## Sergei Natanovich Bernstein on teaching

In 1909 Bernstein wrote an article on teaching mathematics in secondary schools and in universities. The following extracts are taken from [P S Aleksandrov, N I Ahiezer, B V Gnedenko and A N Kolmogorov, Sergei Natanovich Bernstein (Obituary), Russian Mathematical Surveys 24 (1969), 169-176]. First Bernstein's comments about secondary schools:

On considering the problem of how to introduce higher mathematics into the secondary schools, I cannot completely pass over in silence the reasons why it is necessary to study it. It is commonly said that the importance of mathematics in secondary school is the fact that, being a model of correct logical thought, it teaches the student to think correctly. While not arguing against the fact that any branch of mathematics could serve as a shining example of exact logical thought, I have not found (and the experience of my colleagues is probably not any happier) that the teaching of mathematics, which consists only in the reiteration of somebody else's logical thought, develops the ability to reason independently or consistently. In fact, we see that the proofs of theorems and formulae can be completely learnt by heart just as the rules of grammar and poems in an unfamiliar language. Like everything else, it is quickly forgotten and at the very best the student remembers not more than a tenth of the scholarly theoretical wisdom with which his brain is overloaded over many years. Leaving aside the question of how the other sciences should be taught, I think that the primary and secondary schools ought to aim at letting the student obtain knowledge (or skills) which he cannot forget and which blending so organically with his consciousness will be used without hesitation or will power, just as he talks and walks and reads. Guided by this principle, one should not in mathematics expound theories which, by their logical perfection, remain alien to the student's mind and which he will forget without trace. On the contrary the student, should be taught to use independently those methods of mathematical thought that play a specially important part in present day science and technology and in life, and which, rich with varied and useful applications, give plentiful material for interesting mathematical exercises. Apart from arithmetic and the elementary parts of algebra and geometry, the rudiments of differential and integral calculus and analytical geometry would satisfy these requirements.

Here are some of his comments on university teaching:

With the lecturing system in the universities and with the overcrowding of the first year courses, where students are always reckoned in hundreds, the help the Professor can give is almost negligible; due to this the student, unprepared for the new conditions of study and left entirely to his own resources and to his textbooks, is only rarely equal to the task in hand; a significant percentage of the students leave the Mathematics Faculty altogether and a large part of those remaining have not sufficiently thoroughly mastered the foundations of higher mathematics. The specialist courses devoted to branches of science, not yet shaped into a finished, classical form, ought to occupy a central place in university science, since only in these courses can the talents, the originality and the knowledge of the Professor be completely displayed elevating him above the secondary school teacher, and since only here can a delicate critical remark or a profound generalizing idea, casually thrown out by the Professor, bear richer fruits than the systematic supervision of a conscientious but ordinary teacher. Meanwhile our specialist courses as a consequence of the unpreparedness of the students are either almost unattended or have to adjust to the low level of the students, so losing to a considerable extent their scientific interest.