Posted by: Alexandre Borovik | August 21, 2008

Vladimir Radzivilovsky

Another my old post from my abandoned blog — I moved it here because of my recent exchanges with Scott Carter in which I mentioned Radzivilovsky’s name.


If I had not known some of his former pupils, I would treat Vladimir Radzivilovsky’s approach to teaching mathematics to very young children with great suspicion. The following two images are taken from his website: a worksheet of his pupil:

and a photograph of the pupil. Her name is Avital.

Radzivilovsky believes that teaching is an art, not a science. Moreover, teaching, in his opinion, is a performance art and therefore he, unfortunately, does not see the point in putting his ideas in writing. His rare comments can hardly be viewed as recipes. For example, he wrote to me recently:

I use this occasion to formulate some my pseudoscientific proposals for the methodology of mathematical teaching. We have difficulty remembering stuff we do not understand. But the reverse is also true — it is difficult to understand something which still has not settled in our heads. We have a vicious circle. The only way to break it is to repeat the same thing again and again: more we remember — easier to understand. Better we understand, more we remember. […] To a five years old child I draw the unit circle [with formulae for sin and cos] about 20 times, and ask his or her Mom to draw it another 5o times. But the child will know [trigonometry] at the age of 5, not 15.

Please do not take that for a complete description of his method! As I said, I know some of his former pupils. If one uses the criterion set in Matthew 7:16 “Ye shall know them by their fruits. Do men gather grapes of thorns, or figs of thistles?“, Radzivilovsky is a fantastically successful teacher, and his work deserves a most careful study.



  1. Hi Alexandre,
    Radzivilovsky’s teaching reminds me of cognitive therapy where one changes ones beliefs by repeating over and over again the new information until it becomes true. Here he speaks of repeating over and over until one understands. Hmmm intuitively there seems to be a connection.

    Thanks for sharing!

  2. It would be very interesting to know the frequency of these repetitions or how to determine this frequency if it depends on the capabilities of the mind of the student. For example I think it’s not the same if we repet the things we want to learn 20 times a day at a half hour rate or 20 days one time per day, so my questions is how can we determine the period of this repetition if one can do so?

  3. Very interesting!
    I can only say, that in teaching GO (japanese/chinese/korean board game) the similar approach is recommended.
    All japanese professional players recommend two habits:
    1. play a lot of intuitive games – games where you don’t hesitate too much on your move – no more than 10-20 seconds.
    2. try to remember the games you played.
    3. replay games of the model players – world top players, just
    replay in some moderate tempo without analyzing why one or another move had been done, but try to remember the games,
    and again – replay a lot.
    4. tsume-go – counting problems like, checkmate problems in chess.

    the advices seem to utilize the same cognitive properties of the brain, as dr. Radzivilovsky.

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