My Continuing Battle With 3.MD.B.4

     Captain Ahab had Moby Dick. Wile E. Coyote's got Road Runner.  I have 3.MD.B.4.  Teaching kids how to measure to the nearest quarter inch bedeviled me as a grade 3 teacher, and has continued to do so in my role as math specialist.  Recently I described our attempts to tame this unruly standard, and while I was pleased with the results, I felt we needed to look for more ways to attack this beast.  My latest inspiration came from an unlikely source:

I found this book buried in a closet.  

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    The game was familiar.  This past summer, at the Middlesex Math-Science Partnership program at Middlesex County Community College, Dr. Milou had challenged us, playing from 1-20.  He took great pleasure in trouncing all combatants, one after another, until someone in the group figured out the winning strategy.

This caught my eye.  Might this game be employed in the struggle against 3.MD.B.4?  

      I asked Jen if I could use her class to experiment, and she agreed.  First, I taught the kids how to play a basic game, with the first player saying either the number 1 or the numbers 1 and 2, and the players then alternating the count by one or two numbers in sequence from where the other player left off, with the player saying 20 the winner.  I channeled Dr. Milou, beat everyone in class, and left the defeated third graders with the assignment to play the game with each other whenever they could.
   When I came back the following week, they were excited to tell me that they had figured out the strategy: the player who can say 17 is assured of winning.  They hadn't gotten any further, but it was a start, and I urged them to continue playing.  In the meantime, I explained that I had a different game to teach them.  I explained the "fraction variation": same rules, but instead of using whole numbers and counting from 1 to 20, we were going to count by 1/4s from 1 to 5.

I played a demo game on the board.  I recorded my count in red and the student's in blue.  I wanted them to record their counts, in the hopes that we could connect this number line to the quarter inch ruler.

They picked it up quickly.

 

We had to tape two pieces of paper together to get the right length.

This sample, placed under the document camera, provided the teaching points at the lesson's close: I love how these kids wrote a 1 on top of 4/4 and a 2 on top of the 1  4/4.  We also talked about how to write a mixed number, so that 3 and 3/4 would not be confused with  33/4.

 Jen reported that the kids liked this game, and when I did an item analysis on the unit assessment I was pleased:

This is the item I was interested in.  17 out of 18 kids had gotten all three parts correct. 

     But 3.MD.B.4 never sleeps, and I knew we couldn't rest on our laurels.  The conceptual groundwork we were developing here would only help the kids as they moved into the more complex measurement and fraction standards awaiting them in the upper grades.

I took our standard grade 3 ruler and photocopied and enlarged it.

I whited out the numbers and put two on an 11" by 18" page.

Would you look at that!   It's a ruler!  And marked at quarter inch intervals, too.

     My hope is that, if they play enough, the kids can start to notice certain patterns: that the whole numbers live on the longest lines, the 1/4 and 3/4 on the shortest, the 2/4 right in between.  (And I would like to transition the kids to writing 1/2 instead of 2/4, although this is a good way to reinforce these equivalent fractions.)
     My hope is to avoid Captain Ahab's fate, and one day vanquish 3.MD.B.4 once and for all.  So... what's your white whale?
   

Box of Clay

     It's volume time again in grade 5.  Last year Rich and I tackled this important standard with a cereal box project.  This year we added a wonderful item from Illustrative Mathematics called Box of Clay:

  I decided to introduce the task by removing the question and replacing it with a notice and wonder prompt:

This removes the anxiety of having to "figure something out" right off the bat.  The kids have a chance to process the information and enter the task in a comfortable way.

I gave them several minutes to think on their own, then do a turn and talk, then share out with the class.  Most of what they had to say didn't surprise me:

• There are two boxes.
• The first box is smaller than the second box.
• The first box can hold 40 grams of clay.
• The first box is 2 cm high, 3 cm wide, and 5 cm long.

 But then...

• There's something wrong.  The first box is supposed to hold 30 grams of clay, not 40.
• 2 x 3 x 5 is not 40.
• The volume is measured in grams, not cubic centimeters.

   I was not expecting this.  Many kids were completely thrown off by the two units of measure in the problem: the height, width, and length of the box, measured in centimeters, which they dutifully multiplied to get the volume (30 cubic cm), and the weight of the clay, stated as 40 grams.  

   This issue came up again in their "wonderings":

• Why 40 grams of clay?  3 x 2 x 5 = 30.  It's supposed to be 30.
• Why is the unit grams?
• Is a gram the same size as a centimeter?
• What's the volume in centimeters and grams?
• Why are there two different units of measurement?
• What does 40 grams of clay look like?

  Along with:

• What's the volume of the second box?
• What's the volume of both boxes together?
• How much bigger is the second box?
• How many grams of clay can the second box hold? (This was the actual question!)
• How many grams of clay can both boxes hold together?

And...

• It's a box of clay?  Why is the box made out of clay?  

     Before I go on and explain my next move, I want to reflect on just how powerful this experience was.  Rich and I could have just copied the task and handed it out for the kids to work on.  But providing time and space for the kids to notice and wonder before they got to work revealed some very important understandings, and allowed other very telling misunderstandings and misconceptions to surface.  That, in and of itself, proves the efficacy of the "noticing and wondering" protocol.  But even more, the kids generated the actual question themselves, along with several others they could work on, which turned the experience into a quasi-3-Act.
   I explained to Rich that the kids could begin working on the project the following day, but we would need to start with a whole class demonstration.  That morning I rounded up some supplies:

I felt it was important to clear up the misunderstandings about the units.

I started with a small piece, and continued adding until the two pans balanced.  40 grams of clay.

   We discussed that what was contained in a box could be measured in many different ways, but that a useful measure of the size of a box itself was cubic units.  I'm not sure if the distinction was clear to all, but I felt it was a start, and I had them imagine what a box that could hold 40 grams of clay might look like.  Not very big.
    I was about to get them started working out the problem, when a hand popped up in back.
   "If the clay was flattened out, would it still weigh 40 grams?"
   Again, this was not a question I expected.  And, surprisingly, the class could not agree.  Some kids thought that a flattened piece would weigh less.

Nope.  Still 40 grams.

With that out of the way, it was time to get to work.

What do you know!  The actual question from the Illustrative Mathematics task was one the kids had come up with the previous day during the noticing and wondering activity.

Several kids wanted to build the boxes using centimeter cubes.  I could only drum up a small bag.  They had to improvise.

Ultimately, however, they all drew boxes...

  

...and there were issues here too.  Is this box really 2 cm high, 3 cm wide, and 5 cm long? 

There were mistakes and false starts.  The erasers got quite a work-out.

But they persevered, and the boxes of clay took shape.

I'm glad we invested the time in having the kids draw the boxes.  Because when it came time to answer the question, we saw quite a bit of this:

Uh oh.

     With Michael Pershan's ShadowCon talk on hint-giving fresh in my mind, I had to do some thinking.  What could I tell these kids to get them on the right track?  I looked at the commentary that Illustrative Mathematics provided to go along with the task:

This picture provided a helpful clue.

    While looking back at the pictures of the boxes they drew, I could ask something like, "How many of the first size boxes could fit inside the second box?"  And if that didn't work, I could try something more explicit: "The first box is 30 cubic cm and can hold 40 grams of clay.  What if there were two of those boxes?  What would be their volume?  How many grams of clay could two boxes hold?"
I tried it out the following day:

As fortune would have it, our next unit is on rates.  Think this will be useful?

     And as the kids finished working on the main question, they started in on some of the others they had brainstormed during the notice and wonder phase:

Yes, I made a mistake wording question B.  The kids caught it right away.

     Though they are not all created equal, there are many other rich tasks on the Illustrative Mathematics site, and I have encouraged our teachers to look there for ideas and supplemental lessons.  Reflecting on the experience with Box of Clay, I would offer these suggestions:

• Try introducing the task with the question removed and replaced with a notice and wonder prompt.  What you find may surprise you, and will help inform the direction of the task moving forward.
• Take a careful look at the commentary that accompanies the task.  
• Take Michael Pershan's ShadowCon advice.  Try to anticipate potential wrong turns, and plan out in advance what "hints" will put students back on track.

     And there's something else about Illustrative Mathematics that has rocketed it towards the top of my "go-to" list: the project's CEO is Bill McCallum, and Phil Daro and Jason Zimba are listed as Senior Advisors.  Who are they?  Only the leaders of the team that wrote the Common Core State Standards for Mathematics.  Who better to create and select questions, problems, and activities that accurately reflect the standards.  I've got only one question for them:

    Why is the box made out of clay?

Teacher Appreciation Week

   Several years after I first started teaching, an administrator encouraged me to go back to school and earn a post-graduate degree.  I was young and single, he explained, and with no other major responsibilities besides my teaching position, going back to school would never be easier.  I took his advice, enrolled in the Graduate School of Education at Rutgers University, and took the first steps in earning a Master's Degree in Elementary Ed.
    I was allowed one elective, and I chose 20th Century British Fiction, taught by Dr. George Kearns.  This was the primary reason:

It was on the reading list.  I had always wanted to read this book, but was intimidated by its impenetrability.  I felt I needed a strong hand to guide me through the text.

      I will never forget that first class period.  It didn't take long before my excitement turned to anxiety and fear.  As a grad student, I was in a small seminar with other grad students.  But this class was not an elective for them; they were pursuing advanced degrees in the field of literature, and their knowledge far eclipsed mine. Words I had never heard of, like deconstruction and semiotics, and unfamiliar names, like Derrida and deMan, were being thrown around at a rapid clip.  At times I felt like I was listening to a foreign language.  The professor, Dr. Kearns, seemed likable enough, but at some point during that first class I decided that I was in over my head and would need to talk to him about dropping the class.
   Alone in the room with him after all the students had left, I expressed my concerns.  I will never forget what he said.
  "You like to read books, right?" he asked.
  "Yes," I replied.
  "And you have thoughts about what you read."
  "Yes."
   He smiled, and said reassuringly, "That's all you need.  You'll be just fine.  See you next week."
   Dr. Kearns turned out to be one of the best teachers I ever had.  He led us all through Joyce, as well as Beckett, Forster, Woolf, Lawrence, Lessing, Henry Green, and Elizabeth Bowen with humor, grace, and profound insight.  He valued everyone's opinion, mine included.  With his encouragement, I overcame my sense of intimidation, and became a regular contributor to class discussions.  And talk about feedback:

These are the comments he wrote in response to a paper I submitted on D.H. Lawrence's novel Women in Love.  And he did this for everybody.

This meant a lot to me:

Did he hold me to a different standard than the other students in the class?  I'll never know.

     Looking for a picture of him to accompany this post, I was saddened to learn that he had passed away in September of 2010 at the age of 82.  His obituary described him as, "An inspirational teacher, whose brilliant mind, dry wit, and immense learning were informed by great kindness."  That's exactly how I knew him.  I was one of perhaps thousands of students who had sat in his classes, and I am quite certain that he would not have remembered me.  But I remember him.  And on Teacher Appreciation Week 2015 I'd like to say: thank you Dr. Kearns.