The Hard Work of Counting by Groups, Part 2

Last week, I wrote about some first graders’ work on problems about how many fingers were on 4 or 8 hands. This week, I want to share an interaction I had with one child, as the class’s work turned to thinking about groups of 10. Students were working on two types of problems about cubes, organized in towers of 10.

·      Given the number of towers of 10, how many cubes?

·      Given the total number of cubes, how many towers of 10?

When I joined Nik, he had already solved problem 1, about a boy named Max who had 3 towers with 10 cubes each. I was stunned that this first grader had accurately drawn a 10×3 array. Nik was very clear about what the problem was asking, explaining that Max had 10, 20, 30 cubes.

The 2nd problem — about a girl named Rosa, who has 70 cubes, in towers of 10 – seemed to be giving him trouble. He had a bin of black cubes, recently organized into towers of 10 by students, and a bunch of towers of 10 on his desk. On his paper, he had written 10+10=20, 20+20=40, and 40+40=80. When I asked about his thinking, he reached for the eraser. I encouraged him to leave his work, so we could think some more, but he was determined. He erased it all. Then, he started snapping the towers of 10 into one long train.

On his paper, he wrote: “I stacked all the cubes on each other.”

Nik: Now I am going to count them.

On his paper he wrote: “I counted all of them.” and then drew a circle, where his answer would go.

Then he turned back to the cubes — one long train, and two unattached towers of 10. He began to count, from 1, carefully and accurately. At the end of the long train, he said 52. He counted on the two other sticks of 10, got 72, and wrote 72 in the circle. He seemed satisfied and proud.

I complimented his careful counting and asked, “So what did you find out?”

Nik: There are 72.

Me: I agree that you have 72 cubes there. Let’s look back at the problem. How many cubes did Rosa have?

Nik: 70. Huh.

He removed 2 cubes and put them back in the bin. I asked him why he did that.

Nik: There were 72 and we want 70.

I was surprised that he “just knew” that 72 was 2 more than 70.

Me: Ok. Now, what are we trying to figure out?

Nik gave me a puzzled look. He clearly thought he’d solved the problem. We reread the problem together.

Nik: I can count the 10s.

This threw me. He had just combined all the 10s into one long train!

Me: Show me what you mean.

He counted “1, 2, 3…10,” snapped it off, and handed it to me. I placed it on the desk, above the long tower.

Me: So 10 cubes makes 1 tower of 10.

He returned to beginning of the long tower. He surprised me by starting at 11.

Nik: 11, 12, 13…20.

He snapped it off and handed it to me. I placed it with the other tower.

Me: So 10 cubes made 1 tower of 10. And 20 cubes makes 1, 2 towers of 10.

We continued in this way. He counted 21, 22, 23…30; 31, 32, 33…40 and so on. Each time, we placed the new tower with the others and commented on the new total. He began filling in the final sentence, e.g.

Me: So 30 cubes makes…?

Nik: 3 towers of 10.

He finished the last train, “61, 62, 63…70” and placed it with the others.

Me: How many towers of 10 do we have?

He counted them, touching each as he did so: “1, 2, 3, 4, 5, 6, 7.”

Me: So, just like us, Rosa had 70 cubes. How many towers of 10 did she have?

Nik: 7!

He picked up his eraser, erased the 2 in 72, and headed off to recess.

I’ve thought a lot about this interaction since then. How Nik was perfectly happy to keep thinking and working with me. How fluently he counted by 10s for the first problem. That 10×3 array that research suggests is difficult for students older than Nik to draw accurately. How he wrote out his strategy for problem 2 before enacting it. How he counted those groups of 10 starting with 11, 21, 31, 41, 51, and 61. How often I think I know what a student will say and end up surprised!

As we talked, what I kept thinking about was how kids come to understand big new ideas, like ten ones can also be thought of – and counted as – one ten. How kids move in and out of understanding. How they try on new strategies with confidence, and then revert to ones they really trust when they are not sure. How problems that feel similar to us as adults, feel quite different to students, for whom these ideas are still fragile. How every time I have a conversation with a student, about their thinking, I learn something new.