In this chapter, we consider professional development experiences based on the “case study method.” Here teachers analyze and discuss “cases” that are written narratives or video excerpts of events that are used as catalysts for raising and discussing important issues regarding school mathematics reform.
Because several cases currently being used in professional development programs show students working on mathematical tasks, there is some overlap between this category of professional development experiences and analyzing students’ thinking, the category discussed in the previous chapter. However, cases can be used to focus on other educational issues besides students’ mathematical thinking. Furthermore, case discussions more generally have a long tradition in a number of professions besides education. These combined reasons led us to the decision of examining the use of cases in professional development as a separate category.
Theoretical rationale and empirical support
While using cases to develop professional knowledge in education has not been widespread, there is a strong tradition of using cases in other fields, such as law and business. Engaging mathematics teachers in the analysis of practice is certainly consistent with the principle of focusing professional development on the concrete activities of teaching and learning rather than abstractions and generalities. Appropriately selected cases can also be the starting point for all the six teacher learning cycles identified by Simon (1994), as reported in Chapter 3. The guided discussion of examples of practice can indeed provide the stimulus for new constructions of meaning by evoking cognitive dissonance, especially when the cases show a problematic situation. Furthermore, discussing such concrete examples offers teachers an ideal context for reflection and for hearing alternative viewpoints.
Indeed, Barnett (1998) has argued that the public scrutiny of ideas that takes place during a case discussion often leads teachers to new knowledge about mathematics, pedagogy and student thinking. Such knowledge is co-generated by the group in a way that significantly enhances what individuals could have come up with on their own.
Proponents of using cases in professional development have also pointed out that this kind of experience can potentially develop teachers’ habits of inquiry into practice (Barnett, 1998; Schifter, Bastable & Russell, 1997). Empirical evidence in support of using cases comes from research studies evaluating the effects of the Mathematics Case Methods project, a program based entirely on case discussions (Barnett, 1991; Barnett & Ramirez, 1996; Barnett & Tyson, 1993 a&b; Gordon & Heller, 1995; Gordon & Tyson, 1995; Tyson, Barnett & Gordon, 1995). Barnett & Friedman (1997) write that these studies show the following:
Teachers involved in case discussions move towards a more student-centered approach, learn to adapt and choose materials and methods that reveal student thinking, and anticipate and assume rationality in students’ misunderstandings. Moreover, it appears that without being exposed to these ideas in research literature, teachers naturally move towards constructivist views of learning and develop a complex knowledge of students’ thinking processes and underlying mathematical concepts. (p. 389)
While these findings could be attributed to the particular focus for the case discussions that the Mathematics Case Methods project employed (where all cases show classroom vignettes of students grappling with ideas about rational numbers), similar outcomes were found in field-testing the Developing Mathematical Ideas (DMI) program, which also uses cases (personal communication with Keith Cochran, 2001).
Schifter, Bastable and Russell (1997) have also pointed out the value of teachers creating their own cases, not just discussing ready-made ones. In their project, Teaching for the Big Ideas, a number of teachers successfully created cases.
Illustration 5: A case discussion about rational numbers
The vignette we present in this section illustrates a typical case discussion in the Mathematics Case Methods project (Barnett, Goldenstein & Jackson, 1994 a&b). The 2-hour session featured here occurred in a training session for experienced teachers who, although they had not had previous experience with case discussions, expressed interest in this approach and in the possibility of eventually becoming case discussion facilitators. This case discussion was the first for these teachers. The case, called “Beans, Rulers and Algorithms,” is the first in a series of cases about rational numbers that the Mathematics Case Methods project (Barnett, Goldenstein & Jackson, 1994a) developed.
The session began with a brief ice-breaker activity in which participants introduced themselves by saying their name and giving an adjective to describe their personality. Then, participants worked independently on the following problem designed to engage them personally with the key mathematical ideas in the case:
Think about what might be difficult or confusing for a child. Use beans to solve this problem: 1/3 + 3/12.
Teachers then read the case silently. It is a two-page narrative reporting a teacher’s experience in a combined fifth/sixth-grade class working on fractions (Barnett, Goldenstein, & Jackson, 1994a). The students in this class had already worked with equivalent fractions, addition and subtraction of fractions with the same denominator, improper fractions and mixed numbers. They had done so with success, using both manipulatives and pencil-and-paper tasks. The class had then moved to adding fractions with different denominators. The teacher introduced this new situation by providing the students with 12 beans, asking them what part of this whole would correspond to 1/2, 1/3, 1/4, 1/6 and 1/12. She also showed them a ruler, pointing out how each inch is divided in 16 parts and asked students to locate various fractions on the ruler. Using this information and the two tools (i.e., the beans and the ruler), the teacher asked the students to add several fractions, including such problems as 1/3 + 3/12 and 1/2 + 5/16. Once again, the students seemed to understand and had no difficulty with these problems, at least as long as they worked with the manipulatives. However, when the teacher moved to adding fractions on paper a few weeks later, the students seemed suddenly to “switch from understanding the concepts to memorizing a formula,” and mistakes such as 1/6 + 2/7 = 3/13, or 1/6 + 2/7 = 7/42 + 6/42 = 13/42 surfaced. These outcomes puzzled the teacher, and she questioned what the students really understood about adding fractions. She wondered what she should do in the next lessons to help them.
When almost everyone had finished reading this case, the facilitator asked the participants what they thought were the important facts about this case. As participants offered suggestions, the facilitator recorded them on newsprint without comment. A list of about a dozen items was quickly generated, including such information as “it was a fifth/sixth-grade class,” “students already knew how to add fractions with common denominators,” “they had been working on this for some time (but not clear how long),” “they were using manipulatives,” and “they did not understand the process.”
The facilitator then asked participants to work in pairs to generate issues for discussion about the case, requesting that each issue be expressed in the form of a question. She pointed out that issues could be about the mathematics involved in the case, the children’s thinking, aspects of the instructional practice, the materials used or even the language used. She noted that based on past case discussions, some kinds of questions generated more interesting discussions than others. Therefore, she suggested teachers avoid yes/no answer questions, such as “Did the …?” and try instead to express their questions in a more open-ended way, such as the following:
“Why might a student . . . ?”
“What might happen if . . .?”
“What does . . . mean?”
“What if the problem/manipulatives were . . .?”
“What are the benefits/limitations of . . . ?”
She elicited a few examples of each kind of question from participants to serve as models before the group broke into pairs to work on the task.
After about 10 minutes, the group reconvened and each pair shared some of the questions it had generated. Once again, the facilitator recorded all these questions on newsprint with minimal comment, making sure that every pair had an equal chance to contribute and that every voice was heard. The list contained about 15 items that addressed a variety of elements in the case, all using the format for questions suggested by the facilitator. They included very specific questions, such as “What might have happened if they had used fraction bars or paper folding instead of beans and rulers?” to more general ones, such as “How do you make the connection from the manipulatives to the paper-and-pencil process?” and “What does ’basically understand’ mean?” While the majority of questions were about the teacher’s instructional choices and alternative possibilities, some questions looked more at the children’s thinking, such as: “Why might students not understand the concept using beans?” and “Why would they add numerators and denominators?” Other questions focused on the mathematics, for example: “What do the beans represent?”
The group then picked one question for further discussion: “What does ’basically understand’ mean?” In the remainder of the session, teachers discussed just this one question, although several other questions on the list were also addressed in the process.
The discussion began with several teachers trying to articulate what “understanding addition of fraction,” or even “understanding fractions,” meant for them. To help clarify their position, the facilitator occasionally invited them to come to the board and illustrate the point they were trying to make with an example. These examples usually made the discussion more concrete and raised some interesting mathematical questions about fractions and their representations. For example, participants generated new insights about the complexity of using beans to represent fractions. They noted that, depending on the number of beans chosen as the “unit,” one single bean might represent a different fraction. For example, if the unit is 12 beans, 1 bean represents 1/12, but if the unit is 8 beans, one bean (the same bean!) represents 1/8. This suggested to another participant a possible explanation for why students might have added numerators and denominators in the problem 1/4+1/3 when using the beans, as shown in Figure 10.
A participant’s graphical explanation of the mistake 1/4+1/3=2/7
This discussion led several teachers to appreciate the importance of clearly specifying what the unit is whenever using discrete representations for fractions. It also revealed that students might reasonably be puzzled by the fact that the teacher chose different sets of beans as the unit depending on the problem. It also suggested the value of making the reasons behind that choice explicit for students.