I once went to an Inquiry Based Learning conference and attended several lectures about how good teachers employ IBL to teach mathematics. (We learned about IBL through its antithesis….) IBL is certainly not a process of presenting algorithms. The heart of the matter is that students make building blocks out of what they already know in order to personalize new knowledge to turn into more steps.

But what it looked like to me is that you give students a bag of apples, some flour, equipment; then show them a pie (which they are NOT allowed to eat). Students are set the task of figuring out how to make their own pie. A priori, we’re assuming students are interested in pie. And I think teachers are check whether they know how not to chop off a finger with the paring knife. Or maybe losing a finger would be part of the process….

I didn’t learn to make pie that way. For years, I watched my mother make pies and helped eat them. When I was big enough to reach the counter I copied her pie-making steps. Along the way I learned which steps were essential and which could be varied. After awhile my pies weren’t copies of my mother’s pies anymore…

Once upon a time my PhD advisor gave me a problem to solve. He told me to use a certain theorem and pointed out the relevant paper. For days I banged my head trying to figure out how the results in that paper had any bearing on the problem he’d handed me.

Finally I wrote out a careful description of what the problem and the paper’s result each said and showed how they had nothing to do with each other. And I hadn’t solved the problem.

He wrote back. I wasn’t supposed to be using the paper’s result at all. I was supposed to have noticed that the author’s proof technique presented ideas about how to tackle my problem.

This business of sorting facts from techniques…

My students like it when an algorithm is called an algorithm. They like to get to practice using algorithms and they learn to recognize the kinds of problems the algorithm may be used to solve. The good ones eventually get behind the algorithm and make the solutions their own.