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Russian Mathematics Education: Programs and Practices
5.4.3
The textbooks of A. L. Werner and his coauthors
After Alexandrov’s death, his coauthors and collaborators prepared
several new textbooks. Adhering to the same principles on which the
earlier textbooks had been based, the authors attempted to address
certain critical new problems.
One of them consisted in the need to fill out the course in
plane geometry with elements of three-dimensional geometry. We
have already pointed out that without this, the spatial imagination
of students who are immersed for three years in the world of plane
geometry grows weaker (or atrophies altogether). The problem of
overcoming students’ spatial blindness is well known to teachers who
are beginning to teach a course in three-dimensional geometry. Fur-
thermore, since a complete (11-year) secondary education once again
became nonmandatory, it was deemed necessary to provide students
with some rudimentary knowledge of three-dimensional geometry in
basic schools (the nine-year program).
The authors of existing textbooks often merely supplemented their
plane geometry textbooks with one last chapter, which presented the
rudiments of three-dimensional geometry. This in no way solved the
problem of developing students’ spatial imaginations: as before, they
were immersed for three years in the world of plane geometry. There-
fore, during the very first year of competitions, the NTF announced
a competition for a new textbook, Geometry 7, and during the second
year, for a second textbook, Geometry 8–9, in which a systematic course
in plane geometry would be supplemented with elements of three-
dimensional geometry, presented in a visual–intuitive fashion. Both
competitions were won by textbooks written by a working group that
included A. L. Werner, V. I. Ryzhik, and T. G. Khodot, an associate
professor at Herzen University’s geometry department (Werner et al.,
1999, 2001a, 2001b).
The elements of three-dimensional geometry in these textbooks
were presented along with analogous topics in plane geometry: per-
pendiculars in a plane were accompanied by perpendiculars in space,
parallels in a plane were accompanied by parallels in space, the circle and
the disk were accompanied by the sphere and the ball, and so on. Each
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textbook placed special emphasis on the main theme of each course: in
grade 7, the geometry of constructions; in grade 8, the geometry of
computations; in grade 9, the ideas and methods of post-Euclidean
geometry — vectors, coordinates, and transformations.
The same main themes were followed in another series of textbooks
prepared by Werner and Ryzhik on the basis of textbooks prepared
under the supervision of Alexandrov as part of a project to create the
so-called Academic School Textbook (the heads of the project were the
academician V. V. Kozlov, vice president of the Russian Academy of
Sciences; the academician N. D. Nikandrov, president of the Russian
Academy of Education; and A. M. Kondakov, general director of the
Prosveschenie publishing house and corresponding member of the
Russian Academy of Education).
Among the distinctive features of this series of textbooks (Alexan-
drov et al., 2008, 2009, 2010) were their sections on logic and set
theory, as well as their heightened attention to the history of geometry.
The textbooks placed considerable emphasis on issues pertaining to
the language of geometry, providing translations of geometric terms
accompanied by lists of words with the same roots. They contained
numerous illustrations showing various architectural constructions
(“frozen geometry”) and discussed symmetry and its role in connection
with this, and so on.
Ryzhik broke down the problems in the book into sections whose
titles indicated to teachers and students the main form of activity
involved in solving them. Among these titles were the following:
• Analyzing solutions. Students are not only given completed
proofs, which are part of the theoretical course, but also shown
how these proofs are found.
• Supplementing theory. Students are given theoretical propositions
that do not belong to the main theme of the course, but are useful
for solving other problems. Students can refer to them along with
the theoretical propositions that belong to the main theme of the
course.
• Looking. Students are taught to interpret information presented
in visual form, and students’ spatial (two- and three-dimensional)
imaginations are developed.
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• Drawing. Students develop their spatial thinking skills.
• Representing. The problems in this category may be solved
using only visual representations, without boring theoretical
explanations.
• Working with formulas. Important problems that link the courses
in geometry and algebra.
• Planning. Designing an algorithm that leads to the solution of a
problem.
• Finding the value. Ordinary classroom computation problems.
• Proving. Problems involving proofs.
• Investigating. Problems whose conditions or possible results may
contain some uncertainty, incompleteness, and ambiguity.
• Constructing. Construction problems.
• Applying geometry. Problems from outside mathematics that must
be translated into mathematical language.
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