I must thank Luis Armendariz for pointing me to the PDF of this book, Math From Three To Seven, The Story of a Mathematical Circle for Preschoolers, by Alexander K. Zvonkin.
This translation (the original was written in Russian) began with an observation by an American mathematician that Eastern European mathematicians, including Russians, are particularly intense and passionate about mathematics; they enjoy spending many hours delving into mathematics – and they do this as a social activity. The prefator attributes this to a culture of intellectual inquiry, in particular to one aspect: the mathematical circle.
Most math circles have several features that distinguish them from simple math clubs. First, they are vertically integrated: young students may interact with older students, college students, graduate students, industrial mathematicians, professors, even world-class researchers, all in the same room. The circle is not so much a classroom as a gathering of young initiates with elder tribespeople, who pass down folklore.
Second, the “curriculum,” such as it is, is dominated by problems, rather than specific mathematical topics. A problem, in contrast to an exercise, is a mathematical question that one doesn’t know how, at least initially, to approach. For example, “Compute 4 times 3” is an exercise for most people, but a problem for a very young child.
Zvonkin has compiled a log of his activities in a math club for very young students, aged 3 or 4 to 7. The scope of these problems may astound you; it challenges many of the notions of “what math children can understand and enjoy.” I can say from personal experience with my children and grandchildren that typical kindergarten and primary curriculum is much too slow and boring for any bright child. The book also challenges, for some of us, our notion of what math is. Instead of a boring set of workbooks and correct algorithms, Zvonkin treats math as a voyage of discovery, full of many interesting detours.
You may view this book as a collection of interesting problems, and it is that; but it is best viewed as a glimpse into a math culture which has been to shown to be far superior at training young mathematicians than the usual “just the facts” drill and kill method. It is superior because it engages the passions of young folks, and encourages them to think deeply about what they are discovering, rather than simply memorizing rote facts.
I have argued elsewhere that the sooner a learning culture begins, the better; infancy is not too soon. It should not be a matter of flash cards and drills, but of conversation and discovery.