The Chinese Hedgehog 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 Isaiah Berlin. In this essay, Berlin distinguishes unchanging, all-embracing, sometimes two types of thinkers. Briefly, “hedgehogs” 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; Plato, 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, Aristotle, 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 Leo Tolstoy’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.