Throughout the discussion, the facilitator tried not to drive the conversation in a specific direction although she was not neutral either. Rather, she tried to deepen the participants’ analysis and challenge their thinking through a combination of “pulling probes: ”

“Can you show us what you mean?”
“What do other people think about that?”
“What are benefits/drawbacks of this position/idea? Why do you think so?”

 She also used “pushing probes: ”

“What about [counterexample]?”
“Is that always true?”
“What might be the impact on students?”
“What new ideas can you envision for this situation?”

The facilitator also had to interrupt the discussion before the group could reach closure on the original question. She explained that while it is always hard to interrupt a good discussion, it is almost impossible to reach closure on this or any case. However frustrating this may feel at first, it also has the advantage that participants can continue to think on their own about the issues raised in the true spirit of inquiry.

The session concluded with a brief round-robin closure activity in which each teacher identified something he/she was thinking about differently as a result of the experience. Participants also gave feedback on the process by filling a process check form; the facilitator quickly reviewed the results of this feedback before the end of the session so that the group could think about how the process could be improved the next time around.

Illustration 6: Examining an example of teaching mathematics through inquiry

We took the next illustration from the Leadership Seminar in the Making Mathematics Reform a Reality (MMRR) project that we described in Chapter 2. At the beginning of this project, one of the main goals of the Leadership Seminar was to develop a common understanding among lead teachers of what it means to teach mathematics through inquiry and what it takes to put such an approach into practice.

To these ends, the facilitators devoted a 1 1/2-hour session to discussing a vignette of an inquiry lesson. The participants first read a four-page account of a lesson on constructing a congruent triangle given a side and two angles, where the students used creatively what they already knew about triangles and constructions to accomplish this novel task (Borasi, 1995, pp. 44-48).

The facilitators then carefully framed the discussion of this teaching episode. They asked the teachers to refrain from commenting on the quality of the lesson or the suitability of the example for teaching mathematics through inquiry. Instead, they should identify the elements of teaching mathematics through inquiry that were illustrated in the vignette.

As individual teachers shared the elements they had identified, facilitators asked them to explain why they had reached their conclusions and encouraged other participants to challenge these conclusions and ask for further explanation if it seemed necessary. A facilitator then recorded on newsprint the elements of inquiry-based instruction that the group agreed upon.

This exercise produced an extensive list of elements that characterize teaching mathematics through inquiry. It represented the group’s shared understanding of this instructional approach at this point in time. This list was later reproduced for all participants, and they referred to it frequently in later sessions as the group continued to refine its understanding of inquiry-based mathematics as a vehicle for mathematics reform.

Main elements and variations

The two illustrations we offer in this chapter only begin to illustrate the variety of interpretations about what constitutes a case and how cases can be used in mathematics teacher education. However, a number of elements are common to all these interpretations and are thus worth highlighting as characteristic of this kind of professional experience, despite its many variations:

Teachers engage in the in-depth analysis of a shared example of practice. The concreteness of the case enables participants to ground their reflection and discussion of more abstract ideas about school mathematics reform.
Each case is carefully selected to stimulate debate on specific issues. A case is not simply a story, but rather a story with a “point” – although case discussions may sometimes surprise the facilitator by developing in unexpected directions!
Facilitators elicit and explore multiple perspectives and opinions about the cases. One of the main benefits of case discussions is that teachers can benefit from the group interaction to construct meaning and knowledge that goes beyond what they, as individual participants, could have achieved. However, this requires careful facilitation of the discussion.

Within these guidelines, case discussions may differ considerably with respect to both the nature of the case used as a starting point and the nature of the discussion that is orchestrated around the case. Cases may differ along the following important dimensions:

The content of the case. While most cases used in teacher education deal directly with classroom instruction, some feature other aspects of teachers’ and/or students’ practice. For example, there are cases that portray teachers’ interactions with colleagues, teachers’ experiences in professional development settings or even students’ learning as it occurs outside of the classroom.
The format in which the case is presented. The vignette may be presented as a story, in narrative form, or conveyed through a video. Each of these media has unique advantages and disadvantages. Most notably, while videos can allow the direct observation of non-verbal as well as verbal behaviors, they are less flexible than a narrative format and less able to convey background information about the event.
Whether the case is a “stand-alone” or part of a collection. While almost any case can be used in isolation, programs that rely on case discussions as their primary vehicle tend to use carefully sequenced collections of cases, designed to provide teachers with multiple opportunities to examine a complex concept in different contexts. Multiple cases examined in a sequence make it possible to highlight different aspects of a topic each time, allowing for meanings to be constructed and revised over time.
The extent to which the case illustrates exemplary practice. While most cases make no assumptions about the quality of the practice they portray (as, for example, the case on rational numbers used in Illustration 5), some are created specifically to illustrate exemplary – although never perfect! – practice (as exemplified by the case used in Illustration 6).
How “real” the case is. The cases currently available in the literature cover the entire spectrum from faithful representations of real-life events to fictitious situations. Most cases, however, are composites of several real-life events that have been created for the purpose of illustrating specific issues.  
How “pointed” the case is. A case is usually selected or constructed to illustrate specific points. This is especially true in collections of cases designed to help teachers grapple with different topics, such as elementary students’ developing conceptions of numbers and operations. However, Illustration 6 shows that almost any account of practice can become a case if it is appropriately framed for participants.

The other major area of variation depends on how the facilitator organizes the discussion about the case. Important variations can occur along any of the following dimensions:

How the case discussion is framed. Facilitators may determine the specific goals and foci for the discussion in advance and communicate this to the participants upfront, or be more open-ended and willing to set goals together with the participants.
How the case discussion is facilitated. As mentioned earlier, all facilitators should ensure that participants feel free to express their opinions and show respect for others’ ideas. Facilitators should also try to elicit multiple opinions, encourage debate, and invite further articulation of ideas among the participants. However, there are various ways to achieve these goals. Some programs, such as the Mathematics Case Method featured in Illustration 5, expect facilitators to follow a carefully articulated set of practices, while others are less prescriptive about what the facilitator should do.
What activities may accompany the case discussion. While case discussions may occur in isolation, most often they are accompanied by other activities intended to strengthen or extend the outcomes of the discussion. For example, teachers in the rational numbers case discussion (Illustration 5) engaged first as learners in the same mathematical tasks discussed in the case. In this way, they gained a personal understanding of the mathematics involved and began to think about alternative ways to approach these tasks. In other implementations, teachers have been invited to further pursue issues raised in the discussion through follow-up readings, or even mini action research projects in their own classrooms.

Cases can be used in a great variety of professional development formats – including summer institutes, courses, workshops and study groups.

Case discussion facilitators may require different kinds of expertise depending on the content and focus of the case. Whenever the case involves mathematics, a good understanding of the mathematical topic involved is critical to be able to direct the discussion in productive ways. However, cases focusing on leadership and school reform issues more generally may not require any mathematical expertise in the facilitator. Regardless of the content of the case, facilitators can greatly benefit from specific training in conducting case discussions, to learn strategies to set a conducive learning environment and to ask questions that can move the conversation in productive directions without dominating it.

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