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Making Sense of Teaching and Learning Mathematics: Using Cases
Susan N. Friel
The University of North Carolina at Chapel Hill
Meg's Case
Meg, a second grade, field test teacher, is using an activity called
"Enough for the Class," in which students consider whether the number
of cubes in a bag is enough for each student in the class to have one. If it's
not, how many more are needed? If it is, are there extras?
Meg thinks of this problem as a subtraction situation and assumes that her
students will do something like the following sequence of steps:
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Find out how many cubes are in the bag,
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Remove the number of cubes equal to the number of students in the class, and
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Figure out or count how many cubes remain.
She gives them the following problem:
There are 16 blue cubes and 17 red cubes; are there enough for the class?
Students quickly decide that there are enough for the class of 26 students and
begin figuring out how many extra cubes there will be. Meg is taken by surprise
when some of her students solve the problem this way: I can take 10 cubes from
the 16 and 10 cubes from the 17, that makes 20. Then I need 6 more cubes, so I
take away 6 from the 16. Now, I have 26, enough for the class. That leaves just
7 cubes from the 17.
Meg wrote about this episode: "Many children actually did solve the problem
the way I expected. Many didn't.… They showed a lovely ability and
willingness to take numbers apart and put numbers together. They...had made
sense of what was being asked. But they still didn't figure out how many cubes
there were in all! I am not sure what surprises me more-that so many children
don't think explicitly about the whole or the total when solving these
problems, or that it never occurred to me that they didn't have to. (Russell,
1997, pp. 249-250)
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The case approach has long been used in professions such as law, business, and
medicine and it is gaining momentum in the education profession. Cases are
developed to represent reality, to stimulate thought and debate, and, as
teaching instruments, for study, examination, and discussion. Most recently,
narrative case materials have become available that address dilemmas in
teaching elementary, middle grades, and secondary mathematics
(Barnett,
Goldenstein, & Jackson, 1994; Merseth, 1996; Schifter, 1996 a, b;
Schifter, Bastable, & Russell,
1999-a,
1999-b; Stein, Smith,
Henningsen, & Silver , 2000,
Wilcox
& Lanier, 2000).
There are a variety of interpretations of what cases are and what the purposes
of their use might be. Merseth (1996) provides a simple conceptual framework as
an organizing scheme for thinking about cases. The framework divides case
purpose and use into three categories:
Cases as exemplars
This view sees cases as paradigmatic representations of generic problems in
teaching (Copeland & Decker, 1996). The emphasis of these cases is on the
theoretical, the prescriptive, and the model. Their purpose is to develop
knowledge of theory or to build new theories.
Cases as opportunities to practice analysis and contemplate action
This view sees cases as providing opportunities to practice decisionmaking and
problem-solving. Cases are used to present situations from which theory
emerges, rather than explicitly to exemplify.
With such cases, students can, within the confines and safety of a teacher
education classroom, "practice such professional skills as interpreting
situations, framing problems, generating various solutions to the problems
posed and choosing among them."
Cases that focus on problem-solving and decisionmaking typically are based on a
real situation where "an actual instance of practice is presented in much
of its complexity rather than an episode constructed to illustrate a
point" (Merseth, 1996, p. 728).
Cases as stimulants to personal reflection
In this view, emphasis is on personal professional knowledge and the
individual. Reflection is promoted from directly or vicariously experiencing a
situation that puzzles or surprises. This view includes the use of
teacher-constructed cases along with other types of cases. The cases serve as
"the data" and the discussion of the cases articulates possible
"courses of action."
Cases in mathematics education are designed to engage teachers in thinking about
mathematics, pedagogy, and children's thinking.
A teacher's knowledge can "play out" in the classroom in a variety of
ways, but the predominant view of mathematics teaching and learning (NCTM,
1989, 1991, 1995, 2000) emphasizes the following:
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The fundamental goals of school mathematics are to teach students to understand
and reason with mathematical concepts, solve problems arising from new and
diverse contexts, and develop a sense of their own mathematical power.
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Teachers create conditions that allow students to take their own effective
mathematical actions. They play more the role of classroom facilitator than
knowledge source. They model important mathematical actions, coach student
thinking, pose mathematical questions, and stimulate and moderate classroom
discourse.
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Students learn by taking mathematical actions. They represent their ideas, make
conjectures, build models, collaborate with other students, and give
explanations and arguments. In other words, mathematics is learned through
reasoning, and teaching should focus on providing an environment in which
reasoning can take place. The goal is to make the mathematics part of
"one's experience," so the teacher not only seeks to provide
experiences for students but also listens to students in order to understand
what they are experiencing, making modifications in instruction as needed based
on these understandings.
Implicit in the above discussion is a need for a new instructional paradigm for
teaching mathematics. In particular, critical instructional decisions in
teaching mathematics must be re-conceived in such a way that the goal is one of
creating a mathematics classroom that is "a problem solving environment in
which developing an approach to thinking about mathematics is valued...more
highly than memorizing algorithms and using them to get right answers"
(Schifter
& Fosnot, p. 9). There is considerable writing about these
decisions (e.g., NCTM 1989, 1991, 1995, 2000; Shulman, 1987). Below are four
specific areas of focus; case materials address each of these areas with
differing kinds of emphases.
1. Knowing mathematics
What do I know about this mathematics content? What are the central ideas and
processes? What are the relationships to other ideas in mathematics or in other
disciplines? What are the standard misconceptions? In what contexts does this
content appear? What are my purposes for instruction? (as an example, several
cases in Barnett, Goldenstein, and
Jackson 1994 and Merseth 1996 focus attention on both teachers' and
students' understanding of mathematics).
2. Selecting and sequencing tasks
What mathematical tasks, representations, teaching methods, and models do I
choose to use and for what reasons? What adaptations do I make to the children
I teach? How do I want to plan, sequence, and pace instruction? (as an example,
cases in Schifter, Bastable, & Russell
1999-a, 1999-b are
clustered by selected "big" mathematical ideas; teachers reading
these cases begin to understand the extent of what is involved instructionally
to teach these ideas.)
3. Orchestrating discourse
What is the nature of the classroom environment I want to establish? How are
the students grouped? What is the nature of the interactions that occur among
students and teacher? What levels of questioning can be used to probe for
deeper understanding and/or to focus attention on mathematical relationships?
What questions are students asking? Who is doing the thinking? Are students
making intended connections? Are students listening to each other? (as an
example, several cases in Stein, Smith,
Henningsen, & Silver, focus on distinguishing between high and low
level tasks, and what is the nature of pedagogy that supports student learning
through the use of high-level tasks.)
4. Assessing what students know
What kinds of questioning help uncover students' thinking? What are different
strategies I anticipate the students using? What practice activities and
problems do I choose to support student learning? What evidence will I use to
make inferences about students' knowledge of, ability to use, and disposition
toward, the mathematics being addressed? How can I design questions that are
neither too vague nor too directive so that children develop understanding of
the relevant mathematics? (as an example, several cases in
Wilcox & Lanier, 2000, focus on using assessment to make sense of
the nature of students' understanding, to consider what counts as evidence of
that sense-making, and to decide what are next instructional moves.)
Within each of these areas, it is important for teachers to become more
reflective about their practice and to make explicit instructional decisions
from among clearly identified options. It is this process of making such
decisions that that can be called pedagogical reasoning (Shulman, 1987).
Cases serve as a useful tool for developing pedagogical reasoning
There several protocols possible for instructional use of cases. Historically,
the typical protocol is to provide appropriate theoretical readings which are
followed with the reading of a case designed to provoke discussion of teaching
dilemma(s) that require applications of the theory. In using cases, readers
first respond individually (usually in writing) to a series of questions used
to guide analysis of the case. A case discussion follows, and readers write
additional reflections based on these discussions.
Questions Used to Guide Analysis of the Case
1. What's going on in this case? Summarize aspects that you think are important
to understanding it.
2. What are some of the issues that come up for you in this case (vignette or
set of vignettes, video)? What are some of the questions that this case raises
for you?
3. Why do you think it is important to raise the questions, concerns, or issues
you wrote about in question (2)?
4. What are some other ways to teach this lesson (set of lessons)?
5. How would you teach this lesson? What materials would you use? What order
would you do things in? What examples would you give?
6. What else do you need or want to know about this case?
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More recently, in mathematics education, a greater variety of the kinds of cases
that may be used has emerged, with some being quite short as in this paper's
opening example and others longer and more comprehensive. While there is no
standard protocol for how these cases are to be used, current case materials
provide facilitators guides that document suggested protocols. Clearly, the
critical instructional decisions in which teachers need to engage provide the
background for thinking about ways that any given case may be used. For
example, the short example presented at the beginning of this paper might be
used with three or four other similar cases. Each of the cases may be focused
on teachers' expectations about the ways children solve certain kinds of
mathematics problems and on the actual ways that children choose to solve these
problems. Discussion might focus on teachers' understanding of the ways that
children appear to understand mathematics and whether it is "okay"
for mathematics to be understood in these ways.
The culture of teaching presented through cases reflects necessary paradigm
shifts and provides opportunities for close scrutiny of the content and
practice of teaching of mathematics. While anecdotal by their very nature,
cases can be helpful in provoking teachers' examination of their own practice
through examination of others' practice.
References
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