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this was a task chosen because it was more challenging than the last two problems. It became obvious in this task that the struggle with the language of mathematics created a need for more time to solve the problem for these two students. They supplemented each other’s voices as they worked to solve the task.
Finn and Ester began the Pair phase with the formula for the area of a triangle. Finn began voicing the equation, struggled, and Ester finished.
F: Ummmm. It’s half, it’s…so E: Ummm. Half base times height.
Although they voiced “right triangle” and “perimeter”, the connection with the geometric figure was not immediate. Finn said, “so… a right triangle and perimeter” then drew a square on his paper. They both repeated “right triangle” three times before Finn said, “So if it is a right triangle, this thing is 30 cm.” Ester corrected him emphatically, “It is a right triangle. It’s not a square.” Finn hit his pencil, said, “Oh, wait!” and drew an equilateral triangle below the rectangle. As soon as he saw the triangle, he said, “Ohhh,” sat up straight, scratched his elbow and focused on the drawing. Ester said, “It’s a triangle, not a right triangle.” They continued to finish each other’s sentences while they corrected the drawing of the right triangle and both wrote on the diagram.
E: So it it’s F: A triangle E: So it’s 30 cm.
F: The whole thing is 30 cm?
E: Yeah and then one side is 7 cm longer than the other. Yeah I’m guessing that’s 7 and then
136 F: That makes this one…and the other E: Is 30, but
F: No, the other is equal to 23 (subvocalized as he calculated) E: So then what do we do with 30?
Finn and Ester postulated that one side was 7 and began finding the other two sides by dividing the remaining 23 cm in half. They struggled with the calculation and looked for a calculator.
F: That’s the total combined. (pause) That’s the total. So… (Subvocalizes again 23 divided by… Is what? Is… 11.5. So we’ll do ___ plus ___ 25 … equals 22, 23 plus 7 equals… twentyyyy thirty. For some reason…(pause) What’s the area of the triangle? Yeah that’s not right because that’s stuff I figured out about the triangle.
E: Do you have a calculator?
The bell rang to end the Pair phase without further mention of one side being 7 more than another. During T2, Finn scratched his neck and wrote diligently on his paper. Finn continued writing on his paper even after the bell rang to begin the Share phase but listened to other presentations.
Post Interview and Summary with Finn. Finn’s placement in the first session was based on his preference to work alone. However, in the post-observation interview, Finn changed his preference to working with others because they helped him to figure out the steps if he had problems on his own. This preference, he said, depended upon how much he felt he could contribute to the group. “Sometimes I felt like I wouldn’t need their achievements with their grades being higher than me. I felt like I would be the
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underman going into the group. But then other times I would feel like I knew what I was talking about and could solve it.” He preferred to work alone when he felt he understood how to solve it and could get it done. Quiet time was still important to him for thinking through what he was saying, but he claimed that he now only used his book for assigned problems. Finn’s responses about how he used quiet time to work on assigned problems also changed from the pre-interview. In the post-interview, he stated that he used pictures for geometric shapes and he definitely liked being able to ask someone else for help.
Finn was quick to report that TPTS helped him to understand math. The additional quiet time was helpful when he tried to gather thoughts together but was a waste when he completely understood. It helped him to see the steps that other students took to get solutions; it gave him more ways to look at math. He stated that the last part of TPTS, Share, was probably the most important for him because he saw how others used their thoughts. He reiterated in the focus group that TPTS helped him see the ways other students thought and integrate those methods into his own thinking (Figure 4.6).
Figure 4.6 Finn’s feelings about TPTS phases
Think 1 Pair Think 2 Share
Excited to try to come up with the answer. To try to solve it, like a puzzle.
Nervous. Having to share my ideas.
(Long thought and then sigh). Don’t know, it’s a lot. I’m trying to think of a word that would fit. (Long pause) hmm, probably a little confused. Trying to put together their ideas with my ideas to come up with the conclusion of what the answer would be.
Nervous. Having to give my answer so everyone else could understand.
138 Ancillary Profiles
Incomplete data were collected from the last (non-selected) four students to provide a more complete picture of the classroom. For instance, Gabe was included in the results section even though he was not one of the six students originally chosen for the study because his insights often differed from other students. These students were videotaped in classroom settings including during the focal group interview. Gabe, male Caucasian, struggled with mathematics and English
Gabe allowed videotaping but not interviewing so data were limited to classroom sessions and profile information from the administration prior to the study. He told Alberto that geometry was more difficult than algebra. Field notes showed he responded quickly to calculation questions and frequently utilized the back of his book for
definitions and theorems. I suspected that he was at Van Hiele’s visualization level because he was unable to recognize a right angle in session two when the hypotenuse faced the bottom of the page. He quietly studied tasks during T1 in each session. Script from session 3 showed him inquiring about alternative solutions when he paired with Diana, an inquisitive female who considered herself a top student in the school.
Session 3 with Gabe and Diana. The pair went to the front of the class to get a