to the school, he was placed in a triad with his longtime friends, Chad and Isaiah, who liked to banter and challenge his ideas. (I noted Chad in an earlier section with Alberto; this section complements that description.) The administrator described Chad as an unmotivated male who enjoyed the social aspects of school rather than the academic. He spent the last two years in a different school where he claimed he learned nothing. The other partner in the triad, Isaiah, was of average ability but was not one of the six selected
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individuals. The administration said that Isaiah earned B’s and C’s with seemingly little effort. He enjoyed socializing and courting Ester, a female in the class.
The task for session 2 required students to explain a corollary that involved a perpendicular line as a mean proportional between two segments on the diameter of a circle. The book provided no diagram. I include a diagram and explanation to aid understanding of their problem.
B
During T1, all three males read quietly. As soon as the Pair phase began, Bradley laughed, looked at the teacher, and raised his hands. He explained during the VSRI that he believed he had the solution at this point. The other two males said they did not understand so Bradley read aloud to them and drew a picture in his book. He explained by saying, “So this is going to be the same as this.” Bradley believed that it was a very simple problem and that his answer was correct. The other two males continued to question him, pointing to the book, repeatedly using the terms this and that and pointing at lines around the picture. Bradley looked at the teacher, laughing, seeking help to quiet his partners. The teacher moved away. Bradley asked why they did not understand and Chad responded, “Because you circled everything.”
When the teacher returned, Bradley asked, “Is the perpendicular dividing right there or is it everything? This is the diameter.” I coached the teacher that responses
A
D
C
Given BD perpendicular to AC, AD : BD = BD : CD
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should be limited during Pair work and he did not respond. Chad continued pointing and tapping, saying, “So this and this” (moved hand sideways) “are to this” (moved hand up and down the segment perpendicular to the diameter). Suddenly, Bradley stared straight ahead with a blank look and mouth open. He exclaimed, “Oh! I get it now!” His head rotated upward and his eyes moved around the space in front of him. Chad said, “Wait! So am I actually right?” This time, Bradley looked at the book and said slowly, “This is to this as this is to this” as he pointed distinctly at segments. Chad responded, “That… is what I said.” He looked at the teacher, not laughing, mouth slightly open, as if in
amazement. After a moment of pause, the three males began arguing over who had the solution first.
Still in the Pair phase, Bradley and Chad began thinking about the presentation. Once again, Bradley used the this is to this verbiage. Chad lowered then raised his head, placed his hands in a vertical position, and slowly enunciated, “the vertical, perpendicular line…is the… What is the bottom part of a fraction?” His partners answered. Chad
continued, “Denominator, denominator. And the horizontal part of the line is the
nominator [sic].” Bradley looked at the teacher and said, “I’m not saying I’m not having a good time. But from a learning point of view…” and he shook his head to imply that it was not helping him to learn.
Bradley and Chad both tried to talk during T2 until I reminded them to be quiet. Bradley wrote something in his book and Chad looked around the room, appearing disengaged. When Chad learned he would have to present, he panicked and asked Bradley for help but Bradley said he could not talk. Bradley played with his cell phone.
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Chad ran his finger up the perpendicular line segment and tried, unsuccessfully, to get Bradley’s attention.
During the Share phase, Finn and Diana presented prior to Chad. Diana’s
presentation actually involved the right triangle inscribed in the circle for Bradley’s task but the book did not point that out and no one mentioned it. All students except Bradley watched her; he looked at his iPad until the teacher began to speak. As soon as the teacher finished, Bradley returned to the iPad. Diana had an error. The teacher asked her to reconsider and called upon Chad for the next presentation. The teacher saw Chad’s error and spoke with him and the others in the group while Alberto presented. After the teacher addressed the triad, Bradley shook his head affirmatively, pointed to the book, then picked up his iPad again. He was convinced that he had the correct solution but Chad had not presented it correctly.
This session was the focus of the VSRI for Bradley and Chad where they revealed differing insights. I will share Bradley’s perceptions here and will include Chad’s in his own section. Bradley believed that he nearly always solved problems within a few seconds during T1. He claimed to need no writing aids since it was all in his head. He would then sit there and wait. He said that the bantering with his partners in session 2 was merely their effort to confuse him because he already had the solution. He thought they were just pointing to random lines. When shown the clip where he expressed new insight, he said, “I’m sure something probably clicked in my head that was…I was trying to figure out. I might not have been 100% sure about it.” We looked at the clip again and I probed further. He stated that the other two were just trying to, “drive him nuts” and that any new insights came from looking over what he already had. He explained that during
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the Pair phase others usually looked to him, asked what he thought about it, and then agreed - except with these two males. He said that these two always tried to come up with something different. Bradley stated that when he asked other people for their thoughts, they gave no input. “I usually think maybe they assume that I am right so they don’t say anything.”
Asked if he thought others could explain his reasoning after the Pair phase, he said, “They probably could after they heard me explain it.” This was not the case in session 3 when he paired with Ester.
Session 3 with Bradley and Ester. The teacher selected the rectangular cake task