This morning in calculus, a student — in fact, a very good student who is the best in the class — was trying to take the derivative of 1/(fourth root of t3). She has the correct answer but isn’t sure that it is correct; she asks me “Is this right?” I ask her to explain her reasoning. She gives me a correct, well-articulated justification of each step (change to t^(-3/4) and then use the Power Rule). But… she still doesn’t believe her answer. Having a solid, verifiable chain of reasoning connecting the statement with the conclusion isn’t enough. She isn’t satisfied until I say, “Yes, that’s right”.
I’ve often wondered what would happen if I taught a calculus class (say) by teaching total — but consistent — falsehood all the way through the semester. For example, I could teach students that the derivative of f(x)g(x) is f’(x)g’(x); then when I grade their homework, I’d be sure to give full credit for students who use my “Product Rule” correctly and then go on to use the “Product Rule” in applications and even integration. My guess is that a few people would start to wonder once they get, for example, negative derivative values for a function that is obviously increasing. But I also suspect that most students would simply believe whatever gets graded highest.
Then, at the end of the semester, I would reveal the ruse to students: “Guess what? I’ve been teaching you complete lies all semester and nobody ever caught it.”
To my mind, there’s nothing sadder, or more dangerous to the development of students’ minds, than when a student simply must have an authority figure sign off on their work in order for it to have any validity in their eyes. In fact, students in that situation are equating my authority — or the authority of the back of the book, etc. — with validity. Never mind the fact that backs of books, and professors, are often wrong — and never mind the fact that most problems that students will go on to encounter are not cut-and-dry.
To many students, mathematics is not something whose validity is something intrinsic; the validity is extrinsic, coming from without in the form of professors and backs of books. Part of this perception comes from the fact that more students than ever are badly prepared for college in general and college mathematics in particular, and their self-confidence is basically nil. But part of it must also come from the perception, drilled into them prior to college, that things are not true or false on their own, but rather because somebody says so.
I’m coming to realize that a huge part of my job is to kill the dependency of students upon authority before that dependency kills them first. Not that authority is bad; but it should lend validity to something, not determine validity. And as long as students need external validation to tell them “the right answer”, they are not fully educated.