What's the derivative of the arctangent?

I estimate that at least 999 of the first thousand people who read this column won't know. The ones who are curious will type "derivative arctangent" into their browsers and have the answer within 4 seconds and two clicks. (I just did it—the answer came up right away, along with an ad for a Boston hotel across the street from the one I stayed at recently—but that's the subject for another column.)

In a standard calculus class, one would undoubtedly see, and later reproduce, the derivative of the arctangent. Indeed, one's grade would be affected by one’s success or failure at reproduction, which would sit alongside the reproduction of the derivatives for many other functions. But does this reproduction demonstrate a true understanding of the mathematical concept of a derivative? Or, even more telling, does it represent the creative process through which a mathematician discovers new things about the world? I think not.

Our educational system is stultified by an answer-based curriculum. What we need in order to produce creative problem solvers for this new millennium—when answers are always two clicks away—is a process-based curriculum.

I can already see the comments being appended to this column: "In my class, we discuss IDEAS!" Perhaps … but probably ideas that your students are required to turn into answers on an exam at the end of the term.

Let me illustrate the distinction between answers and process with an example from a recent class I taught. The class was about how habitable planets form and included a segment on the remarkable recent discovery that planets are not unique to our Sun, but are commonplace throughout the Galaxy -- several hundred billion in our Milky Way alone. To discover this, one needed to know how planets move around their parent stars.

Johannes Kepler figured this out in 1610 when he posited three laws of planetary motion. In a standard lecture class (of which I have given many), I draw a diagram of a planet orbiting a star, write down the three laws, draw arrows from the diagram to the equations, and then ask "How long does it take for the Earth to go around the Sun?" Someone eventually says, tentatively, "One year?" and, by plugging in the masses and distances involved and keeping track of all the constants, I show that the “Laws” give the right answer. Students dutifully copy all this down in their notes (even though it is in the textbook and, more importantly, is two clicks away on their phones), they reproduce some variant and answer a question on the exam, and that's it. A year later, none of my students could tell you Kepler's three laws, how they were derived, or why they are interesting.

In my Quest University class this year I adopted a different approach. I divided the class into teams of five and gave them a sophisticated computer simulation of planets orbiting their parent stars. The simulation had a dozen free parameters, including the number of observations one could make and the size of the telescope used (and thus the noisiness of the data collected). I also gave each group a dozen suggestions as to how to play with the simulation. Three hours and thirty minutes into a three-hour class, with everyone still there, one group got up and wrote out Kepler’s Three Laws of Planetary Motion. They had derived them empirically, just as Kepler did, from noisy data.

I have no illusions that this class of students will remember Kepler’s Laws a year from now any better than the lecture class students did. But this group has something much more valuable. They have experienced the process of making observations of a natural phenomenon, and learned what was needed for their measurements to be useful, how to seek patterns in their data, how to cast those patterns in mathematical form, and how then to generalize using the math to predict the behavior of another physical system. They have, as one of my colleagues likes to say, been in “the mess” for quite a while, until they thought their way through to a genuine discovery.

Later in the course, there were more simulations and paper models – and fewer hints were required. My students were comfortable jumping into the mess, arguing with each other, pursuing dead ends, failing … They were engaged in process-based learning, not seeking an answer they could find on their phones. This is the kind of education our students require to support a lifetime of creativity.

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