About a month ago I visited an elementary school classroom. The students were working on a problem that (as my memory stored it) was something like this:

*Ricky ate 2 pieces of a pizza. He gave the rest to his friends. How much of the pizza did he give to his friends? Can you write that in lowest terms?*

The students were seated at tables, some working in pairs, some working alone. I asked one of the students if I could sit next to her. She said, “yes” and after I had introduced myself, she told me her name was Abrianna. I noticed Abrianna had already been thinking about the problem and had written something like this on her paper:

I remember thinking that from looking at her paper, it looked like she really understands what’s going on in the problem! I asked her to talk to me about what she was thinking as she wrote (and I pointed to the paper). She said that since Ricky had eaten 2 out of the 8 slices of pizza that he had eaten two-eighths of the pizza. I asked, how did you know the whole pizza had 8 slices? She patiently pointed to the picture included with the problem and said, “See the lines?” “Oh,” I said, “I see them now!” and smiled.

She continued to tell me more about what she had written but when we got to the “equals” sign she said “No, two-eighths is NOT equal to one-fourth.” I was surprised because that seemed to have been what she had written. I started wondering if “equals” meant something different to her than using the “equals sign” and so I asked her, “What are two things that ARE equal?”

She responded by saying, “Two equals two.” When I asked for another example, she said “Five equals five.”

I pointed to the two pizza pictures and I said, “What if you had eaten the pizza that we can see is gone from this first picture and I had eaten the pizza gone from this second picture, who ate more pizza?” She said, “I ate more because I ate 2 slices and you ate 1 slice.”

I tried again and said, “If you ate 6 slices from a pizza cut into 8 equal pieces and I ate 3 slices from a pizza cut into 4 equal pieces, who ate more pizza? She said, “I ate more because I ate 6 and you only ate 3.”

My visiting time in that classroom was over and I moved to observe another classroom but my conversation with Abrianna stuck with me throughout the day and as I talked about it with my colleagues as we walked back, I described the conversation by saying,

I noticed

- Abrianna’s paper included the correct notation.
- A teacher looking at her paper might assume she understood the problem.
- Her teacher would probably give her credit (or a grade) indicating she had the right answer.
- Abrianna used “=” on her paper.

I wondered

- why “=” and “equals” prompt such different responses from Abrianna.
- why does “equals” mean “is identical to” for her.
- why doesn’t “=” mean “is identical to” for her.
- did Abrianna just copy the notation as a way to “do the problem.”

I was reminded again of how important it is to unsilence students’ voices. I hope Abrianna has continued opportunities to talk about her mathematical thoughts!