How, Not What, to Think

BatBoy has for the past year or so been very keen on matchstick puzzles.  I picked up a book of them at Chapters and he often likes to work on them while SpiderGirl has her classes.  One day, we were with a good friend as BatBoy worked on the puzzles, and our friend wanted to engage with him a bit -- which is great!  And she did what people often do when they help someone with a puzzle:  She figured out the answer in her head, and then hinted at what the solution was.  Of course, he then quickly figured it out and went on to start another puzzle. 

I mean, sometimes it is helpful, when you are stuck, to have someone give you a hint so that you can look at the problem from a different perspective to get unstuck.  But used as the primary "helping strategy," I think hints to the right answer lend support to the myth that math is for people who can magically do it; you either have it or you don't. 

Take matchstick puzzles.  Take 3 away to leave 2 squares.  You take three away and it leaves 2 squares, but there is a stick left over.  Why?  Where does the extra stick come from?  Those are important questions!  They lead you to make a better guess the next time.  Or what if you have 1 square, and 1 square with a side missing?  How can you use fewer sticks to make the second square?  That strategy doesn't lead you anywhere?  Is there a square that you aren't seeing?  A different size?  Do you have to tilt your head to see it another way?  You don't have enough sticks left to make your squares.  How can you minimize the number of sticks you have to take away?  More specifically, is there a way to eliminate a square by taking away only 1 stick?  2?

Good questions are applicable to many situations.  They help a student practice general problem solving strategies:  guess and learn from your mistake; isolate and analyse the problem; look at it a different way; see if you can solve a simpler but similar problem and extend it; look at minima and maxima or other boundary conditions, and work from there; if you hit a dead end, try something else.  These strategies help a student learn a way of thinking and increase their ability to apply mathematics.  They are empowering. 

Supporting students in learning these strategies shows that math is very much about process, not just result.  The exchange between "teacher" and "student" also becomes more of a dialogue, where the student has just as much to contribute as the teacher.  Plus, your teacher didn't just do half your puzzle for you if you didn't want them to.  For those of you who still don't see the harm, it's a little like giving away the ending to a good book.  Now, what's the point in reading the rest?  --  I didn't want to know where the squares were; I wanted help to see them myself!