Chapter 15 page 343
Reflecting on Student Thinking
Rachel Williams, a fifth-grade mathematics teacher, regularly uses collaborative groups in her classes,
but she has a lot of questions about whether all of her students are benefiting from collaborative learning,
and, if not, why not.
Rachel wonders how to make students’ talk during group work as effective as possible. She decided to
listen carefully to the students in one group to try to understand what they say during group work and how
what they say is related to what they learn. During the first week of a unit on adding fractions, Rachel
focused on one of her groups, a group with four students who worked together on the problems at the end of
each section. All of the students were average B students in math, and all of them scored between 15% and
25% correct on a challenging pretest assessing their ability to successfully add fractions. She took notes on
what the students in these groups were saying during the week. The groups worked on problems silently part
of the time, and at other times, they would talk about the problems. Although she wasn’t able to write down
every word the students said during the times when she was listening to their talk, she is confident that she
captured most of what they said. The transcripts below are typical of the talk she recorded for each student.
After four days of working together, Rachel had the students take a test very similar to the pretest so that
Rachel could see how much they had improved.
Here are the pretest-posttest gains that each student made:
Student A. 20 point gain Student C. 65 point gain
Student B. 35 point gain Student D. 55 point gain
Here are some representative discussions:
Tuesday.Problem #1. 2/5 + 1/4
A: What did you get on this one?
B: 13/20.
A: OK.
Problem #5. 2/3 + 4/5
D: How about #5?
C: I got 8/15.
D: Wait. That’s not what I got. How did you get
that?
C: You have to find a number that divides by both
3 and 5. And that’s 15. So you make a number
that equals two thirds that has 15 on the
bottom. And that’s 10/15. And 4/5 is the same
as 12/15. And now you have the same number
on the bottom both times, so you can add them.
And you get 10 plus 12. That’s 22. 22/15.
D: I see. You have to find the smallest number that
3 and 5 will go into, and put that on the bottom.Wednesday.Problem #3. 1/4 + 3/8
A: What’s for #3?
C: I got 5/8. Did you?
A: Uh...yeah.
Problem #4. 1/6 + 1/3
B: Why isn’t this one 9/18?
D: Yeah, that’s what I was thinking, too.
C: You don’t have to do 18 here. Six times 3 is
18, but you can just stay with 6, because 3
goes into 6. So you can change 1/3 to 2/6, and
they’re both 6’s on the bottom now, so you can
add them. Three sixths. But you can reduce
that. It’s the same as one half.
B: So that’s how.
D: So we don’t just automatically multiply. See if
one goes into the other first. Are there any
others like that?