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.
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.
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 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 tradeoff 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.
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 = ax^{2} + 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 nearinvention 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.
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 dryerase 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

Michael Ortiz, Ph.D.
Associate Professor of Mathematics
Department of Natural and Behavioral Sciences
Sul Ross State University Rio Grande College
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