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9
Studying the Process of Teaching Mathematics:
Connections Within Subjects, Continuity
and Succession in Education
In Soviet methodology, the teaching of mathematics was traditionally
seen as being connected with the teaching of other subjects and with
establishing and underscoring links between the various topics covered.
Dalinger (1992) collected numerous examples to demonstrate that the
aim of teaching students to view mathematics as a unified subject has
been achieved only to a very small degree. He himself identifies several
groups of possibilities for establishing such links, in particular pointing
out links offered by the subject itself and possibilities that arise in the
course of a teaching activity. As examples of the latter, he mentions a
set of various problems and, in general, the involvement of students
in a type of activity “that would allow them to assimilate the main
components of a concept and its internal conceptual connections”
(p. 29). Dalinger’s study contains much information on how students
solve (or fail to solve) various problems; he analyzes the obtained
data and offers general theoretical and concrete methodological
recommendations.
To some degree, Sanina (2002) continues in the same line of work,
attempting to construct a theory and methodology for generalizing and
systematizing students’ knowledge. While noting that generalization
and systematization may also occur spontaneously, she searches for
forms of working with students that might help many (if not all)
of them to acquire not fragmentary but systematic knowledge. Her
approach to solving this methodological problem consists largely in
constructing special lessons devoted to generalization. She works
on the methodology (and theory) of such lessons, formulating, for
example, the criteria for selecting systems of problems for such lessons
or defining the degree to which students’ knowledge is systematic
and the degree to which students have assimilated generalized knowl-
edge. Sanina also examines the possibilities of constructing special
courses devoted to integration. She writes that her experimental
work on the methodology of generalizing spanned 13 years and
encompassed both diagnostic and formative stages (during which she
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determined how generalization usually takes place and shaped a new
approach), as well as a concluding stage devoted to monitoring and
verification.
Links between topics studied at different organizational stages
constitute a special class of links within a subject. In this context, it
is customary to speak about continuity in education. Turkina (2003)
studies continuity within a framework of developmental education.
She takes developmental education to mean, first and foremost,
education in which attention is concentrated on students and not
on the educational process. Her analyses of existing data once again
demonstrate that continuity is a critical problem: during the transition
from elementary school to the first grades of middle schools (to
use Western terminology), students’ grades noticeably drop, and the
same happens during the transition to a different form of subject
organization in mathematics education (in seventh grade). Among the
theoretical results of her analysis, we should note that she considers
it expedient, in addition to distinguishing between a “zone of actual
development” and a “zone of proximal development” (in which,
according to Vygotsky, education must take place), to identify a “zone
of prospective development,” in which education will take place in
the future. This zone must be assessed and prognosticated in order
to establish continuity in education. Turkina formulates concrete
recommendations for teachers, including the suggestion to create
situations in which students can construct the necessary knowledge and
establish the necessary continuity links on their own. She has carried
out experimental work which, according to her, has confirmed her
propositions.
Magomeddibirova (2004) likewise focuses on issues of continuity,
but she concentrates on the development of a concrete methodology
for achieving continuity as students acquire computational literacy
in studying algebra and geometry, and solving word problems. The
overall conception of the approaches which she recommends includes,
for example, the suggestion that “each stage of education be oriented
around the scope and level of the students’ previously acquired
knowledge” (p. 16).
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