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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:
- Find out how many cubes are in the bag,
- Remove the number of cubes equal to the number of students in
the class, and
- 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:
- 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.
- 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.
- 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
Barnett, C., Goldenstein, D., and Jackson, B. (Eds.) (1994). Fractions,
Decimals, Ratios, and Percents: Hard to Teach and Hard to Learn? Portsmouth,
NH: Heinemann, Inc.
Copeland, W. D. and Decker, D. L. (1996). Video cases and the development
of meaning making in pre-service teachers. Teaching and Teacher Education,
12 (5), 467-481.
Merseth, K. K. (1996). Windows on Teaching: Cases in Secondary Mathematics.
New York, NY: Teachers College Press.
Merseth, K. K. (1996). Cases and case methods in teacher education. In
J. Sikula, T. J. Buttery, and E. Guyton (Eds.) Handbook of Research
on Teacher Education. New York, NY: Macmillan, pp. 722-744.
NCTM (1989). Curriculum and Evaluation Standards for School Mathematics.
Reston, VA: National Council of Teachers of Mathematics.
NCTM (1991). Professional Standards for Teaching Mathematics.
Reston, VA: National Council of Teachers of Mathematics.
NCTM (1995). Assessment Standards for School Mathematics. Reston,
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NCTM (2000). Principles and Standards for School Mathematics.
Reston, VA: National Council of Teachers of Mathematics.
Russell, S. J. (1997). The role of curriculum in teacher development.
In S. N. Friel and G. W. Bright (Eds.) Reflecting on Our Work: NSF
Teacher Enhancement K-6. Lantham, MD: University Press of America.
Schifter, D. (1996-a). What's Happening in the Mathematics Classroom:
Issues of Practice in Teacher Narratives from the Mathematics Reform Movement
(Vol. 1). New York, NY: Teachers College Press.
Schifter, D. (1996-b). Constructing a New Practice: Issues of Practice
in Teacher Narratives from the Mathematics Reform Movement (Vol. 2).
New York, NY: Teachers College Press.
Schifter, D., Bastable, V., and Russell S. J. (1999-a). Number and
Operations: Building a System of Tens,Casebook and Facilitator's Guide.
White Plains, NY: Dale Seymour Publications.
Schifter, D., Bastable, V., and Russell S. J. (1999-b). Number and
Operations: Making Meaning for Operations, Casebook and Facilitator's
Guide. White Plains, NY: Dale Seymour Publications.
Schifter, D. and Fosnot, C. T. (1993). Reconstructing Mathematics
Education. New York, NY: Teachers College Press.
Shulman, L. A. (1987). Knowledge and teaching: foundations of the
new reform. Harvard Educational Review, 57, 1-22.
Stein, M. K., Smith, M. S., Henningsen, M. A., and Silver, E. A. (2000).
Implementing Standards-based Mathematics Instruction: A Casebook for
Professional Development. New York, NY: Teachers College Press.
Wilcox, S. K. and Lanier, P. E. (Eds.) (2000). Using assessment to
Reshape Mathematics Teaching: A Casebook for Teachers and Teacher Educators,
Curriculum and Staff Development Specialists. Mawah, NJ: Lawrence
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