Monday, 12 September 2011

How a piece of paper taught me the exponential function

During my second volunteer block, the 11 applied class was preparing to learn about exponents.  So their teacher handed out some pieces of paper, and casually said to me "here, have one Roy."  So we folded it once, and responded that there were now 2 layers.  Folded it again and confirmed that there were 4 layers.  With the next fold, some students replied 6 layers until others counted the layers and confirmed that there were 8 layers.  With the 4th fold the students had recognized the pattern and by induction got 32 and 64 with no difficulty.  So to prepare for the graphing exercise, they wrote out a table with the values of x, 2 to the x, and the first differences.  While they were graphing the function, the teacher motioned me over, pointed to the first difference column and asked "so Roy, do you understand 'e to the x' now?"

Wow, it took 25 years of using the exponential function to finally understand it.  I could always rattle off a few digits at the drop of a hat, toss you a numeric value courtesy of the limit definition, and celebrate when it showed up in calculus as my labour saving friend.  I could recite the mantra "the slope is equal to the x value" 'til I achieved nirvana, but I could never answer the question "how do you know that?"  This coming from the math student who tries to avoid using the binomial theorem in his work because he doesn't remember how to develop it.  (as an aside, I do recall that it was derived through completing the square, but since I haven't worked my way through it recently ... the binomial theorem only comes out when eyeballing / completing the square isn't working).

This came to mind today after my Math teaching class was done today.  Last week we were asked to shake everybody's hand in the room, calculate the number of handshakes between the 14 people, and tell the group how we knew.  Strangely enough, none of us actually gave the answer 91; I reached out for the summation of 1 to 13 while another selected 14 C 2.  This week our follow-up exercise was to try to come up with the visual representations or methodologies that our students might use.  One group used the matrix technique (cross out the diagonal and those below the diagonal) and the other highlighted the people in colour and crossed out the duplicates.  My reaction to the matrix was 'oooo, subtract the diagonal and divide by 2.'  Other people were attracted to the colours in the other representation and commented as such.  And until the prof mentioned it to us, I hadn't realized that they were the same solution diagrammed in different ways.  To my discredit, I had no use for the colour representation until the link with the matrix was established.

Overall, I wanted to remember these things today for a few reasons.
1)  Students will pay attention to different things ... and if I'm not careful they'll pay more attention to the format than the effectiveness.  Also, if I'm not careful, I'll use a format that doesn't attract their attention.  I was attracted to the matrix solution as it seemed to be a more 'math-like' approach (in my mind).  Unfortunately, matrices have more limited applications and many students won't gravitate to a math-like response to this kind of question.  Others were attracted to the colours in the list ... a solution that seemed a little too laborious a counting and colouring exercise to me.  It's not time effective (in my mind), but it popped off the page for many despite the extra colouring time required.
2)  I'm aware that there will be many solutions to every problem.  With a bit of engineering and a lot of liberal arts in my education background, I tend to be one that gravitates to a solution that many others won't find attractive.  What I'll have to work harder to realize is that there are many representations of the same solution to any problem.

Well, time to get back to my 5 minute micro-teaching planning.  Tomorrow, 9 people in my cohort are going to be learning more about making dill pickles than they could've imagined.  One week from now, it'll be a 15 minute presentation on vampires and choosing sources to supplement an essay's thesis.  Working title:  Don't put Bela and Bella in the same room.

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