Mathematics teaching all too often follows a “monkey see, monkey do” blueprint: the teacher writes down a formula, works a few examples illustrating it, and then assigns a bunch of near-identical exercises to his students, which they solve by mimicking his examples.

This form of teaching is often very popular with students. It is easy to understand why. “Monkey see, monkey do” teaching is very clear-cut and straightforward. You know exactly what to do and exactly what to expect on the test. You are spoon-fed everything you need, so you are spared the effort of doing any actual thinking of your own. The teacher spends every minute of class time explaining things that are immediately relevant to getting a good grade on the next test. And the grading itself is very objective and beyond reproach since every problem has an unequivocal right answer.

As a teacher it is tempting to give in to this crowd-pleasing form of teaching. But it is a temptation we must resist. For “monkey see, monkey do” teaching is bad teaching. The goal of teaching is independent thinkers, not trained monkeys.

Good teaching stimulates thought, raises questions, sparks curiosity. Good teaching challenges. Good teaching draws connections, asks why, sees purpose.

All of those things will be perceived as negative to students in the “monkey see, monkey do” mindset. The teacher is “not teaching,” they will say––that is to say, not spoon-feeding you everything you need to get an A. “Just tell us the answer!” Instead the teacher is asking open-ended questions and demanding explanations, which is unpleasant since it pushes you to think for yourself rather than to follow a predetermined script. In fact, not only will the teacher not tell you the answer, sometimes there isn’t even a straightforward “right answer” at all to these kinds of questions, which probably means that the teacher’s grading is arbitrary and biased (especially since “I always got good grades in math before”). Finally, much of all this messy stuff won’t even be on the test! Which proves of course that the teacher was obviously wasting your time.

“Monkey see, monkey do” teaching demands that the teacher does nothing but churn out examples of replicable model problems that can occur on the test with a 7 in place of a 3. Consequently mathematics courses vastly overestimates the importance of problem types that fit this mould, and vastly underestimates the importance of questions that don’t. The latter category includes every why-question, every question making a new connection, every question involving genuine thought, and every question that ever interested any actual mathematician or scientist. If we want to be engaged in meaningful teaching rather than populistic monkey-training we must resists the above pressures to avoid such questions.

In calculus teaching in particular, the “monkey see, monkey do” model is what leads people to assign dozens of problems on computing limits using l’Hôpital’s rule, or testing droves of artificially contrived series for convergence. These are stupid and pointless problems that are assigned for one reason and one reason only, namely that they are eminently “teachable” in the sense that an uninspired and mediocre teacher can drill it into the heads of uninspired and mediocre students, so as to create a mirage that could pass for learning on superficial examination.

Meanwhile, genuinely interesting problems, like the ones I use in my calculus textbook, are “unteachable” in this sense. They are “one-off” investigations of interesting questions, not replicable ad infinitum as drill problems. Hence each problem requires genuine reflection and creative thought——an unpopular concept with most students (and some teachers). Furthermore they are not artificially contrived to test only one specific technique; instead, since they show real mathematics in action, they draw on multiple ideas and concepts. But this of course requires students to actually understand ideas they have studied previously and be able to use them in a contextually meaningful way——again hardly a way to endear yourself to your students. Unpopular as they are, though, you can ask yourself whether these are skills worth fostering.