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Gathering and Making Sense of Information
We described a number of creative and
novel learning experiences for teachers in the previous
chapters, but some traditional learning experiences still have
much to contribute to teacher learning. Indeed, in several of
the illustrations reported in the previous four chapters,
participants read articles or listened to presentations. In
this chapter, we show how teachers can benefit from these as
well as other forms of data gathering and sense-making,
including action research, as a main venue for learning. More
specifically, we will examine ways in which teacher education
informed by a constructivist paradigm can facilitate
teachers’ learning from
and with texts, videos,
presentations, and even data they have gathered in their own
research.
Theoretical rationale and empirical support
Having teachers listen to experts’
presentations and doing assigned readings has been the
preferred mode of professional development so far at both
pre-service and in-service levels. Interestingly, however, not
much research documents the effects of these learning modes on
teachers’ knowledge, beliefs or practice.
Nevertheless, gathering and making sense
of information continues to be a valuable tool for teachers and
any other learners. This mode of learning can become an
integral part of constructing a personal understanding of
issues and theories that are at the core of school mathematics
reform. Indeed, readings, presentations, and data collection
and analysis can all contribute to teacher education although
they may take on different forms and purposes when informed by
a constructivist perspective.
Recent research on reading, in
particular, can help us begin to reconceptualize how making sense of information can become an active and socially constructed
process. Reading researchers have argued that reading does not
need to occur as an isolated, or even individual, activity
(e.g., Harste & Short, 1988). First, reading should be
purposeful. In other words, teachers should read either to
address questions that they feel the need to know more about or
because their concerns could not be resolved through
discussion. Reading can also be a catalyst for other
experiences. Indeed, reading can fulfill many functions while
teachers inquire into any topic (Siegel, Borasi & Fonzi,
1998). Readings can provide background information, raise
questions for further inquiry about a topic, synthesize
different points of view, and offer models for teachers’
own practice. Research also teaches us that reading is not a
passive or straightforward matter of decoding or extracting
information from text (e.g., Pearson & Fielding, 1991;
Rosenblatt, 1994). Rather, readers always construct meaning in
interaction with the text, their own background and interests,
and their purposes for reading the text. Furthermore, such
construction of meaning can be even more productive when it is
augmented by interactions with other learners, so that
different interpretations can be shared and discussed.
Reading researchers also argue for
expanding our notion of what constitutes a text (e.g., Bloome
& Egan-Robertson, 1993; Green & Meyer, 1991), noting
that the principles of reading outlined above also hold true
for other “texts,” such as videos, presentations or
electronic media. Indeed, teachers can benefit from actively
constructing and negotiating meaning not only through written
texts but also videos they watch together or independently,
information they gather on the Internet or presentations made
by an expert or a colleague.
In addition to benefiting from
information others provide, teachers can gather their own data
to illuminate issues of particular interest to them. Teachers
can gain from participating in many forms of research, but
“action research” is especially promising as a form
of professional development (Holly, 1991; Eisenhower National
Clearinghouse, 2000). Action research is defined as “an
ongoing process of systematic study in which teachers examine
their own teaching and students’ learning through
descriptive reporting, purposeful conversation, collegial
sharing, and reflection for the purpose of improving classroom
practice” (Eisenhower National Clearinghouse, 2000,
p.18). Action research thus offers an ideal way for teachers to
learn more about teaching and learning mathematics and to apply
the results immediately to their own practice, although
conducting full-blown action research studies is not the only
way that teachers can benefit from gathering and analyzing
classroom data.
Illustration 9: Using a variety of
resources to rethink the teaching and learning of geometry in
middle school
The experience captured in this
illustration took place in the Leadership Seminar that was one
of the components of the Making Mathematics Reform a Reality
(MMRR) project we described in Chapter 2. After several
teachers had participated in the first year of the program,
they wanted to make more radical changes in their teaching.
During the first year, they had attended a Summer Institute
introducing them to an inquiry approach to mathematics
instruction and then implemented an illustrative inquiry unit
on either tessellations or area in their own classrooms. Their
experiences with the tessellation and area units made them
aware of the inadequacy of traditional approaches to teaching
geometry in the middle school curriculum. Although they felt
that the next logical step would be to revise their
school’s geometry curriculum, they were not sure how to
proceed. In the usual process for rewriting curriculum,
teachers sat around a table, and based on the current textbook,
discussed what contents should be covered at each grade and
how. The teachers suspected that this process might at best
eliminate some repetition in the existing curriculum, but that
it was not likely to help them reconceive the entire middle
school geometry curriculum.
After some lead teachers shared these
concerns in the Leadership Seminar, the facilitators decided to
use this opportunity to lead the group in a systematic
rethinking of the teaching and learning of geometry in middle
school. Such an experience could serve as a model for lead
teachers interested in replicating a similar process with
colleagues in their own school. An even more important goal for
this experience, however, was to familiarize the lead teachers
with the resources offered by relevant research studies and
exemplary instructional materials, so they could use these
resources well in the future.
The group inquiry started with a few
readings about geometry. As a homework assignment, participants
read two mathematical essays from the book On the Shoulders of Giants (Steen, 1990). One essay focused on the concept
of “Shape” (by Senechal) and the other on
“Dimension” (by Banchoff). As part of the same
assignment, participants reviewed the NCTM Standards (1989) for
geometry in middle school.
In the group discussion that resulted,
the lead teachers analyzed the meaning and rationale of each of
the NCTM geometry standards in light of the “big
ideas” of geometry presented in the two essays. This
discussion enabled participants to enhance their understanding
of the mathematical concepts presented in the two essays and to
consider implications for instruction. For example, some
teachers said they found it very helpful to think of geometry
as the study of “shapes,” especially as they had
come to realize the connection between the geometric properties
of a shape and its possible functions. This realization helped
them frame in a more meaningful way the study of geometric
figures for their students. It also helped them change their
instructional goals because they agreed that students should
learn strategies for identifying the attributes of any
geometric figure, not just memorize a pre-established set of
properties for a few standard figures.
Although very helpful, this activity did
not immediately result in a plan for what to teach about
geometry, and how, at different grade levels in middle school.
The facilitators then suggested that the group look at the
choices made by two of the comprehensive middle school math
curricula funded by the National Science Foundation, the Connected Mathematics Project and Mathematics
in Context. In both cases, groups
composed of mathematicians, mathematics educators, and teachers
had grappled for years with the same question: What should
students learn about geometry in middle school? The
facilitators argued, then, that the group should capitalize on
all the thinking that had gone into the development of these
exemplary curricula.
However, it turned out to be difficult to
extract from the curricula the choices that the authors had
made about what geometry content to cover and how, and the
rationale for these decisions. Although the background
materials accompanying each of these curricula did address, to
some extent, these choices and how they were made, the
information was not specific enough for the group. It soon
became clear that the group needed to examine the individual
geometry units in each curriculum.
To make this task less daunting and
time-consuming, the group divided up the responsibilities. Each
participant, including the facilitators, agreed to review one
or two units from each curriculum to identify what was taught
and how and to present their findings to the group. To ensure
consistency, the facilitators proposed some guidelines for the
review and report on each unit and then modeled a presentation.
A 3-hour session was then devoted to the
geometry unit presentations. To get a sense of how topics in
each curriculum were sequenced, participants presented the
units in the order they were intended to be taught. As each
unit was presented, a facilitator recorded on newsprint the key
ideas about geometry that the unit addressed. At the end of the
presentations, the teachers had a detailed list of the geometry
content that each curriculum covered.
The group then compared these lists to
identify similarities and differences between these two
Standards-based curricula and the traditional middle school
geometry curriculum. Many teachers were amazed at the richness
of the lists describing the new curricula when compared with
the traditional middle school math curriculum. They were struck
especially by the emphasis in both of the new curricula on
three-dimensional geometry and spatial visualization, topics
they rarely covered but that were highlighted in the geometry
essays they had read. On the other hand, they were puzzled by
the presence of some new topics, such as Euler’s formula
and graph theory in the Mathematics
in Context curriculum.
The facilitators then suggested they seek
a mathematician’s help to examine further the relative
importance of the topics on the lists. The facilitators met
independently with Dr. Sanford Segal, a research mathematician
on the faculty at the University of Rochester, to share the
group’s lists and ask whether he felt comfortable
commenting on the mathematical significance of the topics
listed. They also shared some information about the
group’s background and goals to help him prepare his
contribution.
Dr. Segal then joined the group for a
2-hour session in which he presented his comments on the
relative importance of items on the lists from a mathematical
stand-point, and then he answered questions. His presentation
and the follow-up discussion further confirmed the critical
role of spatial visualization in mathematics, and hence the
importance of developing this skill in middle school through
appropriate learning experiences. On the other hand, Dr.
Segal’s personal position on the relative importance of
graph theory and transformation geometry challenged the need to
introduce these topics at the middle school level.
Overall, all participants, facilitators
included, emerged from this inquiry with a much deeper
understanding of what the “big ideas” in geometry
are and a greater appreciation for the complexity of making
good choices about mathematics content at any grade level.
Continued
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CHAPTER 8