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Discussing Cases
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.
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.
Continued
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CHAPTER 6