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.
Continued
