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!

With the substantive conversation that occurred you would not have know that she didn’t fully understand. In middle school students get cloudy with the language of reduce, lowest terms, simplify. The conversation is a formative assessment, but also deepens understanding.

I’m reminded of Piaget’s concept of conservation. Depending on how old Abrianna is, it wouldn’t be unusual for her to struggle with this notion. In case you’re not aware of (or don’t recall) this concept: Conservation is what allows adults to see that, when water is poured from a bowl into a tall, skinny class, the water will have a much different height even though there’s still the same amount. According to Piaget, this ability typically develops somewhere between age 5 and age 11. So it’s possible that, by tying fractions so intimately to real world examples (instead of focusing on fractions-as-division), we may be working against cognitive abilities of children.

Also, it’s not technically accurate to say that “equals to” means “identical to”, especially with the Common Core emphasis on teaching fractions in older elementary years in terms of the denominator being a unit. That is, 2/8 is two 1/8 units, while 1/4 is one 1/4 unit — those two have the same measure (and hence are equal) but are not mathematically identical.

Paul, you’ve made a good point about Piaget’s work in children’s cognitive development. My best guess is that Abrianna is 9 years old since she is in a 4th grade classroom and so she falls within the age 5 to 11 range that you point out.

What fascinated me about the conversation I had with Abrianna is that she seemed to be saying to me that when you say “something equals something else” what you really mean is that “something is identical to something else” since her two examples of when two things are equal involved 2 (2 equals 2) and 5 (5 equals 5). For me two things that are identical are equal but two things that are equal are not necessarily identical.

We moved in our conversation to discussing who would get to eat more pizza (the person who gets 3/4 of a pizza vs. the person who gets 6/8 of a pizza). She was adamant that the person eating six slices of pizza would be getting more than the person eating three slices of pizza. I didn’t include this in the post above but on another paper Abrianna showed me a pizza divided into fourths and then that same pizza divided into eighths. She confidently could do that (draw the lines) and she could talk about how each of the 1/4s had two slices that were 1/8s.

I wish I had had more time to talk with her to try to find the disconnects or if there was some way that we were talking (and the words I was using and that she was using) were the root of what I think was a misconception. It really was too short for me to completely uncover what might have been going on.

My simple conclusion was that Abrianna had “learned” notation that she didn’t yet completely understand. My hope is that she’ll keep talking and thinking and understanding more!

~Suzanne

I’m with Abrianna on this one. It seems to me that she understands at least at some level that there are different degrees of “equality” depending on what attributes you’re interested in. This is the essence of abstraction.