The Hard Climb toward Jaw-dropping, Mind-bending . . . Equations!

by Gerald Proietti, Ph.D., teacher of Geometry, Latin, and Greek

“No forced study abides in the soul.” Plato’s Socrates used these words in presenting a cardinal principle of education: that it must develop in a spirit of freedom, not compulsion. At the same time, Socrates emphasized that a rigorous training of the mind takes a great deal of work on the part of the student; so a strong education also requires that students develop a “love of labor” in their studies—an appetite for hard work when the learning becomes difficult.

Cultivating this work-appetite is one of the things that makes teaching an especially challenging art. Besides diverse forms of small encouragements, it requires careful pacing so that the work leads to discoveries that would not have been possible without that work. As often as possible, the joy and the ongoing radiance of those discoveries should be the primary reward for all the struggle, and the work should feel invigorating. The study of mathematics provides beautiful examples of how the long path of striving can be punctuated by very distinct insights, including surprising moments of perspective-altering insights. In working with students in geometry, one often gets to see eyes pop, jaws drop, and other signs of minds orienting themselves to new dimensions of thinking.

Plato suggested that intuitions about equality or sameness lie at the root of our powers of reasoning; but it was writers like Euclid, in the Elements of Geometry—still today the foundation of our understanding of plane geometry—who documented how deeply the intuition of “equal” versus “not-equal” is fundamental to our mathematics, and how our thinking can build on this intuition. Mathematics begins with the simplest kinds of equalities and progresses to ever more complex equations. Equations reveal definite relationships between things that we may previously have thought to be unrelated or even incommensurable.

In his most famous allegory, Plato’s Socrates speaks of a way “upward” from the deep cave of customary perception into the daylight of real understanding. He proceeds to outline an education that uses mathematical training as a preparation for the later challenge of rigorous thinking about principles of ethics, psychology, and the dynamics of human communities.

Abraham Lincoln—who studied Euclid—in a speech to farmers at the Wisconsin State Fair (1859), related “thorough work” in farming to America’s potential to remain a nation of free-spirited, self-governing citizens. He spoke of education itself as “cultivated thought” and urged that it “can best be combined with agricultural labor, or any labor, on the principle of thorough work . . . [whereas] careless, half performed, slovenly work, makes no place for such combination.”

At LCA, we study Euclidean geometry, including the thorough, logical reasoning; in all of our studies the faculty strive to cultivate this love of thoroughgoing work; and this, too, is a labor of love.