Logic... Not Magic

I was reading a post on Dan Meyer's blog the other day titled You can't break math. There were a number of aspects to that post that resonated with me, but I came across the passage below that really made me think about the subject I teach in a slightly different way, this passage is..

One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand.

I started thinking about my own experience with my own mathematics education and in many ways the approach I took to my first few years of teaching. I realised that when I was learning maths in school, it was presented as magic rather than logic, and that is the way I presented it in my first years of teaching.  My teachers didn't intend to present it in this way, and neither did I.  

Part of the beauty in mathematics is in the patterns that emerge and in the certainty we have in our conclusions.  However in the highly formulaic way that many classrooms still operate, more time is spent practicing how to use the formula under the guise "trust me it works", rather than spending the time to get students to develop a sound line of reasoning where that formula is the only logical conclusion.

The formula is simply the highly refined end point of a lot of thinking about a particular mathematical idea.  That formula is the point at which all the uncertainty in their argument has been stripped away and what remains is the pattern that has emerged.  When we present that formula without the thinking behind it, without developing that understanding prior to presenting the formula, this is when when it appears that we have 'pulled a rabbit out of a hat', it is something that has come from nowhere.

However it is important to remember that even magic is not magic to everyone, this has become abundantly clear to me through watching a lot of the TV show Penn and Teller Fool Us.  For those who do not know the show, a range of aspiring magicians come on to a show and try to fool a world famous pair of magicians.  Most of the time, these magicians cannot fool them. Even though the trick has been previously unseen, these magicians strongly understand the principles of magic, concepts such as misdirection and sleight of hand, so even if the trick is unseen, they can unpack the thinking that may produce that result.  Their knowledge of these concepts is so strong, and so flexible, that they can apply them to any trick and to create new tricks of their own.  This is what I want for my students in relation to their mathematical knowledge.  More than just understanding the individual tricks, I want them to understand the underlying framework of the mathematics they are studying so that they can apply it in unfamiliar and sophisticated ways.