You're not doing math

I had an interesting conversation with a student after class today. The essence of the conversation was that he did not receive full credit on a problem on his exam even though he arrived at the right answer. My reason for not giving him full credit was that he did not demonstrate how he arrived at his answer and that the work that he showed did not make sense.

The problem gives some relationships between various angles of a triangle and asks the students to compute the angles. This student's first step was to divide 180 by 3 because "there are three sides on a triangle." Now, it happens to turn out that one of the angles of the triangle is 60 degrees, but the fact that 180/3 = 60 has no bearing on this. Nevertheless, the student continued to insist that this makes perfect sense.

Upon further attempts to argue with me, he reached the point where he declared that I'm not giving him credit because he "did not do it [my] way." To this, my response at the time was that he did not receive credit because he did not demonstrate how his calculations give him the correct answer.

Afterwards, I decided that I had the option of taking the much more confrontational route, which is the "what makes you an expert in math?" route. Am I not the teacher in the class, the one who is being paid to determine what is and what is not math? Did I not spend 9 years of my life studying math? And yet he believes what he is doing is, in fact, math. And because he thinks it's math, I should therefore defer to his understanding of math? How absurd. Does this happen in other fields? Do physics students try to convince their physics professors that they know physics better? That doesn't make sense (but I'm perfectly willing to believe it happens).

Whether this second option is any better doesn't seem like a good debate. Obviously, it's more edifying to me to simply pull rank to shut him down, but it does little for the student besides pulling further into the grave he has dug himself by being unteachable. My basic conclusion is that I really don't care what this student thinks. He's wrong and he's unwilling to admit it. So how can you work with such a student?

I offered him a challenge: I changed the numbers slightly and challenged him to show me how dividing 180 by 3 is a correct first step towards arriving at the answer. If he can do that he will get back his two points on the test. I changed the numbers so that the answer will end up with fractions, so he's not going to accidentally stumble across the right answer. I gave him until next class to come up with something, so it should be interesting to see what he says. Part of me thinks that he's simply going to walk away from the class, which is perfectly fine by me.

There's a principle in play here that is part of Gershon Harel's formulation of learning mathematics. It is the "necessity principle" (if I'm remembering right). A mathematical concept will be accepted as true by a student until he discovers that it is necessarily false. Necessarily false means that the two ideas are inherently contradictory and so one must eliminate one of them in order to maintain a logically coherent understanding. The purpose of the challenge is to force the student into a position where he is unable to solve a problem using his random techniques. Until he realizes that his toolbox is insufficient for the task at hand, he will never look to new tools. So now I just sit back and wait to see what he comes up with.