March 9, 2011
15:1
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch03
110
Russian Mathematics Education: Programs and Practices
Fig. 3.
depicted in Fig. 3. This is what the best style of mathematics consists
of: taking something that is not obvious and making it obvious by
means of a clever construction, technique, or argument. (Alexandrov
et al., 1992, p. 139)
Alexandrov formulated several principles for teaching geometry
(following Werner, 2002, p. 166):
• Since one of the aspects of geometry is its rigorous logical
character, and since the students of grades 7–11 are already
capable of grasping this logical character, the course in geometry
must be presented sufficiently rigorously, without logical gaps in
the basic structure of the course.
• Since the second basic aspect of geometry is its visual character,
in the teaching of geometry every element of the course should
be initially presented in the most simple and visually intuitive way,
using that which may be drawn on the blackboard, demonstrated
on models, on real objects, as far as possible.
• Further, despite its high degree of abstraction, geometry arose
from practical applications and is put to practical uses. Therefore,
the teaching of geometry must unquestionably connect it with
March 9, 2011
15:1
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch03
On the Teaching of GeometryGeometry in Russia
111
real objects, with other disciplines, with art, architecture, and
so on.
• A textbook aimed at ordinary secondary schools must not
contain in its basic part anything that is extraneous, of secondary
importance, or of little significance in the main body of the text.
• But since the abilities and interests of the students are quite
varied, such a textbook must contain supplementary material,
aimed at students who are stronger and have a greater interest
in mathematics.
• Geometry must be presented geometrically. It contains its own
methodology: the direct geometric methodology of grasping
concepts, proving theorems, and solving problems. The synthetic
methodology of elementary geometry must not be squashed in
school-level instruction by any coordinate-based methodology,
vector-based methodology, or any other methodology. The direct
geometric methodology is simpler, more important, and more
natural for the purposes of a general secondary education and
corresponds to the very essence of geometry. It is needed by
anyone who deals with three-dimensional objects.
• The school course in geometry must be connected with contem-
porary science, must include, as far as this is possible, elements of
contemporary mathematics. In addition, the course in geometry,
as a logical system in which everything is proven, is impor-
tant for developing the rudiments of a scientific worldview,
which demands proofs rather than references to authoritative
sources.
• But since there is simply no such thing as absolute rigor, a certain
level of rigor must be selected and established, and this level of
rigor must be maintained through the entire course. The course
must not have logical gaps, at least in its basic sequence of
topics. Otherwise, it will lose its systematic aspect, the logic
of the exposition will become blurred, and students will be
exposed not to a unified science — geometry — but only to its
fragments.
In this way, the three foundations of the textbook, according to
Alexandrov’s way of thinking, were supposed to be visual explanations,
logic, and connections with the real world and practical applications.
Such a conception of the author’s task led Alexandrov to present
March 9, 2011
15:1
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch03
112
Russian Mathematics Education: Programs and Practices
many sections in a new way. For example, the fundamental object that
he chose was not the straight line, as was the norm in other school
textbooks, but the segment, since it was precisely this that people dealt
with in practice. For the same reason, the traditional axiom of parallel
lines — which states that through a point outside a given straight line,
only one straight line may be drawn that is parallel to that line —
was replaced by the axiom of the rectangle. This axiom postulates
that it is possible to construct a rectangle whose sides are equal to
given segments (the possibility of such a construction is confirmed by
everyday practice).
Alexandrov used the congruence of segments — visually apparent
and “testable” — to define other concepts, including the congruence
of figures. Untraditional definitions (although equivalent to traditional
ones) were given in the textbook for other concepts as well — for
example, the similarity of triangles.
All of this made the presentation shorter and more visual. At the
same time, although Alexandrov’s approach was based precisely on a
deep understanding of the classical tradition, the novelty of many of
the ideas scared off some teachers when they suddenly discovered that
from now on they would have to teach the congruence (equality) of
triangles in a way that differed from what they had been accustomed
to for years.
Do'stlaringiz bilan baham: 
