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!