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American Mathematical Monthly Vita Mathematica: Historical Research and Integration with Teaching.
Edited by Ronald...
Vita Mathematica: Historical Research and Integration with Teaching. Edited by Ronald Calinger
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Cilt:
104
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english
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The American Mathematical Monthly
DOI:
10.1080/00029890.1997.11990667
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May, 1997
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The American Mathematical Monthly ISSN: 00029890 (Print) 19300972 (Online) Journal homepage: http://www.tandfonline.com/loi/uamm20 Vita Mathematica: Historical Research and Integration with Teaching. Edited by Ronald Calinger Hardy Grant To cite this article: Hardy Grant (1997) Vita Mathematica: Historical Research and Integration with Teaching. Edited by Ronald Calinger, The American Mathematical Monthly, 104:5, 471478, DOI: 10.1080/00029890.1997.11990667 To link to this article: https://doi.org/10.1080/00029890.1997.11990667 Published online: 10 Apr 2018. Submit your article to this journal View related articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uamm20 money concerns, some have to do odd jobs, some have children. Of course, all this interferes with their study. It would be much better for them to have started to solve nontrivial problems years ago. This inconsistency is especially noticeable when one reads the initial Chapter Zero, intended for students of ages 1011. "The problems of this chapter have virtually no mathematical content," naively claim the authors. Actually the problems of this chapter require the most fundamental abilitythe ability for abstract thinking. This ability should by no means be taken for granted; it develops as a result of careful and wellthought schooling. Let us remember that most teachers of circles in Russia were not professional teachers. In most cases they were university students, inspired but unexperienced. Their teaching was successful due to the sound preparation provided by the national educational system. In my young years the Russian educational administration seemed very stupid to me, but now I see how efficient actually it was. Don't ask me how does this square with the tyrannical Soviet rule and the ailing Russian economy because I don't have all the answers. Some people ask whether there are competent enthusiasts in America who could and would te; ach classes of creative problem solving. The answer is "yes, of course," but this is not the right question to ask. The right question is whether the educational system can teach the basics of mathematics so that children will be able to attend such classes. REFERENCES 1. 2. Boris A. Kordemsky, The Moscow Puzzles, 359 Mathematical Recreations. Dover Publications, 1992. H. Steinhaus, Mathematical Snapshots. New York, Oxford University Press, 1969. Department of Mathematics University of the Incarnate Word 4301 Broadway San Antonio, TX 78209 toom@thecollege. iwctx.edu Vita Mathematica: Historical Research and Integration with Teaching. Edited by Ronald Calinger. Washington, D.C., Mathematical Association of America, 1996, 350 pp., $34.95. Reviewed by Hardy Grant That mathematics has an "image problem" with large segments of the general public has been obvious for a long time. The stock indictment is all too familiar, for it has echoed repeatedly in the cultural creations that reflect and shape our collective values. Since Aristophanes, mathematics and its devotees have been lampooned, gently or savagely, as narrow, austere, mechanical, passionless, excessively cerebral, cold. Examples abound. Jane Austen's Emma (1816) considers at one point that the romantic events unfolding around her must impinge on "the coldest heart and the steadiest brain," on a linguist, a grammarian, "even [!] a mathematician." The "Master Mathematician" in Oscar Wilde's The Happy Prince (1888) "frowned and looked very severe, for he did not approve of children dreaming." Have perceptions changed in our own time? A few months ago, the 1997] REVIEWS 471 parents in the enjoyable comic strip Sally Forth, weighing possible summer destinations for their 9yearold, threatened her with "Algebra Camp"("just outside Geek City") as the absolute antithesis of a place where a kid could hope to have some fun. Readers of the MONTHLY could multiply these instances at will. The problem runs very deep, and does not look like going away any time soon. Is there in fact a cure? Countless adults blame their distaste for mathematics on bad teaching, so the schools would seem the natural place to start. But what to do there? For many the best hope remains the use of history in the mathematics classroom as a "humanizing" corrective. The Danish educator Torkil Heiede goes so far as to say, in his contribution to the volume under review, that to teach mathematics "without its history" is to teach it "as if it were dead." The book's origin reflects the now widespread sharing and institutionalization of such sentiments. Many of the papers assembled here began as talks given in the summer of 1992 at either the Quadrennial Meeting of the International Study Group on the Relations between History and Pedagogy of Mathematics (" HPM"), at Toronto, or the Seventh International Congress on Mathematical Education ("ICME"), at Quebec City. According to Ronald Calinger, the volume's editor, other papers are the fruits of a call sent out by him to "historians of mathematics." The book's subtitle, "Historical Research and Integration with Teaching," captures well enough such focus and unity as this wildly diverse collection may be said to possess. Calinger identifies the primary audience as "mathematics teachers, research mathematicians, historians of mathematics, and historians of science" (p. vii). Predictably, these thirty articles vary enormously in subject matter, in level of presentation, and in putative audience. Someone, presumably the editor, has gamely tried to partition them according to theme, but the subsets' boundaries are rather slippery, and indeed I would argue that perhaps half a dozen contributions have ended up in the wrong boxes. I shall adopt a somewhat different grouping in what follows. I should add that any attempt to summarize so many papers in so brief a compass must be lamentably selective and superficial, and due apology is hereby extended both to their authors and to potential readers. A preliminary word about the book's "production values" may be in order. The index is good, and in many articles the bibliography alone is worth the price of admission. Many marvellous illustrations have been culled from inaccessible sources. The writing is mostly very competent, if occasionally clumsy and seldom inspired. Surprisingly or not, some of the authors for whom English is presumably a learned language write it better than do some presumably native users. Alas, these pages will not reassure those readers (I am one) who fear that in our time professional standards of editing and proofreading are in steep deCline. Thus (to offer a sampling that is very far from exhaustive) "data" is stubbornly treated as singular (pp. 92, 335), and "phenomena" too for good measure (p. 332); two verbs on p. 94 disagree with their respective subjects; several references of a paper to itself have been left as "p. ???";and computer gremlins have at several places been allowed to substitute commas for occurrences of "e."(One might have thought that "Dieudonn," (pp. 9, 10) would rouse the sleepiest proofreader; but apparently not.) It is only fair to add that the passages involving technical mathematics seem to have been treated with commendable care. Not much in these pages will jolt readers out of their seats, or spark revolutions. Few seeds of potential controversy are here sown. But one paper, David Rowe's fine survey of trends in the historiography of mathematics, deals at some length with old disagreements that still smolder; I shall come back to that paper, and those issues, toward the end of the review. The author here whom I should most 472 REVIEWS [May enjoy debating face to face is Ubiratan D'Ambrosio, who offers a spirited defence of an "ethnomathematics" conceived so broadly as to coincide, so far as I can see, with all of cultural anthropology ("the study of techniques developed in different cultures for explaining, understanding, and coping with their physical and sociocultural environments"). Regrettably, a definition so vague and idiosyncratic seems likely to take the paper out of the mainstream discussion of its ostensible subject. One of the book's great delights is its several ventures along relatively untrodden byways of our mathematical heritage. Complete with charming cartoons (by the author?), Beatrice Lumpkin's article cleverly interweaves the history of the rule of false position (which she traces to ancient Egypt) and the career of the outstanding mathematics educator Benjamin Banneker (d. 1806). Peggy Kidwell tells the absorbing story of the various objects, especially geometric models, that saw service as teaching aids in 19thcentury schools and universities. Kidwell links these longago educators' enthusiasm for such objects to their belief that "it was more useful to show pupils the properties of surfaces and solids using models than to offer formal proofs of those properties"a notion that has not lost its validity. Karen Dee Michalowicz rescues from obscurity Mary Everest Boole (the mountain is named for her uncle), wife of George Boole and a distinguished writer on, and teacher of, mathematics in her own right. Those who still doubt that women have "come a long way" in our subject may like to ponder a despairing remark made by Mary's father in 1842, when she was ten. If she could go to university, he said (which she could not), she "would carry everything before her ... But what could a girl do learning mathematics?" This article includes twoplus pages of short quotations from Mary Boole's writings on the art of teaching, a little treasury that I wouldn't trade for ten years' worth of "aims and objectives" from my local Ministry of Education. Vita Mathematica contains a number of papers on "straight" history of mathematics, without overt or obvious pedagogical application. Jens H0yrup traces from old Babylonia to the Renaissance the career of a single problemto find the side of a square from the sum of its perimeter and area; this, he says, turns out to belong to a nonscholarly tradition of practical geometry, whose interaction with the contemporary "literate" mathematical culture he discusses at length. Wilbur Knorr's theme is the ancient practice of the "method of indivisibles" and its influence in the age of Cavalieri. Knorr argues that an "indivisibilist heuristic" predated Archimedes, and that earlymodern versions represent "a reconstruction of the lost heuristic by geometers who were impatient over the demands of formal demonstration." These two papers are substantial monographs, with the detail and rigor of argument and documentation that we expect from their authors. Shorter and slighter, but perhaps on that account better suited to "integration with teaching," are an overview of traditional Chinese mathematics by Frank Swetz, a discussion of combinatorics and induction in medieval Hebrew and Islamic mathematics by Victor Katz, and Barnabas Hughes' description of the earliest correct algebraic solution of cubic equations, by Master Dardi of Pisa (c. 1350). All three of these essays are solid and useful. Zarko Dadic well summarizes the work of the Croatian scientist Marin Getaldic (15681626). Getaldic wrote inter alia a treatise De resolutione et compositione mathematicawhich is to say, the use in mathematics of the method of "analysis and synthesis." Some day that volume, and DadiC's paper here, will be sought out by the author of one of the great unwritten books of the western intellectual tradition, a history of analysis and synthesis in their two millennia of life, not just in mathematics but in philosophy and earlymodern science as well. 1997] REVIEWS 473 The other purely historical essays in Vita Mathematica take the reader much nearer to our own time. Roger Cooke gives an admirable account of Sofya Kovalevskaya's "mathematical legacy," in particular her "discovery of a physical configuration for which the equations of motion of a rigid body about a fixed point under the influence of gravity can be integrated in closed analytic form." William Aspray, Andrew Goldstein, and Bernard Williams entertainingly recount the rise of theoretical computer science and engineering, in both their intellectual and social aspects but with emphasis on the latter, specifically the supporting role of the National Science Foundation. Other papers treat of the history of mathematics education. Hans Niels Jahnke, in one of the book's best essays, relates the complicated 19thcentury history of "algebraic analysis"the most familiar manifestation is probably Lagrange's recasting of the calculus in terms of power seriesand explains how its vogue had the remarkable effect of excluding infinitesimal methods from the curricula of German gymnasia for several decades. Jahnke argues that pedagogically this was no bad thing, for it entailed a stress on "concrete" problemsolving as opposed to blind application of algorithms. Susann Hensel's topic is the mathematical education of engineers in late19thcentury Germany. As she says, some of the questions then under debate are hardy perennials: how "pure" and rigorous should the mathematics in "service" courses be, and should they be taught only by mathematicians? Ronald Calinger focusses on the early years (1861 ff.) of the famous mathematics seminar at Berlin; the midwives at its birth were Kummer and Weierstrass. Calinger weaves into his tale a good deal of the mathematics of these two giants and of their contemporaries. The papers in yet another group get closer to the volume's professed goal of integrating history with teaching. Fred Rickey makes the case in general terms, describes particular techniques in his own practice, and calls attention to such resources as the HPM newsletter and the MAA's thriving email discussion group on the history of mathematics. (Rickey is too modest to mention here that the latter is his own creation.) Torkil Heiede relays the welcome news that the Danish government has mandated the use of the history of mathematics in "upper grades." One longs to know more: who had the ears of the bureaucrats, and what arguments were used that might be transplanted to less progressive jurisdictions? A repeated theme in these pages is the value of history in promoting a view of mathematics as a process, rather than merely a product, of human striving and discovery. This contrast explicitly guides Evelyn Barbin's persuasive advocacy, with historical examples, of a problemsoriented approach to teaching. The benefits, she says, include a better understanding and tolerance of pupils' errors. Similarly Manfred Kronfeller, after providing a short history of the function concept, draws pedagogical lessons that include the realization that student errors may actually mimic those of the great masters. Indeed, Kronfeller says, in teaching any set of ideas one should follow as much as possible their historical evolution, for one can "assume" that aspects elucidated earlier in history should be easier for pupils to grasp. Some empirical evidence in the same direction is offered by Peter Bero, who conducted in Slovakia a survey of young people's perceptions of the continuum. (The nodding proofreaders here provide the funniest of the book's innumerable typos: the age range of Bero's respondents is said (p. 304) to be "1 to 18." So what does the playpen set make of Zeno?) Bero reports that these elementaryschool and gymnasium students conceive the continuum in ways "often similar to those displayed by ancient Greek mathematicians." One thinks of the biologists' catchy 474 REVIEWS [May old saying that "ontogeny recapitulates phylogeny"the individual's development retraces the evolution of her species. Other authors seek to base teaching on the direct use or creative reworking of primary sources. Richard Laubenbacher and David Pengelley describe an upperlevel honors course that presents students with "mathematical masterpieces," ranging in time from Archimedes to John Conway, and asks them to function as "critics" in the sense in which this term is used in the arts. Israel Kleiner outlines a course, for teachers, built around the use of quotations about mathematics, supplemented by a "very concise" chronology. Very instructive here are a number of pairs of mutually opposing quotations, which give a vivid sense of the complexity of issues and of their inherent drama. This potential for drama is taken to its natural conclusion by Gavin Hitchcock, who has written stageable dialogue in which figures from the history of mathematics debate their respective stances. In the first of these playlets Simon Stevin touts irrational numbers against the qualms of Michael Stifel; in the second, set in 1827, George Peacock and Augustus de Morgan try to make William Frend (himself a mathematician) accept multiple and negative roots of equations. Given the right actors and audience, these exchanges could "come off the page" with real theatrical effectiveness. Several authors present case studies that extract pedagogical morals from specific historical episodes. John Fauvel's point of departure is a wonderfully curious "tree" diagram, from a book of 1808, which shows graphically the various factors that led to the abolition of the slave trade. Fauvel's agenda here is threefold. He argues that (i) "graphical representation and modelling" have been neglected by historians of mathematics, (ii) they have also been "devalued" by teachers, in comparison with "prose text or algebraic symbolism," and (iii) they can and should be used to "empower" students, that is, to encourage in students the "knowledge and belief that they can use and create mathematics to influence their way in the world." Marie Fran~oise Jozeau and Michele Gregoire describe the meridian measurements in France (179299) that led to the first definition of the meter, and tell also (a bit cursorily, to my regret) of an imaginative reenactment, near Paris, of part of that labor by a group of modern high school students. Jim Tattersall sketches the history of attempts to estimate the total number of people who ever lived, and proposes a classroom exercise to the same end. This piece revives an ancient wheeze which is the best joke in a book not brimming with humor. A class of students was asked how they knew they were going to die, and one replied brightly that "it was because so far most people have." Several other papers (in addition to Tattersall's) draw their case studies from the history of the calculus. Judy Grabiner summarizes the sharply contrasting approaches taken to the calculus by Maclaurin (geometric) and by Lagrange (algebraic), respectively. That is a familiar story, but Grabiner goes on to trace the difference to differing cultural influences, and sets out the lesson for teachers: the tendency of students to adopt diverse approaches to mathematical problems is both natural and legitimate, especially where "nontraditional" backgrounds are involved. Martin Flashman suggests that instructors give an extra historical dimension to the textbook account of the Fundamental Theorem of the Calculus by presenting in detail the wholly geometric version proved by Isaac Barrow just before the watershed work of Newton and Leibniz. ManKeung Siu considers "integration in finite terms" from its first rigorous treatment by Liouville (1830s) to the modern classroom. He gives splendid expositions both of the mathematics itself which is far from elementaryand of its history, and in both aspects he 1997] REVIEWS 475 maintains an exemplary balance between superficiality on the one hand and excessive detail on the other. And Siu is equally good on the pedagogical issues. He has a fine sense of the kind of question that better students are likely to raise, and he specifically addresses the impact of computers on the teaching of his chosen topic. For my money his paper is another of the volume's highlights. I come finally to the excellent essay, already cited above, in which David Rowe depicts "new trends and old images" in the historiography of mathematics. In particular he discusses the notorious debate that climaxed in the late 70s over the "geometrical algebra" commonly credited to the Greeks. Many results stated and proved in geometrical language by Euclid can easily be "translated" into elementary algebraic identities; but does such restatement distort the Greeks' own point of view? So argued the historian Sabetai Unguru, whose views then evoked rebuttals from mathematicians of the stature of Andre Weil, Hans Freudenthal, and B. L. van der Waerden. Rowe expounds this particular dispute with applaudable fairness, and then sets it in a larger context, centering his discussion around some provocative views voiced by the same Andre Weil in a famous lecture in 1978. Weil declared on that occasion that for mathematicians the "first use" of the subject's history is "to put or keep before our eyes 'illustrious examples' of firstrate mathematical work." Therefore "the craft of mathematical history can best be practiced by those of us who are or have been active mathematicians or at least who are in contact with active mathematicians." Moreover, Weil urged, it is appropriate, indeed necessary, to interpret the mathematical ideas of past cultures in modern terms. For example, "it is impossible for us to analyze properly the contents of Books V and VII of Euclid [on ratios of magnitudes and on number theory, respectively] without the concept of group and even that of groups of operators, since the ratios of magnitudes are treated as a multiplicative group operating on the additive group of the magnitudes themselves." The tendency exhibited in this last quotation is probably more seductive in mathematics than in any other subject. For in mathematics, as the same quotation shows, many ancient ideas can be fitted with perilous ease into modern abstract frameworks to which they are wholly foreign in conception and in spirit. Perhaps then it was no great surprise to read recently a hint that mathematics may have acquired among outsiders a certain reputation for this kind of thing. In the obituary of Joseph Needham that she wrote for the June 1996 issue of Isis, Francesca Bray tossed out the aside that "projecting modern meanings onto ancient terms [is] a practice that often passes unnoticed, I am told, in the history of Western mathematics." "Often," indeed; for the temptation is chronic. But not, in fact, "unnoticed." It was just this tendency that Sabetai Unguru protested back in the 70s, and now David Rowe takes up the cause. Still addressing himself specifically to the views of Andre Weil, quoted above, Rowe explains with great civility what "disturbs" historians about viewing ancient ideas through modern lenses. He associates Weil's vision of history with a Platonist philosophy, which makes mathematical truths independent of time and of historical milieu. As he says, this orientation disposes one to "present the development of mathematical ideas as a steadily unfolding search for Platonic truths that transcend the particular cultural contexts in which these ideas arose" (p. 10). But he raises at once, in the completion of the sentence just quoted, the obvious objection, that one can write history in this way "only by discounting the rich variety of meanings that accompanied" those ideas in their actual concrete settings. With this objection I agree entirelybut I would push the argument a bit further. There is a facet of the issue that most discussions mention scarcely if at all. 476 REVIEWS [May I suppose that those who credit the ancients with this or that modern idea think that the ascription amounts to generous praise. For all their limitations, the argument goes, these forerunners had the Right Stuff, they were spiritually so close to us that they seem (in Littlewood's memorable phrase) like "fellows of another college"; and wouldn't they rejoice in our good opinion of them? Well, maybe so. It might just be, however, that the mathematicians of bygone ages would also, or even rather, wish to be studied and judged for what they were, in their own proud individuality and distinctiveness. In the writing of history within the traditional academic discipline of that name, the goal of an empathetic portrayal of the past on its own terms is now a commonplace. This is not at all to say, of course, that anticipations of, influences on, the present are not worth notice. But the aim is the polar opposite of the deliberate backward projection of modern modes of thought and feeling. Rather, the quest is for what William Blake would call the "minute particulars," the specific and defining uniqueness, of each time and place and people of the past. Historiography so conceived has at its very heart a moral imperative, a scrupulous respect for, and honoring of, those who went before. At its highest, it isdare one say?like an act of love, which would know and possess yet ultimately leave singular and autonomous. That makes its practice at once a noble and a formidably demanding enterprise. George Steiner once wrote, in a different but parallel context, that "there can be no other thanks [to the creators of our cultural heritage] than extreme precision, than the patient, provisional, always inadequate attempt to get each case right." That challenge is present no less in the historiography of mathematics than elsewhere, despite the (for some) supposedly eternal character of the truths unveiled. Indeed, in the last analysis this timelessness is a red herring, for the most passionate Platonist must concede that mathematical discovery is the work of individual minds in historically conditioned settings. So who should write the history of that discovery? Where the object of study is the technical progress made in the 20th century, Andre Weil's claim that only mathematicians need apply seems incontestable. In most fields of current mathematics, the dynamic of the development is so overwhelmingly "internal," and the barriers of specialist knowledge so forbidding, that the prospective historian cannot hope to succeed without something very close to the active researcher's intimate grasp of, and "feel" for, the subject. The scarcity of people qualified in this sense must be the prime reason for the deplorable underrepresentation of the 20th century in the journals and conference programs devoted to serious history of mathematics. But if the historiography of 20thcentury mathematics must be left to the practitioners, the pasteven the relatively recently pastis a different case. Roger Cooke says acutely in this volume (p. 177) that, thanks above all to the massive modern trend toward abstraction and generalization, To reconstruct precisely the present state of any nineteenthcentury mathematical topic is in a sense impossible. No mathematical problem is understood exactly as it was understood . . . a century ago. Then how much wider still the gulf between us and the still more distant past! The farther back in time, the more foreign and elusive must be the "mindset" that the historian would seek to know. Moreover, and crucially, increasing remoteness from the present increases also the role of "external" factors in mathematical activityand so diminishes, in proportion, the place of purely mathematical skills 1997] REVIEWS 477 and knowledge in the aspiring historian's stock in trade. Where ancient mathematics is in question, the historian needs little if any technical grasp beyond the very elementary levels (quadratic equations or whatever) achieved by the civilization under her gaze. The resources that must sustain her are in other directions entirely. The importance of the relevant linguistic competence is obvious. The whole of the surrounding cultural and social matrix is germane and must be mastered. Perhaps the insights of modern anthropology can be brought to bear, as by Geoffrey Lloyd in his superb studies of ancient Greek science. Above all, the successful explorer of mathematics' distant past must bring the precious gifts of imagination and of empathy that alone give any hope of access to alien minds. In this terribly difficult undertaking the research mathematician as such has absolutely no privileged status, no claim whatever to special authority. The good things in the book under review, and there are many, make their own valuable contributions to the history of mathematics and to the creative use of that history in our classrooms. Thus they may serve also toward fulfilling the hope, articulated here by Fred Rickey, of educating "a general population with a much better feel for what mathematicians do and why it is important." In that urgent task Vita Mathematica will not set the world on fire, but it should light some candles; and every bit helps. 539 Highland Avenue Ottawa, Ontario K2A 218 Canada hgrant@freenet.carleton.ca Contributed by Russ Hood, Rio Linda, CA 478 REVIEWS [May