educators, textbook authors, and ministerial curriculum designers remained largely among themselves, with universities and learned societies serving at best as alibis. In these circles, an old and nefarious prejudice seems chronic, namely that research mathematicians have nothing to offer in educational matters. Thus it appears that the ministerial apparatus is a self-stabilising system with its own dynamics and rarely open to constructive suggestions. Experience with it is apt to confirm many a university mathematician in an attitude of resigned disinterest. The few laudable exceptions to this statement do not suffice to negate it. 3.3 New Directions. A more positive impact on the educational scene, especially as regards mathematics, comes through outside initiatives by the private sector (cf. http://www.mint-ec.de) and various support programs offered by foundations (cf. http://www.nat-working.de). Implicitly, these programs contain a belief in progress through paradigmatic change: research scientists transmitting science in schools, professional teachers on leave to act as serious participants in university research teams, students in mixed teams sharing learning experiences with teachers and having direct contact with professional researchers, and so on. Emotional benefits and social contacts are, of course, part of the plan. The goal of the most ambitious private sector initiative, known as MINT (M = mathematics, I = informatics, i.e., computer science, N = natural science, T = technology), is to identify schools which are particularly open to new ideas in mathematics and science teaching, and to give them special support and public status. Today almost 100 schools carry the challenging, non-permanent designation as Centers of Excellence for MINT. In these schools, teachers as well as students are given special opportunities to develop their mathematical interests; inservice training courses, for instance, can be organized at a high level. Moreover, the sponsors aim at enhancing the teachers’ all too often low esteem of their own work by providing generous furnishings and equipment. At the moment, even the professional societies – who began the schisms – seem to come around to a new way of thinking. They realize that, all across Germany, the problem is too massive and too serious to allow small groups any chance of effecting socially significant changes. The separations of the past are now to be at least partially reversed. Nevertheless, a certain hesitancy and fear of being reabsorbed is palpable among mathematics educators, while mathematicians seem to welcome all the pedagogical help they can get. References . Atiyah, M.F. (1978). The unity of mathematics. Bull. London Math. Soc. 10, 69 - 76. Bass, H. (1997). Mathematicians as Educators. Notices of the American Mathematical Society (AMS) 44 (1), 18 - 21. Berliner, D.C. (2001). Educational Research: The hardest science at all. Educational Researcher 31, 18 - 20. Courant, R.; Robbins, H.; Stewart, I. (1996). What is Mathematics? An Elementary Approach to Ideas and Methods. Corby: Oxford University Press. Enzensberger, Hans Magnus. (1999). Drawbridge up - Die Mathematik im Jenseits der Kultur. Nattick, MA: A K Peters, Ltd. Hersh, R. (1997). What is Mathematics, Really? New York: Oxford University Press. Horup, J. (1994). In measure, number and weight: studies in mathematics and culture. Albany: SUNY. Klein, F. (1908). Elementarmathematik von einem höheren Standpunkt. Berlin: Springer-Verlag. Neuauflage. Ralston, A. (2003). California Dreaming: Reforming Mathematics Education. Notices of the American Mathematical Society 50 (10), 1245 - 1249. Schoenfeld, A. (2000). Purposes and Methods of Research in Mathematics Education. Notices of the American Mathematical Society 47 (6), 641 - 649. Stiff, L.V. (2002). Working together to improve mathematics education. Notices of the American Society 49 (7), 757. Thom, R. (1973). Modern mathematics: does it exist? In A.G. Howson (Ed.), Developments in Mathematical Education. Proceedings of the Second International Congress on Mathe-matical Education. (pp. 194 - 209). Cambridge: Cambridge University Press.

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