8 1/2 by 12 Inch Paper: Cardinal vs Ordinal Counting in Third Grade

One of the advantages of my new job this year is that I sometimes (not always, not even most of the time, but sometimes) have the luxury of not feeling rushed and not being in charge of the whole class, and I can spend a long time watching just one or two students as they work through something.

That is what I did on Friday, when I worked with two third graders who were tasked with finding the perimeter of a piece of paper (an 8 1/2 x 11 piece of paper, for those of us in the know).

When I first came upon Sonia and Freya, they had carefully marked the inch lines from their ruler along the length of their piece of paper, and labeled the marks from 1 to 12.

It's twelve inches long, they told me confidently.

Interestingly, they had already measured the other dimension, and told me (correctly) that it was 8 1/2 inches long.

See how they did the two sides differently?

I asked them to show me how they counted each side. What ensued was fascinating. Each time they counted, they got a different measurement. Sometimes they counted the inches themselves, by putting their fingers in the middle of the inch (cardinal counting). When they did that, and got 11 inches for the length, they would say something like, "Oh! It's 11 inches long!"

I would then say, "Oh, okay. So it's 11 inches long. Show me how you can use the ruler and get 11 inches." And they would line the ruler up with the 1 at the edge of the paper, count using the numbers on the ruler (ordinal counting, but thinking of 1 as the beginning of the first inch, instead of the end of the first inch), and get 12. "No," they would say, "we were right after all. It's 12 inches long."

"Where is inch Number One?" I asked. They pointed to that little tiny tick mark at the edge of the paper.

When I asked them to show me how they got 8 1/2 inches for the width, they did this:

Look at that! When they started from the "wrong" end of the ruler, they got the right measurement, because they didn't use the numbers on the ruler at all. They counted the inches themselves (see that pencil pointing in the middle of the inch, rather than at the number?) and didn't let the ordinal numbers on the ruler confuse them.

When I asked them to turn the ruler around and re-measure this side, they got 9 1/2 inches. At some point, I grabbed color tiles and, once I had them confirm that each tile was 1 inch long, we lined them up along the edge of the paper and counted them. 8 1/2 inches. Then we added the ruler to see how it compared to the color tiles.

Again, they lined the 1 up with the edge of the paper, but they counted correctly because they were counting tiles.

This whole conversation went on for at least 45 minutes, so I don't have a detailed record of what I asked and what they said. What I can tell you is that they measured the dimensions of that sheet of paper over and over and over again. Sometimes they got 11 inches. Sometimes they got 12 inches. Each time, they were convinced they were right, until I would remind them that the last time they had reached a different total, and ask them to show me again how they knew for sure. Then they would get the other answer.

After many, many counts, they concluded that the paper was 8 1/2 inches long by 11 inches wide, and added up all the sides to find the perimeter.

Their next step was to find the perimeter of something else in the room. They chose a copy of Strega Nona which, coincidentally, was also 8 1/2 inches by 11 inches. And... once again they placed the 1 on the ruler at the end of the book and told me the book was 9 1/2 inches wide.

Again we lined up the color tiles and counted them (8 1/2), then added the ruler.

I can't remember what led to them lining the ruler up correctly this time. What I do remember is that Freya continued to insist the book was 9 1/2 inches long. (Can you see in the picture how her tiles continue past the edge of the book? She attempted to make her tiles equal what she thought the measurement should be.) I remember touching that first tile on the right of the picture and asking her, "What number tile is this?" "0," she answered. (Which would have resulted in a measurement of 7 1/2 inches if she had followed this line of thinking!)

"Zero?" I asked. "If you were counting the tiles, you would touch this tile and say zero?" (A leading question if there ever was one.) "Oh, no," she answered. "It's number one." Eventually, they talked each other into the correct measurement and seemed convinced.

At times, Sonia and Freya seemed to see the ruler as some kind of magical tool that gave them an answer but that didn't have to make sense. It was an object that was separate from their own counting of the tiles. I see this often among the students I work with: small comments like "Well, you can't ..." or "I have to..." as they name rules that either someone has taught them or they have made up for themselves. I have taken to saying, over and over again, "Math is supposed to make sense. There aren't rules that you have to follow just because. Whatever you do in math, you should do because it makes sense."

This is the same thing I say to beginning readers: "Whatever you read is supposed to make sense. If it doesn't make sense, you need to go back and re-read it until it makes sense." It's an idea that is not intuitive to beginning readers. Like these beginning measurers, they think that people magically know what the words on a page say, and that the words on the page are disconnected from each other and don't have to make sense.

I have written before about the importance of letting students take their time to do something over and over and over again before they trust it. (To me, trusting it means noticing the pattern or procedure, believing that it always works every time, and then beginning to use it automatically.) This year I am relearning that as some of my second graders repeatedly count groups of 10 by ones instead of counting them by tens. They have a healthy skepticism about numbers. Does this always work?

Sonia and Freya needed to do just that -- measure over and over again, and have someone (me) there with them to point out that their measurements were different each time, and ask how they could be sure the most recent measurement was correct. I was surprised, but also fascinated, that it was this challenging for them. It made me wonder:

• How does their understanding of the numbers on the ruler, and the difference between cardinal and ordinal numbers, connect to their understanding of the number line, which they have been using to add and subtract numbers in the hundreds? 
• Do they see the ruler as something completely different from a number line? 
• What questions would I need to ask to figure out if they think about the number line in the same way they think about the ruler?

It is so tempting to tell students how to do something. "You have to line the very end of the ruler up with the end of the paper," I've said to students in the past. And then they learn that as a rule, until they come to a test question like this, or find themselves in a real life situation where they can't start at the end of the ruler. When I have the time, I remind myself not to jump in -- to watch as they say "one" at the edge of the paper -- to wait, and then to ask questions. Moments of dissonance like these will help solidify their understanding, which is wobbly now, more than if I told them a rule. Meanwhile, watching their thinking and their questioning is such a treat.