Analyzing Students’ Thinking
In this chapter, we examine the type of
professional development experience in which teachers analyze
student thinking as revealed in students’ written
assignments, think-aloud problem-solving tasks, class
discussions and clinical interviews. Within this kind of
professional development sessions, teachers learn to observe
various types of student mathematical activity and to interpret
what they observe, with the ultimate goal of enhancing their
students’ learning opportunities.
Theoretical rationale and empirical support
In Chapter 1, we discussed the research
evidence that supports teachers learning about students’
mathematical thinking. We argued that doing so can help
teachers develop not only a knowledge base about
students’ conceptions and problem-solving strategies that
they can use in planning instruction but also skills for
listening to students and interpreting their thinking.
Professional development that helps
teachers analyze students’ mathematical work is a logical
vehicle to achieve these goals. First, it is consistent with
the professional development principle that teachers should
engage actively in concrete activities close to their own
practice, not just abstract discussions. Second, according to
Simon’s (1994) Learning Cycles model, analyzing student
artifacts creates the context necessary to start a learning
cycle focusing on students’ thinking. As groups of
teachers examine artifacts together, they can engage in active
learning, experience cognitive dissonance as different
interpretations are proposed and construct new meanings. Third,
examining students’ work and thinking is precisely what
we want teachers to do as part of their everyday teaching
practice. Therefore, engaging in these tasks with the guidance
of an expert is a valuable way to learn to do the same tasks
independently (Collins, Brown, & Newman, 1989).
Research shows that analyzing student
thinking can promote instructional practices that result in
higher student achievement. Evidence supporting this claim
comes from several research studies on outcomes of professional
development programs for elementary teachers based on a
Cognitive Guided Instruction (CGI) model (e.g., Carpenter &
Fennema, 1992; Fennema, Carpenter, Franke, Levi, Jacobs, &
Empson, 1996), as well as research conducted by the Integrating Mathematical Assessment (IMA) project involving middle school students
(Gearhart, Saxe & Stipek, 1995).
Moreover, comparison studies between
teachers who had participated in CGI training and those who had
not showed that CGI teachers were using some highly effective
practices in their classroom teaching:
[T]eachers who had been in CGI workshops
spent more time having children solve problems, expected
multiple solution strategies from their children, and listened
to their children more than did control teachers. (Fennema,
Carpenter & Franke, 1997, p.194)
Case studies of teachers who participated
in CGI programs (Fennema Carpenter, Franke, & Carey, 1992;
Fennema, Franke, Carpenter, & Carey, 1993) also show that
these teachers gained a better understanding of student
thinking and expressed views about the learning and teaching of
mathematics consistent with the goals of school mathematics
reform. We attribute these results mainly to the
teachers’ analysis of student thinking, as this was the
key professional development activity used in the CGI programs.
Illustration 3: Building a classification
of addition/subtraction problems from the analysis of a
videotaped problem-solving session.
This illustration depicts a typical
2-hour-long session in a CGI program. We adapted this vignette
from the description provided in Fennema, Carpenter, Levi,
Franke & Empson (1999). In this session, teachers viewed a
videotape of a first-grade child solving four word problems.
The goal was for them to identify different types of problems
involving addition and subtraction. From this activity, the
teachers were able to reconstruct the “Classification of
Word Problem Chart” (shown later in Figure 9) that is
part of the research model informing the CGI program.
The session opened with teachers
discussing three mathematical word problems:
1. Lucy has 8 fish. She wants to
buy 5 more fish. How many fish would Lucy have then?
2. TJ has 13 chocolate chip
cookies. At lunch she ate 5 of these cookies. How many cookies
did TJ have left?
3. Janelle has 7 trolls in her
collection. How many more does she have to buy to have 11
trolls?
These problems represent different types
of addition and subtraction problems. At first glance, problem
1 seems to involve addition, and problems 2 and 3 seem to
require subtraction. However, problems 1 and 3 can also be
characterized as having to do with “joining” two
sets, while problem 2 is about “separating” an
original set into two subsets. Characterizing problems in this
way suggests that subtraction may not be the only approach to
solving problem 3, for example.
After the participants had a chance to
solve the three problems on their own, the facilitator
initiated the discussion by asking, “Which of these two
problems are most alike and why?” Besides noticing that
problems 2 and 3 involved subtraction, a teacher also commented
that problem 3 would be harder for his/her students. After a
brief discussion of this point, the facilitator introduced the
videotape, in which a first-grade child, Rachel, solves the
same three problems. (The videotape is available in the CGI
professional development support materials available from the
authors.)
The facilitator invited the participants
to watch how the child solved these problems and to think about how the child perceived
these problems in terms of similarity, difference and level of
difficulty. Rachel’s approach surprised the teachers, as
Rachel solved the third problem by “joining,” while
most teachers had solved the same problem by subtraction. In
the ensuing discussion, problems involving a joining action
were distinguished from ones involving a separating action. To
clarify the difference, the facilitator asked teachers to write
a problem of each kind and then to share and discuss these
problems with the group. In the course of the discussion,
participants also agreed that problem 3 must have been more
difficult for Rachel because “the child just can’t
go step by step through the problem and do what it says.”
The facilitator then introduced the next
segment of videotape, in which Rachel solves yet another
addition/subtraction problem:
4. Max had some money. He spent
$9.00 on a video game. Now he has $7.00 left. How much money
did Max have to start with?
The follow-up discussion on the
child’s solution of this problem led the group to realize
that this problem, too, could not be easily solved “step
by step.” In addition, this problem could be even harder
to approach because the child would not know where to start.
Building on these observations, the
facilitator pointed out that addition and subtraction problems
may vary not only according to the type of action involved in
solving them (i.e., “joining” or
“separating”) but also according to where the
unknown appears in the story. After some discussion, the leader
suggested that the following variables could be used to
organize the four problems:
A. Involve joining
B. Involve separating
and,
i) The unknown is introduced at the end
of the word problem
ii) The unknown is introduced in the
middle of the problem
iii) The unknown is introduced at the
beginning of the problem
The group then used these variables to
create the 2x3 matrix reproduced in Figure 9. When the matrix
was completed, the leader also introduced the “official
names” used in the CGI project to refer to each of these
six types of addition and subtraction problems (highlighted in
boldface in Figure 9).
Figure 9
CGI classification of word problems chart
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The session concluded with further
discussion about each type of problem.
Continued
