Tag: algorithms

Do as I DO…

Do as I DO…

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.

Thoughts thought along the road

Thoughts thought along the road

I meant to ask my students yesterday what a number is.

I did that once with a class of College Algebra students who had appointed as spokeswoman a classmate who dared to ask me if I would please explain something “in regular English.”  I told them I would try, but first I needed them to tell me in regular English not what a number does, but what a number is.  They clustered up and went to work; discussing, giggling, yelling; digging in to the chore.  My objective was complete when the reassembled horde reached agreement: “That’s HARD!”

Several years ago, and several years later, I griped, “It is impossible to remediate mathematical deficiencies for university students who don’t understand order or magnitude.”

More and more, university level remedial math is relegated to online drill systems which do a pretty good job of conditioning students to perform logically consistent symbolic manipulations.  Any student with a tolerance for shoving symbols into formal patterns does a good job.  I’ve taught many students who’ve cleared those remedial hurdles, who arrive in my room for their next course which is often “the last math they need.”  These students clamor for detailed algorithms by which to arrive at the right answer: a symbol attained by careful traversal of a necessarily arduous path, in the end a trophy divorced from context.

It is students who pass through that sort of training that tell the world and their children, “I NEVER used algebra again in my life once I graduated.”