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
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).
CGI classification of word problems chart
The session concluded with further discussion about each type of problem.