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On Mathematics Education Research in Russia
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a system for teaching future teachers principles for selecting bodies of
problems for elementary school students.
Since the mid-1980s, the importance of differentiated mathematics
education has received increasingly great emphasis; therefore, ques-
tions have arisen about how to prepare teachers for a new education sys-
tem that includes classes of different levels, and in particular, questions
about preparing teachers who are capable of teaching in classes with an
advanced course of study in mathematics. Ivanov’s (1997) work grew
out of his experience in organizing and teaching a pedagogical major
at the St. Petersburg University’s mathematics department, oriented
toward preparing highly educated mathematics teachers. The aim of
his study is “to identify opportunities for combining the fundamen-
tal and research–scientific preparation of student–pedagogues in the
mathematics departments of classic [nonpedagogical] universities with
their professional preparation as future teachers in specialized schools”
(p. 2). The author formulates and offers supporting arguments for sev-
eral theoretical principles that he follows in his methodological designs.
Among them is the principle that education is cumulative, according
to which relatively small quantities of acquired information at certain
stages may produce structural changes in the system of knowledge and
intellectual development; or the principle that education is polyphonic,
according to which it is possible to organize the education process in
a way that integrates various content-methodological lines; and so on.
The author introduces the notion of a cluster of concepts, propositions,
and problems, with which he explains how many interconnected
concepts may be discussed that are related to the same topic. Ivanov’s
theoretical work generalizes his practical work, which has included
writing a course in school mathematics that emphasizes investigating
numerous connections and parallels with more advanced courses.
The work of Petrova (1999) is also concerned with the problem
of preparing teachers for specialized mathematical schools (and even,
as she herself writes, schools for the mathematical elite), but she is
more focused on the pedagogical and methodological sides of this
preparation. For example, in her view, this preparation must include
special courses with in-depth study of school-level mathematics, as
well as courses that integrate psychology, general pedagogy, and the
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Russian Mathematics Education: Programs and Practices
methodology of mathematics teaching. The system developed by
Petrova (1999) also requires students to write final theses on topics
connected with in-depth instruction in mathematics. Her theory about
how a system for student preparation should be designed, on which all
of these proposals are based, relies on numerous studies of system-
based approaches in general and in pedagogy in particular. She also
offers her own diagnostic system for assessing whether future teachers
have reached the requisite level; this system has shown that students
who have been prepared within the framework of the system proposed
by her are better prepared to teach an in-depth course than ordinary
students.
Drobysheva (2001) is concerned with the broader issue of preparing
teachers who are capable of implementing differentiated education.
She notes that, at present, all teachers must be able to conduct
differentiated education, i.e. education that takes individual abilities
into account; meanwhile, neither the theoretical nor the practical side
of such preparation has been systematically worked out. She posits that
students’ distinctive individual characteristics may be of two types: the
first type consists of characteristics which, in her opinion, can be taken
into account without adjusting the content (characteristics connected
with attention, temperament, character, etc.), while the second type
consists of those characteristics which it is impossible to take into
account without recoding the content (different types of perception,
different types of memory, different forms of reasoning, etc.); thus, the
teacher must be prepared for such recoding. Drawing on an analysis
of existing literature, Drobysheva describes the components that such
teacher preparation must include, specifying that it is not enough, for
example, to offer a list of studied topics, but that it is necessary in some
measure to describe the relevant body of educational materials. Her
theoretical program has been embodied in a monograph which she has
written, as well as in concrete materials and courses she has developed.
She has also developed diagnostic materials, which show that if prior to
experimental teaching in a given group not one student possessed the
skills required to carry out differentiated work, then after experimental
teaching such skills were possessed by practically all students (except
the very few who had to be dismissed from the group anyway).
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In concluding this section, we should mention the dissertation of
Silaev (1997), which is devoted to preparing teachers to teach the
school course in geometry. The author offers his own idea of how
such preparation may be improved, “based on an understanding of
such preparation as a synthesis of preparations in courses in geome-
try, elementary geometry, and mathematics teaching methodology”
(p. 8). He formulates the principles according to which the relevant
instructional–methodological toolkit must be designed. It is notewor-
thy that his theoretical analysis includes an examination of foreign
findings; moreover, he notes that “the teacher’s ability to carry out
a critical analysis … of foreign findings concerning teacher preparation
constitutes one of the factors in the improvement of methodological
preparation” (p. 14). Unfortunately, it is impossible to determine from
the author’s summary which countries’ experiences he has analyzed and
which must also be analyzed by the teacher. In addition, the author
investigates how cognitive techniques are formed when the teacher
solves geometric problems. He offers a schema of the formation of
such techniques, which involves identifying the technique’s logical
structure, studying its basic characteristics, determining the main types
of geometric problems connected with the technique, and so on. He
has embodied his theoretical ideas in a number of methodological
manuals, video courses, and problem books.
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