Sunday, November 25, 2007

The Right Combination - A Metaphor for Teaching and Learning Mathematics?

If you were looking for a challenge here in higher math using combinations and permutations, sorry to disappoint you! I felt compelled to write this essay after watching my wife patiently attempting to teach one of my children how to open a combination lock. She doesn't think of herself as a teacher, but, she is, and, in many ways, far more skilled than I ever was.

One of the rites of passage for many middle schoolers is mastering the intricacies of the combination lock for their lockers, somewhat akin to elementary schoolers learning how to tie their shoes. Do you remember the frustration you felt the first few times you tried to solve the puzzle of these locks? Do you recall your euphoria when it magically opened? Consider all of the 'skills' involved and think of the parallels to mastering the algorithms of mathematics:

(1) Fine motor skills required to precisely turn the dial and stop at the correct number
(2) Memorizing the 3 numbers in sequence
(3) Understanding the difference between Right and Left when rotating the dial and retaining the R-L-R sequence
(4) The absolute discipline and precision required - close is not good enough
(5) The dreaded second step of the process needed in going 'past zero'
(6) The extreme feelings of frustration from failing repeatedly and the inclination to give up, yet driven to continue
(7) The elation felt in getting it the first time all by yourself, only to be followed by despair when you can't seem to duplicate the feat!
(8) The feeling of accomplishment when you can do it almost every time without anyone helping you
(9) Is there any substitute for independent practice in achieving mastery here?
(10) How important is motivation here in driving the child to continue in the face of adversity?

What about the challenges faced by the 'instructor' here? If you were the one who helped someone succeed, did you find it frustrating or did you have 'unlimited' patience? Did you have to practice it yourself first and think about breaking this 'automatic' process into simple discrete steps? Did you have to try different verbal instructions (for example, using 'down' and 'up' vs. 'left' and 'right') or different techniques of one approach failed? Did repeated demonstrations in front of the child suffice? Did the child say, "Let me do it by myself?" If you've helped several children learn to 'unlock' the combination, did you use the same approach successfully with each child? Are some youngsters simply unable to 'solve the problem' at that time and need to be given a key lock instead as an accommodation? Is making this concession detrimental to their self-esteem and eventual development or is it reasonable at that time? Will some of these youngsters be able to succeed later if given the opportunity to try again (when developmentally ready)?

Is there a metaphor here for teaching children mathematical algorithms? By the way, can you think of others skills or concepts involved in opening the lock that I overlooked? Pls share!

Now, parents, extrapolate this 'teaching' process to dozens of unique math students every day with a myriad of different algorithms over the course of a school year? Anyone can teach, right?

I realize some of you will see the flaws in this metaphor and will point out all the differences between opening the lock and solving a mathematical problem? I know the parallel is far from perfect but this is something that just struck me and I had to put my thoughts down. You know, like a journal, a diary, a blog... Your thoughts?