The Standard Formerly Known As 4.2.3D.2.a

   From the Common Core Math standards:

     For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.”

    Agree with the conclusions or not, there's no doubt that promise has been kept, at least at the elementary level here in the Garden State.  If you don't believe me, compare the old grade 3 standards to the new ones.  Many of the skills and concepts I taught as a third grade teacher are no longer there.   One that did survive the purge was a confounding little number we grew to know quite well as 4.2.3D.2.a.  It has changed its name to 3.MD.B.4, but retained all of its deviousness.  Can you guess what it is?

     If you guessed measuring to the nearest quarter inch, you are correct!

     Seems not much has changed since my third grade days.  Despite our best efforts, which this year included counting by 1/4s in counting circles, the kids had enough trouble with 3.MD.4 to keep the folks over at Math Mistakes busy for weeks.  I'll propose the following reasons; feel free to add your own:

• Kids count 3 hash marks between each inch and express their response in thirds.
• Kids do not know how to write fractions and mixed numbers correctly.
• Kids think that when they are asked to measure something to the nearest 1/4 inch, their answer has to include a fourth.
• Kids are unable to connect what they know about fractions and number lines to the actual ruler.

As I talked it through the issue with our grade 3 team, I decided to throw out an idea I had been turning over in my mind.   The teachers felt it had potential, and when I got back to my room I started working up a model:

I started with square inch grid paper...

...cut out the squares and labeled one side.

I took a square inch, folded it in quarters, labeled them, and cut them out.  They were small and hard to work with.

What might happen if, when asking a kid to measure something to the nearest quarter inch, you gave the kid the square inch and quarter inch pieces and said something like,
    "Use these to measure.  If you need the little quarter inch pieces, use them.  If not, then don't."
    Would they do this?

Not enough.

Too much!

Just right.

      We decided to find out.  Theresa grabbed a student who had difficulty with the skill on the assessment.  I gave her the square inch and quarter inch pieces and let her give it a try.

The student used the pieces to measure the pin, the key, and the pencil.  Theresa said that, although the small quarter inch pieces were not easy to manipulate, the student was able to use them to measure correctly.  They also helped her understand how to write the mixed number measurements.

From here it's a natural move to...

...and then finally:

     We scaffold many concepts with manipulatives.  We use base ten blocks to help children understand place value concepts and regrouping, and tiles to help children compare and work with fractions.  Why should measurement be any different?  Rulers need scaffolds too!
    Did we beat 3.MD.B.4?  The jury's still out.  We need to get the square and quarter inch pieces into the hands of the teachers, then into the hands of their students.  We need to redouble our efforts at counting by 1/4s, connect that work to number lines and rulers, and make sure the kids have an understanding of how to both say and write the fractions and mixed numbers.  If we do all that, I think we can give 3.MD.B.4 a run for its money.  It may even have to change its name again!

No First Graders Were Harmed Playing This Game

     Up until several years ago, our district ran a vibrant remedial summer program, euphemistically called "Summer Academy".   I spent many hot and humid New Jersey mornings in classrooms, both with and without air conditioning, attempting to find ways to get third, fourth, and fifth graders interested and engaged with a math curriculum that, during the school year, had left them dazed and confused.
     I knew what skills I needed to focus on, but was left to my own devices to figure out some new approaches.  This being pre-MTBoS, I ran to the nearest bookstore:

This became my curriculum, and remains an invaluable resource to this day.

 When the first grade teachers came to me last month with a request for a game to practice basic addition facts, I turned to an old Summer Academy favorite.

This comes from pg. 73, an activity in the Probability and Statistics chapter.

Here's the sheet I made, saved from my Summer Academy days.

     I made copies of the mats, put sets of 11 counters, along with dice, in plastic cups, and trudged down the hall to Jen's first grade classroom.  She had set up a center day, and "The Two-Dice Sum Game" was one of the stations.  I sat down on the carpet with the first group of kids, and asked them to make sure they all had 11 counters in their cups.  I quickly explained the rules, and we began placing our counters on the mat.

Every kid's mat looked like this. 

Here's how I set up my mat:

The kids eyed this somewhat suspiciously, but they said nothing and neither did I.

     I won.  We played another round.  I won again.  And this scene was repeated with two more small groups of first graders as they rotated through the station.  There was some minor grumbling regarding the fact that I always won, but I didn't make a big deal out of it and they seemed to take it in stride.
   I left the game with Jen, and took a look back at the notes I had taken.

• Many kids, even after being told that they would receive 11 counters, said, "I only have 11.  I need 12."  It seemed to be some sort of optical illusion: they would count and re-count and somehow the 11 counters, one counter per number, filled up a mat that had a string of numbers ending with 12.  Weeks later, after many times playing, there were still kids that insisted they needed 12 counters.
• Some kids had to search for the sum in the number line running across the bottom.  For example, one child, who rolled a 5+ 5 and knew the sum was 10, started at 2 and scanned up the line until she reached 10.  Others knew that 10 was going to be more towards the right end of the number line and were able to find it easily.
• Some kids used the number line on the bottom of the mat to add, for example starting at 5 and counting up 4 more to reach 9.  Very resourceful!

    What I had learned seemed interesting, but more than anything, I was left with a sense of guilt.  Did I take advantage of the kids by not explaining that certain numbers had a greater probability of turning up than others?  I had played the odds and won.  They had no idea that odds were involved and lost.  Could I learn something from this?  Could they?  I was curious to see how many of them would continue placing one counter on each number even after repeatedly seeing me arrange my counters in a different way and regularly win the game.  I decided to find out, and asked Jen if I could come back and set up shop again.  She agreed, and my experiment was on.  Over two days the following week, I played this game again and again with the kids in her class.

I continued with minor variations on this arrangement...

...while many students soldiered on with this one, encouraged by comments from their classmates such as, "Don't take a chance, put one on each!" and, ""Make a straight line, then we can beat Mr. Schwartz!"

Then, late on Day 1 and into Day 2, kids started to break away from convention:

and my favorite:

     They were learning through experience that placing one counter on each number might not be such a good idea.  And I had some very close calls, but remained undefeated.  Until one boy, waiting patiently as I laid out my counters, put his in the exact same spots on his mat.
     Student: I'll win at the same time that you win.
     Me: How do you know that?
     Student: Because our mats are exactly the same.

    Smart kid.  Before I left, I gathered the class together and asked them why they thought I always won.  Their responses:

• You know a lot about math and we don't know a lot about math. 
• You know what numbers the dice are going to roll.
• We don't play as much.  We forget how to play and you always play.
• You keep putting cubes on numbers in the middle and you keep getting the middle numbers.
• You always put cubes on the same numbers.
• You put cubes on numbers that you usually roll.
• We have 11 cubes.
• You practice a lot, because practice makes perfect.

   I found their thinking fascinating.  Although some ascribed my success to reasons that had nothing to do with probability, others showed an intuitive sense that something mathematical was happening, even if they didn't have the vocabulary to accurately explain what that something was.
    Now the first graders are playing this game:

     Will they be able to use the results to inform their decisions on where to place their counters when they play The Two-Dice Sum Game again?  At the end of last year, I made a commitment to spend more time in the primary grades, listening, questioning, observing, and learning from both the students and their amazing teachers.  Other responsibilities have kept me away from their classrooms for the past few weeks, but I am anxious to jump back in and find out.  Stay tuned!

Standing On the Corner of Open Middle and Intellectual Need

   One of the many treasures awaiting those who explore the MTBoS is Robert Kaplinsky and Nanette Johnson's wonderful site Open Middle.  I've started to encourage teachers to take a look at the tasks he has collected there, and experiment with using them in class.  Several weeks ago the 5th graders tackled this gem:

An example of one done together in a guided group.  

Once the kids started digging in, I realized how rich this task really was.  So many possibilities, so much strategic thinking; you could feast on this for days.  Here are some of the questions that arose:

• Are some fractions easier to place than others?  Why?  Can we use that to carefully pick our fractions, rather than throwing numerators and denominators together randomly?
• What about improper fractions?  Allowed?  
• How can benchmarking help us accurately place the fractions?

   After working through some together, Rich and I decided to set up a center where the kids could collaborate and try some on their own.  I created a sheet for them to record their work, but also asked them to come up with a consensus answer on the whiteboard.
    One group started off with these 5 fractions:

     "Decide how to arrange these fractions, and get them up on a number line on the whiteboard.  I'll be back in a few minutes to see how things are going,"  I told them.
     I went off to check in on some kids who were working in another center.  When I came back, here's what I saw:

Nothing.

    Me:  "What's wrong?  How come there are no fractions up on the number line?"
    Them: "We're having trouble."
    Me: "What kind of trouble?"
    Them: "We're pretty sure about 0/8, 1/9, and 2/6.  But we can't decide on 3/4 and 5/7.  We're not sure which comes first."
     I asked them to elaborate.  They knew that 0/8 was equal to 0, and that 1/9 was smaller than 2/6 because 1/9 was smaller than 1/6.  They knew that both 1/9 and 2/6 were smaller than 1/2 because their numerators were less than half their denominators.  And they knew that both 3/4 and 5/7 were greater than 1/2 because their numerators were more than half of their denominators.  Yet the inability to make a comparison between 3/4 and 5/7 left them paralyzed, staring at an empty number line.  It was time to step in.
     Me: "First off, I love your thinking about 0/8, 1/9, and 2/6; how you're looking at the relationships between their numerators and denominators.  And you're not the first to be stumped by 3/4 and 5/7.  Those two fractions are hard to compare because they are really close together.  Here's a way you can compare them."

I threw this up on the whiteboard.

     Them:  "They are only 1/28 apart.  That's not very much."
      Me: "Carry on."

This student remembered to benchmark the 1 whole...

...and this one benchmarked 1/2 as well.

     As I reflected on what had happened, it occurred to me that showing the kids how to compare two fractions by converting them each into equivalent fractions with common denominators within the context of the Open Middle task was a good example of filling an intellectual need, a concept I was introduced to several years ago in this Dan Meyer post.  The kids knew the skill; they had been finding common denominators and generating equivalents, but in the context of adding and subtracting fractions.  It had not occurred to them to use it to help them with this task.
     "For students to learn what we intend to teach them," write Evan Fuller, Jeffrey M. Rabin, and Guershon Harel,  "They must have a need for it, where 'need' means intellectual need, not social or economic need."  Many of the Open Middle tasks are activities that Fuller, Rabin, and Harel would describe as "problem-laden": mathematical activities that stimulate intellectual need, and that's what makes them worthwhile.  The kids needed a strategy to compare 5/7 and 3/4, not because it was going to be on a test, or because it would come in handy later in their lives when they grew up and got jobs, but because it would help them clear a path through the open middle of an intellectually stimulating task.  In the real world of the elementary school math classroom, the teaching and the needing don't always intersect.  But when they do, they create a nice corner on which to stand.