# One what?

Recently, I have been fascinated by the resurfacing of the counting and cardinality standards in third grade.  Third grade is when students start exploring multiplication and division. They have just spent three years grappling with how to count, add, and subtract whole numbers.

They started the journey in Kindergarten wrestling with the fact that one digit could represent multiple items.  Minds were blown by this concept.  Just when they started to apply it, they came across the number 10.  What ?!? two digits – one and zero – can represent ten items.  That number “1” now means “1” ten?!? Craziness.  Eventually, these same students developed deep understandings of some important units in our number system: one, ten, 100, 1,000, 1/2.  They learned how to take these units apart and put them back together. They used them to represent and understand their world.

Now, they enter third grade and we start presenting situations where 1 can mean a group of any number of items AND you can have multiple groups of those items.  Enter the parallel progression.

In kindergarten, students learn to count a group of objects, they pair each word said with one object.  When these same students start to learn multiplication, sometimes, they revert back to counting by ones, especially if the objects they are counting are presented in an array that is composed of numbers that make skip counting challenging.

Watch Jack:

When students were in Kindergarten, they learned that the last number name said in counting tells the number of objects counted. When students work with arrays, they may take some time to construct the understanding that a 7×7 array that is subdivided contains the same amount of squares as a 7×7 array that is not subdivided. They wonder, do they really both have 49 squares in them?

Take a look:

I asked this student which one of these boxes of truffles was easiest to determine a total. His answer is a reflective and clear statement about why this progression is challenging

I am fascinated by how students progress from using and understanding units of 1 “item” to using and understanding units of “1 group”.

When Jack and I were talking about how the two subdivided arrays were different, he commented on the different numbers he used to do the subdividing. I asked him, “What does that 3 mean? ” He struggled to come up with the words to describe it. He tried “three slots of truffles in a line at the top”, but he recognized that his words weren’t capturing his meaning.  I decided to focus on the middle array since that was the one he proclaimed was easiest because he could skip count.  He could “see” and use  the groups in the middle array.

Listen to him. You can hear him revising his thinking about “items” and “groups”.

After students are comfortable using skip counting, they usually start thinking about and using repeated addition.  This is when they start grouping their groups.  Listen to Ellen describe how she knew the amount of truffles in these arrays:

I have talked a lot about how I think the Counting and Cardinality progression resurfaces in third grade. Now, let’s look at the Operations and Algebraic Thinking progression as it applies to third grade. If you haven’t read the progressions documents that are on Bill McCallum’s website, you really should. I reread them all the time.  They really help me understand the standards.

Here are some quotes from the third grade section of the OA progression.  See if you recognize any of these in the videos you just watched:

• “In the Array situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the situation. But rows and columns depend on the orientation of the array. If an array is rotated 90 degrees, the rows become columns and the columns become rows. This is useful for seeing the commutative property for multiplication”

The progressions describe three levels of representing and solving multiplication and division problems:

•  “Level 1 is making and counting all of the quantities involved in a multiplication or division situation.”
•  “Level 2 is repeated counting on by a given number, such as for 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern.”
• “Level 3 methods use the associative property or the distributive property to compose and decompose. These compositions and decompositions may be additive (as for addition and subtraction) or multiplicative.”

I want to understand more about how children use what they have constructed about counting, addition, and subtraction and apply it to make meaning of  multiplication and division situations. Bill McCallum says, “These skills and understandings are crucial; students will rely on them for years to come as they learn to multiply and divide with multi-digit whole number and to add, subtract, multiply and divide with fractions and with decimals.”

I wonder if we give them enough time to really explore the connections between these concepts.  I also wonder if we give them enough time to defend and question the ideas they are developing.

I also wonder:

• Would these students have used the same strategies if I showed them this picture:
• How could I help Erin see her addition statement, “I know 5+5 =10 and 10+10 =20” as a multiplication statement, “2 groups of 2 groups of 5”.  Is Erin thinking intuitively about associativity?
• Are certain properties more intuitive to certain kids or is it about what learning opportunities they are exposed to?

I wonder.  I wonder. I wonder. What do you wonder?

# How Many?

Listen. This is the sound of sense making.

I taped this audio recording at the end of the day.   The three boys were waiting to picked up. They came over and said, “What are you doing Mrs. Caban?” I told them I was reading a blog post about decimal quick images (thanks Kristin Gray).  They asked me if they could do some more problems like we did this morning. So, I showed them the one above.  It was so cool to listen to their voices collectively buzz as they wrestled with unit conversion.

I started the morning with these boys in their fifth grade math class.  Mrs. Gordon and I have been co-planning and co-teaching this week. We are introducing operations with decimals and we decided we want to anchor the unit in the student’s prior knowledge about decimal fractions.

Yesterday, Mrs. Gordon did count around the circle. First, they counted by tenths. We had planned that she would intentionally ask each student how he/she wanted Mrs. Gordon to record what they said – at least in the beginning. We were curious about how many of them were actually picturing decimal fractions and how many were picturing decimals. We knew that how we recorded what a student said would influence how the subsequent students in the counting sequence responded. We also decided she would record in rows of 10.  Some interesting things happened.

The boy who started the circle told Mrs.Gordon to write “one over ten”.  The tenth person said, “ten tenths or 1 whole.”  Mrs. Gordon said the students seemed to enjoy the challenge of changing from mixed numbers to decimals to improper fractions and back to decimals again.  They noticed that the halfway point of each line was always composed of 5 tenths.

Then they counted by hundredths.

Today, when I went in, we wanted to see if we could get the kids to establish some relationships between tenths and hundredths.  We knew we wanted to try to switch the unit as we counted – start with tenths, move to hundredths, move back to tenths. We decided to record the counting sequence on a series of blank number lines that were segmented into ten sections.

When we got to 1 and 1 tenths, I asked everyone to pause.  I told them we were going to switch to counting by hundredths. I asked Liz if she wanted any suggestions from her classmates or if she wanted to try it herself.

Liz said, “Well. I think it will be 1 and 1 tenth and 1 hundredth.”

We wondered, “What do we call that?”

“Well,” said Liz “I think there are ten hundredths in 1 tenth so that would be like having 1 and ten hundredths plus another hundredth which is 1 and 11 hundredths?”

I wish I had written 1.10 underneath 1.1 on the recording sheet,  as Liz was speaking, so she could see the ten hundredths that she had so elegantly described.  We continued to count by hundredths until we got to one and 17 hundredths. Then, I switched us back to tenths. There were audible gasps.

“Whoa.”

At this point, we could tell that the transition back to tenths would be tricky for a good number of the kids. There were some who thought they could just add one tenth to the tenths place, but they struggled to convince some of their peers.  One student was stuck.  I asked him, “Do you think you could use a number line?”

He drew a number line and figured out that he could decompose the tenths into ten hundredths.  Then, he broke up the ten hundredths into 3 hundredths – to get the nearest tenth and then added 7 hundredths to arrive at 1.27.  Yay! We were really hoping someone would consider decomposing.  He still had to count be one hundredths to arrive at his solution, but he was able to see the .3 and .7 chunks after he counted. I decided to add his representation to our chart. You can see it above.

I wish that I used a bigger number line so everyone could see the hundredths. I also wish we grouped the kids in smaller circles so everyone could engage in more counting. So… we made three big number lines to use later in the week.

I have been thinking a lot about productive struggle.  What does it really mean to struggle productively? This is what I found when I Googled the words:

: achieving or producing a significant amount or result.

That seems pretty straight forward.  How about “struggle”?

Yikes! I would like to think we can avoid violence in math class.  Could the purpose of productive struggle in math class be to “strive to achieve or attain something significant in the face of difficulty or resistance”?

This week, I  facilitated Math Learning Labs with our elementary school teachers. It might have been my favorite week of school so far. Learning labs are a time for us to learn and grow together. Last week, I wrote about my experience with the Kindergarten teachers. Today, I get to share what happened with the third grade teachers.

Once a week, since early September, I have been co-teaching with Mrs. Watkins.  We have been trying to help her third grade students build stamina for problem solving. You can read more about that here. Since the grade level meeting was going to take place in Mrs. Watkin’s building (we live in a small rural district so each elementary school takes a turn hosting a meeting), I wanted the learning lab to meaningful for her. We discussed what kind of experience would be beneficial for the teachers AND also for Mrs. Watkin’s students. We decided that we would like to give the students an opportunity to justify their thinking and the teachers an opportunity to support productive struggle.

Here is what we came up with for an agenda:

We started by reading  two blog posts about the Hundreds Face Challenge, one written by Malke Rosenfeld and one written by me.

Then, I gave the teachers a collection of statements about productive struggle. I got most of these statements from NCTM’s recommended teaching practice, Support Productive Struggle in Mathematics. I added some of the statements that Mrs. Watkins and I had discussed with her students.

I asked the teachers to arrange the statements so they showed a progression or “road map” through productive struggle. This is what they did:

Then, we jumped right into building our own Hundreds Faces.

Some people started by counting out a collection of rods that equaled 100 and then building and adjusting as they went.

Others, started a design and kept track as they went:

Everyone was able to build and justify a hundreds face:

After we built our hundreds faces, we shared our thinking and reflected about how our approaches were similar and different. I shared how I had figured out that if I could rearrange my Cuisinaire rods into a 10 x 10 square  than I could prove that I used the equivalent of 100. I wondered, would you always be able to make a 10 x 10 square out of any hundreds face?  Could you make a 25 x 4 rectangle?  They wondered these things too. We agreed to explore these questions further during our next unit.

Next month, we will be using Muffle’s Truffle, one of the mini units in Cathy Fosnot’s Context for Learning Math series, to kick off our third grade exploration of  the relationship between arrays, addition, multiplication, squares, and rectangles. This will be a perfect time to explore our questions further.

As we revisited  our maps of productive struggle. I asked, “did you experience any of these statements while you were building your hundreds face?” The teachers responded with many connections:

“We all solved problems.”

“We had to find our own mistakes.”

“We had to prove and justify our thinking.”

“We asked a lot of questions.”

Then, they started to have a really interesting conversation about whether the math leverages the creativity or vice versa.

Have a listen:

After this conversation, I asked the teachers to each make their own individual map of productive struggle. We didn’t have a lot of time, but I was hoping they would transfer whatever meaning they had constructed to a sketch.  We each thought of our journey through productive struggle differently:

After we shared our maps, we brainstormed some questions that we could have in our back pockets as we interviewed the third graders about their hundreds faces. Here are some that we came up with:

• How do you know if this is equal to 100?
• Can you show me?
• Is there another way we can figure if it is 100?
• What does that (having too many or too few) mean?
• What do you need to do next?

Finally, we went into third grade to see whether or not the students could justify their hundreds faces.  As we circulated the room, the kids were eager to share all their strategies with us. Unfortunately, most of the video footage from this experience is full of so much math talk that it is hard to discern who is saying what.  Fortunately, there is a TON of wonderful math talk happening in this classroom. Here is a sample of an exchange between a teacher and student:

S:  “Right now I am at 81.”

T:  “and you have to get to how many?”

S: “100”

T:  “so how many more do you need?”

Below, you can see some sample faces with their justifications. Can you find the math in the pictures?

Right before we were wrapping up with the kids, our district curriculum coordinator managed to capture her conversation with a student who was navigating his way through some big ideas about area, geometry, and multiplication.  Take a look:

If the purpose of productive struggle in math class is to “strive to achieve or attain something significant in the face of difficulty or resistance”, was Marshall engaged in productive struggle?   What did he produce that was significant?  I noticed that he produced a pretty significant statement regarding the classification of the shape that he created:

“It’s a rectangle!”

I wonder which was more significant; the statement he made or the resistance that got him to a place where he could claim it? Think about the questions/statements that Nancy contributed to her conversation with Marshall:

• What is this?  Can you tell me about what you made here?
• How do you know it is a square?
• Rephrase:  “So, anything with four edges and four ends…”
• How do you know that those are equal edges?
• Prove it to me.
• Oh, you are measuring with your hands. What if I said show me with the rods?

Now, think about what would have happened if Nancy had never asked the first question. What if Marshall left class thinking he had built a square?  Teachers often admit to me that their biggest concern about allowing students to experience disequilibrium is that the student might leave the classroom confused about something.  What if they leave thinking the wrong answer is actually the right answer?  What if they leave my classroom not knowing all the answers?

I usually respond that the difference between the struggle being productive and unproductive is the teacher’s level of awareness. If you know your students have partially formed understandings then you can revisit and explore these with carefully planned questions and problems. But, how do we know what our students don’t know?

We ask.  Then, listen. I mean really listen.