Was ever a human activity preached so differently from how it was practiced, taught so clumsily, learned so grudgingly, its light buried beneath so many bushels, as mathematics? It would take a separate book to count the ways this is so, much less to explain their intricate causes. Here we will survey the eccentric history of math teaching only enough to sketch a context within which The Math Circle rose.
Mathematics, as we have said several times already, responds to our architectural instinct—but different people, at different times, have understood this instinct differently and differently pictured how best to cultivate the response to it. This is why, from the Sumerians to the present, math teaching has swung like a pendulum between opposite extremes: between Mary and Martha—salvation by grace or by works: between math for itself, that is, and math as the servant of science; between math as formalism and math as intuition; between math as a body of discovered, infallible truths and math as an evolving product of human invention; between math as the material of thought and math as a way of thinking; between math for the millions and math for the elect—to name only a few of its polarities.
One constant in all this flux, which we’ll take as a measure of how desperate the state of math teaching has variously been, is the approach that from the earliest written records has pinned down the bottom of the range: “If it looks like this, do that to it.” Decorated with the elevated name of Algorithm, but commonly called Cookbook Math, it relieves the student from any need for thinking, and substitutes Truth by Authority for what could be dangerous encounters with reason. To find examples of it you need only open just about any door to a math classroom anywhere in the world, at any time.
This example goes back to Egypt and the Rhind Papyrus (dating from about 1800 BC). Its third problem asks you to divide 6 loaves among 10 men, and just gives the answer: 1/2 + 1/10 to each (with the exception of 2/3, the Egyptians only had fractions with numerator 1). But now you were supposed to check the answer, which you did by repeated doubling: 2(1/2 + 1/10) = 1 + 1/5. Doubling this, you get 2 + 2/5—but 2/5 isn’t in your vocabulary, so you must resort to your handy “division table,” which—without benefit of explanation—gave the results of dividing 2 by every odd integer from 3 to 101. Here you would have found that 2/5 = 1/3 + 1/15, so that you now have 2 + 1/3 + 1/15. Doubling this gives you 4 + 2/3 + 2/15—and again, the last fraction needs to be deciphered, so once more you unroll your division table to find that 2(1/15) = 1/10 + 1/30. Your grand total has become:
4 + 2/3 + 1/10 + 1/30.
That’s the result of doubling the supposed answer, 1/2 + 1/10, three times, making it eight times as large. But we want it to be ten times larger, so add its double (1 + 1/5) to this result, giving you
4 + 2/3 + 1/10 + 1/30 + 1 + 1/5.
You now look this up in your extensive addition table (!), and learn that the answer is 6—as desired.
Harder than seeing—as we would—that the answer is 6/10? Very much harder? Head-splittingly hard, and done in a cloud of unknowing? And yet this blind algorithm was used, around the Mediterranean, for two thousand years.
Should a student have asked why you do this, a common answer would be: “It works.” The weight of tradition might have been added: “This is how it’s always been done.” Enthusiasts might, at a stroke, have replaced tradition by eternity: “This is how it is done.” Should a student have asked why it works, few people would have understood the question, and it’s unlikely anyone could have answered it.
The cookbook approach can’t be pardoned, but it can be understood. In earlier days, when you didn’t get to keep the book and your slate was wiped clean of one precept to make way for the next, you had to memorize. And if you were leaving school at thirteen to go into who knew what line of work, you’d better have been prepared for all, with a smattering of weights and measures, money exchange, gauging, and ready reckoning. It must have been like preparing young pilots for the Battle of Britain: hardly time to learn that if you do this the plane will dive, and if you do that the guns will fire—and no time at all for a course in aerodynamics.
But the cookbook approach also suited—and still suits—people who have gone into teaching math because it offers a quiet haven, where the lines are always neatly drawn and the equations beautifully lettered. Here are the instructions for preparing your homework folder that preface an old American geometry book:
To the outside of the cover must be attached a label about 1½ by 2½ inches, filled out like the model below:
This label may be neatly lettered in any style fancy may suggest but the order of entries must be the same as here shown. Before the first work is presented, attach the label to the cover, with its upper edge even with the leather back. Name and both city and home address must also be written or lettered neatly on the inside of the cover.
It is significant that the book presents the postulates of Euclidean geometry in exactly the same tone as these directions; they are equally valid, equally arbitrary. Do you catch, in the ritual purification of all this, a whiff of the obsessive-compulsive? These acolytes of the cleanliness that is next to godliness have cut out for themselves a semi-abstract space in the midst of life’s chaos. The truths, the diagrams, even the lettering itself are impervious to calamity. This is how triangles must be in heaven, with vertex A always in the lower left-hand corner.
Romantics may look toward once and elsewhere for a Golden Age of Mathematical Tuition, when scholars studied patterns drawn in the sand and the stars, and lived in harmony with a thoughtfully cultivated nature. Where would this have been? In the Middle Ages, students began with the Trivium (not the three Rs, but grammar, rhetoric, and logic) and then graduated to the Quadrivium, where along with astronomy and music they studied arithmetic and geometry. Surely that would be a Golden Age curriculum. But much of arithmetic was spent in learning to cage numbers, as if they were wild animals, in ill-fitting cells (evens, for example, were led into the overlapping categories pariter par, or powers of 2; pariter impar, evens times 2; and impariter par, odds times 2). Most arithmetic, up to the fourteenth century, was taught by Bede’s finger-reckoning. Division tended to be limited to certain numbers: Bede said you had to be able to divide by 59, in order to get the 29.5-day lunar month to fit the 59 half-day calendar (though in fact you could carry out this division using “Greek Tables,” as he explained). By the sixteenth century, it was also important to know how to divide by 12, 20, and 28, for reasons of commerce.
If you were avid for mathematics, where could you go? In the middle of the sixteenth century an Italian named Ferrerius taught mathematics in a Scottish monastery, using Boethius’ Euclid for his advanced class. It contained statements of theorems—no proofs—from the first four books, then rules for measuring lengths, arcs, and volumes. He taught arithmetic from Sacrobosco’s De Arte Numerandi: the rules were stated, no examples were worked out, and “the student was left to make a kirk or a mill o’t,” as a Scotsman, writing four hundred years later, put it. No wonder, then, that we read in the Ludus Literarius of 1612: “You shall have scholars almost ready to go to the University, who can yet hardly tell you the number of pages, sections, chapters and other divisions in their book to find what they should.”
R. W. Southern, in The Making of the Middle Ages, says: “If a man wanted to study mathematics or logic he might have to wait for a chance encounter which sent him to a distant corner of Europe to do so: and in most cases, it is probable that the chance never came.”
And out of Europe? Ancient Indian and Chinese mathematics were full of many correct and spectacular algorithms but without any indication of how they had been arrived at, or why they were correct, or what the limitations on them might be. Little chance, then, that they would have been taught other than as recipes. The great Islamic mathematicians were devoted to the Greek tradition of proof—but there is no indication that it was passed on in the schools. The art of reckoning on your fingers and on the abacus flourished among traders, which may have set a paradigm for the culture: the rules obviously worked, what need to inquire into their workings? Were curiosity aroused in this one or that, surely he would find his way to the House of Wisdom, or to a teacher who could satisfy his fervor. So among us, we learn to drive without needing to know how an internal combustion engine works, but can learn all the physics we need, if so inclined.
It is possible, of course, that somewhere in medieval or ancient times, East or West, mathematics was taught as a reasoned structure, founded on axioms and built up by logic, but that the texts have all since disappeared; or that something this important, or sacred, would not have been committed to writing (as the workshop secrets of other crafts were transmitted to apprentices by word of mouth). Possible … but improbable enough for Occam’s razor to shave such an apology away.
If the Golden Age doesn’t lie in the past, perhaps it is just around the corner we’ve been careening toward ever since the Middle Ages. Again, we can take the prominence of cookbook texts as our guide. From the Statutes of Charterhouse School, 1612: “It shall be the Master’s care to teach the scholars to cipher and cast an accompt, especially those that are less capable of learning and fittest to be put to a trade.” Unlikely, then, that these skills were taught to the “less capable” other than by rote (evidence too of how far back the tradition of C. P. Snow’s Two Cultures goes). And you certainly couldn’t learn by reason what so much of school mathematics was devoted to: memorizing the different measures for different goods. Try keeping straight that while pounds and ounces were used for leather, tallow, soap, flour, bread, candles, resin, hemp, flax, and some Baltic goods, you needed lasts, sacks, weys, todds, stones, clones, and pounds for wool; pints, quarts, pottles, gallons, pecks, bushels, coombs, and quarters for grain, salt, sand, fruit, and oysters; and gallons, anchors, rumlets, and barrels for wine—but gills, munchkins, pints, quarts, and gallons for beer—to begin with.
The virtues of cookbook math were extolled in Thomas Dilwater’s The Schoolmaster’s Assistant, of 1743. This popular book went through forty-five editions in Great Britain, to 1816, and seventy in America, through 1832. Cast in question-and-answer form, Dilwater explains: “Children can better judge of the Force of an Answer than follow Reason through a Chain of Consequences.” A book published at the time had proofs in it—and never reached a second edition. Another gave “reasons” in small print and suggested that they need not be read.
By 1800 textbooks were written for the teacher, not the pupil: “The text contains only such matter as is intended for the pupils to copy into their account books. By adopting this plan … teachers are enabled to reject the explanatory explanations without trouble.”
A significant change comes with a growing realization—not that children might be capable of thinking, but that their teachers might not be. By 1926 answers were printed in the back of the book. The 1939 Beacon Arithmetic was published with a full teachers’ guide, with “reinforcement” for teachers “who might be unfamiliar with the material.” It was used until 1955, although by that time the prices in its word problems were wildly out of line with the economics of the day.
It would be unthinkable now to issue a textbook without a separate book of answers, and a guide for teachers, and course plans, and chapter tests, and pedagogical supplements—not only because these satisfy an ever greater need, but because publishing is big business, with all the add-ons this entails. By the mid-1970s textbooks were produced primarily by foundations and publishers, rather than teachers or mathematicians. Some of these total packages were innovative, some traditional, with the consequence that a child changing schools often moved into an entirely different understanding of what mathematics is. Hence the clamor now, in reaction, for national curricula: a new Big Business.
Where do we stand? In 1966, 20,000 students earned math BAs in America: 3.8% of all BA degrees. By 2001 there were 11,000: and that was just .9%. The drop happens before students ever reach college: 11.4% of high-school graduates in 1975 were aiming for math and science in college; this had shrunk (despite the enormous new field of computer sciences, and despite the ongoing efforts to raise grades by dropping standards) to 4.4% in 2001.
In Great Britain the number of secondary school students taking A-level mathematics declined from 85,000 in 1989 to 54,000 in 2002, even though the courses had been restructured to be less demanding. Perhaps it will make Americans feel better to learn that (according to a 2005 report from the Manchester Institute for Mathematical Sciences)
British mathematics postgraduates with a PhD from a British university are now largely unemployable in British universities. The level of research output, which British university departments are required to demonstrate in order to obtain adequate levels of funding … can now only be achieved by sucking in increasing numbers of older and more experienced researchers from overseas. Mathematics departments have no choice but to appoint the best applicants, and at present British applicants stand little chance of being shortlisted.
On the other hand, Americans will feel little better than the English when they look at the results of a comparison made in 2003 (through a straightforward test) among what were judged to be the mathematically most able fourteen-year-olds in the United States, England, Hungary, Italy, New Zealand, Singapore, and a few other countries. Fixing the average score at 500, an “advanced benchmark” was set at a score of 625. While 13% of all the students tested scored at or above this higher level, only 7% of American students did, and only 5% of those in England.
Let’s turn over this sorry tapestry and look, as mathematicians should, for its structural elements.
The teaching of math is always in crisis, thanks to what we might call, in an example of itself, “abstract recursion”: the rising spiral in which the observed becomes a concept, which is then treated as an observation from which a new concept is derived. In math teaching, this means that as broader and broader generalizations come into view, those who see them try, mistakenly, to show them at once to those who don’t, without providing the necessary intermediate steps that alone will make them meaningful. Having failed to get their ideas across, those teachers then retreat to the research that engrossed them anyway, for which they are amply paid in self-esteem and the respect of their peers. Those who have soared won’t want to waste any more time teaching fledglings to fly. Meanwhile, those teachers who don’t see the greater abstractions won’t know where math is now going, and so will teach it with outdated goals in view. Learning to divide by yet another number is no longer as exciting as it was in the sixteenth century and, in a climate that has itself grown more sophisticated, will seem pointless. And so two cultures arise within the field—research vs. teaching—with consequences as pernicious as those from the split between the sciences and the humanities.
As a result of all this, the means which teachers focus on go astray—or replace the ends. “First get the basics down pat” turns into “Practice makes perfect,” and then “Practice is perfection.” It’s as if a teacher of English were to say “You’ll love reading—here’s a phone book!” Frustration at the failure to perfect or even to master these means leads to their fraying into a bizarre variety, which calls up sweeping reforms—that fray in their turn—and the all but organic cycle begins again.
You can see this as far back as ancient Greece, in the decay of elenchus to catechism. Plato’s ends were deep and his means devious. One of them was to give to his dramatic character Socrates (a single, though long, remove from the historical Socrates) a method he called elenchus. This amounted to eliciting the truth from someone who didn’t know it, by asking pertinent questions—on the assumption that he need only be woken to what he already knew from a prior existence. This method annoys Plato’s readers (as he meant it to), because the truth seems not so much drawn from as thrust into the victim’s mouth.
The classic example of elenchus is in the Meno, where an ignorant slave boy is led to discover a line segment whose length is 
. Elenchus is certainly an ingredient of our approach to teaching math. We try harder to avoid ventriloquism, confining ourselves (more or less) to asking a fruitful question and trusting collegial conversation among our students to bring up the logical consequences and sort them out.
What is remarkable is the speed with which eliciting answers turns, in most settings, into a question-and-answer quiz. Elenchus takes too much time, too much patience on the part of the teacher, too much endurance on the part of the student. With a religious model in mind, and a sense of the immutability of mathematical truths, it is enticing to conclude that, as in the catechism, the ritually repeated question must elicit the ritually repeated answer. Answers are what matter, and these are in The Book, and teaching is preparation for the time when you will be tested. All of this coincides with the great shift from tutor to teacher: from what may become too intimate a relation between individuals to what edges away into an impersonal tension of one with many.
Because it is so fraught to teach your own children (so much emotion is invested in their success), sending them off to study with others has a long tradition. Monastery, Madrassah, and Yeshiva; the Masai huts where the secrets of adulthood are passed on by elders; putting your son, in the Middle Ages, as a page in some other noble’s family—we take this unquestioningly in stride. Even the Hensley settlement of backwoods homesteaders, perched high in the Cumberland Gap a hundred years ago, wrote away for a schoolmarm from the valley to come up the broken track and live among them (is it significant that until modern times, girls tended to learn what they did within the family?).
Nowadays teaching usually takes place in the Classroom: that institution so common, for so long, among us that its peculiarity has yet to be fully appreciated. A teacher and students are thrown together—almost always by external forces—and must associate on unequal but intense terms for long volleys of brief encounters. Nothing in family life prepares us for this style of association that we have come to take for granted. The wholly different rules of engagement in office life afterward, or in the small societies of our choosing (sports, church, club) make us forget the bizarre conventions we once lived so fully by, and thought and longed and loved and hated within.
A student cohort develops along neolithic lines (scapegoats, heroes), with the alien Other of teacher putting in sudden and electrifying appearances. Many a teacher lives and dies for each least of them (“I was heartbroken by their final exams”), and then for the next year’s. Teachers may be airily casual around the coffee machine in the lounge, since it wouldn’t be professional to admit to others (hardly even to themselves) how much their hopes have become entangled with those in their passing care. These emotions rarely spill over into outright love or hate, but—standing in loco parentis—most teachers live in a haze of emotional perplexity, and most students catch enough of the confused signals to grow, intermittently, often reluctantly, confused themselves.
The components here span a peculiar terrain of human relations. Students may confide to a teacher what they can’t bring themselves to tell anyone else, because the teacher is, in the end, a concerned but safe stranger (think of confessions made to gas station attendants). Teachers know that they will be measured by the success of their students, so their self-interest alone involves them in their students’ work. In the course of the class, however, what they want is their students’ admiration. At one extreme of temperament, they relish their dictatorship of a banana republic; at another, the handing on of a loved tradition and of the knowledge stored within it.
The explicit aim in a class—to master its subject—can wobble and fall away under the pressure of competing human needs: to get on with each other under these extraordinary circumstances; to establish and maintain a hierarchy and a way of speaking. A distance between teacher and student, casually called “professional,” grows up to correct this wobble and re-establish the commerce of life, where we help others without having to invite them home. If artificial mannerisms develop, it is, after all, on the ball bearings of manners that society glides forward.
That style of speaking soon begins to take over. Teachers learn to purse their lips and shoot laser rays of disapproval from gimlet eyes; they address invisible figures at the back of the room, and use “we” to mean “you.” A silent contest of wills can decide who is called on, who ignored. Without any deliberate intention, many a teacher will neotenize his or her students to an age where rigid forms of response are accepted and everyone has a nice day. The tone of exchange becomes more important than its content, and a class may end up without any class at all.
How does mathematics, in particular, fare in this constricted environment? Often very badly, since of all subjects, teachers feel least competent in math. Against a background of disdain for teachers in general (“those who can’t do, teach”), those who are assigned—often against their inclination—to keep order in these least liked courses even come to describe themselves as “sad old teachers” (rather than staying young with the young, they feel the age gap widening away, as they catch fewer and fewer trendy references). They certainly can’t afford to reveal their ignorance; access to hidden truth is their only source of power. Where knowledge is absent, form enters: x is always the unknown, a is always the leading coefficient, and the bar of the square-root sign must end with a hook over its right-hand-most content. Weighty issues are debated in the faculty lounge: should proofs be in flow-form or T-form? Should answers be boxed or underlined? In the sodality of the defeated, they may well console one another from time to time with the reflection that they have lived life at the chalkface, not in an ivory tower.
Crosswise to the perennial crisis, opposite tugs from the multiple polarities we listed at the start of this chapter deform the texture of teaching, locally and at large.
If math is the servant of science, then its study should begin and end with real world problems—posed from the start as messily as real world problems generally are, with extraneous data and ill-defined givens; and solved with error terms and approximations. But if math is derived from Understanding (the real meaning of Mathesis, as science is from Knowledge, Scientia) we must clear the paths to and within it of rubble, and let the mind enjoy the pure play of form. One of the earliest expressions of this opposition can be read into its Greek beginnings: is mathematics abstracted from the world (Aristotle), or is the world extracted from the forms that forever precede it (Plato)? Does form follow function, or function form? Schools and colleges that offer math courses designed—but never with quite enough content—to parallel the science courses taken simultaneously fall in the first camp; those that stress the unapproachable quality of mathematical questions, methods, and standards of proof, the second. (While the French make this qualitative judgment explicit, with pure math at the top of their tree, it is implicit in most academic communities.) Inevitable dissatisfaction, in either case, periodically moves just about any institution—and at times the general fashion in math teaching—from one extreme to the other.
Proponents of mathematics as a formal enterprise have tended to hold the balance of power over the centuries, whether the forms were taken seriously in themselves or as a proxy for meaning. In epochs when geometry has been in the ascendant, intuition has overtly or covertly displaced formalism as the motive power: “seeing” a relation preceded its proof in value as well as in time. Worries about lack of rigor always swing the balance back to the formal and its spokesman, algebra; although geometric proofs can be fully as rigorous as proofs carried out in symbols, the latter bear a heavier impress of abstraction, and so read as somehow more sacred. But even when formalists are in the ascendant, they are always under attack from those who fear that their approach will desiccate mathematics by drying out the juices of insight. At the same time, new phenomena, in pure as well as applied contexts, urge us to find at least first approximations of explanation, even though these might lack the ultimate seal of rigor’s approval. And just as in sports, the irrepressibly playful in us holds back from letting legalism bring invention to an end.
The “math is discovered”/“math is invented” dichotomy used to be innocent, the latter view naturally accompanying times of great mathematical ferment, the former, times of consolidation—and both leaving their traces in the teaching that scrambled to keep up. Now this opposition has been co-opted by rather boring political factions. Conservatives need an intellectual fixed point: God may no longer exist, but Math is still omniscient. Even if you never understood it—perhaps especially if you never understood it—it has the comforting status of divine absolutism. If the social world is in terrifying flux and morality has lost its bearings, at least Math is fixed and unambiguous.
The New Right of 1985 called what was then the New Math (investigation, problem-solving) “the latest stage in a disastrous process that has seen school mathematics drift toward becoming a low-level empirical science.” The New Left sees in math an easy and early opportunity to make independent discoveries, untrammelled by the immense data-gathering needed in other fields—but then, ashamed of its elitist past, would put all mathematical ideas, however misguided, on a Politically Correct par, where sincerity displaces rigor as the touchstone, and such lurid flora as Ethnomathematics sprout. It is just common sense, of course, to take into account the context of life and thought in, for example, ancient India, when studying the rise of number theory there; or the Greek context, when looking at how geometry and deductive proof arose in its midst. The silliness starts with claims that mathematics follows from rather than underlies cultural phenomena; that rather than being universal it is fundamentally different in its Eastern and Western varieties, and speciates with national differences; and that one can only fully understand and enjoy the mathematics of one’s physical rather than spiritual ancestors. Here too you will find that recently fashionable reinterpretation of Karl Popper’s “falsifiability” criterion as a desirable end: mathematics should be in perpetual revolution, with any hint of its having unearthed a morsel of Being discarded as antiquarian thinking. So Right and Left misread as centrifugal or centripetal the rising spiral of abstract recursion.
Is mathematics the material of thought or a way of thinking? The first hardly differentiates it from the other sciences, taking it as “out there,” to be discovered. Mathematics as a way of thinking received its greatest impetus from those Formalists who sought to logicize mathematics, giving logic and math alike the common language of set theory. They rode high, as far as teaching went, in the 1970s, with that New Math, which left parents unable to help their offspring decode Venn Diagrams, and teachers at a loss to decide which solution sets were complete (though it was rather fun rolling the Well-Formed-Formula dice, and filling in truth tables).
Which is the proper clientele of the mathematical enterprise? Lancelot Hogben’s 1937 Mathematics for the Million wasn’t the first attempt at popularization. For that you have to go back at least as far as AD 100 and the Introductio Arithmetica of Nichomachus of Gerasa, which entertained schoolchildren for a thousand years with its numbers dressed up as fanciful animals. And Cocker’s Arithmetic, published in 1678, was so popular for the next century that “according to Cocker” entered the language as a synonym for “accurate.” And why shouldn’t everyone have access to this language, deep in the human brain?
But the view of mathematics as the province of the elect has always rivaled the view of it as a universal birthright. Its origins lie, perhaps, in the ancient world, where priests kept to themselves the knowledge of how the algorithms worked, whose husks alone they transmitted to what was itself a small group of scribes. Perhaps an inner circle of Pythagoreans, in Greece, knew the dark secret about the irrationality of ; certainly what mathematics they did teach to their initiates wasn’t meant to spread beyond them. You needed a credit in geometry to enter Plato’s academy—almost as if it were a talisman. Old ways in olden times? It would be interesting to study these days the elitist hierarchies in Gifted and Talented groups, where the deep talk only to the profound, and the profound talk only to—themselves. Are there quiet networks of funding and coaching for young students identified as having genius potential? Are mathematical Olympiads nodes in such networks?
Attempts at makeshift syntheses of these opposite views are by no means recent. The 1868 report of the Taunton Commission, in England, concluded that “mathematics” should be taught to children of the upper-middle class and professional classes; “arithmetic and the rudiments of mathematics beyond arithmetic” to the mercantile class; “very good arithmetic” was for the sons of smaller tenant farmers, small tradesmen, and superior artisans; while working-class children were to get elementary arithmetic. We wince at this now—but we let the mathematical fate of people be sealed by their performance, in adolescence, on a three-hour multiple-choice exam.
We’ve gauged the lower reaches of math teaching by the extent to which it consisted in having students do no more than memorize recipes, and said that this dreary waste of mind and time has often been dressed in the fancier togs of “mastering algorithms.” Perhaps this was a bit unfair, since algorithms condense great efforts of understanding into usable formulae, facilitating the next move upwards in abstraction. The mortal error comes in ignoring the meaning, history, and context of these efforts—skipping what, how, and why these algorithms facilitate—and simply presenting them as facts, no different from figures or from rules of symbolic manipulation. This stunts thought by demoting math to the level of incantatory magic, and accelerates the decline from shaping questions to memorizing answers. At its extreme, it takes the edge off the mind’s finest capacity: to size up the unknown.
We can imagine two sorts of defense being made out for the cookbook approach to teaching. The first is that people desperately need a degree of awe in their lives: awe of authority (as represented by the hieratic figure of the teacher); but even more, of holy writ. Cultures with sacred texts often think it important to commit them to memory, or at least to attend their recital, whether one follows the words or not. What matters is being in the presence of The Word, in the language in which it was originally spoken. So with the Mass in Latin, the different parts of the liturgy resounding together in Orthodox churches, the spoken Talmud, the chanted Koran, or the Tudor English of the Book of Common Prayer. We have our literary talismans too: any Greek can intone the first few lines of the Iliad or Odyssey for you, any Russian the beginning of Eugene Onegin, any American the first sentence of the Declaration of Independence. Generations of English students say “To be or not to be, that is the question,” and “Romeo, Romeo, wherefore art thou Romeo,” even if they think “wherefore” means “where.” The theory (if that isn’t too strong a word) behind the vast majority of math teaching has been to dig it up from the page and plant it in the memory, where it may root, flourish, and propagate—though, more likely, rot and wither. Never mind: you will have been in The Presence, and therefore saved; you will pass the test, and therefore make your way in the world. Even better, you will know that your worth has been successfully measured against a platinum rod in the only Bureau of Standards that matters.
The other argument is secular. A good teacher knows the limits of a student’s curiosity. The young love patterns but don’t much care about explanations or justifications; watching them unfold and repeat, catching the lilt of their song, is enough. Why distract with trying to prove what slides so easily into memory? FOIL (First, Outside, Inside, Last) is all you know and all you need to know about binomial multiplication, SOHCAHTOA about trigonometric functions.
Sally tarries by the gate:
Four times seven’s twenty-eight.
That’s from the 1839 Marmaduke Multiply, with its pretty woodcut illustrations. Sally and her gate may have no more relevance to four times seven making twenty-eight than they would to four times twelve making forty-eight, but picture, rhyme, and product enter the mind of childhood together. Older students have begun to gravitate toward what will be their lifelong interests, which might just possibly involve mathematics as a means—but hardly ever as an end. Why then punish the majority with proofs of what they need only apply? It suffices for most people that someone knows the reason why (this harks back to the prior argument); the student’s “I’ll take your word for it” is the antiphonal response to the teacher’s “‘Shut up,’ I explained.”
These two defenses of teaching math by recipe stem from the world-weary conviction that people just can’t and shouldn’t be bothered with going back and back in explanation. Being educated means learning things, and “minus times minus is plus,” like the rules for which and that, can be explained, or made vivid, but the point is being able to get them right. Claims that stepping back better lets you leap forward, and that knowing the truth will set you free, bounce harmlessly off impatience, and our confidence that trial and error will see us through.
Upper-level math courses, from high school on, begin to leave the cookbook approach behind and teach math more as theory developed in a sequence of theorems, with the theorems presented as the conclusions of proofs. Now, you would think, Mind has come into its own. But no; the common practice has to be seen to be believed. The instructor (teacher, teaching assistant, graduate student, lecturer, professor) enters stage left, spends a few minutes on administrative details, then approaches the board and spends the rest of the class reciting definitions, theorems, and proofs—and writing them out. Everyone silently copies all this down. When a student once asked the instructor if she might photocopy and hand out the notes, he answered in puzzlement, “Then what would I do during the class?”
Some people may teach like that because they have just come from, and are just about to go back to, their research, and know the material so well that they can painlessly rattle it off in this way. Some are merely carrying on the tradition in which they were taught; reading is, after all, what “lecture” means. Others believe that the greater the variety of ways in which students can encounter these difficult ideas, the better, and watching them unfold right there before their eyes has a dramatic force to it; and besides, questions will be addressed in problem sessions, or in conversations among the students themselves, afterward. A more serious justification is that the particular order of the material the instructor chooses, the particular proofs, the variations on definitions and lemmas, gives his particular take on the theory and a point of view that gets across in little asides, pointed emphases, and throwaway lines. This is how the touches of the workshop, the style of the master, and the secrets of the guild, are communicated.
Behind these various reasons, we suggest, lies a deeper explanation. You begin to detect it in the relish with which an instructor performs his work, writing with increasing brio as the class goes on. He is performing a piece of music. To know, to understand, a corner of mathematics is to love it, and to find ever new depths and implications in it each time you think it through. This writing up of the ideas is as much for his own enjoyment as for that of the audience, but it is crucially for the audience too: these are the song-lines of the tribe, the map sung into existence for each apprentice then to be able to follow on his own. Such, at least, is the fervent (if often unacknowledged) belief in the mathematical community.
It is a wonderful and inspiring undertaking—for the performer. But is it in fact the ideal way for the novice to learn his craft and the craftsman to become an artist? Put aside the niggling problem that should you lose for a moment the thread of exposition, there is hardly a hope of picking it up again: Theseus has disappeared around who knows how many turnings of the labyrinth. Put aside the cramped writing, the errors in notation, the sudden change in tempo from the hour’s adagio beginning to its accelerando end. No matter how thoughtfully the definitions are phrased, nor how well the material is calligraphed nor how beautifully sequenced, the route that should have been discovered is no more than being retraced. The student will have to unlearn and reinvent it to make it his own.
Song-lines at one extreme, cookbooks in the vast middle—and what are the practices that fade into cookbook, or rival it, along the youngest margin? You’ll find one attempt after another, especially recently, to lure children into math by making it fun. “Manipulables” replace memorizing times tables; a pattern is discovered, and then another; shapes are folded; bells rung; numbers dance. All this has the welcome effect of not giving fear and loathing even a look-in: games become the arena where minds encounter math. The problem usually is, however, that these encounters stay superficial—a decorative rather than an architectural instinct is catered to. No topic is dwelt on long enough to open up its depths, and the repetition that used to go into mastering formulae is replaced by the safe repetition of harmless games. Proof never puts in an appearance, for fear that it will be discouragingly hard (and so waits menacingly in the wings, or is finessed altogether). But when children ask why, it ill serves them to answer: because it’s nifty.
This benign approach has a second source in political correctness—as a canny fourteen-year-old explained to us. He showed us a word problem from his final exam: “A charity dinner was held for 1,500 students and adults. The adults were charged $15 per ticket, the students $12, and $21,000 worth of tickets were sold. How many adults attended, and how many students?” We said that this looked like a problem that might be solvable. “It’s solvable, all right,” he answered, “but that’s not the point. Look at how it’s put: ‘a charity dinner.’ You’re not allowed any more to talk about people making a profit.” Of course! How naïve of us! The young must be screened away not only from difficulty and perplexity but from the fallen world of adults. It isn’t that abstraction is replaced by reality, but that both are replaced by an “under fourteen”-rated video game. This systematically condescending attitude toward people in general and the young in particular has taken its toll in math teaching. By stifling how we imagine others and sugarcoating their differences, we injure imagination itself; by feigning compassion while enjoying the moral superiority it makes us feel, we nurture hypocrisy, which lames our relation to truth; by keeping students from ever feeling frustrated, we prevent them from having revelations. When, however, in The Math Circle, we accept any conjecture our students offer, this doesn’t mean we endorse it, nor are we saying that your truth is all right and so is mine. Rather, the taking now obliges us to discover together whether this insight solves the problem—and the game’s afoot.
As if all these internal conflicts weren’t enough, a medley of outside interests, at odds with one another, further distort the teaching of mathematics. These include pressures downward from working mathematicians; sideways from science, business, and industry; and in all directions from social concerns.
Although mathematics is deployed over a great expanse, its practitioners know its anatomy well enough to trace the connections among its major systems. Neophytes, they argue, should learn the progressively richer communities of number first: the naturals, integers, rationals, reals, and complex numbers, along with the operations on them. This leads the student through arithmetic to algebra, with geometry studied next or simultaneously (depending on the emphases in the mind of the person you’re talking to). Now calculus, the fourth great system, opens up, and beyond it the spiral returns to arithmetic as number theory, and to the abstract algebra of groups, rings, and fields; and then (or concurrently) to the spectrum of geometries, and topology; and calculus rethought as real and complex analysis. The rudiments of logic, absorbed within some or all of the previous studies, resurfaces now in set theory, which is meant to slide under them all; and category theory arches overhead. This organic progression lays down the clearest possible guidelines for studying mathematics from the year dot.
The science lobby answers: this is a curriculum for pure mathematicians, and less than 1 percent of the students who start to climb will ever see the view from the top of that tower. What’s needed instead is the ability to make sense of scientific data and to think your way through real-world situations: knowing how to model a problem; understanding statistics and probability; being at home in calculus and, at its upper limits, with differential equations.
“And what percentage of students, these days, will be scientists?” ask the representatives of business and industry. This elitist curriculum ill prepares people for the workplace, where what you need is common sense sharpened by the discipline of mathematics (following rules to results, and knowing how to test them; doing word problems). You need the sense of order in your life and thought that arithmetic gives—and all these innovations model only chaos. Teach them to add and subtract! Get students to learn how to use their calculators! If they’re going to get ahead in the world, teach them programming!
Meanwhile, teachers have to think about accreditation from local, state, and national agencies; about climbing career ladders and winning merit pay. Their students are exhorted not to be left behind—rather to be accelerated, while not standing out from their group. Haven’t we here the problem of Rousseau’s Emile: the education of the free individual is contrary to the education of the useful citizen? Should math therefore be taught as a liberal art or as a practical skill? Should the pain that its students and teachers suffer be respected, and the teaching of it accordingly cut down, or back, or out altogether (now that computers can do it all for us)?
And the mathematicians answer, as Plato did, that one studies mathematics to turn the soul’s eye from the material world to the subjects of pure thought, and an a priori knowledge of eternal objects.
And the scientists reply …
These accumulating pressures periodically erupt in a survey that shows just how low the standards have fallen, and then (writes George Walden), “everyone slips into a well rehearsed posture … the unions call for cash; the teachers sulk; the press trumpets its outrage; another segment of the middle classes grits its teeth and stumps up for private education.” And a committee meets to frame yet one more curriculum reform. Who is on this committee? In one case, in 1987, it was “nine mathematics educators, three school heads, four educational administrators, two academics, one industrialist, and one member of the New Right.”
Most reforms address how math is taught, but a notable exception would change the math itself. Let’s call this movement Muscular Mathematics; Back to Basics is a typical example. Enough of investigation, experimentation, discovery, and invention, say its proponents; enough of theorems and proofs, and “maybe it’s the other way”: replace ambiguity by certitude! Anything can be learned and made rock-solid by drill, and reinforced by Mad Minute quizzes, in which the aim is to get a hundred calculations perfect. Algorithms decode the universe, and minds are made to memorize them.
If you ask teachers what reforms they favor, they speak again and again about altering the way they themselves were taught. Students at a teachers’ college were asked how they felt about math: “totally devastated”; “really useless and frustrated”; “stupid”; “humiliated”; “ashamed”; “inadequate”; “small”; “like crying”; “slow and thick”; “a failure”; “an idiot”; “shattered”; “depressed”; “bored.”
Graduates of the college, who were now student teachers, were asked the same question: “It was all too fast for me—I couldn’t keep up.” “I just learned the rules in order to pass the examination.” “By the time I got to the final year of my schooling, the gaps in my knowledge were so wide I gave up.” “I don’t have the basic mathematics knowledge to risk giving children very challenging work.” “The word math makes me have a panic attack.”
And what did veteran elementary school teachers answer? “Embarrassed”; “struggling”; “failing”; “terrified”; “demoralized”; “pressured”; “frightened”; “out of my depth.”
When happy and successful teachers, who feel comfortable with the material, are asked for their secrets, they tend to talk geographically. Some prefer being in front of the room, where all their students can focus on them; some like leading from the rear, giving the students a sense that they are actually in charge; some like wandering about, helping this one here, consulting with another there, now leading, now following. What has location to do with it? Clearly nothing, since all positions work equally well; but like the knight who was given a dragon-taming formula, the teachers succeed in part because they are confident that they will.
What do teachers know anyway? The spirit of reform takes their ignorance as its premise (“In England,” says a 1991 report, “the public acceptance of a national curriculum followed from a growing belief that teachers were both incompetent and politically suspect.” The inspiration for and implementation of No Child Left Behind is based on the view that education is too important to be left to teachers.) School boards realize that their schools are failing, and since it is obviously impossible to replace all the math teachers, they buy a curriculum designed to retrain and support them. But just as every Torah becomes a Talmud, so every curriculum writer begins with a list of crucial topics, adds charming problems to tease attention and enthusiasm awake, and ends by writing out step-by-step instructions that explain nothing, yet leave nothing to the teacher’s initiative or imagination.
Because the school board is answerable to the voters, there must be scientific means to test a curriculum’s validity; these ways are, of course, statistical. What works for the majority must be used for all—hence no more individualized instruction. Yet what actually happens in a classroom? There are some students whose intuition is geometric, others algebraic. There are eye learners, ear learners, and hand learners. There are those who need to work it all out for themselves in silence, and those who need company and conversation to stay focused. These are independent variables, so you’ll find all sorts of combinations sitting in front of you. The teacher’s job is to get everyone to master the ideas by finding the right combination for each of these complicated locks—but if she has to “follow the curriculum” rigidly, her hands are tied.
The situation is, to be sure, much more complicated than this. What tone should you take to correct mistakes? Two researchers in 1976 found that in upper-income classrooms, praise was “negatively related to student learning gains”; in low, students prospered from a warm, supportive approach. There’s the culturally enhanced distinction between male and female styles of learning and taking part; there’s the tenth-grade ambiguity splitter: some hate it and opt for clarity and authority; some suddenly wake to the attraction of metaphor and to the appeal of nuanced answers. Each teacher of any experience will have a catalog much longer than this of the ways students differ in their outlooks and needs.
The inevitable solution of reformers—explicit for at least the last forty years—is to make curricula teacher-proof. Threads leading to this conclusion in fact trail back at least as far as the 1920s, with those answer books and manuals for teachers and handy guides: meant to support, they end up crippling by keeping a person from exercising his mental muscles. Why not then make a virtue of this necessity and relieve the teacher altogether from having to stand up? Is anything more pathetic than a teacher, supposed to be guiding children’s self-motivated investigations, who comes to rely on problems gleaned from old teachers’ magazines, who hands out dog-eared cards with outdated information and says “Here’s something new to think about”?
The next step is to suit textbooks carefully to the ability level of the classes they are to be used in, so that the teacher needn’t have that responsibility. The School Math Project, for example, published “Y” texts for the ablest, “G” for the slowest. They differed not just in the difficulty of the math but in the attitude taken toward it. To paraphrase from a 1991 Open University Course: the “G” materials were centrifugal: lightweight in form and written in explicit, everyday language. They were mildly sensational, and pictures outweighed text. The jokes in them were nonmathematical, and even antiacademic. The “Y” were centripetal: the language was esoteric, the jokes cerebral. The cover pictures celebrated the enigmatic and erudite (contour maps of the human face, Escher prints). “The mundane inevitably enters the domains of both series,” this course concludes, “but whereas, in the Y books, the everyday world is sacrificed on the altar of mathematics, in the G series the everyday appears as an effusive apology for the tentative intrusion of the academic.”
Such efforts take a considerable burden off the shoulders of a struggling teacher, but opportunities to interfere still remain. Why should the towered sorcerers (who, for all their powers, can’t be present in all classrooms at once) let their apprentices wreak havoc with the wands put in their clumsy hands? This thought is taking us straight toward the pet goat. You may recall that President Bush was reading the story of a girl and her goat when he was stunned by the news of 9/11. It was written for second graders by Siegfried Engelmann, as part of his Direct Instruction method, which (according to a New Yorker “Talk of the Town” of July 26, 2004), dictates “every word of every lesson, including which words of encouragement teachers may and may not use.” This article quotes from the Direct Instruction website: “The popular valuing of teacher creativity and autonomy as high priorities must give way to a willingness to follow certain carefully prescribed instructional practices.” And it quotes Engelmann himself: “We don’t give a damn what the teacher thinks, what the teacher feels. On the teachers’ own time they can hate it. We don’t care, as long as they do it.” Direct Instruction’s principles have naturally been applied to mathematics too. According to Engelmann, says the article, his is one of the few methods that has been consistently shown to improve student achievement. This brings it in line with the government’s program, which requires that only scientifically based educational programs be eligible for federal funding. Perhaps there has been a paradigm shift in the basis of science.
Even when designing less extreme teacher-proof curricula, an inevitable consequence is that the texts become learner-proof too; the problem-writers so want to guarantee that their unseen students will succeed that they can’t leave them to figure out relations for themselves, but merely check them. Problems intended to foster discovery are given in such small spoonfuls that you needn’t see the idea behind them at all (or even realize there was an idea) in order to answer each question in the sequence. Only the most aggressively obtuse student will fail to see that the sequence of hints converges to the answer.
Since the teacher has been written out of these scenarios—made superfluous even before being made redundant—the teaching machine follows as night follows day. Your patient and nonjudgmental friend is quietly blinking in the corner, awaiting your command. Here is one company’s description of its paragon:
The Company is currently completing the development of its first game, designed to teach Algebra I. The product seamlessly integrates math equations and concepts into an exciting first player action adventure scenario that engages, encourages, and ultimately pulls students past the pain of learning. The game awards points for efficient problem solving, demonstrated improvement, and collaboration with other students (when played in multiplayer mode). It also deducts points for simply guessing. The parent sponsored points are integrated into the proprietary rewards system that allows students to exchange their points for merchandise and other rewards from strategic retail partners.
With this beside you in the wilderness, what more—other than perhaps a keg of beer—could you ask for?
From this account you may think that curricula, like water, run only downhill—but there are movements against this flow, among which we count The Math Circle’s predecessors. Perhaps their common ancestor is Rousseau, for whom the task was to develop what already lay in the glory-trailing child (and so gave rise to the false etymology of “education” as a “leading out”). Mathematics is free: if only the individual were, we would be living in an earthly paradise.
Johann Heinrich Pestalozzi (1746–1827) and, over a century later, Maria Montessori (1870–1952) developed Rousseau’s image of the child along different lines. Pestalozzi sought to join in education what Rousseau had broken apart: the free individual and the citizen in society. He spoke of education “as a form of action which allows each person to recognize his own individuality and make a creative work of himself.” He was repelled by systems, and focused not on theories of education but on the particular child’s immediate concerns, which he addressed along the three routes of heart, head, and hand. The heart gives us our sense of union with our fellow men; the head, detached reflection; and having felt and thought, our hands now let us act to create ourselves—but all three must be equally involved in any educational undertaking.
Montessori saw the importance of focusing the child’s attention and having hands shape what the mind sought for, the abstract becoming vivid in tangible geometric shapes. Her work with orphans made her realize that in an increasingly fluid society, schools must take on what was once the province of parents: bringing up children to comprehend others and to be comprehensible themselves. Her classrooms, like Pestalozzi’s, fostered a collegial spirit, as do ours.
The need for deeper and more satisfying approaches to mathematics than could be found in schools led, in the last century, to all sorts of math clubs, math camps, math programs, and math contests, all with wildly differing agendas—some embroidering on, some part of, the fabric of mathematics. The University of Utah math circle meets during term time, and makes a particular effort to integrate both local high school students and their teachers with the faculty and students of the university’s math department. There are distance learning programs, where problems are set and corrected via computer. Serious summer programs, such as the late Arnold Ross’s at Ohio State, David Kelly’s at Hampshire College, Glenn Stevens’s PROMYS at Boston University, and the peripatetic MathCamp, immerse their generally adolescent students in several weeks of intense classes, lectures, problem sessions, discussions, and camaraderie. Like music camps, these draw and wonderfully develop students who are already thoroughly devoted to the art. The relationship between the students and their instructors, who are often only a few years older, adds humanity and life to what might even be a fairly standard classroom format: exposition, problem sets, and presentation of results.
One very different approach to training young mathematicians was what came to be known as “Texas Topology,” taught by R. L. Moore from 1920 to 1969, at the University of Texas.
“I’m afraid,” said the Bishop, in an ancient Punch cartoon, “that your egg is bad.” “Oh no,” answered the deferential curate, “parts of it are excellent.” The Moore Method was R. L. Moore’s egg. It too was excellent, in parts.
Moore began by interviewing each applicant. Any student who had already taken a course similar to the one he was applying for was ruled out; the aim was to have the whole cohort start from scratch, ignorant of established terminology, notation, methods, and results. In The Math Circle we always choose a topic that none present are familiar with (easier to do, certainly, with younger students), but we take whoever applies. Another difference from The Math Circle is that Moore also ruled out black students, and, they say, foreigners and Jews. On at least one occasion he refused to begin a class until a black woman in the room left.
Once in the course, the students were put on their honor not to read anything about the topic being studied. It says something about Moore’s own sense of how honorable they might be that he removed all pertinent books from the university library. We, too, hope that students won’t bring in ideas they’ve found online or learned from their parents. It adds an extra fillip to work around such difficulties on the spot: keeping up a collegial tone while getting across that the pleasures of discovery outweigh those of borrowed glory.
At Moore’s first class meeting, the eminent mathematician, Paul Halmos, writes: he “would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: ‘Mr. Smith, please prove Theorem 1. Oh, you can’t? Very well, Mr. Jones, can you? … ’ If no one could, class dismissed.” Well, not exactly. Moore wouldn’t talk about the theorem thus left in suspended animation, but around it: about logic, or kinds of proof, or even history or just plain gossip—but hints toward the proof might be hidden in what he said.
Moore’s whole effort was directed at the often nerve-racking business of proving, which is why he would hand out, at the beginning of a course, the sequence of theorems that were to be tackled (and by proving them, master the subject these theorems marked out). Here, too, our approach differs, letting students work as mathematicians do, by first shaping a conjecture from inchoate data.
Presently, says Halmos, Moore’s students were proving theorems and watching the proofs of their fellow students like eagles. You weren’t allowed to let a mistake get past; it was your duty to point out mistakes to the presenter and demand a correction or offer one. Students, he says, were quickly ranked by quality, and once the order was clear, Moore would begin by calling on the weakest student. This prevented the best from holding the floor, and made for fierce competition, which Moore encouraged. “Do not read,” says Halmos, describing the ethos of Texas Topology, “do not collaborate—think, work by yourself, beat the other guy. Often a student who hadn’t yet found a proof of Theorem 11 would leave the room while someone else was presenting a proof of it—each student wanted to be able to give Moore his private solution, found without any help.”
Needless to say, the bitterness of this competition goes dead against our grain. What’s wrong with a vision of the little society of a classroom (or any society, for that matter) as a trading place for insights?
Unquestionably, Moore and his method turned out a vast number of PhDs in mathematics, many of whom became noted researchers and teachers. “Moore never missed a chance to praise students for their accomplishments,” writes Peter Renz, “and worked tirelessly to bring out the best in all his students.” One of Moore’s biographers, Steve Kennedy, says that “his real genius lay in his ability to inspire people to do more than they themselves thought they could,” and remarks that despite the competitive air, Moore’s students admired, respected, and even loved one another. “Partly of course,” he adds, “this is the foxhole phenomenon—we always remain attached to folks with whom we’ve shared an arduous, stressful trial.”
Halmos again: “Moore felt the excitement of mathematical discovery and he understood the relation between that and the precision of mathematical expression. He could communicate his feeling and his understanding to his students, but he seemed not to know or care about the beauty, the architecture, and the elegance of mathematics and of mathematical writing.” Moore was intolerant, he says, of every part of mathematics other than his own, seeing algebra and analysis as competitors and enemies; and made his students much less well educated and useful than they could have been.
And yet … and yet … Moore shared the same admiration that we, and all sensible people, do for the old Chinese proverb: “I hear, and I forget; I see, and I remember; I do, and I understand.”
The large, dysfunctional patriarchy—what a very American story! A story, too, that calls Ty Cobb to mind: perhaps the greatest hitter of all time, who let it be known that he filed his cleats, so that the fielders would stay out of his way as he slid, spikes high, into the bag.
We chose “The Math Circle” as the name for our approach to honor what we understood to have been a loose tradition in the Soviet Union of informal, often covert, meetings of students with teachers (their name, “circle,” also made us think of those Decembrist philosophical circles, where conversation and debate flowed in amicable surroundings: free minds at play in an otherwise dangerous world). The image we had was of evenings of fervent discussion in someone’s apartment, one person stirring the soup while another watched by the window for signs of the KGB. This picture may not have been wholly accurate. In fact these circles may have had a Hungarian origin, in 1894, with competitions that came to be called The Hungarian Nursery—and a spirit of competition has informed them since. The contestants work in classrooms under supervision, the Society selects the two best papers, and the awards—a first and second Eôtvôs Prize—are given to the winners by the president himself at the next session of the society. Notice, however, that even in Hungary some have had doubts about the value of competition. In Gábor Szegú’s 1961 preface to the Hungarian Problem Book, he said that although competition was a powerful stimulant, it was not necessary for a lively mathematical culture, and that the students who were so deeply involved in the long and hard work of solving the problems were probably spurred on not by the possibility of a medal, but by “the attitude which rates intellectual effort and spiritual achievement higher than material advantage”—something that couldn’t be fostered by decrees from on high or more and more intense mathematical training: “the most effective means may consist of transmitting to the young mind the beauty of intellectual work and the feeling of satisfaction following a great and successful mental effort.”
In the Soviet Union, at any rate, math circles went back at least as far as the great mathematician Andrei Nikolaevich Kolmogorov (1903–87).
In addition to his own vast research (it was said that it would be shorter to simply list the fields in which he hadn’t done significant work), he was deeply committed to teaching young students. When he taught math he didn’t lecture but just posed problems, gave the students complete freedom to attack them as they saw fit, and as his biographer said, was “always waiting to hear from the student something remarkable.”
Despite the intensity of his own passion for mathematics, he encouraged his young students to venture into other realms, introducing them to “literature and music, joining in their recreations and taking them on hikes, excursions, and expeditions.” His idea was that students shouldn’t be intellectual clones of their teacher but should develop their own personalities in life and in their work. He didn’t even mind if they didn’t become mathematicians, as long as they retained the principles he taught: a broad outlook and an unstifled curiosity.
Russian math circles and math competitions meshed imperfectly, like large and slightly worn wooden gears. In Moscow, at least, both were organized by the same people, though the students involved with one needn’t have been (but usually were) involved in the other. In 1934 Moscow State University (MSU) started a series of lectures for high school students and a series of math publications aimed at them; in 1935 the first Moscow Math Olympiad was held, with 314 participants, and the next year a math circle was organized at MSU. Students were bored by professors’ long lectures and students’ clumsy presentations. In the late 1930s this changed radically: an MSU student named David Shklyarskiy organized his section as a short lecture, followed by students’ discussing solutions to problems they had worked on. The problems were of varying degrees of difficulty and could have something to do with the lecture—or not. Some were mini-research investigations. Occasionally Shklyarskiy asked students to find elementary solutions to problems for which none were known, and finding these solutions sometimes extended over years. The extraordinary success of his students at the 1938 Math Olympiad, where they won half the prizes, led to Shklyarskiy’s format being adopted nationwide and maintained into the 1960s. By then some math circles focused on problem solving, and others on discussing a weekly topic (these were called “mathematical etudes,” with an attractive nod to music).
The body of mathematics may be universal, but it wears the clothes of a time and place. In these Russian math circles students always called the teacher by his first name rather than his patronymic: rare in Russia for someone addressing an older person, and the teacher always called students by their last names, as if they were already mathematicians. At the first meeting of a math circle, the various teachers would give short advertisements for their courses, promising everything except success in the Olympiad. “In our section,” one would say, “we will do everything that previous teachers have already talked about, but also … ” Section size varied from three to more than a hundred, and if a section had too many students in it, the presentation was simply made more difficult, encouraging students to leave. Why does this alternating attraction and repulsion remind one so of Dostoyevsky?
The Moscow Math Olympiads attracted thousands of students from all over the Soviet Union. Its two rounds extended over four spring Sundays: in round one the students spent four or five hours attacking four to six problems; the next Sunday there was a public discussion of solutions and common mistakes; then the successful candidates—the 30 to 50 percent who had solved at least two of the problems—went on to round two, where they were faced with another four to six problems, but of much greater difficulty. On the final Sunday, the winners were announced and presented with their prizes: not cups or plaques, but autographed copies of small math pamphlets.
In the spirit established by Kolmogorov, a student who hadn’t solved a single problem might yet be a winner, if his work showed genuinely mathematical originality, thoughtfulness, and rigor. In 1945, for example, solving a relatively simple geometry problem depended on recognizing that a line which doesn’t meet any of a triangle’s vertices cannot intersect all three of its sides. The Olympiad committee thought of this as something that students could take for granted. But the fifteen-year-old Roland Dobrushin got to this point, then wrote: “I have spent a long time trying to prove that a straight line cannot meet all three sides of a triangle at internal points, but I couldn’t do it, since I realize to my horror that I do not know what a straight line is!” For this confession, he was awarded first prize, since this is indeed left as a mystery in Euclid’s geometry.
All those students from every part of the vast Soviet Union! Let’s lift the veil of anonymity and look at one. In 1963 Tatiana Shubin was a thirteen-year-old in Alma Ata, in Siberia, when a math circle was set up at her school. There was already a dance circle, an art circle, a theater circle, and so on. The math circle lasted for only a few sessions, because all the students did was to read short biographies of prominent mathematicians. The next year Tatiana saw, in the Komsomol’skaya Pravda, an announcement about the first stage of the all-Siberian Math Olympiad with a collection of problems, arranged according to grade levels; readers were encouraged to send in their solutions. Tatiana tried them and was invited to the next level of the competition, which was to be held in her hometown. The participants were seated in different classrooms, according to grade level, and were given several problems to solve in three or four hours—with the inevitable proctors vigilantly patroling the aisles. The top scorers from this competition were invited to a summer school in Academgorodok, a town newly built in a pristine forest, across the Ob’ river from the great city of Novosibirsk.
The summer school lasted a few weeks. Tatiana went to daily lectures—some given to the entire camp by leading research scientists, some more specialized and held in smaller auditoriums by younger and less famous people. “I was really taken,” she said, “by a series of two or three lectures in which relativity theory was explained—mostly by means of diagrams. The clarity of it was simply breathtaking.” There were also problem-solving sessions led by graduate students, but no activities were compulsory, and many students preferred simply to roam the streets of this town of science and scientists, sheltered by the magnificent Siberian forest.
At the end of the summer there was a final competition, and those who succeeded were admitted to the Physics and Mathematics Boarding School, which, because it was a department of the Novosibirsk State University, escaped the scrutiny of the Ministry for the People’s Education and the severe limitations it placed on education elsewhere. Tatiana started studying there in the golden autumn of 1965.
It was all very different from a standard school. There were big lectures, given by university professors, and small problem-solving sessions led by graduate students. In other schools, every student in the nation was taking the same set of courses and using the same textbooks. Here the state curriculum was ignored. In math courses everything was proved rigorously, and students were expected not only to remember the theorems and their proofs, but also to devise their own proofs in the problem-solving sessions. Their homework exercises were like Olympiad problems, and required a lot of hard thinking. For physics, the students took part in the real work going on in the labs, in a building with an impressive brass plate at the door, inscribed “The Siberian Branch of the Academy of Sciences.”
Tatiana could have stayed there until she graduated, but after a year her parents wanted her to come back home. That time at the Physics and Mathematics Boarding School—its freedom, its intensity, its remoteness, farther than Academgorodok, in a world long since disappeared—shimmers still under the dark pines.
