The Chinese and the American Fox: An Invitation to Dialogue Mark Saul

The fox knows many things, but the diffused, moving on many levels, seiz- hedgehog knows one big thing. ing upon the essence of a vast variety of experiences and object for what they –Archilochus are in themselves, without, consciously The title refers originally to a fragment by an ob- or unconsciously, seeking to fit them scure ancient Greek poet, made famous in an essay into, or exclude them from, any one by . In this essay, Berlin distinguishes unchanging, all-embracing, sometimes two types of thinkers. Briefly, “” are self-contradictory and incomplete, at people who concentrate their efforts on a single times fanatical, unitary inner vision. set of ideas, while “foxes” are those who bring a The first kind of intellectual and artistic variety of ideas, from widely different areas, to personality belongs to the hedgehogs, bear on their work. the second to the foxes…we may, with- In Berlin’s own words: out too much fear of contradiction, say that, in this sense, Dante belongs to …the words can be made to yield a the first category, Shakespeare to the sense in which they mark one of the second; , Lucretius, Pascal, Hegel, deepest differences which divide writ- Dostoevsky, Nietzsche, Ibsen, Proust, ers and thinkers, and, it may be, human are, in varying degrees hedgehogs; beings in general. For there exists a Herodotus, , Montaigne, Eras- great chasm between those, on one side, mus, Moliere, Goethe, Pushkin, Balzac, who relate everything to a single central Joyce are foxes.1 vision, one system, less or more coher- ent, or articulate, in terms of which they While Berlin’s immediate purpose was literary understand, think and feel—a single, criticism—he used it to examine ’s universal, organising principle in terms ideas about history—his metaphor has influenced writers in the history of ideas in a wide range of of which alone all that they are and say areas. has significance—and, on the other This note applies Berlin’s idea to the situation side, those who pursue many ends, described by Liping Ma2 with regard to American- often unrelated and even contradic- style and Chinese-style elementary mathematics tory, connected, if at all, only in some curriculum documents. Ma argues that the very de facto way, for some psychological structure of American curricula, as laid out in or physiological cause, related by no these documents, leads to flaws in the system of moral or aesthetic principle. These last education. But, seen through the lens of Berlin, one lead lives, perform acts and entertain ideas that are centrifugal rather than centripetal; their thought is scattered or 1Isaiah Berlin, The Hedgehog and the Fox. An Essay on Tolstoy’s View of History. Elephant Paperbacks, Ivan R. Dee, Publisher, Chicago, 1993. Originally published 1953 Mark Saul is director of The Center for Mathematical by George Weidenfeld and Nicolson, Ltd. No place of pub- Talent Program at Courant Institute of Mathematical lication listed, pages 3–4. Sciences, New York University. His email address is 2Liping Ma, “A critique of the structure of U.S. elementary [email protected]. school mathematics”, Notices of the American Mathemati- DOI: http://dx.doi.org/10.1090/noti1108 cal Society, Vol. 60, No. 10, Nov. 2013, pp. 1282–1296.

504 Notices of the AMS Volume 61, Number 5 can say that there are corresponding flaws in the The Structure of American Curricula: Where Chinese system. And, I argue, a synthesis of the Is the Center? positive aspects of both will achieve more than Let us set aside these comments for a moment, either of us has achieved separately. and accept the notion of coherence as central to For what follows, it is important to note that Ber- the discussion of a curriculum. It is my view that lin—and most other writers—do not see any value Ma’s note makes the hedgehog-like assumption judgment in the division into hedgehogs and foxes. that arithmetic is central to elementary math- Berlin himself (above) names important hedgehogs ematics, then demonstrates the coherence of the and foxes, and it would be difficult to say that one Chinese curriculum she discusses. But if another set of these thinkers is in any way better or more assumption is made about the central concept for important than the other. The same list shows that a curriculum, then the coherence may vanish, or Berlin’s classification does not correlate with the may reduce to a foolish consistency. I will argue field in which the thinker is working. Any field can below that Ma’s comments flirt with this danger. profit from either sort of contribution. To illustrate why I question the assumption of From Berlin’s point of view, we can characterize the centrality of arithmetic in elementary educa- the Chinese curricula described by Ma as largely tion, let us consider some third-grade students. hedgehog-like, while American thinking is much (These sketches are in fact drawn from life.) Stu- more fox-like. If, like Berlin, we withdraw from the dent A can add two-digit numbers in the standard notion of judgment, we can distinguish strong and way: 46+35=81. She can do this consistently, weak points in either way of thinking. And, if we accurately, and fluently, and can do the same look for ways to synthesize these viewpoints, we for three- and four-digit numbers. Student B un- may be able to strengthen both efforts. derstands what addition means, but has trouble Before commenting on this possibility, I note remembering to “carry the one”, or makes similar that the criterion of “coherence” used by Ma to errors in computation. So he may write: 46+35=71. judge curricula is a problematic one. For example, Student C has his own idiosyncratic way of add- we could construct a marvelously coherent curricu- ing two two-digit numbers (for example, he might lum centered around the goal of learning the Eng- –24 reason that 47+47 is six less than 100 because 50 lish names of rational numbers between, say, 10 and 1024. Children would be taught to recite these +50=100, and so on). Perhaps it is not as efficient numbers correctly, complete with Latin or Greek as student A’s. And perhaps it doesn’t quite gen- prefixes, and using the word “and” in the correct eralize to three-digit numbers. But it serves him places. The skill would be easily demonstrable, for many two-digit numbers, and he clearly knows both on a written test and at the dinner table. And, what he’s doing, having invented the algorithm. in the age of teraflops and nanoseconds, a case On the other hand, student A makes mistakes could be made that it is useful. But central? like adding 46 degrees in New York to 35 degrees in Less flippantly, we might construct another Chicago, getting 81 degrees, which student B does marvelously coherent curriculum around mastery not make. Or student A says that the perimeter of of a certain set of arithmetic algorithms, termed a rectangle with adjacent sides of lengths 46 and “standard”. (Whether or not this term is appropri- 35 is 81, while student B, says it is 142 (recalling ate is here irrelevant.) Students would learn to his error in the last paragraph). That is, student A multiply three-digit numbers fluently and accu- makes errors in logic while student B makes errors rately, would perform long division in an approved in computation. And student C makes errors in nei- format, and would subtract the bottom number ther, but has to re-think the problem when he must from the top, using algorithms tailor-made and add three-digit numbers with paper and pencil. honed by historical forces for computation in the The assumption we make here is that the errors base ten system, itself a product of long intellec- made by student A are much more significant than tual evolution. those made by student B or C. The empirical evi- These skills would be easily demonstrable on a dence for making this assumption, drawn from my written test, if not at the dinner table. Are they use- own classroom experience and that of colleagues, ful, in this age of ubiquitous hand-held calculators? is that the errors of student B or C are much more We can debate that question at another time. The easily remediated than those of student A. point here is that the answer to that question does It follows from this assumption that the central not depend on the coherence of the curriculum, goal of elementary mathematics education—and but on values external to the curriculum. Values perhaps of secondary mathematics education—is are cultural, not mathematical. Values, like taste an understanding of logic (not formal or symbolic or morality, are not subject to proof (the central logic, but the intuitive and essential idea of a criterion of mathematics) or refutation (the central chain of implications). The study of arithmetic criterion of science). So whether or not a curricu- algorithms, when well taught, then becomes a lum is useful is not a question answered by looking tool for the delivery of what I see as a deeper skill: at internal coherence. reasoning from one statement to another, a skill

May 2014 Notices of the AMS 505 that cannot be duplicated, or even aided, by the Likewise, the American curriculum—as de- use of a calculator. scribed by Ma—can now be seen differently. Placing implication at the center of elementary An Alternative View of American Curricula mathematics, we can read American curricula and Using the alternative assumption of the central- standards as delivering this same central concept, ity of logic in elementary curriculum, we can re- but using a variety of examples: arithmetic, geo- examine some of Ma’s points. metric, statistical, probabilistic, and so on. Viewed Most of Ma’s discussion is an examination of syl- this way, the American “strand” structure pointed labi, and standards documents, rather than about out by Ma becomes a flexible asset, and not a wild live classrooms. The hedgehog-like assumption is liability. made right at the start of the discussion, in noting Has this asset been exploited in published that older American textbooks of arithmetic took American curricula? I would argue that the answer Euclid’s Elements as the standard for a unified is “sometimes”. But I would likewise argue that the exposition of their subject, and that more recently, Chinese curriculum “sometimes” uses arithmetic Chinese texts, among others, have continued and to deliver more profound mathematical ideas. Evi- refined this tradition. dence for this is lacking in Ma’s work (she concen- But it is clear, from this very choice of mod- trates, here and elsewhere, on successful instances of Chinese teaching, and in fact those are probably els, that there is something deeper than simply the ones we can learn the most from). My own work arithmetic which is the standard. Euclid, too, was in China with gifted students indicates otherwise: in some ways a hedgehog-like thinker. He has a that they are often poised to think out of the box, wonderfully articulated paradigm of mathemati- but must be invited. That is, I have found that when cal truth, using geometry as the central topic. His Chinese students are faced with new and unusual rigorous treatment of arithmetic (including the problems, they often balk, intellectually. They try arithmetic of irrational numbers) and number to think where they have seen the problem before. theory was based more or less entirely on geomet- Sometimes they apply familiar techniques or pat- ric intuition, as was his treatise on optics that has terns of solution inappropriately. However (my 3 come down to us (but not as part of the Elements). work is almost entirely with gifted students), when His algebra remained two-or three-dimensional invited to invent their own solutions, they respond because it, too, is expressed geometrically. The quickly, and often succeed brilliantly. The point solution of equations of higher degree, and an is not that they cannot think outside the box. The efficient algebraic notation, were never developed point is that their curriculum does not invite this. by the Greeks. Rather, it cultivates a deep and reliable mastery of The fact that one can base a development of a specific set of techniques. arithmetic on the same methods Euclid used for Chinese colleagues have commented on this geometry suggests that Euclid’s basic concept is phenomenon. One said, “I don’t worry about my deeper. It consists in the idea of an axiomatic sys- [Chinese] students passing a test. But I want to give tem, or the underlying concept of implication: one them opportunities to do something new.” I have statement or set of statements implying others. If often been asked, by Chinese colleagues, “How can we take this wider view of what Euclid is about, I introduce creativity into my classroom?” This is indeed of what mathematics is about, Ma’s analysis itself a hedgehog-like question, as if there were an looks somewhat different. algorithm for inducing creativity. Colleagues I’ve The Chinese, as described by Ma, can be now worked with in China have often questioned ex- seen as using an exposition of arithmetic as an actly the highly structured curriculum that Ma de- example of an axiomatic system. On the pedagogi- scribes. Perhaps the most striking comment on this cal level, the Chinese elementary curriculum uses level is about Chinese Nobel prize winners in the statements from arithmetic to build a notion of im- sciences. A list of these will show that none of them 5 plication. A student who knows the “compensation did their work in Chinese institutions. Some might law for addition” that Ma mentions4 can reason see this as a narrow comment on graduate educa- that if 8+8=16, then 9+7 must also equal 16. That tion and research institutions, but the implication if, well taught, the Chinese curriculum delivers that many Chinese educators draw is that they not simply the facts and algorithms of arithmetic, need to look at how the twig is bent. but also ways to reason about numbers and state- I might add that I have found this phenomenon ments about numbers. Poorly taught, of course, in work with American students as well. They too, it deteriorates into recitation and memorization. 5See http://en.wikipedia.org/wiki/List_of_ Chinese_Nobel_laureates, accessed September 2013. 3 See http://www.math.cornell.edu/~web1600/ One laureate is listed as affiliated with the University of Terrell_OpticsOfEuclid.pdf (accessed September Hong Kong. All the others worked in American or Euro- 2013). pean institutions. Two winners in peace and literature 4Ma, Op. cit., page 1286. were educated in China.

506 Notices of the AMS Volume 61, Number 5 often need to be invited to think outside the box flexibility(!). As we find new ways to approach the that their curriculum has put them in. This is partly basic logical structure of mathematics, a “strands a consequence of the very nature of curriculum: a structure” will allow us to incorporate these new set of skills and concepts to be mastered. The strik- approaches easily into classroom technique. For ing aspects of my work with Chinese (and Japanese, example, the American curriculum has looked at and Taiwanese, and Malaysian) students are two: probability, statistics, and computer algorithms the occurrence of the phenomenon in very able to find ways to approach the idea of implication. students, and the alacrity with which they accept It is important to understand that we are talk- the invitation to break outside the box created by ing here about Ma’s concept of structure, and not their training. about implementation of structure. Perhaps this or Ma brings up another point in her work, one not that American curriculum has not been successful directly connected to her central thesis. We can in flexibly introducing new ideas, or has intro- look at curriculum in yet another way (examine yet duced them inaccurately. The argument against another strand!): the way in which it is perceived by “strands”, against fox-like thinking, would have the student. This aspect of curriculum is not dis- to be that it actually implies failure in introducing cussed by Ma. It is perhaps the most fox-like way of new ideas. But the argument against the Chinese looking at education, in that it has, as of now, the curriculum could equally be made that it pre- least structure. As Ma points out, the structure of vents—not just makes difficult—the introduction mathematics has been worked out over thousands of new approaches. The American structure has of years. On the other hand, we are just beginning doors, where the Chinese structure—as described to formulate and test theories of learning, either by Ma—has walls. of mathematics or of other subjects. And we have almost no theories of teaching, which cannot be Accumulation of Knowledge directly inferred either from theories of learning In fact, American curriculum writers do accumu- or from the structure of mathematics. late knowledge. We now know better than we did Ma mentions the fact that many of the American thirty years ago how to teach combinatorics in curriculum documents she reviews include strands elementary school. We have had a number of suc- that are not properly mathematics, involving at- cesses in introducing concepts from computer sci- titudes towards mathematics, “number sense”, or ence into elementary education. Is this mathemat- problem solving. Should these in fact be part of ics? Do these topics relate to the drawing of logical a curriculum document? This is a difficult judg- inferences? I would argue that the relationships are ment call. The question becomes: is teaching and certainly there, but are still being worked out. That learning mathematics informed solely, or even is, knowledge is still being accumulated. centrally, by the nature of mathematics itself? I And, I would argue, the multi-strand nature of do not wish to answer, or assume an answer to, American curriculum encourages the accumulation this question here—it is much too involved. But I of knowledge. A more open structure to curriculum would argue that a fox-like way of thinking about allows us to introduce new materials in ways that curriculum would include at least a consideration are consistent with the old. Whether a student of these other factors, even if merely to reject them draws implications from a statement in arithme- or distinguish them from mathematics “proper”. tic, in geometry, in statistics, or whether they use arithmetic equations or a computer programming The Critique language, becomes a detail of implementation, Ma’s article goes beyond a description and charac- rather than a source of confusion. terization of specific curriculum documents. She levels three charges against the “strand” structure Incoherence of the American curriculum: instability, lack of We can say the same about what Ma terms “inco- accumulation of knowledge (of teaching and learn- herence”. No matter how we structure our curricu- ing), and incoherence among concepts. In this next lum, we need teachers who are trained to see how section—the least constructive in this essay—I mathematics coheres. A narrow focus on arithme- argue that (a) many of these charges vanish if we tic may easily lead to an incoherent view among look at curriculum as logic-centered, rather than teachers. Ma herself mentions that only 10% of her arithmetic-centered, and (b) many of these same sample of Chinese teachers really had a profound negative characteristics can be found in the “core understanding of arithmetic.6 This statistic seems structured” arithmetic-centered curriculum, as Ma to indicate that concentrating on arithmetic leads describes it. We look at these charges one by one. to an incoherent view of mathematics for most of the teachers exposed to it. Instability Incoherence in a single-strand curriculum can Ma asserts that the “strands structure” leads to occur on the student level as well. We have mentioned instability in the curriculum. From the fox’s point of view, we might just as well say that it leads to 6Ma, Op. cit., page 1295.

May 2014 Notices of the AMS 507 that a well-taught arithmetic-centered curriculum strong deductive powers allow students fluency in may be capable of delivering the concepts of im- arithmetic. A faulty delivery of either style of cur- plication and logic. (Whether this transfer is easy riculum will not serve students well, and there is or difficult to implement is an empirical question, no “teacher-proof” way of designing a curriculum. not examined by Ma.) But let us take a case where And here is where the notion of a synthesis a student has successfully learned about logic from becomes useful. My feeling is that we must look at a study of arithmetic. Such a student may or may not see that its application to counting problems both types of thinkers, hedgehogs and foxes, and is part of mathematics. Or that we can harness the find ways to use both. For example, we can find same sort of logic to a study of geometry. That is, a foxes to design curricula with many strands, then narrow focus on arithmetic can create a discontinu- hedgehogs to polish each strand. A conscious ef- ity in the student’s experience, just the fault that fort to fit them together would effect a synthesis Ma attributes to a “strand” structure of curriculum. of the two views. Implementation and Synthesis Ma herself points out that many Chinese are open to working this way: they are looking at Ma’s central point, as I read it, is that a strand structure of American curriculum, whatever its American curricula to see what they might learn. strengths, encourages poor implementation. I Where Ma bemoans this phenomenon, I celebrate see an equal but opposite danger with a core- it. Reciprocally, the success of her book Know- structured curriculum. Especially in America, with ing and Teaching Elementary Mathematics is one our emphasis on testing and accountability, the indication that some of us are also open to new temptation will be strong to teach students the insights. Let us hope that this sort of synthesis can mechanics of arithmetic without its meaning. And be accomplished successfully. this is certainly possible. There is no evidence that a mastery of standard algorithms or of the notion I would like to thank Susan Addington, Judy of place value delivers to students power in logical Roitman, Douglas Clements, Yvonne Lai, and Peter deduction, any more than there is evidence that Shiue for their input into this article. Marc Yor (1949–2014) Jean-François Le Gall and Jim Pitman

Marc Yor, one of the most distinguished probabi- Université Pierre et Marie Curie, where he remained lists in the world in recent decades, died suddenly until his retirement on January 1, 2014. on January 9, 2014, near his home in St. Chéron, Marc Yor is world renowned as a prolific re- France, at the age of sixty-four. He was born on searcher in the theory of probability and sto- July 24, 1949, in Brétigny-sur-Orge, France. After chastic processes. He wrote over 400 research studying at the École normale supérieure de articles and ten research monographs. Most of Cachan, with thesis work under the supervision the research articles and several of the mono- of Pierre Priouret, he quickly became a researcher graphs were written jointly with one or more at the French Centre National de la Recherche Sci- coauthors from a list of over 100 collaborators entifique (CNRS), then in 1981 a professor at the from all over the world (see http://zbmath. org/authors/?q=ai:yor.marc) including many Jean-François Le Gall is professor of mathematics at the of the most prominent probabilists of the era. Université Paris-Sud Orsay. His email address is jean- During the 1980s and 1990s, Marc Yor largely [email protected]. took over from Paul-André Meyer the mantle of Jim Pitman is is professor of statistics at the University of responsibility for development of research in California, Berkeley. His email address is pitman@stat. probability in France. He was an influential editor berkeley.edu. of the Séminaire de Probabilités, founded by Meyer First appeared in the IMS Bulletin, February 15, 2014 in 1967, over a span of twenty five years. In this (http://bulletin.imstat.org/2014/02/marc-yor- capacity he set a new tone for the Séminaire as a 1949%E2%80%932014/). Article by Jean-François Le Gall diverse compendium of contemporary research in and Jim Pitman, with thanks to Jean Bertoin for his advice. probability, with a focus on work done in France, DOI: http://dx.doi.org/10.1090/noti1128 but also welcoming contributions from abroad.

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