Humanistic Mathematics

Mathematics is like a flight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process of sharpening our perception so that we discover patterns that are everywhere around… To share in the delight and the intellectual experience of mathematics — to fly where before we walked — that is the goal of a mathematical education. [2]

What is humanistic mathematics?

To some people, humanistic mathematics is simply mathematics concerned with the humanities, with literature or art or philosophy. According to this view, humanistic mathematics is a matter of content rather than of teaching style.

To others, however, humanistic mathematics means the recognition that mathematics that has a human orientation.

Humanistic mathematics is not just "friendly" math or "touchy feely" math. It is mathematics with a human face because there is no mathematical discipline without a human face. Stories of mathematicians are "color"… But stories about what historical individuals saw when they looked on the mathematical universe at historical points of time are not color. They are mathematics. No one can learn mathematics without being inculcated into this tradition. [3]

Every mathematical theory is, at heart, a story. Most of these stories have their roots in the dawn of civilization and the awakening of mathematical consciousness. Each is driven by human personalities. Each is situated in the culture of its time. Difficulties once regarded as insurmountable come, through the revolution of centuries, to be regarded as imaginary. Achievements are brought about through men and women driven by diverse impulses: curiosity, ambition, love of beauty, hunger for power, desire to be of use. Mathematics is a human endeavor.

When I went to graduate school, I encountered a mathematical world radically different from the one I'd encountered in grade school or even in college. Instead of learning from textbooks, which present theorems and techniques as easily digestible pulp, we learned from humans. We talked to each other and to our professors. We read papers — small contributions to incomplete theories — written by working mathematicians. We told stories. We drew pictures, made models, and shared parlor tricks.

Mathematics is not just a universe of mathematical objects or formalisms or constructions; at the very least it includes a point of view on that universe. This is the essence of humanistic mathematics — mathematics requires a perspective, and a human perspective is the only perspective we can get…

Pure mathematics is ultimately humanistic mathematics, one of the humanities, because it is an intellectual discipline with a human perspective and a history that matters. There is no final answer to the question: what is important in mathematics? We can only ask what is important in mathematics to human beings, with given abilities and limitations at a given point in their mathematical development. The discipline of pure mathematics is much more like geography than it is like physics. [3]

To present mathematics as a collection of computational techniques is to present it as something that is neither interesting nor useful. It can safely be said that no one has ever encountered a textbook exercise in real life. The average person does encounter situations in which mathematical reasoning would be useful, but, unless they've mastered the art of reason and developed the audacity to think creatively, all the exercises they've completed will be useless.

It is very hard to get a sense of the depth, liveliness, power, and breadth of mathematics from any ordinary experience with mathematics in school. I believe that most students who really master the subject matter… have some other channel for learning mathematics, outside the classroom…

For people who are not already tuned in to the mathematical style of discourse, it is especially important to teach in a way that is not watered down, but that begins from a person's real experience. [2]

It is my philosophy, therefore, that the best way to make mathematics accessible and useful to students is to transcend accessibility and utility by presenting math as a humanistic discipline, with a history and an orientation, while employing a variety of learning experiences.

Although, in a practical sense, there is only so much time in any degree program, and the humanistic approach may sacrifice some "height" to which the students attain, that "loss" is merely the trade-off for an increased breadth, which is at least as important, particularly for students who desire to become educators themselves. Furthermore, this broadening merely counteracts the modern emphasis on the rapid acquisition of calculus techniques to the exclusion of all else.

The acceleration of the curriculum has had its cost: there has been an accompanying trend to prune away side topics… The shape of the mathematics education of a typical student is tall and spindly. It reaches a certain height above which its base can support no more growth, and there it halts or fails. [2]

We've come to view mathematics education as an assembly line: each new topic or technique merely leads to the next topic or technique. But what is it all for? Are we actually assembling anything worth having? How many students could answer that question? How many teachers?

Let's take parabolas as an example. Parabolas are typically encountered as the graphs of equations like

y = ax² + bx + c

The typical high school or college student learns the quadratic formula and various techniques for graphing. They solve problems about farmers who want to build pens by their barns. Eventually they may model projectile motion while pretending that air resistence doesn't exist. And that's all they ever learn.

This approach, which is utilitarian rather than humanistic, thus pares the study of parabolas down to those topics that lead into calculus.

The humanistic approach, on the other hand, begins at the tomb built by Minos, the legendary king of ancient Crete. It consults the oracle at Delphi, takes counsel with the Academy of Plato, and rebuilds an altar to Apollo to end the plague at Delos. It pores over the Conics of Apollonius, where it encounters the section cut by a plane parallel to the generatrix. It follows Archimedes' computation of area through geometric series and near-invention of integral calculus in his Quadrature of the Parabola. It envisions and draws a parabola as a locus of points or an envelope of tangents. It casts whispers in a rediscovery of the reflective properties first used (legend has it) to defend the city walls during the seige of Syracuse by the Romans and employed today in telescopes and satellite dishes. It sits with Galileo under house arrest, reading his Concerning the Two New Sciences to discover how the parabola was seen to model projectile motion before coordinate geometry or calculus were invented. And, yes, it characterizes the parabola in terms of quadratic equations, deriving the quadratic formula, which appears in Descartes' Geometry but has been discovered time and again by mathematicians in different cultures down through history.

In short, the humanistic approach to parabolas at once more rigorous, broader, and more interesting than the approach that has taken over the academy.

Some instructors might object that they don't have the leisure to take such an approach, that their duty is to make sure their students know certain techniques before passing onto the next stage. They're a bit like Charlie Chaplin working the assembly line in the classic silent film Modern Times, tightening bolts on widgets without knowing what they're for, unable to stop twitching their wrenches even after the whistle has blown. It may be that they are compelled to teach in this manner, but those who have the freedom to try something different should, because students are human beings, not widgets.

To teach mathematics as a humanity means nothing less than to teach that it possesses the awesome power to influence and change our lives, and to teach that we who use and foster it, must subject it to constant study and scrutiny. [1]

One last thing remains to be said. The approach I use plays out at a small regional college divided into three campuses, each separated from the others by sixty or seventy miles of brush country. All the classes I teach are taught using some kind of distance learning technology. Technology can very quickly dehumanize the classroom if not used appropriately. Let it be said, therefore, that I am committed to using any kind of learning technology — stick drawings in dirt, chalk and dry-erase markers, books, calculators, teleconference systems, course management software — for the sole purpose of increasing and improving the human contact between the student, their peers, and myself.

References

1. Davis, Philip J. "The Humanistic Aspects of Mathematics and Their Importance." Humanistic Mathematics. Ed. Alvin M. White. Washington, D.C.: Mathematical Association of America, 1993. 9 – 10.
   
   
2. Thurston, William. "Mathematical Education." Notices of the AMS 37 (1990): 844 – 850.
   
   
3. Tymoczko, Thomas. "Humanistic and Utilitarian Aspects of Mathematics." Humanistic Mathematics. Ed. Alvin M. White. Washington, D.C.: Mathematical Association of America, 1993. 11 – 14.