The Newspaper Problem

DISCLAIMER:  I procrastinate on something until I feel like it’s “perfect.”  Then I eventually realize it will never be perfect and either (a) concede and move on with it anyway since it’s too good to pass up, or (b) toss it aside if it wasn’t a worthwhile venture.  In the realm of my blog, I’ve decided to choose option (a) after months of waiting around.  The website is still (yeah, yeah, I know: it always will be) a work in progress.

POST:  I suppose that the name of my site provides an appropriate context for my first blog post.  When all is said and done, I have seen myself grow immensely as a mathematician and as a teacher by being open and forcing myself through turbulent waters (habits of mind…?).  I want my students to experience mathematics this way, as I only have as an adult.  I want them to experience actually doing mathematics.  A recent experience I had provides a good illustration.

I was on a flight, and decided to whip out my current favorite book, Measurement by Paul Lockhart.  I was intrigued when I was asked to find coordinates for the point a third of the way along a line segment.  A fairly complex process ensued.  Well, I was aware of its complexity anyway; I don’t know if a new problem-solver would be.  Here’s my work; I basically only had a newspaper handy.

plane_problem

So many facets of what I did made me think about my teaching:

  • Writing on newspaper:  I love problems, and I’ll find a way to solve them!  Interesting problems are good.  On another note, I also used Desmos to visualize some things.  Choosing the appropriate tool is handy, too.  In the end it was a number line on my paper that created the light bulb.
  • The disorganization:  I was venturing into the unknown, not sure what the best path was to take (the standard methods for the midpoint weren’t screaming anything obvious at me).
  • The multiple representations:  I tried building patterns with specific coordinate pairs.  Interesting patterns were coming out, but still nothing immediately evident for a generalization.  So I changed my approach to making tables, to Desmos, and then finally to a number line.  Perseverance and different strategies/points-of-view are important.  I kinda wish I had someone else there working on it with me.  I don’t think the guy next to me wanted to.
  • The scribbles:  I made mistakes and I didn’t care if what I was doing was inefficient.  It’s what came to me, and I went with it.  Had I shared with someone else, I would likely have seen things in an enlightening way that I hadn’t thought of (collaboration is great).  But even alone, the environment was just right for exploration and growth: open, encouraging, and absent of creativity-stifling criticism.
  • The systematization (thought I might have made up that word, but nope; I checked):  Yes, it looked a bit disorganized.  But, like my closet, there was a clear system that I was building.  It was actually organized and precise.
  • SO MUCH MATH happened along the way to the answer!  Sure, I walked down different paths awhile before I realized they didn’t lead where I was going at the moment.  But what happens when I need to use those paths later to get somewhere else (possibly related)?  I may not have ever found them again had I not explored them in the first place.  Plus, I wouldn’t have found my way had I not gone down the wrong path a few times and made some decisions to go back.

My point here is that this is a perfect example of what I view the process of doing mathematics to be.  This is my goal: to provide a stage in which my students can have these experiences as often as possible while still operating within the framework of the necessary system and content.

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