Summer in New Jersey, or What in the World is Sitzfleisch?

It's summer here in the Garden State.  That means:

These are amazing.  No tomato in the world comes close.

White, yellow, bi-color,,,it's all so good.

And of course:

Healthy doses of "the Boss".

     Summer also means it's time for two weeks of PD with over 60 math teachers from all over central New Jersey at the Mid-New Jersey Math-Science Partnership (MSP) on the campus of Middlesex County College. I've taken advantage of this PD opportunity 2 out of the past 3 years, and its quality has been uneven.  But judging from our first few sessions, I think that this year's session will be different.
   One of our first assignments was to take a look at several recent articles and compose some reflections and thoughts.  Elizabeth Green's piece in the New York Times Magazine section has gotten lots of attention. In fact  Dan assigned this for summer reading last week.  We were also given copies of Jordan Ellenberg's Times op-ed, "Don't Teach Math, Coach It".  Ellenberg advocates using games, both classic (chess, board games, cards) and new (an app called DragonBox) to teach kids sitzfleisch, which in its traditional sense means the ability to sit still and concentrate for an extended period of time, but which Ellenberg defines as, "the ability to focus on a complicated skill for the length of time it takes to master it."  Because games can be addictive, Ellenberg believes they can play an important role in building sitzfleisch, which in turn can promote attentiveness, doggedness, and perseverance, all good qualities any learner should develop.  Perseverance even pops up in the Common Core's first standard for mathematical practice, "Make sense of problems and persevere in solving them."
     I've blogged about games before.  From Ultimate-Tic-Tac-Toe, to an in-depth exploration of Factor Captor, to having the kids create their own games.  But I've always thought of them as skill reviewers or problem-solving experiences, not as sitzfleisch builders.  Here's one that Ellenberg recommends:

This game is called Rush Hour.  The object is to move the cars around so that the red car can exit the parking lot.  Ellenberg claims that this game is about search algorithms; I've used this quite a bit this year, especially with one or two especially impulsive third graders.

 
      I don't know anything about search algorithms, but it was gratifying to see the kids grow in their ability to stick it out until they find a way to solve the puzzle.  Often the solution requires them to take two steps forward and one step back before they can move forward again, itself an important problem solving skill.
    Here's another favorite:

Mancala is a go-to game.  Easy to learn, and it doesn't take very long to complete.  But you need to be able to think ahead and play out moves in advance in order to be successful.  And the colored stones are cool.

     Again, I have seen impulsive kids, who start by randomly choosing a cup and dropping stones, learn how to think before they act.  This a behavior that can be transferred back to the classroom and used in a traditional learning and practice environment.
     Here at MSP Dr. Milou has encouraged us to explore on-line game sites that promote computational fluency, like arcademics.com and sumdog.com.  They build sitzfleisch too.
    So fire up the BBQ, put up the corn, cut up a big Jersey beefsteak tomato, put on some Bruce, and go forth and play a game!

Summer's here and the time is right...

"The world would be a great place if we could play Factor Captor all day long."

     Factor Captor is an Everyday Math game designed to help students practice identifying factors and multiples, and explore prime and composite numbers.

The rules are not as complicated as they look.  

Here's a close-up of the board.   Jeff and I decided to use Grid 1 (Beginning Level).

   We introduced the game to the fourth graders in October, at the beginning of our first multiplication unit.  It took a few days, but eventually the kids got the hang of it.  We encouraged them to use multiplication tables to find factors, and made a solemn vow not to use the words prime or composite.  We just let them play and hoped that at some point they would come to the realization that some of the numbers had more factors than others.

      One day, about a week after we had introduced the game, I was walking around the room with my iPad  and decided to record some kids playing a round.  I got about 5 minutes of video, and later watched it during lunch.  Knowing how much they like to see themselves on TV, I played it for the class the next day.  As the kids watched the action, something occurred to me: why don't I use it like a coach would use game film?  So before the next move was made, I hit the "pause" button.
    "What do you think the next move will be?" I asked.  There were lots of opinions as to what the best next move was, and I made sure that each one was backed up by some type of explanation.  The idea that there was some strategy involved, which some kids had already begun to intuit, was brought out into open discussion.  And after everyone who wanted to had a chance to express their opinion, I just hit "play" and we saw what actually happened.  The kids loved it!  So we continued in this way for several turns, and came back to it the next day with more video.
    This is when I began to think that maybe we should push this game a bit more.  At the very least the kids were getting practice with some multiplication facts (3 x 9 = 27 was pretty much automatic by that point), they were also adding (as they kept score), and of course continuing to build some understanding about factors and multiples.  Besides, they were still having fun with it.
  A look at a random notebook page where two kids had kept score of a game they played gave me another idea:

How easy would it be to recreate a game based on a copy of the score sheet?  

     Jeff and I tried this ourselves one day at lunch.  It wasn't so easy.  Some of the numbers on the page represented a first number selected, others represented factors or factor totals selected in response, and others represented running cumulative score totals.  We weren't sure how the kids would respond, but we decided to give it a try.
     This got frustrating for some as there were often several false starts, and kids were confused as to what the numbers actually meant.  But the majority were able to piece together at least the beginning moves of the game.
    Of course next came the reverse: Given a picture of a board from a game that is already in progress...

...can you recreate the score sheet?

     By this time the unit was nearing its conclusion, the assessment was just days away, we still hadn't introduced the terms prime and composite, and Jeff was starting to panic.  So we decided to have another class discussion about Factor Captor strategies.  It was centered around numbers that were "good to pick".  By this time the kids knew that 11 and 13 were good because they had no other factors except 1 and themselves.  In fact, in most games, 11 and 13 were the first numbers selected.  This is how we organized the concept of a prime number.  Other numbers might be more or less "good to pick", but they had other factors besides 1 and themselves.  So that's how we organized the concept of a composite number.  Jeff was relieved.  And the unit drew to a close.
   At this point we would have put the game away.  The next unit dealt with fractions, and so did the one after that.  No need for factors and multiples, prime and composite numbers anymore.  But Jeff and I decided we wanted to continue to explore the game with the kids, and that these explorations could take place in and around other lessons and activities.
  Circling back to our conceptions of primes and composites in relation to the game, and the strategy involved in playing, I thought it would be interesting to see if the kids could classify the numbers on the board as "really good", "kind of good", or "not so good", and give a reason to back it up.  The fact that they would have to cut up a board appealed to my sense that if you want to understand something, it's a good idea to take it apart.
 

Jeff and I liked this explanation.
These kids understood that the classification would have something to do with the point differential.

Here's what we were driving at: what's the point differential?  These kids nailed it for 27.

   After spending several days working on this project, we decided to let them play the game again, this time using their charts.  We asked them to keep in mind whether or not their charts were helpful, and whether some numbers might be classified incorrectly.
     My supervisor came to observe me in December.  I wanted to use the observation as a chance to showcase some of the different projects and activities we had been exploring in grade 4.   We started with a homemade estimation 180 activity, and then divided the class in half.  Some were working on the highway sign project, and others playing factor captor and working on their strategy posters.  He liked the game, and noticed how engaged and excited the kids were.
  "The world would be a great place if we could play factor captor all day long!" he said wistfully.
  It's now June.  And Jeff and I have made it a point to integrate the game into our guided group routine at least twice a month. This has provided another opportunity for us to reteach, conduct formative assessments, and have the kids revisit the skills and concepts.

As they have played, their strategy posters have been amended, edited, and revised.

Some have needed more paper!

 
     Jeff and I agree that the strategic use of games like Factor Captor, and the Pie Eating Contest, must be an integral part of our program going forward.  These games have the potential to be repurposed, taken apart, and mined for their rich content.  They have too much going for them to be played just once or twice and then put away.

Lights, Camera, Estimate!

Readers know that Andrew Stadel's estimation180 site has had a tremendous influence on our teaching practice.  Thus far we've concentrated our efforts in grade 4.  We have of course mined Andrew's site, and Theresa and I have created our own activities, but in order for us to build up a stock-pile to use with other grade levels, we've needed some help.  After reading about Jonathan Claydon's amazing "Estimation Wall", I realized we had a work-force right here in house: the fourth graders! So I had Jeff put his Newhouse degree to work setting up our own "Estimation 180 Studio".  He helped me design a proposal sheet:

We made these available for the kids to describe their tasks.

Jeff and I felt that it was important that the student(s) who wrote the proposal also compose and take the actual shots.  Here is a final product:

The question.

The reveal.

In this proposal, the student visualized the tissue box, "with one sticking out."

Many students want to use video.  Unfortunately, our studio is not very high tech (No-tech, really.  It's my room and an ipad.) And we do not have Andrew's videography skills.  For example, when I explained to the student who submitted the tissue proposal that it would take too long to film her counting out all 144 tissues, she said, "You can just speed it up!"   Well, no.  At least not yet.

We decided to alter the proposal, using a post-it rather than a notebook as a referent.

We have managed to pull off some videos.  "Erasers in a Cup" took four takes!

IMG 0967[1] from Joe Schwartz on Vimeo.




IMG 0957[1] from Joe Schwartz on Vimeo.

We have more in the production pipeline, and one of my goals this summer is to improve my video skills.  Involving the kids in the process is a win-win: they get a chance to create, we build up a bank of tasks to use with other grade levels, and everybody walks away happy!

Boogers and Bloody Noses

I suppose I should no longer be surprised when a simple estimation180 activity explodes into a full blown lesson.  Or lessons.  But ones filled with tissues?  A definite first.
 Several months back, Theresa and I decided that we needed to start getting the third graders into the "estimation180" routine.  We wanted to get them comfortable with making number lines, coming up with "too lows", "too highs", and "just rights", and justifying their decisions.
 So a few weeks ago, in Jane Garvin's third grade class, I threw this one up:

After giving them some time to get their work down, and doing some discussion, I showed them the reveal:

Some of the kids were surprised.
Kids: Only 10?  What a rip-off!
Me: Well, yes there are ten.  But it says on the package that there are Ten 3-ply Tissues.  What do you think that might mean?
Kids: ???????
Me: (as I searched for the classroom tissue box) It means that each tissue (and here I pulled one out of the box) is actually made of thin tissues stuck together.  This tissue is actually made of two thin tissues stuck together (I separated them and held up the thin tissues).  So this is a 2-ply tissue.
Kids: !!!!!!!!
Me: Why don't you think the company puts them in the pack that way?
Kids: Because if you blew your nose with one of those thin tissues the snot would come right through!

Now we were talking third grade language!  I taught this grade for many years, and knew that anything related to boogers would be met with great interest.  About as "real world" for a third grader as you could get.  I pushed on.

Me: So now that you know what a 3-ply tissue is, how many tissues are in the package?  Turn and talk with a neighbor.
Kids: (after some discussion) 30!
Me: How do you know?
Kids: Because 3 x 10 = 30!

That was certainly fun.  But I knew there was something else there, and it was related to multiplication, an important grade 3 topic.  What if I gave kids their own personal pack of 10 3-ply tissues, and asked them to separate them all?  They'd have 30 1-ply tissues to mess around with, and could arrange and rearrange them into different combinations.
 I asked Jane if I could steal her class for a period to try it out.  She agreed, and this is what we came up with:

Jane put her class into partnerships.  I reminded them about our tissue estimation activity, and explained that they were going to do a project with tissues.  I gave them no oral directions, just the sheet and a pack of tissues.

They got right to work.  When I tried it out, I had a hard time separating the tissues, but the kids did it with no problem.  One boy made a 15 2-ply tissues, folded them back up, and tried to stuff them back into the packet.  Here you see 3  10-ply tissues.

One student slipped over to her desk and pulled out a multiplication table.  What was she looking for?  

     Time was running out, and Jane helped me collect all the tissues.  Once all the kids were back in their seats, I asked them to talk about some of the different combinations they had found.  Here's what they came up with:

The only one they missed was the 1 x 30, which I was able to pull out of them with some leading prompts.    

Then Jane asked them an interesting question:
   "What do you think of a 30-ply tissue?"
The kids laughed; clearly it was overkill.  A 30-ply tissue made no sense.
    As the kids filed out the door, a quiet boy showed me the back of his sheet.
    "Is this OK?" he asked.  "Would 3 8-plys and 1 6-ply work?"
   "You tell me," I prompted.
   "Well 3 x 8 is 24, then there would be 6 left.  24 + 6 is 30."
    I gave him a high five, and turned to Jane.
   "I have to come back,"  I said.  "We're not finished here."

I returned about a week later.  I thought I would start them off with another estimation activity.  So we looked at this:

The reveal gave us a chance to practice a newly-acquired skill: multi-digit multiplication!  The kids got to work on finding how many single ply tissues there were.  Some multiplied 85 and 3 using the traditional algorithm, others used partial products.  This is a good example computation being a means to an end, not an end in itself.

 
My plan was to move the kids towards using counters rather than tissues.  We discussed the difficulties they encountered working with the tissues: they crumpled easily, they flew around and got mixed up with the tissues from neighboring groups, they were difficult to count because they got stuck together...the kids came up with an impressive list.
      "What could we use as a stand-in for tissues?" I asked.
     "Counters!" they volunteered quickly.

     Next came this question  What about 3 8-plys and 1 6-ply?  Is that an OK way to arrange the tissues?  They took a few minutes to debate this at their tables.  Some kids were bothered that not all the groups were equal; but most felt it was a perfectly reasonable option.  Time to get to work.

They worked out their combinations on poster paper.

I asked them to include a picture and a number model.

Again, the period was drawing to a close and we wanted to wrap up.  Thinking about the question Jane had asked regarding the 30-ply tissue, and the boy who had folded up 15 2-plys, I asked each group to share one of their combinations, and how practical the tissues would be if they came packaged that way.
   "I have to come back one more time," I said to Jane.  "Then we'll be done.  In the meantime, get the tissues back out!"

Several days later I was back.  This time we started with the third and final activity in the series:

     What I really wanted to accomplish in this third session was to focus on the first part of Standard for Mathematical Practice #3: "Construct viable arguments...".  I also wanted them to do a little writing, something I know I need to incorporate into lessons on a more regular basis.  So after giving back the posters and allowing them to do a little more exploring, Jane and I gave them the following task:
 
We know that personal tissue packs come with 10 3-ply tissues.  The tissue company has asked you to come up with a different way to package them.  Your group must come up with a recommendation to present to the company.  

     Jane gave each group their 30 1-plys back, and we told them to fold them up the way they wanted, as if they were going to put them back in the package.  They also had to write some sentences justifying their choice, and then present their decision to the class.  The groups came to consensus, and organized and folded their tissues.   After the previous lesson, I think they knew this was coming.
     Jane and I could hardly wait to see what the kids would come up with.  We weren't disappointed; but we were surprised at the extent having a bloody nose haunts the minds of third graders.  Here were some of their combos:

• 2 10-plys (for bloody noses) and 5 2-plys (for stuffy noses)
• 2 5-plys (for bloody noses) and 10 2-plys (for "normal snot")
• 2 4-plys (for runny noses) and 2 11-plys (if you have the flu)

     The kids enjoyed sharing their proposals, and Jane and I had a hard time holding back our laughter. This time I remembered to have them give us some reflective feedback, which was overwhelmingly positive.  They described the activity as, "fun", "imaginative", and "exciting".  And who wouldn't love hearing those words used to describe math class?

Test Prep I Can Believe In

As we wrapped up a unit on volume, and with the state standardized test looming on the near horizon, Rich and I decided to try out a project involving cereal boxes.

The kids brought these in from home.

First, they needed to find the volume of their box.  Since we figured they could use some practice multiplying mixed numbers, we decided to alter the requirement: instead of measuring to the nearest centimeter, the kids would need to measure the length and width of the base of their box to the nearest 1/8 inch.

We decided that they should use the rulers they were going to have to use on the state test.  The 7-inch rulers were not up to the task of measuring the heights and the widths of the boxes, but Rich and I decided that was OK...

...they would figure it out.

     I was most excited to use the opportunity to revisit an area model for multiplying fractions and mixed numbers, so we had the kids outline the area of the bases on square inch grid paper.

7 1/2 by 2.  We had originally thought they would do it on blank paper.  But we decided the grid paper was better because it more effectively expressed the idea that the fractions were pieces of actual squares.

At some point I had the idea that the model might be made more powerful if it was color coded by each partial product.  I felt it lent a visual aspect that was missing in the original design.  Then they had to multiply the area of the base by the height of the box to come up with the volume.

Rich's student teacher, Shannon, made a chart to to help the kids keep track of the data. (Note to self: next year have the kids create the chart themselves).  This would be used to compare findings, and also to provide them with referents for the dimensions of a standard sized cereal box.

This continued for several days.  They got a lot of practice working with fractions, multiplying, dividing, and measuring, and the fun was just beginning... because now they would have the chance to create their own cereal, and design the box.

Drawing a three dimensional shape on paper presented some challenges.  This student's ingenious use of two rulers gave Rich and I the idea that protractors could be used to make the representations even more accurate.  Duly noted for next year.

Nike is into everything these days.

Love the mascot!

For the One Direction fan in all of us.

Check out the expiration date.

This is a "mini-box".  Each group had to come up with one that was half the length, width, and height of their standard sized box.

We also had them come up with the dimensions of a jumbo box by tripling the length, width, and height of their standard sized box.  There was quite a bit of calculation involved here, but we felt it was good practice, and it was embedded in the project.  Multi-digit multiplication and addition, multiplying and dividing fractions and mixed numbers, adding fractions, converting improper fractions into mixed numbers...lots of opportunity for review, formative assessment, and on-the-spot reteaching.  Turns out that there were quite a few kids who forgot how to do many of these operations.  Go figure.

     The kids loved it.  There was a high level of engagement, great collaboration and problem solving, and a nice blend of computation, measurement, and geometry.  As we put the projects away (we're not done with them yet!), and headed into the week of the NJASK, my one thought was: That's the kind of test prep I can believe in.

Teacher Appreciation Week

I am grateful for Mr. Tom DiGanci, my social studies and history teacher at Watchung Hills Regional High School.

He taught me that there was a world out there much larger than the one I could see.
He taught me that history was alive, and encouraged me to take my place in it.
He knew that we had to be engaged in order to learn.
He played Moody Blues records on a beat up school record player before class.
And he always had time to listen.
Thank you, Mr. DiGanci.

And thanks, Christopher Danielson.

Another 3-Act

Fresh off their mile walk around the perimeter of the school, the fourth graders took a crack at another 3-Act.

ACT 1:

How much would it cost to spread grass seed on the field?

The kids are quite familiar with this field.  They've been playing on it for years.  

 

We thought it would be a good idea to give the kids a visual.  Jeff found a video for the afternoon class to watch.

ACT 2:

What do you need to know?

What the AM class wanted to know.

What the PM class wanted to know.

Here's what we gave them:

I made copies of the screenshot for them to have at their desks.

Jeff likes the Pennington grass seed.

  
ACT 3:

Let's Get to Work!

Some kids wanted to work on the task on their own, in their math notebooks, and I gathered a few kids back at the whiteboard and we tackled the problem together.  I tried to let them take the lead, facilitating and questioning to keep them on the right track.

We've done a lot of math and we're not even finished yet!

     The kids in the AM class had some trouble getting started.  I suggested a table, which I began and they finished.  An interesting discussion then took place around what we teachers would call "interpreting the remainder", but what they called, "Do we buy 15 bags or 16 bags?"  Ultimately they came down on the side of 16, after learning that the store would not open up a bag and give them just the amount they needed, and after overruling one boy who wanted to save $50 and leave some of the field unseeded.

The kids in the PM class started right away, using something like a "trial and error" method.  After a discussion much like the one that took place in the AM class, they also decided to buy 16 bags of seed.

And also came up with a price of $800.

I believe that the most important piece takes place in Act 2.  It's so strange, and so unlike what we normally ask kids to do in class:  just answer the question.  Yes, they're going to get to do that, but first they need to figure out what they need to know.  We can learn so much about their understanding of concepts from the information they request!  And some of that information we will provide (like the measurements of the field, and the grass seed specifications, although the project would even be better if the kids did their own work on Google Earth and researched grass seed on the Home Depot website), but the rest they'll need to get for themselves (with a little nudge or redirection when needed).  And we've got lots of computation, but as a means to an end, not an end in itself.
  There are some things we need to think about:

• How will kids be held accountable for their work?
• What exactly do we want to assess?
• How involved should we be in the problem-solving process?    

 But I think we've made a good start. Now where can we come up with $800?