Mathematics as Art: The Missing Standard
James R. Rogers
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.—Bertrand Russell
As mathematics educators, every one of us perceives, at least in some small measure, the essential beauty in mathematics that distinguishes it from all the other disciplines. Consequently, each of us bears the classic professional obligation to share with students some appreciation of this magnificent facet of mathematics’ nature. Yet, too many teachers by their words and deeds these days seem to have lost sight of this important educational goal. Of course mathematics should be valued for what it lends human experiences and how it supports human experiences, but it must also be valued as a human experience itself, worthy of at least as much honor as any other. Mathematics is the queen of thought, and her realm is one of culture’s richest domains. When we neglect to teach mathematics for mathematics’ sake as a part of the curriculum, we deny our students access to the queen’s aesthetic treasures. Moreover, when we omit appreciation of mathematics in the practiced curriculum, we relegate in student consciousness the majestic queen of thought herself to the lowly status of mere servant to other disciplines.
Ironically, what serves as today’s greatest guide lamp to mathematics education also acts as our students’ greatest blinders to mathematics itself—showing a part of its nature, but hiding a part of its nature. The authors of the NCTM Curriculum and Evaluation Standards for School Mathematics have done a good job of emphasizing the connections and patterns of mathematics. They have made sure that teachers make sure that students understand and use these connections and patterns. However, they have failed to complete their presentation properly; they have failed even to suggest the inherent “amazingness” of the connected structures or the obvious charm of the patterns. Nonetheless, when students say, “I love math,” or “Math is great,” they are expressing the pleasure taken in the very beauty of such structure and pattern.
At another level there is the joy that comes from successful exploration in mathematics and its uses. The Standards authors naturally have promoted mathematical discovery and problem-solving, but they have overlooked the exultation that these achievements generate. One of the greatest thrills of my life was the revelation that a single formula—the Universal Rate Formula, Rate X Base = Product—could be used as a reasoning guide to explain and solve a multitude of commonly-occurring applied mathematics problems. This, of course, is the family of all rate and ratio problems, whose members vary vastly in both topic and complexity and range from fundamental multiplication to differential calculus. Finding the thread of that simple formula’s applicability running through the fabric of that much utilitarian mathematics literally sent chills along my spine. All mathematicians, all mathematics teachers, and all mathematics scholars have felt the exhilaration of such discovery and problem-solving success, but the Standards is silent about the experience.
Indeed, the Standards document lacks many words whose ideas are essential to a proper education in mathematics. Generation after generation, it is the fascinating nature of things mathematical that brings many a bright mind to the discipline in the first place and then keeps that mind interested long enough for it to make contributions of worth not only to mathematics itself, but also to the other disciplines that mathematics supports. Where in the Standards is that word “fascinating” or, for that matter, “intriguing,” “lovely,” “sublime,” “splendid,” “exquisite,” or “magnificent?” Mathematics is indeed every one of these and more. How often do we in the field say to one another that an expression or a function is nice? We all understand the pleasing quality that this word implies, but descriptions like it are not found in the Standards. Furthermore, we all remember doing, or at least seeing, a proof of a substantive mathematical statement that was so simple, so powerfully direct that only the word “elegant” could adequately describe it. Where is such appreciation in the Standards?
After nearly ten years, it is time to amend or replace the Standards document. What I call for is a new separate standard—Mathematics as Art—to be added to the existing standards—Mathematics as Problem-solving, Mathematics as Communication, and Mathematics as Reasoning—and to be a complement to them. This proposal is not a call for other topics of study to be added to what may well be an already-overloaded curriculum; it is rather a call for a change in how we view whatever the content of the curriculum is at any given time.
The Standards’s current focus on utilitarian mathematics is obviously important, but it is just as obviously shortsighted. In mathematics education as in the world beyond it, ignoring art may keep from diminishing the efficiency of getting practical things done practically, but it also makes for a dull and joyless experience. On the other hand, teaching mathematics appreciation, as teaching any art appreciation, heightens student awareness, brightens student mood, and stimulates student creativity. In addition to these advantages, it has been my experience that teaching appreciation of mathematics in each of its courses improves the motivation of students to continue taking further courses in the subject.
In summary, mathematics as art is important. It derives its value from the pleasure it bears us while inspiring us to learn, grow, and create. It deserves to be a part of what we teach. The NCTM Standards document in its present form is insufficient to aid our students in their need to experience and explore the beauty in mathematics. I therefore propose developing and adopting Mathematics as Art as a formal standard in order to initiate a renaissance of mathematical aestheticism within the field of mathematics education. The primary goal is that teachers and students will see and revere, as mathematicians do, the artistry that is displayed in mathematics. The intended result is that more students will enjoy more mathematics more—aesthetic mathematics and utilitarian mathematics.
Recently, humanity worldwide paused to celebrate its proof, after nearly 360 years, of the universally understandable Last Theorem of Fermat. Despite the fact that this theorem lacks extensive applicability in the everyday world, even humanity’s most practical-minded elements seemed genuinely to appreciate Andrew Wiles’s feat. In particular, those students who had been properly educated to value such things shared a bit in Wiles’s ecstasy, while other students merely muttered, “So what?” The very fact that the world cared demonstrably about the result seems evidence enough that society in general does not consider the value of pure mathematics to be that of trivial pursuit. In short, if the world appreciates mathematics for mathematics’ sake, should not mathematics teachers, of all people, do so as well? Should not the Curriculum and Evaluation Standards for School Mathematics or its successor, the former being arguably the most influential document in modern history on learning and teaching the subject of mathematics, address this appreciation and advance its perpetuation? I say it should.
REFERENCES
National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation
Standards for School Mathematics. Reston, Va.: NCTM, 1989.
Rogers, James R. HiMAP Module 15: A Uniform Approach to Rate and Ratio Problems:
The Introduction of the Universal Rate Formula. Lexington, Mass.: COMAP, Inc., 1988.
Russell, Bertrand. “The Study of Mathematics.” In Philosophical Essays, 73-74. London,
England: Longmans, Green, and Co., 1910.
This commentary is dedicated to the honor of Dr. Ward C. Barnes for the excellence of his mathematics and of his teaching at the University of Mississippi 1959-1993.