Davel
23 4 92
9 4 36
9 4 36
5 4 20
36 36 20 92After the work of these three students was on the board, I asked, “Are there any
similarities among these ways of solving the problem?” The room was quiet for a
short while. Most of the students seemed to be thinking.
JULIO: Two used multiplication. The number 4 appears in the work of
Fleurette and Davel. [another long silence]
ALEJANDRO: You can see the work of Fleurette’s equations in Lucia’s skip
counting method.
TEACHER: What do you mean?
ALEJANDRO: You can see the 10 4. [Goes to the board and points] 4, 8, 12,
16, 20, 24, 28, 32, 36, 40. When she skip counts by 4s and gets to 40, she
skip counted 10 times.
TEACHER: Can you mark that with a marker?
ALEJANDRO: This is 10 4. [He circles the numbers from 4 to 40.] Then she
does another 10 4 .[He circles 44, 48, 52, 56, 60, 64, 68, 72, 76, 80 with
another color.] 40 and 40 makes 80. Then these 3 are the 3 4: 84, 88, 92.
[He circles these 3 numbers with another color]. Lucía counted by 4s 23 times.
But if you want to do it faster, you multiply.Alejandro made a very strong connection between both strategies, but I
wasn’t sure everyone understood his explanation. I continued the conversation
by asking Lucía to rephrase what Alejandro had said. Lucía did not completely
trust multiplication. She was not using it consistently in her work, or she skip
counted first and then wrote a multiplication equation to match it. I purpose-
fully asked Lucía to rephrase Alejandro’s idea to help her articulate the con-
nection between both strategies. Had she understood what Alejandro said?
Although Lucía explained Alejandro’s idea in her own words, it was clear that
she was still working through it. Yet, to hear other students say out loud what
she was considering silently validated her ideas and at the same time clarified
her thinking.
Students don’t always say what I would like to hear the first time I raise a
question, but I keep asking the question. It is not necessarily that students don’t
have the language to answer these questions. Their initial silence might mean
that they had never thought about finding similarities among strategies, and that
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