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!

Sunday, October 19, 2014

The Bestest Things in the World

    One of the things I love about being an elementary school math specialist is getting to work in primary grade classrooms.   They have lots of cool stuff down there!


    With Mr. Harris as my inspiration...

... I thought I might try them with the first graders to explore visual representations of complements of 10.  I am fortunate to work with some amazing first grade teachers who allowed me to experiment.
     First we let the kids play around:

They went wild.  "These are the bestest things in the world," exclaimed one girl.  A boy remarked, "This is better than having an i-pod!"  They made necklaces, bracelets, key chains, swings, snakes, and handcuffs.  They linked them together in patterns, measured them, and sorted.  All without any teacher direction.

After about ten minutes, I called them all together.  I showed them a chain that I had made and asked:

    I asked them to turn and talk to a partner, and when I called on a student, I asked him to share what his partner noticed.  They had quite a bit to say, including observations about the colors, the total number of links (10), and their arrangement.  Some children saw 2 red, 6 yellow, and 2 red, others 4 red and 6 yellow.  Others saw 5 and 5: 2 red and 3 yellow and 2 red and 3 yellow.   I explained that they were now going to make chains of 10 links, but that they could use no more than 2 colors.

This was a popular arrangement: alternating colors 5 and 5.

Everyone was able to make at least two chains.  I added another stipulation after reviewing the results from the first class: once you had used a combination (5 of one color and 5 of another, for example), you could not repeat.  This was because, out of 44 chains, 35 were 5 and 5 combinations, 5 were 10 and 0, and 4 were 6 and 4.   I wanted representations of all the complements of 10.

The next step was to record the chain on a piece of paper.  The first class drew only the links, while the second class recorded the number of each color used.

     This took about an hour.  I collected their work and came back several days later with...

...strips of paper to make paper chains!

The kids had to look at their original picture and create a paper chain.  We asked them to attach an index card with a number model describing how they made 10.

The completed chains made for a colorful display, and a reminder of the complements of 10.

Turns out the project had many benefits:

  • Linking and gluing promoted fine motor skill development.
  • Students had practice sorting as well as creating patterns.
  • The process of turning their pictorial representations of the chains they had created with the plastic links into paper chains required an attention to detail.
  Now that the students have the procedure down, they can begin to compose and decompose other sums.  I know there are more links hiding in closets and collecting dust on shelves.  Let's put them to use!

Monday, October 6, 2014

The Math Message is Dead! Long Live the Math Message!

     I see this in many classes I visit:

It's a Math Message!

     Back when I was in teacher school we called this a "sponge activity".  You might call it a "do now". Either way, it's meant to be something kids can work on for a few minutes as they transition into math class.  Teachers have it up on the board, and the kids copy it down in their math notebooks. Maybe it's a skill review, maybe the introduction to a lesson, maybe something to keep the kids occupied while the teacher gathers his thoughts and plans together.  Whatever it is, it has to go away.

Here's an example from the 5th grade manual.  Typical scenario: 4 kids are finished in 5 seconds, 5 kids can't find a pencil, 3  have no notebook, 7 are trying to multiply 37 x  62,  and the rest are praying that the teacher won't call on them. 

Here's one from grade 3.  For classroom scenario, see above.
Another gem from grade 3.  This question may have been interesting to its writers, but I have seen it cause many 9 year-old eyes to glaze over.

Here's a sample from grade 2.  How long do you think it will take a second grader to get all that money out of his tool-kit?  Paging Mr. Stadel.

Finally one from grade 4.  How long should the teacher wait for everyone in the class to be prepared to read the numbers aloud?  

     I confess to having used these on a regular basis when I was a classroom teacher.  But stepping back now as an observer, I have come to the conclusion that we must re-imagine what a Math Message can and should be.
     Last year the fourth grade teachers agreed to scrap the traditional Math Message in favor of daily estimation180 tasks, something I've blogged about repeatedly (here, here, and here for example).  Theresa and I encouraged the third grade teachers to replace their traditional Math Message once a week with estimation180 tasks, some of which were created and produced by the fourth graders.  We even got the fifth graders into the act, with tasks from the estimation180 site along with ones Theresa and I created to tie into the curriculum, including a series on volume:

volume1movie from Joe Schwartz on Vimeo.

and another starring a balance scale:

1marker from Joe Schwartz on Vimeo.

     While we were happy with the results, we felt that this year we needed to expand the repertoire.  So we've started to experiment with two new "do now" tasks in the hopes that we can add them into the Math Message rotation.  Neither are revolutionary, but we feel they represent a vast improvement over what they're replacing.
    The first is "Always, Sometimes, Never".  Here's one Rich and I tried out:

Seeing this up on the board for the kids to mull over made me happy.

     The kids got to work in their notebooks, and while they may not be so great at explaining, defending, or proving their positions yet, we do see levels of engagement, thinking, and excitement that we rarely see during the traditional Math Message.
    Many students felt it was "always", though several students did mention decimals, one brought up negative numbers, and, "Is 003 a 3-digit number?" was a question that got lots of debate.  One student even tried to rewrite the statement to make it Always True.  It was a mathematically rich and engaging 5-7 minutes, it reinforced concepts we're teaching this unit (and that we revisited during our mid-workshop interruption when we looked at the problem 8.4 - 5.73), and completely blew the doors off finding out how many days older Amy is than Bob.
     The second is to ask the kids to do some noticing and wondering.   I tried this out in second grade, when I used the technique to introduce a game.  I've encouraged teachers to use 101qs as a source of interesting pictures that can inspire kids to notice and wonder.  And it wouldn't require much work to turn this:

Into this:

  What conversations might this Math Message provoke?  What type of thinking might it inspire?  We don't need to use our imaginations.  We just need to go forth and give it a try.