What Teachers Know

Here are two parallel tales about teaching. Mathematics and flying share the tradition of using teachers who have a "knack" for the subject, which is great for students who have the knack.

It as around 1990. I was teaching at a small college in New York State. The Space Shuttle was flying a high-inclination orbit, so I did some calculations and determined that it would be visible overhead. Cool, I thought, and made a flier that explained the trigonometry. I invited people to join me to watch. The problem was that it was mid-Winter with temperatures around 10F/-10C, and the pass would be at 2am. Only one student joined me. It turned overcast and we saw nothing, but it was worth the try. He went on to get a PhD in Mathematics.

I offer to take all of my instrument students on a long cross-country, the kind involving an overnight and some real weather and real terrain. Most decline; in fact, only one has really gone through with it. He did a great job. The best part was after he botched the ILS approach into Pendleton, Oregon. We landed for a snack and a discussion, and he shook it right off and flew a great ILS into Portland. He went on to become a captain at a regional airline.

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What do teachers know? More and more I think that teachers know nothing, not in the sense that we don't know our subject, but in the sense that we don't know our students' minds. The point of the two stories is that I really managed to get into the heads of two students, but they were like me as far as goals and dedication. Not everyone is like me, or even like you, which is how it should be. But how can we teach people who see things so differently? A private pilot student can become a good pilot even though he may never get an instrument rating, probably won't become a commercial pilot, and will almost definitely never get an ATP. A calculus student can become a good scientist or engineer or economist, even though he may never take any more mathematics, probably won't become a math major, and will almost definitely never get a PhD. Everyone has unique tastes and talents and desires and backgrounds. So by definition the teacher's goals and motivations are, or were, different from the students. What worked for you may not work for them.

Some of you may disagree, but there is an art to algebra. Algebra is actually the simplest thing in the world, much simpler than baking a cake (which I am doing as I write). There are four operations, a fixed rule for deciding in which order to do them, and an associative law that guarantees that everything is unambiguous. Algebra is simpler than music, which has twelve tones, not just four, and all of the complications of rhythm and dynamics and feel. But both simple systems produce an endless range of expression, which is part of what makes them art. There is also art in knowing when to factor, when to simplify, and when to just leave it alone. There is art is in seeing that "3+0" carries more information that just "3", even though they are equal: the former fits into a pattern, while the latter is just a number.

But algebra is different than flying, right? Algebra is abstract, and flying is visceral. That is of no consequence: what matters is that in each case the operation is easy, so learning means learning to perceive, learning where to look and when to look and how far out to focus and to what to expect when you get there.

The teacher's disadvantage is in never seeing what the student sees. There are exceptions, like the two students above, but they began with goals like mine. They're easy to teach. The work and the art are in teaching the student who is completely unlike you by learning and speaking the student's language.

Here's an example. In a Cessna 172, the nose pitches up when flaps are added. "Don't let the nose come up," I say as the student reaches for the flap lever, "Hold this attitude." I watch the windshield view change from grass to clouds, but the student cannot perceive it until the airspeed decreases, which is already too late. How could I have said it differently? Should I insist on rote ("When you lower the flaps trim nose down one turn of the wheel.") That may be enough! It might not have been enough for you, but it may be enough for this student, for now. The commercial student needs to understand the pitch change and explain its aerodynamics; the instrument student needs to apply this knowledge to keep the proper pitch attitude during configuration changes; the ATP candidate needs to correlate this knowledge with other effects of control manipulation.

You need to need to know more than the student; that's why you became an instructor.

I'm afraid that we end up teaching people to do all kinds of things the same way that we teach children to talk: we allow them to hang around and listen, and as they try we offer critiques. It works pretty well for talking, but while a Chomskyite might claim that we are pre-wired to learn talking, and some people appear to be pre-wired for mathematics, nobody claims that we are pre-wired to learn flying. While my twins might have learned to talk listening to me and their mother converse over dinner, my twin-engine students are less likely to learn from watching me and my wife try to synchronize our engines. (If you know what I mean...)

Knowing that just watching isn't enough, we throw stuff at our students. "Here's the theory," we say, giving one each more or less according to individual taste. "Here's a drawing of what it looks like," although we only produce a cartoon. "Here's a silly mnemonic," we say, reciting "True Virgins Make Dull Company." We make up colorful reminders: "Tail in the weeds!" I say before a short field takeoff, meaning to use the full length of the runway.

(In calculus class all these efforts go for naught when the student can only think about how attractive the girl next to him is.)

The airline-oriented flight schools and the military count on their students working hard to make up for any deficiencies in the teaching, but that model won't work in general aviation. Besides, it's inappropriate. So as teachers we must toe the line between discipline and laisser-faire. Most important of all, we have to offer the student what we have to offer and then listen.