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"Now you understand that you'll never see this problem again, don't you?" I said, after a particularly productive problem-solving session at the MLC whiteboard with a group of students.

And then the world ended.

At least, I would have gotten the same reaction from the students if it had. They were all staring at me with wild expressions of undisguised horror. It was such a sudden and strong reaction that I almost turned around just to check if the four horsemen of the apocalypse were riding through Hub Central at us.

They weren't, of course. It was just that I had effectively pinpointed something fundamentally wrong with their way of studying maths.

You see, when we were doing the problem, I noticed that they were all furioulsy writing down everything I had done, and asking me about the exact wordings. This worried me a little, but I continued working with them until the problem was done. And then they were all staring at the whiteboard with looks of intense concentration on their faces, but they were not the looks of people trying to understand what I had done. No, they were the looks of people trying to remember what I had done.

It occurred to me that they were trying to remember how to do a problem like this if they ever saw one again. I suddently had a vision of them poring over the exam solutions from past semesters trying to remember the solution to each one, only to find that many of the questions in their real exam were not at all like the ones in past exams. I couldn't knowingly set them up for that sort of failure, so I said the statement that ended their worlds.

Then I proceeded to explain how at least a quarter of all the things we lecturers ask them to do are things that they've never seen before, but we are confident they'll be able to figure it out from the information we've given them. Then I asked them what this problem could teach them about the concepts in their course, and what it could teach them about how to solve problems they've never seen before.

I think perhaps we could say this sort of thing to students a little more often when we show them the solutions to problems. Then perhaps they won't react so strongly when they realise that maths isn't just a list of different types of problems and how to solve each one!