Disputed Themes in Mathematics, 2012
Contrast (Order and Chaos), M.C. Escher, 1950
by Mircea Pitici
The world of mathematics is a dissenter’s paradise. Although mathematical reasoning binds the mind to rigor and constrains it to obey rules of inference and to accept semantic conventions shared by the community of its practitioners, the world of mathematics at large, in society and in our imagination, is replete with diversity, disagreement and discontent. Every trenchant statement pertaining to the nature of mathematics or to its social role can be reasonably qualified, nuanced, complemented, or even opposed by another compelling statement, to the extent that the ‘true’ state of affairs concerning mathematics will always be murkier — but also subtler and richer — than the initial take on it.
For instance, we hear often that mathematics is the ‘science of patterns’; indeed various kinds of patterns (numerical, geometrical, syllogistic, structural, etc.) are a prominent feature of mathematics, but that is only part of the story — since mathematicians also study the absence of patterns, phenomena of randomness and chaos. We often hear that mathematicians solve problems; no doubt many of them solve problems, but mathematicians also think up highly abstract theories that seem unrelated to our physical environment yet prove to be applicable decades or even centuries later [1].
Likewise, mathematics appears to be mainly a cognitive activity, best practiced in solitude and in our minds — but at close inspection we find important socio-cultural aspects that are inseparable from it. I can list dozens of other similar polarities related to mathematics. Mathematics is ‘easy’ and agreeable for some people, but it is ‘hard’ or even incomprehensible for others; it deals with discrete objects (numbers, points, lines, sets) but also with the notion of continuum; it deals with finitude but also with the infinite; with certainty but also with uncertainty, probability, and chance; with the most general ideas but also with particular cases… and on, and on …
One source of difficulty in sensing the interpretive side of mathematics resides in its cryptic mode of expression and transmission. The apodictic symbolism of mathematics obscures the multitude of peculiar meanings each of us assigns to mathematical results and their cognates. Despite its universal character, mathematics is not the same for all of us; to the contrary, it is different for each of us. The personal aspect holds not just for our holistic perception of mathematics but also for the details, for the mathematical ‘facts’ — and starts early on, at the time of our first encounters with mathematics. We differ not only in how much mathematics we ‘know’ (or we think we know), or how well we master certain mathematical competencies; we also differ in how we (mis)interpret, (mis)understand, and (mis)apply mathematics. We also differ in what we overlook when we refer to the mathematical competencies we claim and we also differ in how we envision the connections among those competencies.
I consider myself fortunate to have had a mathematical upbringing that led me to this syncretic, encompassing, eclectic perspective. More recently, I was lucky to find the right opportunity to illustrate it. A few years ago I started editing at Princeton University Press The Best Writing on Mathematics, an annual series that makes easily accessible interpretive writings on mathematics. Three volumes are available so far and I am working on the next. These books are non-technical, highly accessible to non-specialist readers interested in various aspects of mathematics, but at the same time they are instructive for mathematics learners and for professionals. Reading the collection of 2012 writings, I noticed several polemical themes — some old and enduring, others relatively new.
Let’s start with mathematics education. In the middle of the summer heat Andrew Hacker, a political scientist at Queens College of the City University of New York, published an incendiary opinion piece in The New York Times deploring that eight million American high school students and freshmen undergraduates are “struggling with algebra. . . [and] many are expected to fail”.
“Why do we subject American students to this ordeal?” asks Hacker and makes the case that “we shouldn’t [do it]”, not just with the algebra but also with “the usual mathematics sequence, from geometry to calculus”. Hacker contends that algebra is “the major academic reason” for the high rate of school drop outs, that it “is an onerous stumbling block for all kinds of students” and that defending it is “unsupported by research or evidence, or based on wishful logic”. His strongest charges are that “making mathematics mandatory prevents us from discovering and developing young talent … [forces] potential poets and philosophers [to] face a lofty mathematics bar,…blocks further attainment for much of our population… [and] we’re actually depleting our pool of brain power” [2].
As far as I know, Hacker’s article is the most vehement among a slew of skeptical positions toward the effectiveness of widespread mathematics instruction in American schools, colleges and universities [3]. The assertions I just quoted indicate a high level of indignation but also show unfairness to the many students (of “all kinds”) whose lot is (and has been) improved by the chance of learning mathematics beyond the basics. Hacker has some legitimate concerns about making advanced mathematics mandatory but his discontent is exaggerated. Despite his cries of danger and lost opportunities for discovering talents, exposure to mathematics “potentially” (as Hacker would say) benefits everyone.
An artist’s talent or genius will never be “impeded” by seeing, knowing, and using geometry. Well taught, mathematics remains a “great equalizer” — as it was famously called in the 1988 movie Stand and Deliver. Its role in general education indeed needs rethinking but that does not warrant the bashing it receives in Hacker’s text. To his credit, toward the end of the article Hacker pleads for practicality and sense-making in mathematics instruction and points in the right direction for continuing the discussion. He proposes teaching more of what he calls “citizen statistics … [that] would familiarize students with the kinds of numbers that describe and delineate our personal and public lives” and courses that use more elements of the history and philosophy of mathematics. On these points much more can be elaborated. Hacker’s proposals are within a long tradition that used to prevail in American mathematics instruction during the first few decades of the twentieth century and still has good virtues [4].
Let’s shift the subject slightly. Almost all people living in the developed world (and even more people living in the highly populous and rapidly developing part of the world) have gone to school, for up to sixteen years — and some for longer. Billions of us have taken countless quizzes, tests, and exams, and have been assigned a plethora of scores, marks and grades. Exams and grades are ubiquitous in these times. Even after finishing school many of us continue to be rated by superiors, peers, clients, instructors, etc., based on the implicit assumptions that assessment is relevant.
Perhaps assessment reveals mental powers of memorization, or it measures our diligence in completing certain tasks, or it indicates how we might perform on certain jobs, or it makes the competitive professional life fairer. Yet Andrew Gelman of Columbia University and Eric Loken of Penn State University write [5], with reference to grading in statistics courses but suggestive more broadly for school assessment in general, that most grading is beset by inconsistencies and flawed on multiple counts: exams are “low in both reliability and validity,” effective learning and performance cannot be measured accurately because knowledge and skills are only tested at the end of a course/job but not at the beginning, multiple biases are involved in assessment, etc. Gelman and Loken draw some implications from these observations, concluding that the contradictions they highlight in grading methods lead to ethical questions which, if not properly addressed, leave statisticians (and by implication all those who assess others) under the suspicion of acting similar to “cheeseburger-snarfing diet gurus” who do not follow the advice they give. So much for the supposed objectivity of numbers and letter grades!
We now move to another topic but remain close to the academia. It is widely accepted that among mathematicians and related professionals there are many more men than women. The explanation and eventual remedies for the under-representation of women in mathematical sciences (and by extension in quantitative sciences and engineering) are matters of controversy. I mention here three of several recent positions. Wendy Williams and Stephen Ceci of Cornell University review — and reject as unlikely or secondary — many explanations that hold currency in the debates surrounding this issue [6]; they surmise that the overwhelming factor that accounts for the big gender difference in mathematically-based professions is women’s decision to abandon research careers, often driven by the need to care for children and the family or by their preference for ‘softer’ sciences (biology, psychology).
But Theodore Hill of the Georgia Institute of Technology and Erika Rogers of the California Polytechnic note [7] that women’s choice/preference conclusion does not really clarify the matter and offer a radically different explanation; they propose that the gender gap in mathematics and ‘hard’ sciences is a consequence of gender differences in creativity, with men taking more risk, showing more curiosity, and being more playful with ideas — all important creativity factors and equally important in pursuing mathematically oriented careers. Diane Halpern of Claremont McKenna College and her colleagues offer another explanation; they hypothesize that the source of the discrepancy in gender representation in mathematically intensive professions resides in different cognitive strengths, with women generally excelling in verbal skills and event memory and men excelling, on average, in mental manipulation of objects and quantitative skills [8].
Cathy Kessel of Berkeley, with a well informed article [9] on the unreliability of the statistics that underpin most of the research and the comments on the gender gap in mathematics issue; she shows that the numbers and the proportions have changed considerably over the last three decades but few researchers take notice of the changes; she also opines that citation and editorial practices, persistent confirmation bias, and ignorance of statistical issues in measurement, compound and result in faulty research denoting — as she puts it — “qualitative illiteracy”.
Before I conclude this brief overview of disputed themes on mathematics, let me refer to two articles that focus on specific theorems.
One of the mathematical results easiest to grasp for non-mathematicians is Jordan’s Curve Theorem; it states that a planar loop that does not self-intersect divides the plane in two regions, the “interior” and the “exterior” of the loop. The theorem is given short shrift in most courses, despite its topological importance, generalizations, and not-at-all simple proof; rigor seems, in this case, superfluous. What can be more obvious, more trivial, more unexceptional than this statement, you might ask? It is evident that drawing a closed curve we inevitably separate the plane in two regions. Not so evident, say Fiona and William Ross of University of Richmond [10]. They present a brief history of the theorem, hint at some tricky cases that defy intuition and, most remarkably, illustrate the non-obvious character of the theorem with arresting drawings penned by Fiona Ross.
Finally, I mention an interesting contribution to historical aspects of a famous result in elementary geometry, Napoleon’s Theorem. “Napoleon’s Theorem” states that given any triangle — ABC on the figure above — the centers of the three non-overlapping equilateral triangles constructed on its sides also form an equilateral triangle — LMN on the figure. (Recall that an equilateral triangle is a triangle with all sides of the same length.) Starting in 1825, this result has been published in many books, for a while with no mention on the emperor’s name; but, since a first connection between the theorem and the name Napoleon appeared in 1911, the name became entrenched, with disconcerting and amusing ramifications.
In a recent article [10], Branko Grünbaum traces in charming detail the history of this misnomer that became commonplace in most recent geometry books.
These few examples of more-or-less controversial issues surrounding mathematics do not amount to a comprehensive list. You can find others in the volumes already available in The Best Writing on Mathematics series. For now, I note that to report on these disputes on mathematics I have not used formulas, mathematical symbols, or Greek letters; I just wrote about them, in plain English. Lately, the renewed appreciation for the explanatory virtues of natural language in popularizing mathematics has led to a flourishing of narrative-based works on mathematics. This trend is represented by remarkable authors, such as Ian Stewart, Steven Strogatz, Keith Devlin and many others. Not only authors who write for a wide audience use natural language to convey the richness of mathematics and the reward of learning it but also foremost mathematicians, philosophers, and artists — some of them contributors to the series I edit. I made the volumes in The Best Writing on Mathematics series purposefully eclectic, aiming to corrode the stultifying over-specialization that prevails in most fields of research related to mathematics. Each new volume is a meeting place of diverse perspectives on mathematics. With each volume I am trying to put together the book I like to read.
Notes:
[1] See the group of short pieces titled “The Unplanned Impact of Mathematics” published in Nature 475(July 14, 2011): 166-69 by Peter Rowlett (editor) and rfeprinted in The Best Writing on Mathematics 2012 edited by Mircea Pitici (Princeton, NJ: Princeton University Press, 2012).
[2] Hacker, Andrew. “Is Algebra Necessary?” The New York Times July 28, 2012.
[3] See also Underwood Dudley’s “What Is mathematics For?” in Notices of the American Mathematical Society 57.5(2010): 608-612. Reprinted in The Best Writing on Mathematics 2011, edited by Mircea Pitici at Princeton University Press.
[4] For a succinct but well informed history of ideas in American school mathematics see A History of Mathematics Education in the Twentieth Century by Angela Lynn Walmsley (Lanham, MD: University Press of America, 2007).
[5] Gelman, Andrew, and Eric Loken. “Statisticians: When We Teach, We Don’t Practice What We Preach.” Chance 25.2(2012): 24-25.
[6] Williams, Wendy, and Stephen Ceci. “When Scientists Choose Motherhood.” American Scientist 100(2012): 138-145.
[7] Hill, Theodore P., and Erika Rogers. “Gender Gaps in Science: The Creativity Factor.” Mathematical Intelligencer 34.2(2012): 19-26.
[8] Halpern, Diane F. at all. “Sex, Math, and Scientific Achievement: Why Do Men Dominate the Firelds of Science, Engineering, and Mathematics?” Scientific American Mind 27(2012): 26-33.
[8] Kessel, Cathy. “Rumors of Our Rarity are Greatly Exaggerated: Bad Statistics about Women in Science.” Journal of Humanistic Mathematics 1.2(2012) (available online).
[9] Ross, Fiona, and William T. Ross. “The Jordan Curve Theorem is Non-trivial.” Journal of Mathematics and the Arts 5.4(2011): 213-19.
[10] Grünbaum, Branko. “Is Napoleon’s Theorem Really Napoleon’s Theorem?” The American Mathematical Monthly 119.6(2012): 495-501.
About the Author:
Mircea Pitici lectures at Cornell University and is Editor of the annual series The Best Writing on Mathematics.
