|
On the Teaching of GeometryGeometry in Russia
101
(rigid motion) began to be characterized as “congruent.” It seems
unlikely that the introduction of one new term by itself could have
exceeded students’ capacities sufficiently to warrant discussions about
their suffering (which were not unusual for the pedagogical periodicals
of the time and indeed are not unusual today). On the other hand, the
introduction of a new term always creates certain difficulties, and if it
could have been avoided, for example, by specifying the precise mean-
ing that was being ascribed to the old term, then fighting so hard for the
new term, and turning it into a rallying cry, hardly seems worthwhile.
What probably happened to be more important was that many
proofs turned out to be fundamentally new and unfamiliar. For
example, Kiselev and his followers had proven the classic theorem that
the diagonals of a parallelogram
ABCD bisect each other (Fig. 1) by
examining the triangles
AOD and BOC (O is the point of intersection of
the diagonals). It is not difficult to see that these triangles are congruent
(or “equal,” to use the term of that time), from which everything
immediately follows.
Kolmogorov’s approach was to examine the midpoint
O of the
diagonal BD and point reflection with respect to this point. Since it
was stated at the outset that a point reflection maps a straight line to
a parallel straight line, and since it is clear that point
B, under such
reflection, is mapped to point
D, while point D is mapped to point
B, it was possible to conclude that, under the point reflection being
examined, the straight line
←→
AD is mapped to the straight line
←→
BC (as
the only straight line which passes through point
B and is parallel to
←→
AD). In an analogous manner, it was proven that the straight line
←→
AB
is mapped to the straight line
←→
DC . Thus, it was concluded that, under
Fig. 1.
March 9, 2011
15:1
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch03
102
Russian Mathematics Education: Programs and Practices
the given point reflection, point
A is mapped to point C, which proves
that
O is the midpoint of the diagonal AC.
Kolmogorov’s proofs, which were in their own way beautiful and
vivid, were nonetheless often difficult to grasp. In addition, if a student
using Kiselev’s textbook had the impression that all of the propositions
to which reference was made were completely proven (whether this
impression was correct or not is another matter), then Kolmogorov’s
textbook did not foster such an impression, if only because it attempted
to set a much higher level of rigor than Kiselev’s textbook did.
Discussing the axiomatic approach, Kolmogorov (1968) wrote:
But in schools it has become common practice merely to indicate
“examples of axioms.” The actual list of these examples of axioms is
usually laughably short. Apparently, the students are never asked to
analyze a proof by identifying all of the axioms on which it is based.
Meanwhile, such an exercise should be insistently recommended: the
proof of theorem
T relies on theorems T
1
and
T
2
, the proof of theorem
T
1
relies on axioms
A
1
and
A
2
, while the proof of theorem
T
2
relies
on axiom
A
3
and theorem
T
3
, and so on, until only axioms remain.
(p. 22)
It may be objected, however, that such an exercise is quite difficult
for ordinary public school students if they are dealing with a theorem
that has any substance. Even more significantly, such an exercise might
give rise to a misguided notion of geometry as a subject in which there
is a strange ritual of explaining what is obvious at great length for
unknown reasons (this is especially the case if, as unfortunately often
happens in Western textbooks, the theorem being examined is a very
simple one, consisting of one or two steps).
The first chapter of Kolmogorov’s textbook Basic Concepts of
Geometry formulates and enumerates 15 propositions. Nine of them
are axioms. Five are proven; one is illustrated. Of the five proofs of the
propositions, four are one step away from the axioms on which they are
based, and only one (the derivation of a formula for distance between
points on a coordinate line) contains more than one logical step.
The textbook Geometry 6 (Kolmogorov, 1972) contains 38 separate
propositions in all, over half of which are not proven.
March 9, 2011
15:1
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch03
On the Teaching of GeometryGeometry in Russia
103
We will not discuss the other methodological innovations that
provoked criticism — such as the approach to defining vectors in
Kolmogorov’s textbook and the accompanying textbook by Klopsky
et al. 1977 — or, on the contrary, met with success (such as the
replacement of a separate problem book with sections on “Questions
and Problems” in the textbook itself). Making the course at once more
rigorous and more simple, which was Kolmogorov’s goal, is not an easy
task. Kolmogorov and his coauthors took many revolutionary steps.
Possibly, given many years of further work, many difficult spots might
have been smoothed over. At least, Kolmogorov (1984) himself later
wrote:
The question of when it is proper to begin talking to students
about geometry’s logical structure should be discussed again. The
experience of working with different versions of geometry textbooks
over the past decade has shown that doing so at the beginning of
sixth grade is premature. (pp. 52–53)
But no more time was allowed for correcting, rethinking, and
revising. A major campaign (Abramov, 2010) effectively resulted in
the setting of a new agenda: to create new textbooks with the aim of
replacing Kolmogorov’s.
Do'stlaringiz bilan baham: | 
