Tuesday, October 28, 2014

PD Like It Oughta Be

    It started with an e-mail, and ended as a powerful professional development experience.  Along with five other members of the MTBoS...

....I was invited by Andrew Stadel to participate in the trial run of a new 3-Act task he had developed. I would be required to collaborate with a partner, and despite the fear of having my math inadequacies exposed, I put my faith in Andrew and agreed.  (You can do the 3-Act too!  Just follow the above link.)
    We started off working alone.  He sent us the Act 1 video to watch, and asked us to respond to the following prompts:
  • What do you notice?  What do you wonder?
I shared my responses in a google doc, and it was collected there along with all the others:
  • You're the only one on the playground. You pushed that swing really hard, but not hard enough to wind it all the way around the bar.
  • Will the swing make it all the way around this time, or will you need to push it again? (I'm thinking it will). Is that as hard as you can push? I'm wondering how long the chain is and how high the cross bar is off the ground.

     Andrew commented on our responses, and then popped the question:
     How many total times will the swing wrap around the pole?
     First we had to make our estimates, and provide some reasoning.  My too low was 2 wraps, my too high was 10 wraps, and my "just right" was 6 wraps.  Why?  I had watched and re-watched the video, and it looked to me like 1/3 of the chain was wrapped and about 2/3 was left to go.
     Following 3-Act protocol, we needed to request the information we felt we needed in order to be able to figure it out.  I wanted to know two things: How long is the chain?  How much length does each wrap take off the chain?  Again, our responses were collected in a google doc.
     Andrew's next e-mail contained the information for Act 2:

     We were asked to come up with a representation for our solution, then set up a Google Hangout and share with our partner.  Mine was Shauna Hedgepeth, known in MTBoS circles as Hedge.  I found out through her blog that she is a K-12 math coach in Mississippi with classroom experience teaching 7th grade, Algebra 1 and 2, and AP Statistics.  I knew I would have to up my game.  We set a date and time, and the clock started to tick.

My first instinct was to draw a picture.  After all, isn't this what I always tell the kids to do?  On my first attempt at a solution, I rounded off the chain length to 7 feet and divided that by 2 1/2" , the diameter of the pole.  (The diameter was actually 2 7/16" , but again I rounded).  I got an answer of 2.3 wraps, which I knew had to be wrong.  Why?  Because it made no sense!  The swing had already wrapped around twice and there was still lots more chain to go.

   A day or so after this first, failed attempt, I realized that I needed to convert the chain length into inches before doing my division.  I felt pretty good about having figured that out, but when I tried it (see second attempt), I got an answer of 33 wraps, which again I knew had to be wrong!  How did I know?  Because of my estimate.  33 wraps was way too many.  It wasn't reasonable.  It was time for another approach.

A search around the house turned up a poster tube.   I found a chain in the garage. 

I wrapped the chain around the tube over and over again.

       It didn't help.  No matter how many times I worked with the model, I could not figure out what I was doing wrong.  I would have to confess to Hedge that I could not come up with an answer that made sense.  But I knew enough about the MTBoS community to know that it would be a safe place for me to fail.  I was confident that the faith I put in both Andrew and Hedge was not misplaced.
   The day before our scheduled Hangout, I happened to glance down at the tube.  I saw it from a different perspective:

I stared at it for a few moments.

    Suddenly I felt something click.  "The chain isn't wrapping around the diameter of the pole, it's wrapping around the circumference!  You need to divide the chain length by the circumference, not the diameter!"

After a quick search for the circumference formula, I came up with an answer of 11 wraps.   I will say here that, except for the very basic computation, I did all my work with my cell phone's calculator.  Remind me again why I'm spending so much time teaching kids how to multiply and divide decimals like this with paper and pencil?
    11 wraps was more like it.  My intuition told me it was reasonable, but it was still higher than my estimate of 6 wraps.  Was there something I was missing?  I went back to my model.  My chain was 28 inches long, and the circumference of the poster tube was 7 inches.  Using my formula of dividing the chain length by the circumference,  the chain should wrap around the tube 4 times.  However when I tried it, the chain wrapped around only 3 times.  It was easy to see that the chain's metal links prevented it from laying flat against the tube.  I would have to take that into consideration.  But how?

I set up a ratio.  My thinking went: If a chain that should wrap around 4 times actually wraps around 3 times, how many times will a chain that should wrap around 11 times actually wrap?  8.25 times!

   Closer to my estimate of 6!  I felt good about my work, and looked forward to meeting Hedge.  I was a little nervous, but we hit it off right away; she thought I might make fun of her accent, and I thought she might ask me, "What exit?"  We spent time sharing our work and our solution strategies, and as it turned out she came up with an answer close to mine.  She had drawn a diagram too, and I admired the way she went right to the formula.  She liked my use of the poster tube and the chain.
   The time flew by.  The fact that the child who had grown up and lived for so long with such a tortured relationship with math had just spent a half hour talking with a math teacher in Mississippi about how to solve a math problem was nothing short of amazing.  It was a testament to the redeeming power of the MTBoS.
    For me, Act 3 was anti-climactic.  The fact that the swing wrapped around 7 times and not 8 didn't bother me in the least. I felt successful.  The final debriefing took place in a Hangout that spanned 5 time zones and the entire country, from New Jersey to Mississippi to California to Hawaii as Hedge, Sadie, and I shared our experience with Andrew.  For myself, I had 3 big take-aways:

  • The estimate I made helped me discover my mathematical errors.  I knew that my first two answers could not be correct.  They were not reasonable given what I had seen in the Act 1 video.  I needed no math to figure that out; only my intuition.  This reinforces my belief in the power of building estimation skills through estimation180 type tasks.  The big move will be to get students to generalize that kind of thinking to all their problem solving experiences.
  • The anxiety I felt about having to share my work with someone I did not know, and the fear that I could be totally wrong and might embarrass myself, was alleviated by the fact that I was sure I was working within a community that was safe.  This reinforces the importance of building safe communities within our classrooms, where there is a culture of collaboration and the fear of failure is banished.
  • Most importantly, Andrew gave me the gift of time.  I'm sure that the others came to a solution quickly, but I needed days to process this task.  I needed pictures, physical models, time to fail and fail again.  The "a-ha" diameter vs. circumference moment was an exciting moment of discovery, a moment that would have been robbed from me had I been working under a tight deadline.  Unfortunately time is a luxury we do not often have in school.  It made me wonder: How many exciting, "a-ha" moments are robbed from students simply because they do not have time enough to think things through?  
     It was an enriching project, one that was important for me to experience.   As a wise man once said,
     "You never really understand a person until you consider things from his point of view...Until you climb inside his skin and walk around in it."  I'm so used to being in "teacher mode" that it's easy for me to forget what it's like to be a student.  This was a powerful reminder.  Another gift from Andrew.
     Weeks later, a third gift came in the mail:

My very own Estimation 180 t-shirt.  Thanks Andrew!

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