(1) Session s featur e vivid , spontaneou s communicatio n betwee n student s an d
teachers, i n which eac h studen t i s treated individually , i f possible.
(2) The process begins at a rather earl y age: usuall y durin g the 6th grad e (ag e
11-12), an d sometime s eve n earlier .
This boo k wa s written a s a guide especiall y fo r secondar y schoo l student s an d
for thei r teachers . Th e ag e o f the student s wil l undoubtedl y influenc e th e styl e of
the sessions . Thus , a few suggestions :
A) We consider i t wrong to hold a long session for younger student s devoted t o
only one topic. W e believe that i t i s helpful t o chang e th e directio n o f the activit y
even within on e session .
B) I t i s necessar y t o kee p goin g bac k t o materia l alread y covered . On e ca n
do thi s b y usin g problem s fro m olympiad s an d othe r mathematica l contest s (se e
Appendix A) .
C) I n discussin g a topic , tr y t o emphasiz e a fe w o f th e mos t basi c landmark s
and obtai n a complet e understandin g (no t jus t memorization! ) o f these fact s an d
D) W e recommen d constan t us e o f non-standar d an d "gamelike " activitie s i n
the sessions , with complet e discussio n o f solutions an d proofs . I t i s important als o
to use recreational problem s an d mathematica l jokes. Thes e can be found i n [5—7,
16-18, 26-30] .
We must mentio n her e our predecessors—those wh o have tried earlie r to creat e
a sor t o f anthology fo r Leningra d mathematica l circles . Thei r book s [32 ] and [43] ,
unfortunately, di d no t reac h a larg e numbe r o f reader s intereste d i n mathematic s
education i n secondary school .
In 1990-91 the original version of the first par t of our book was published by the
Academy of Pedagogical Science s of USSR as a collection of articles [21] written b y
a number o f authors. W e would lik e to thank al l our colleague s whose materials we
used whe n workin g o n th e preparatio n o f the presen t book : Deni s G . Benua , Igo r
B. Zhukov , Ole g A . Ivanov , Alexe y L . Kirichenko , Konstanti n P . Kokhas , Nikit a
Yu. Netsvetaev , an d Ann a G . Prolova .
We also expres s ou r sincer e gratitud e t o Igo r S . Rubanov , whos e pape r o n in -
duction written especially for the second part o f the book [21] (but never published ,
unfortunately) i s included her e a s the chapte r "Induction" .
Our specia l thank s g o to Alexe y Kirichenk o whos e hel p i n th e earl y stage s o f
writing thi s boo k canno t b e overestimated . W e woul d als o lik e t o than k Ann a
Nikolaeva fo r drawin g th e figures.
§2. Structur e o f th e boo k
The boo k consist s o f this preface , tw o main parts , Appendi x A "Mathematica l
Contests", Appendix B "Answers , Hints, Solutions", and Appendix C "References" .
The first par t ("Th e Firs t Yea r of Education") begin s with Chapte r Zero , eon-
sisting o f test question s intende d mostl y fo r student s o f age s 10-11 . Th e problem s
of this chapter hav e virtually n o mathematical content , an d thei r mai n objective i s
to reveal the abilities of the students i n mathematics an d logic . Th e rest o f the first
part i s divide d int o 8 chapters. Th e first seve n o f thes e ar e devote d t o particula r
topics, an d th e eight h ("Problem s fo r th e first year" ) i s simpl y a compilatio n o f
problems o n a variet y o f themes .
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