I received this note recently:

I teach 1st and I’m also in … M.Ed program for K-6. I’d be interested in having you come to our class. Let me know if that’s still possible this year.

Can I make time to talk math with young children? Of course I can! Pretty soon, we had agreed to a date and time, and to combining two first-grade classes, and also to a kindergarten visit.

It was a delightful time full of manic energy from me and the children alike.

Let me tell you about some lovely moments.

## Halves are hard

I am presently working on the teacher guide for a book, *How Did You Count?* That book is geared towards third-through-sixth graders. I think of it as *How Many? *meets Number Talks.

I have been curious what younger children will make of some of the ideas, but the numbers I ask children to play with are too big for the youngest mathematicians to have a reliably good time with them.

So I shot a simplified version, and tested it out with these first graders. How many oranges? How did you count them?

The class generally agreed on six. Some counted as two threes; others as four and two more. Nobody told me they counted by twos. Several counted one at a time.

I expected a lot more one-by-one counting to be honest. It was lovely to hear a diversity of ways of seeing things here.

Here’s the next picture we looked at.

There was a near consensus that there are still six oranges. The cutting seemed to be immaterial to these children. Fascinating!

I did hear some claims of four and five, but mostly they were focused on the sameness of this arrangement rather than the difference in the amount of orange.

In one individual conversation I got to listen in on, a teacher asked a student to notice that these were halves. “Two halves make a whole,” she said. The child proceeded to look hard and to think, and finally said “twelve,” having presumably counted by two, six times.

## Diamonds are hard

Another side project has me testing ways of turning the vehicles conversations into fun board books for three-and-four year-olds. These books begin with things that are very much not examples of the thing, and move towards things that share characteristics of the thing, ending with a call to find lots of examples in a culminating image.

For example, I asked kindergarteners whether each of the following was a square, and how they knew, if not. I asked them to count the squares in the final image.

That blue rhombus was especially difficult to describe. I had been told that the penny and the potato were each lacking four sides, so I noticed for the children that this shape did have four sides. They were not persuaded, but they also struggled to state exactly why. There was a lot of gesturing; fingers being used as sides. Productive struggle.

The three-sided figure *was* a square to at least some children; it was just missing its roof.

We had a fabulous time counting the squares, and I was very surprised that the interior blue and orange squares were counted before the squares that frame them. They insisted on counting the whole photograph as one of our squares.

## What counts?

The classroom teacher in the first grade classroom wondered whether we should count the bowl in the lower left of this image as a circle, since (1) the whole thing isn’t in the photograph, and (2) we don’t really know whether the missing part MAKES it a circle; maybe that part does something strange in real life.

These children were very much persuaded that we should only count the full circles we can see.

Also, a very perceptive child pointed out the little tiny circular dot in the center of the plastic cap at lower right.

## It’s been too long

I haven’t been in classrooms much in the last few years. These young ones were late-spring afternoon, post-recess energized! During one *talk with your neighbor* break, I wondered whether as a group we had the stamina to do much more. My host teacher was having none of it! We shall persist! And so, together, we did.

I realized too late that “raise your hand for yes; don’t raise for no” is a bad instructional decision in several ways; especially if the answer to many of your questions is “no”. This is why we our thumbs go both up and down, so that’s how I asked the second group to respond. Magically, I could tell the “no”s from the non-responses.

All in all, a lovely way to spend an afternoon. I learned a lot, and the children were brilliant.

They always are.