This is regrettable. In the ancient world, the mathematical disciplines were honored among the arts essential to the education of free men, and to the road that leads to wisdom. But contemporary approaches to math make this almost impossible to see. Math textbooks and the standardized tests that guide them aim at forming excellent calculators, leaving very little time for exploring why the rules for calculation work, and why anyone would want to be calculating in the first place.
Institute President Michael Van Hecke, who teaches math to seventh and eighth graders, believes that learning to calculate well can contribute in important ways to the development of virtue. “To solve problems consistently, students have to learn to be orderly and to pay attention to detail," he explained. "They have to develop logical thought processes. When you proceed carefully, if you arrive at x=7, it's undeniable.” He insists that students write their steps with the equals signs vertically aligned, so that they can see for themselves the growing simplicity of the problem. Teachers tend to focus on “covering” what is in the book, on processing data, on getting tests passed, and so miss these opportunities for developing logic and attention.
Attention is particularly important in word problems. Students learn to recognize that details are very important. Mike facilitates this by encouraging students to use their imaginations to turn word problems into story problems. “I will ask a student on Monday, 'How was your weekend?'," he said. "When he gives me a one or two word answer, I insist on hearing more details, to tell the story of his weekend. Other students pay attention to the story. This helps them to flesh out what is presented to them in word problems. They are becoming attentive readers, thinkers, and listeners.” Mike compared the analytical abilities necessary for successfully solving word problems to Latin exercises, in which students learn to attack every word to give complete accounts of form and syntax.
Michelle Orhan teaches middle school math at St. Jerome Academy in Maryland. In a breakout session at a Catholic Classical Schools Conference, Michelle explained how she takes time out every week from calculating work to engage students in puzzles and projects that develop their sense of wonder and provide opportunities for discussion. Every Monday, she puts up a puzzle that the students find challenging but intriguing; each one must answer by Friday. Some she makes up herself; some she draws from resources such as Stella's Stunners. These get the kids thinking and talking with one another at lunch and in hallways about how they might be solved. She doesn't demand they all reach a solution, but they have to show that they tried, and explain how they tried, to tackle it.
She also sets them architectural challenges. “Here's some spaghetti and a marshmallow – build the highest tower you can which will still hold the marshmallow,” she instructs them. Students work in teams, and then compare their effort with those of other teams. Sometimes before, but always during the middle and afterwards, Michelle has them discuss what they think might work, and what they learned as they tried to carry out their ideas. Generally students discover that triangular bases work the best, and they learn to build with the marshmallow, as opposed to adding it on top at the end. They then discuss why things unfolded this way. Some students get very excited about these projects (often not those who are best at calculating), and their enthusiasm is infectious, like the team that discovered that the Eiffel Tower's square base provided a useful model.
She introduced her seventh graders to tessellations – spaces filled by the same image repeated over and over – and had them study some of M.C. Escher's work. She then had them create a Sierpinski triangle. First they had them make an equilateral triangle. They tried using protractors, but found the angles were always off a degree or two. Finally they discovered that circles can be used to make sensibly perfect equilateral triangles (just like in Euclid's first proposition). Then they made a two-story tall equilateral triangle, and filled it in with smaller and smaller triangles. She had them show it to the first and second graders and explain the process. The seventh graders were patient explainers, and the younger children picked up key ideas very quickly.
Weekly and quarterly puzzles and projects make math class a time for wonder, imagination, manipulation, and discussion. Michelle, who was not trained as a math teacher, has discovered since taking up the math classes a great passion for teaching. “I feel like I cast magic spells of enchantment on the students," she said. She believes that math teachers need to develop their own creativity by feeding their sense of wonder. She recommends using films and books to fuel the fire, such as “Flatland," "The Dot and the Line," Here's Looking at Euclid, and A Beginners Guide to Constructing the Universe. (I would add Paul Lockhart’s Measurement to that list.)
John Stebbins teaches AP Calculus at St. Augustine Academy. He wants to prepare his students to be productive members of society by helping them be successful at the collegiate level. The AP portion of the course trains students to be excellent calculators, which John thinks is good to a point. Becoming comfortable with calculating techniques allows students to understand what is going on in mathematical arguments, where precision is called for and is accessible. He laments that his graduate courses in mathematics presented theories only in abstract terms, which were too vague to give clarity and confidence. Examples illuminate concepts, and careful calculation makes the examples understandable.
But John really looks forward to May, when the course is done and he can focus on introducing his students to the marvels and beauties of higher level mathematics. The simplicity of the solution for the sine of a sum of angles is unexpected and delightful. Imaginary and complex numbers (which employ i=square root of -1) are cool. For example, all complex numbers have 2 square roots, 3 cube roots, 4 fourth roots, and so on. But what are imaginary numbers? Do they correspond to anything? Yes! Or at least they admit of intelligible geometrical figures on imaginary planes. They invite play – what happens if you raise e, the natural logarithm, to an imaginary power? Crazy, but you get amazing results if you follow the ordinary rules of calculus. They consider what it might mean. John doesn’t know himself, but the question shows his students how math can illuminate theology, where we use also words that say things that must be true, but often we don’t know what we are talking about. The universe, even the mathematical one, is vastly greater than our best minds.
The search for meaning in math is crucial. Over the centuries, math has developed because rules which make sense with ordinary things, like multiplying rabbits, get used in areas that we don’t understand, like multiplying irrationals. The rules continue to work, but what are we talking about now? “Trigonometry is introduced to students through right triangles but its rules get extended to perpendiculars in circles and angles beyond 90 degrees, beyond 180 degrees,” John said. “What does it mean then? That trigonometry is not about triangles essentially, but about cyclical motion, which is found everywhere in nature. You can even begin to see that the Pythagorean theorem is just a small part of the law of cosines. The concept is vastly bigger than you originally thought.”
John works hard to help his students do well on the AP test, but he finds it mostly a constraint. The AP is a feather in the cap for a school, but it doesn’t leave much room for marvels and beauties, like the Mandelbrot set. If he could, he would broaden the topics covered; their calculating ability would decrease, but their wonder would be stoked.
And then there’s Euclid. In the lunch line at Thomas Aquinas College recently, I met a happy freshman. She had visited the College for a few days before deciding to apply. I asked her what has been the biggest surprise for her. She responded immediately, “I love math!” That was a delightful answer, and brought back happy memories of many Academic Retreats, where humanities teachers have found that the session on Euclid was their favorite, completely contrary to their expectation. Her reason, however, was novel: “In high school math, I would always have to check the answers, because I never really knew whether I was right. With Euclid, I can see and understand the steps and know that I am right.” And that is a beautiful feeling I wish everyone had at least once in their lives.