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Using Case Methodology for Professional Development
in Mathematics Teaching
Su-Wei Lin1 Marie Cheo2
National Hualien University of Education1 National Academy for Educational Research2
Abstract
Current mathematics education emphasizes active teacher and student participation in discussion to negotiate mathematical meaning. However, teachers were not exposed to innovative curricular areas and teaching methods in their formative education, nor given adequate professional development opportunities. This study used cases to familiarize teachers with the curriculum innovations. We discussed teaching strategies with teachers in math, and then used the videotapes of their teaching as cases for discussion at professional development workshops. In this study, many of the teachers had only used traditional lecturing method in the past, therefore, participating in the learning was a great challenge to their teaching, and a clash with many of their
teaching beliefs and beliefs about mathematics learning. They also found the necessity of reflecting upon their math content knowledge and pedagogical content knowledge.
Teaching is an art. Effective teaching can inspire kids’ thinking. Playing the role of posing problems and guiding students’ learning, teachers are the key persons who control the tempo and quality of teaching. Current mathematics education emphasizes active teacher-student interaction to negotiate mathematical meaning. It is a process in which both teacher and students are part of the learning community that engage in social interaction to construct mathematical representation, language, procedures, and meaning. In this way, teachers transfer the control of learning to students gradually. However, this approach is unfamiliar to most teachers, so the purpose of this study is to collaborate with classroom teachers to increase students’ autonomy in mathematics learning, and to explore how it can be implemented in the elementary school
classroom. We also want to understand this as a process in school teachers’ professional development.
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Choosing Case Methodology
Professional development associated with the new math curriculum in Taiwan has largely been traditional workshops in which one after another expert gives individual lectures on educational theories and teaching methods, interspersed with workshops in which teachers produce math teaching plans or assessment type
worksheets following the direction of experts. Discussions are rare, often because they are considered too time consuming, and also because many people believe that experts know more about curriculum and teaching than teachers do. Discussion has not been a common mode of teaching and learning in Taiwan. Teachers have been schooled in the traditional lecture and recitation format.
Thus, we want to explore the issue with schoolteachers. Moreover, when it comes to create the social interactive climate in mathematics classroom, which is unfamiliar to most of us, we all need a common experience, going through the process of math interaction together, and working out our common meaning of social
interaction in math teaching.
We decide to use case methodology because cases can present theoretical and practical knowledge simultaneously. Since most math teachers have never
atmosphere and the setting of interactive math inquiry from seeing the events in action, we chose to develop cases that include videos. Narratives would not be able to give the first hand impressions that videos can give.
More important, we wanted case discussions to focus teachers on specific aspects of teaching and learning mathematics. Videotapes have long been a part of
professional development in Taiwan, but most of them have been rehearsed simulations of exemplary classroom teaching intended to serve as a model for
teachers to emulate. These exemplars were supposed to be self-explanatory, and it was assumed that teachers would imitate the master teacher. However, most observers paid more attention to aspects that they already understood, such as classroom
management, or delivery of the subject matter. They would fail to notice the subtle teacher-student interactions crucial to the new aspect of teaching that we hope they might learn.
Sykes & Bird (1992) had described four uses of cases that Merseth (1995) developed into a conceptual framework with three categories of cases: cases as exemplars, cases as opportunities to practice analysis and contemplate action, and cases as stimulants to personal reflection. In this study, we are concerned with Merseth’s second and third categories of cases. We want to create cases not as exemplars of ideal math teaching, but as situations that help teachers reflect upon specific details that will deepen their understanding of social interaction in mathematics teaching. In other words, we are developing cases that would provide teachers with opportunities to think about how to negothate mathematical meaning with their students, and to examine their own teaching in collaboration with their colleagues.
Developing the Cases
We adopted the Four Phase Framework of the Collaborative Inquiry Process suggested by Bray, Lee, Smith, & Yorks (2000). The Framework is shown in figure 1. Twenty-one teachers accepted our invitation to collaborate as co-researchers and executors of our rationale over a period of three years, together with some math educators and researchers. We considered possible teaching approaches, worked out teaching plans, conducted the classes, and reflected collectively on the results. We met
regularly in small and large groups, collected teaching plans, conducted interviews, and examined the literature. We analyzed documents and audiotapes from our own meetings, and examined the videos from our teaching efforts.
Our cases are developed in four broad phases that are iterative, and overlap each other in time: understanding math teachers’ concepts about social interaction;
observing math classroom teaching and discussing the teaching; collaborating with math teachers on teaching plans and trial testing the teaching; trial testing and revising the cases.
1. Understanding math teachers’ concepts and difficulties about social interaction.
First, we asked teachers attending our mathematics workshops to fill in an open-ended questionnaire, describing what they think about the function of social interaction in math classroom, how they would strengthen the interaction between teacher and students, and what the difficulties would be in the curriculum. On the other hand, we conducted focus group discussions in which teachers, master teachers, supervisors, and school principals would share their views on social mathematics interaction according to the questionnaire that they filled in. After sharing their ideas in small groups, each group produced a concept map for social mathematics interaction.
2. Observing math classroom teaching and discussing the teaching.
We formed a research team with researchers and classroom teachers. The number of teachers fluctuated from one semester to another due to constraints of their teaching schedules. The researchers went to the schools to observe the classes taught by the teachers. Before and after the classes, we asked the teachers to describe their original expectations, and to reflect upon their teaching in the observed sessions: whether she/he had planned for math teaching, whether she/he observed the interaction among the students, how she/he might change the
teaching strategies to encourage more interaction, and so on. Some of the class sessions were videotaped and reviewed in the discussion. All classroom sessions and discussion sessions were videotaped. These observations and discussions helped us make choices and decisions regarding the nature of active teacher and student participation in math discussion. They helped us form images and representations of the cases that we might create later on.
3. Collaborating on math lesson plans and trial teaching.
We added more schoolteachers to our research team for developing and trial testing our cases. We worked together on lesson plans, then the schoolteachers would teach the lessons. These classroom trials were professionally taped and transcribed. Members of the team would watch the videotapes and transcriptions. Ultimately, some of the edited videotapes would serve as cases for the next phase of the production.
4. Trial testing and revising the cases.
During this phase, we showed the cases to teachers who had attended our workshop in mathematics teaching, and led them in case discussions. The researchers facilitated the discussions with teachers. The discussions were audiotaped and transcribed. We then reviewed the transcriptions, revised the way the cases were used, and gradually developed facilitation guides for the case discussions.
PHASEⅠ: FORMING ‧ Initiating/Co-Initiating a Group ‧ Obtaining Institutional Consent ‧ Establishing a Physical Context ‧ Ensuring Diversity
‧ Orienting the Group
‧ Developing the Inquiry Project ‧ Framing the Inquiry Question ‧ Designing the Inquiry Project ‧ Transitioning to Collective Leadership ‧ Reflecting on Group Process PHASEⅡ: CREATING
‧ Agreeing on a Constitution for Collaboration ‧ Repeating Cycles of Action and Reflection to
Generate Learning Ⅰ. FORMING A COLLABORATIVE INQUIRY GROUP Ⅱ. CREATING THE CONDITIONS FOR GROUP LEARNING Ⅲ. ACTING ON THE INQUIRY QUESTION Ⅳ. MAKING MEANING BY CONSTRUCTING GROUP KNOWLEGE FORMING (OR REFORMING) PHASE Ⅲ: ACTING
‧ Putting Plans and Design Into Practice ‧ Keeping Reflective Records
‧ Respecting Group Ownership of Ideas ‧ Questioning Honestly
‧ Practicing Dialogue and Reflection
PHASE Ⅳ: MAKING MEANING ‧ Capturing the Group’s Experience ‧ Understanding the Experience ‧ Selecting a Method for Interpreting
Experience
‧ Constructing Knowledge ‧ Avoiding Flawed Meaning Making ‧ Guarding Against Groupthink ‧ Checking Validity
‧ Celebrating Meaningful Collaboration ‧ Communicating in the Public Arena
We have been trying to achieve our goal through collaboration and discussion. Researchers guided the discussion. Teachers tried to understand teaching materials and presented ways to improve their instruction through discussion. The following section presents one episodes of the discussion.
Mr. Wang: My students do not quite understand the diagram on page 89.
Researcher: page 89？Which part do you say that your students do not fully understand? Mr. Wang: 9/2 ÷ 3/5 turns to be 9/2 × 5 ÷ 3, try to draw a linear section diagram to explain it. Researcher: It happens to be taught by Ms. Hsu, we will ask Ms. Hsu to talk about this part later.
Ms. Hsu, how is the situation of your students in class in learning reversed multiplication? Ms. Hsu: I find that there is a big problem here. When we taught students to draw a linear section
diagram in grade 5, we were not informed to pay attention at the following points: such as, what is divided by? What number will be multiplied in a reversed manner? We only read from the diagram.
Researcher: I am concerned about the part ‘divided by 5 is to multiply 1/5 or divided by 1/5 is to multiply 5’.
Ms. Hsu: Since the computation is done by using the unit fraction of the divisor, thus, as a dividend is divided by 1/5, it is required to immediately connect with multiplying 5. If a dividend is divided by 1/4, one has to immediately connect with multiplying 4. One must bear this
connection in mind. Therefore, I want to remind my colleagues to let their students practice it as much as possible before teaching this.
Researcher: What to practice as much as possible?
Ms. Hsu: When a dividend is divided by a proper fraction, it has to time...
Ms. Hsu: What should we do as a dividend is divided by a whole number? When the students were at the 5th grade, we only taught them to draw line section diagrams without displaying the division number sentence, so the students did not know the connection. At that time, the
students drew in a mess and they hated it desperately. So did the teachers. We did not know that we should handle it in this way.
Researcher: In other words, teachers feel that though the students seem to learn well, they don’t fully understand the concept in detail.
In the discussion, teachers would present their perplexities in instruction, exchange teaching experiences and discuss students’ problem solving strategies. Those teachers met every weekend to exchange opinions in teaching materials, ways of teaching and students’ problem solving strategies. As
that is conducted in a setting that promotes investigation and inquiry into the problems of mathematics teaching seems to hold promise for assisting preserve teachers in becoming inquiring reflective
mathematics teaching. Creating learning opportunities in which the learner could engage in reflective thinking about mathematics teaching and learning is essential.
Summary
This paper describes ongoing work to produce cases that would help teachers understand how to teach mathematics. Discussion is not a common mode of teaching and learning in Taiwan. Teachers have been schooled in the traditional lecture and recitation format.
In this study, many of the teachers who tried to interact with their students in learning
communities had only used traditional lecturing methods in the past, so participating in this study was a great challenge to their teaching, and a clash with many of their teaching beliefs and beliefs about mathematics learning. They also needed to reflect upon their math content knowledge and pedagogical content knowledge. In this study, we created an environment that allowed the pre-service teachers to interact with knowledge, beliefs, attitudes, experiences and expectation to develop their interpretation and understandings of mathematics. We feel that there is a need for further collaborative research, and research into the professional development of teachers.
Reference
Bray, J. H., Lee, J., Smith, L. L., & York, L. (2000). Collaborative inquiry practice: Action, reflection, and making meaning. CA: Sage.
Cheo, M., Chao, C. C., Wu, R., and Huang, M. T. (2005). Inquiry into inquiry. Paper presented at the Conference on Educational Research, Sanhsia, Taiwan.
Merseth, K. K. (1995). Cases and case methods in teacher education. In J. Sikula (Ed.), Handbook of research on teacher education, 2nd ed.. (pp. 722-744). New York, NY: Macmillan.
Short, K. G., Harste, J. C., and Burke. C. L. (1996). Creating classrooms for authors and inquirers. 2nd ed. Portsmouth, NH: Heinemann.
Sykes, G., and Bird, T. (1992). Teacher education and the case idea. In G. Grant (Ed.), Review of research in education (Vol. 18, pp. 457-521). Washington, DC: American Educational Research Association.
Thier, H.D, with Daviss, B. (2001). Developing inquiry-based science materials: A guide for educators. New York, NY: Teachers College Press.
