Late last year, Joe Schwartz wrote a very interesting post about the difficulties one student is having with counting and skip counting. I recall hearing a theory that many later math difficulties trace back to when a student missed solidifying the concept of one-to-one correspondence and some related concepts of counting (but don't have a citation or reference). As a result, I was thinking about how our little J's understand these ideas.

In particular, I wanted to try out Marilyn Burn's little game from one of her comments with J3:

Ask him to put out 8 cubes on a paper. [I chose 8 because when I remove one, the child won’t be able to know how many by subitizing.]

Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]

Say: Watch as I take away one cube. Remove one cube and place it on the table.

Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]

Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table.

Ask: How many cubes are there on the paper now? (6) [This is the same as the previous question, a way to check if the child still needs to recount.]

Say: Watch as I put one cube back on the paper.

Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]

Sometimes I repeat again removing a cube and asking: Can you tell me how many there are without counting? Some kids shake their heads to indicate they can’t, others say they’ll give a guess, some are able to.

# J2

Before I had a chance to try out the question sequence with J3, I had some time alone with J2. He was sorting colored pencils, so we used those as counters. Overall, he breezed through the questions, but there were two amusing points:

- After separating out 8 colored pencils, I asked how many he had. His response showed that (a) he believes in conservation, so he
*knew*there would be 8, but also (b) he is used to me doing something tricky, so he wanted to*verify*that there were still 8. - His method of verification: split them into two groups of 4, an amount he could recognize by subitizing, not counting.

He asked me why I was asking these questions and I told him it was related to his understanding of hierarchical inclusion. We talked briefly about what that means and he was delighted by the term, so ran off to ask J1, "how is your understanding of hierarchical inclusion?"

# J3

My counting time with J3 came during dinner. She was eating cucumber slices, so we used these as counters. This turned out to be a mistake, since

*conservation doesn't work*

**! In other words, whenever I asked her how many slices were in our cluster, she would pop one in her mouth and smile, knowing that she was teasing me.**

*with edible counters*Mainly, though, I was able to verify that she doesn't yet have the concepts that allow short-cuts to the questions in Marilyn's sequence and needed to recount to get all the answers.

Incidentally, I started the activity by telling her that we were going to count something together. She immediately grabbed this coaster and then accurately counted the circles to 37.

For me, the entire experience was a really interesting illustration that counting actually requires a complex collection of sub-skills.

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