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On the Teaching of GeometryGeometry in Russia
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Let us begin with the latter. It is not difficult to see that Kiselev’s
textbook, which has to this day been considered a model of rigor and
deductive logic, contains propositions that turn out to be simply false
because of what may be called linguistic sloppiness. For example, it
contains the following theorem: “The three altitudes of a triangle
intersect at one point” (Kiselev and Rybkin, 1995, p. 108). Meanwhile,
generally speaking, an altitude is a perpendicular dropped from a vertex
of the triangle to the side opposite it or its extension. Obviously, in an
obtuse triangle, the altitudes do not intersect at one point; rather, it is
the straight lines that contain the altitudes that intersect at one point.
Moreover, generally speaking, from a certain point of view, virtually all
theorems that involve areas and volumes are meaningless. Say, consider
the statement “The area of a triangle is equal to one half of the product
of its base and height.” How can one multiply a base, i.e. a segment?
One should refer, rather, to the length of the base.
Kolmogorov (1971) wrote: “Traditional geometry textbooks are
weighed down by the extreme polysemy of their definitions and
notations” (p. 17). It turned out, however, that avoiding such polysemy
completely is very difficult, while using symbolic notations overburdens
the teaching of the course and, most importantly, alters somewhat its
direction. The student in effect has to learn a new language and then
to pay attention to subtleties of notation — making sure to distinguish
between AB,
←→
AB ,
−→
AB, and other expressions, instead of focusing on
geometry itself. Of course, no one would deny that it would be good
if all students acquired a command of precise mathematical symbolic
notation, but usually the time that teachers have at their disposal is
limited and they must choose what to spend it on. Russian textbooks
subsequently simplified symbolic notation, writing simply
AB, verbally
indicating what was meant or even expecting students to understand
what was meant from the context.
Precise definitions are indispensable in mathematics (as in any
other science). Moreover, they are vital in everyday life [recall the
example cited by Vygotsky (1986) of a child who said that someone
had once been the son of some woman but was not her son any
longer: the child had formed his definition of “son” spontaneously
and associated it with a certain age — thus, an adult could not be
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Russian Mathematics Education: Programs and Practices
a son!]. The problem, however, is that to give a rigorous definition
of, say, a polyhedron is very difficult (Alexandrov, 1981); meanwhile,
students already have an intuitive notion of it, which is sufficient
for solving certain problems, including quite substantive ones. This
intuitive notion may be made more precise when necessary, and
various relevant details may be mentioned explicitly, which can itself
be useful, but striving to give a complete and precise definition
of a polyhedron is probably not useful (at least attempts to do so
in Russian textbooks have not met with success — teachers and
students have usually simply skipped over them). As Alexandrov
(1984b) emphasized: “The purpose of definitions is not for students
to memorize them by rote, but to make students’ understanding more
precise. We must try to achieve not empty memorization, but effective
learning, i.e. learning that allows students to apply what they have
learned” (p. 45).
Consequently, in dealing with any new concept, the authors of
textbooks — and teachers as well — must confront the question of
whether working toward a precise definition of this concept is justified.
In a very large number of cases, such a definition may be given without
difficulty (here, we will not discuss the question of how this should be
done, but merely point out that, almost always, the precise definition
of a concept must grow out of working with the concept rather than
precede it). Nonetheless, it should be borne in mind that even the
great mathematicians of the past sometimes worked without having at
their disposal definitions that we would consider precise according to
today’s standards (for example, of a limit).
Attempts to sustain high standards of deductive logic, approxi-
mating the standards of modern science, can hardly be considered
successful. Schools have rejected them — theorems that were too
difficult were simply not proven in practice, and as a result the
level of deductive logic fell rather than rose. The school course in
geometry is not a course in the foundations of geometry. The high-
est level of deductive logic that is feasible in the classroom is the
one that should be aimed at, and this should be done by giving
teachers and students difficult problems — difficult but not impos-
sible. The balance of mathematical and pedagogical considerations
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will be different in each situation and depend on numerous social
circumstances.
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