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Schisms, breaks, and islands - seeking bridges over troubled water
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unify. He and his colleague Isadore Singer, have just (2004) received the prestigious Abel Prize for "building new bridges ...". A more recent Fields Medalist, Timothy Gowers, even sees "two cultures" within mathematics itself (wwww.dpmms.cam.ac.uk/~wtg10/2cultures.pdf): the vast and weighty tradition versus newer and less prestigious branches such as graph theory. Needless to say, all of this is again divided into pure and applied areas, which are, however, not as far apart as they may appear.
Though aware of their own diversity, mathematicians by and large tend to regard math
education as a rather dull but straightforward affair (cf. Fischbein (1990)). 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 further 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 Schisms
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 (cf.
http://www.gdnae.de
), which had still included medicine (whence the last letter of its acronym).
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) and thereby prepared their gradual drifting away from mathematics. In this manner, simple common interest groupings can eventually lead to the formation of new disciplines, with separate aspirations, assessment criteria, administrative structures, and degree granting status.
Until well into the second half of the twentieth century, the main qualification for a secondary
school teacher was to pass the "State Exam", a natural milestone on the way to a doctorate. It was taken by the majority of future mathematicians, if only as employment insurance, because academic jobs were rare and often unavailable even to the most talented. Some of the great mathematicians of the nineteenth century began their careers as school teachers (e.g. Weierstrass) and sometimes even stayed there (e.g. Grassmann). While this system provided high school teachers with a strong background in university mathematics, it left them to work out their own ways around pedagogy and school mathematics. After the Second World War, as attendance of secondary schools expanded to a much more general public, these haphazard methods were seen to be insufficient, and lecture courses in mathematical didactics became a useful, ultimately mandatory, part of teacher training.
With its necessary presence in academe established, the new science soon discovered fresh
fields of exploration which were not necessarily concerned with helping prospective teachers find their way into the class room. Though pedagogy could arguably be improved by any serious reflection on it, mathematical competence requires practice more than theorizing, which is the central activity of any academic discipline.
1.3 Breaks
For individual careers of teachers these organizational and social separations are deepened by the mind-boggling discontinuities already described by Felix Klein (1908) -- one of the few major mathematicians of his time to worry about education: ’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
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| | Authors: Toerner, Guenter. and Hoechsmann, Klaus. |
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unify. He and his colleague Isadore Singer, have just (2004) received the prestigious Abel Prize for
"building new bridges ...". A more recent Fields Medalist, Timothy Gowers, even sees "two
cultures" within mathematics itself (wwww.dpmms.cam.ac.uk/~wtg10/2cultures.pdf): the vast and
weighty tradition versus newer and less prestigious branches such as graph theory. Needless to say,
all of this is again divided into pure and applied areas, which are, however, not as far apart as they
may appear.
Though aware of their own diversity, mathematicians by and large tend to regard math
education as a rather dull but straightforward affair (cf. Fischbein (1990)). 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 further 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 Schisms
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 (cf.
http://www.gdnae.de
), which had still included medicine (whence the last letter of its acronym).
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) and thereby prepared their gradual
drifting away from mathematics. In this manner, simple common interest groupings can eventually
lead to the formation of new disciplines, with separate aspirations, assessment criteria,
administrative structures, and degree granting status.
Until well into the second half of the twentieth century, the main qualification for a secondary
school teacher was to pass the "State Exam", a natural milestone on the way to a doctorate. It was
taken by the majority of future mathematicians, if only as employment insurance, because academic
jobs were rare and often unavailable even to the most talented. Some of the great mathematicians of
the nineteenth century began their careers as school teachers (e.g. Weierstrass) and sometimes even
stayed there (e.g. Grassmann). While this system provided high school teachers with a strong
background in university mathematics, it left them to work out their own ways around pedagogy
and school mathematics. After the Second World War, as attendance of secondary schools
expanded to a much more general public, these haphazard methods were seen to be insufficient, and
lecture courses in mathematical didactics became a useful, ultimately mandatory, part of teacher
training.
With its necessary presence in academe established, the new science soon discovered fresh
fields of exploration which were not necessarily concerned with helping prospective teachers find
their way into the class room. Though pedagogy could arguably be improved by any serious
reflection on it, mathematical competence requires practice more than theorizing, which is the
central activity of any academic discipline.
1.3 Breaks
For individual careers of teachers these organizational and social separations are deepened by the
mind-boggling discontinuities already described by Felix Klein (1908) -- one of the few major
mathematicians of his time to worry about education: ’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
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