Foreword
This i s not a textbook. I t i s not a contest booklet . I t i s not a set o f lessons fo r
classroom instruction . I t doe s not giv e a series of projects fo r students , no r doe s it
offer a developmen t o f parts o f mathematics fo r self-instruction .
So wha t kin d o f boo k i s this ? I t i s a boo k produce d b y a remarkabl e cul -
tural eireumstanee, whic h fostered th e creation of groups of students, teachers, an d
mathematicians, calle d mathematical circles , in the former Sovie t Union. I t is pred-
icated on the idea that studyin g mathematic s ca n generate the sam e enthusiasm a s
playing a team sport , withou t necessaril y bein g competitive .
Thus it is more like a book of mathematical recreations—excep t tha t i t is more
serious. Writte n b y research mathematician s holdin g university appointments , i t i s
the resul t o f thes e sam e mathematicians ' year s o f experienc e wit h group s o f hig h
school students . Th e sequence s o f problem s ar e structure d s o tha t virtuall y an y
student ca n tackle the first fe w examples. Ye t the same principles of problem solving
developed i n th e earl y stage s mak e possibl e th e solutio n o f extremel y challengin g
problems late r on . I n between , ther e ar e problem s fo r ever y leve l o f interes t o r
ability.
The mathematical circle s of the former Sovie t Union, an d particularly o f Lenin-
grad (no w St. Petersburg , where these problems were developed) ar e quite differen t
from mos t mat h club s in the United States . Typically , they wer e run no t b y teach -
ers, bu t b y graduat e student s o r facult y member s a t a university , wh o considere d
it par t o f their professiona l dut y t o sho w younge r student s th e joys o f mathemat -
ics. Student s ofte n me t fa r int o th e night , an d wen t o n weeken d trip s o r summe r
retreats together , achievin g a closeness and mutua l suppor t usuall y reserve d i n our
country fo r member s o f athletic teams .
We are fortunate t o be living in a time when Russians and Americans can easily
communicate an d shar e their cultures . Th e development o f mathematics educatio n
is a n aspec t o f Russia n cultur e fro m whic h w e have muc h t o learn . I t i s still ver y
rare to find research mathematicians i n America willing to devote time, energy, an d
thought t o th e developmen t o f materials fo r hig h schoo l students .
So we must borro w fro m ou r Russian colleagues . Th e present boo k is the resul t
of such borrowings. Som e chapters, such as the one on the triangle inequality, can be
used directl y i n America n classrooms , t o supplemen t th e developmen t i n the usua l
textbooks. Others , suc h a s th e discussio n o f grap h theory , stretc h th e curriculu m
with gems of mathematics whic h are not usuall y touched o n in the classroom. Stil l
others, such as the chapter on games, offer a rich source of extra-curricular material s
with mor e structur e an d meanin g tha n many .
Each chapte r give s examples o f mathematical method s i n som e of their bares t
forms. A game o f nim, whic h ca n b e enjoyed an d eve n analyzed b y a third grader ,
vii
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