Ildar Safuanov, Naberezhnye Chelny, Russia
GENETIC APPROACH IN THE SOVIET MATHEMATICAL EDUCATION
The principle of genetic approach in teaching mathematics requires that the
method of teaching a subject should be based, as far as possible, on natural
ways and methods of knowledge inherent in the science. The teaching should
follow ways of the development of knowledge. That is why we say: «genetic
principle», «genetic method».
Probably, the first who used the expression «genetic teaching» was prominent German educator F.W.À.Diesterweg (1790-1866) in his published in
1835 «Guide to the education of German teachers»: «...The formal purpose
requires genetic teaching of all subjects that admit such teaching because
that is the way they have arisen or have entered the consciousness of the
human …Though a pupil covers in several years a road that took millenniums for the mankind to travel. However, it is necessary to lead him/her to the
target not sightless but sharp-eyed: he/she must perceive truth not as a ready
result but discover it. The teacher must supervise this expedition of discoveries and, hence, to be present not only as a mere spectator...
The correct method of teaching is not only the external form imposed to a
subject: it arises from its nature, makes its essence. If the method really matches
a nature of individual pupil it matches also the essence of a science...
The method of teaching of each subject should match its source or principle… Otherwise the method will be arbitrary, borrowed from the outside, not
following from the nature of a subject, and rather being contradicting prescription …» (Diesterweg, 1962).
Earlier the similar ideas were put forward by the philosopher Hegel (1959, p.
15): «Separate individual should also substantially pass through stages of
formation of universal spirit, but as the forms already left by spirit, as stages
of the road already developed and smoothed... Things that in earlier epoch
concerned spirits of gentlemen are reduced to knowledge, exercises and even
games of teenagers, and in pedagogical successes we acknowledge the history of education of the whole world as though outlined in a compressed
sketch». Certainly, ideas of genetic principle had been expressed prior to
Hegel, too. For example, much earlier G.W. Leibnitz (1880) expressed a similar
idea: “I tried to write in such way that a learner could always see the inner
foundation of things studied, that he could find the source of the discovery
and, consequently, understand everything as if he invented that by himself”.
In 20th century F.Klein (1987) used the term «Genetic discourse» in his famous book «Elementary mathematics from the higher point of view»: “One
could characterise the way of teaching that dominates in majority of our
schools by words «visually» and “genetically”. It means that subject matter
develops gradually on the basis of well-known, visual representations. This
is the radical difference from the logical and systematic method of teaching
that dominates at universities”.
Thus, the genetic approach to mathematics teaching has a long history.
The genetic teaching of mathematics is opposite to formal teaching based on
merely logical order of the presentation of concepts and theorems.
The weakness of the formal method of teaching was evident for Russian
educators as early as in the middle of 19-th century. At that time very popular
in Russian gymnasiums and schools was the method of arithmetic teaching
developed by German educator Grubbe, and his method was based on rot
learning , mechanical repetition and formal memorisation. The first Russian
educator who protested against Grubbe’s formal style of teaching mathematics was famous writer L. Tolstoy. He taught elementary mathematics to children in his village Yasnaya Polyana. L. Tolstoy argued that learning of mathematics should be based on the practical experience of pupils. Prominent
mathematicians Lobachevsky, Ostrogradsky and other looked for more effective and natural ways of teaching university mathematics.
In the end of 19-th century and in the beginning of the 20-th century, research in the field of mathematics education rapidly developed in Russia.
Several Congresses of mathematics teachers had been organised. One of the
leading figures in this development was Semyon Shohor-Trotsky, author of
mathematical textbooks for elementary mathematics. He invented the «Method
of expedient tasks» which was essentially similar to genetic method. He wrote
in one of his methodical guidebooks for mathematics teachers:
«The true method consists in that we should put a child in conditions at
which human mind started inventing arithmetic, we should make him a witness of that invention. But it is not sufficient today. Nowadays we should
aim at putting a child in conditions at which she/he would become not only a
witness but rather the active participant of that invention» (Shohor-Trotsky,
, p. 7).
After the October Revolution the method of S. Shohor-Trotsky was rediscovered by prominent mathematics educators A.V.Lankov and
K.F.Lebedintsev as a concrete-inductive method.
Generally, in twenties of the 20-th century, when post-revolutionary Russia
searched for new effective methods for mass education, the attention of the
mathematics educators of was attracted to the genetic approach.
In the “Minimal curriculum of uniform labour school of the 2-nd level” (Leningrad, 1925) it was recommended “to widely use the genetic method of
teaching” (p. 37). This tendency was also reflected in guidebooks, widespread at that time, on methods of teaching mathematics.
Prominent mathematics educator N. A. Izvolsky demonstrated his original
and deep understanding of the genetic approach (not reduced to the historical one) in a book «Methods of teaching geometry» published in 1924:
“In the usual course of teaching neither the text-book, nor the teacher do not
make anything in order to answer (in some form) the question about the
origin of the theorems. Only in rare instances we see exceptions: some teachers in this or that form pay their attention to the origin of the theorems; for
the pupils of this teacher the geometry course accepts other character and
ceases to be the mere set of the theorems. Moreover, sometimes some of the
pupils, independently of both a text-book and the teacher, half-consciously
come to the idea that a theorem has appeared not because of the wish of the
author of a text-book or the teacher, but rather because it gives the answer to
the problem that has naturally arisen during the previous work... Perhaps this
idea of the development of the content of geometry does not reflect to a great
extent the historical path of this development, but this view is the answer to
the naturally arising question: how the development of the content of geometry could be explained? For the teaching of geometry to have such view of
the subject-matter is extremely valuable...” (p. 8).
Izvolsky expresses the essence of the genetic approach by the following sentence: “A view of geometry as a system of investigations aiming at finding
answers to the consequently arising questions” (p. 9).
The authors of textbooks A.V.Lankov and K.F.Lebedintsev used this approach.
In particular, A.V.Lankov (1925) refers to the biogenetic law (p. 40). Close
to the genetic method is offered by K.F.Lebedintsev (1925) concrete-inductive method originating, in turn, from the method of expedient tasks introduced by S.I.Shohor-Trotsky.
Such prominent mathematics educators of the post-war period as V.M.Bradis
and N.M. Beskin also applied the genetic approach in methods of teaching
geometry.
V.M.Bradis, considering the genetic principle in the teaching of mathematics, wrote:
“The experience of the teaching definitely shows that the quality of mastering of a mathematical subject matter will essentially win if each new concept, each new proposition is introduced so that its connection with things
already familiar to the pupil is clear and the expediency of its study is visible.
For pupil, most convincing justification of each new concept and proposition is a practical activity close, whenever possible, to their experience «
(Bradis, 1949, p. 44-45).
N.M.Beskin (1947) wrote: «... It is necessary to show geometry to the pupils
not in a complete, crystallised but in the process of development. The method,
which we recommend, is called genetic. This method makes each pupil the
active creator of geometry: we put before her/him a problem, the process of
its solving gives rise to separate theorems and entire sections of geometry».
Thus, we see that in Russian and Soviet mathematics education many researchers, mathematicians as well as mathematics educators, strongly contributed to the development of the genetic teaching of mathematics.
References:
Beskin, N.M. (1947). Didactics of Geometry. M.-L. Uchpedgiz (Russian)..
Bradis, V.M. (1949). Methods of teaching mathematics at secondary school. M. Uchpedgiz
(Russian)..
Diesterweg, F.A.W. (1962). Wegweiser zur Bildung fuer deutsche Lehrer und andere
didactische Schriften. Berlin.
Hegel, (1959). Works, v. 4. M. (Russian).
Izvolsky, N. A. (1924). The didactics of geometry. SPb (Russian).
Klein, F. (1987). Elementary mathematics from the higher point of view. M. Nauka (Russian).
Lankov, A.V. (1925). On the history of progressive ideas in Russian didactics of Mathematics. M. (Russian).
Lebedintsev, K.F. (1925). Introduction to the modern didactics of mathematics. Kiev, GIU
(Russian).
Leibnitz, G.W. (1880). Mathematische Schriften. Berlin.
Shohor-Trotsky, S. (1915). Didactics of arithmetics. SPb. (Russian).