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Engaging in Mathematical
Experiences-as-Learners
In the type
of professional development experience we describe in this
chapter, teachers engage as genuine learners in mathematical learning experiences. While the
nature, content and duration of these learning experiences may
vary considerably, they all model effective instructional
and/or learning practices promoted by school mathematics
reform. Reflection is a critical part of these activities
because it helps teachers analyze the experiences in light of
their own beliefs and practices.
Theoretical rationale and empirical support
The benefits of teachers experiencing
mathematics as learners go well beyond the important, rather
obvious one, that teachers learn more mathematics. Research
shows that teachers’ beliefs about mathematics and about
teaching mathematics are formed mostly as a result of having
been students in traditional mathematics classrooms (Thompson,
1992). Since traditional mathematics is informed by pedagogical
beliefs and practices that are radically different from those
promoted by the current reform efforts, many teacher educators
argue that before classroom teachers can change their beliefs,
they must have personal experience of alternative pedagogical
approaches (Brown, 1982; Schifter & Fosnot, 1993).
Further support for the value of
experiences-as-learners for teachers comes from research on the
learning of complex tasks. As we discussed in Chapter 1,
Collins and his colleagues (1989) identified modeling as the
first of three phases in the process of learning a complex
task. When the complex task is learning a novel approach to
teaching mathematics, we believe that facilitated
“experiences as learners” activities offer an
especially effective vehicle for such modeling. First, teachers
observe an expert mathematics teacher educator teach
mathematics in a non-traditional way. Second, because teachers
participate in this instructional experience as learners
themselves, they are in a unique position to examine how their
students may feel about the new approach. As a result, they
are in a better position to evaluate its potential advantages
and drawbacks.
Simon’s “learning
cycles” model of teacher learning, which we described in
Chapter 3, clarifies further the multiple roles that this type
of activity can play in a professional development program. In
Simon’s first phase of the learning cycle, teachers must
participate in situations that engage them actively as learners
and that evoke cognitive dissonance. In this way, they are
stimulated to construct new meanings. In the second phase,
through sharing and discussing these constructions with a
group, teachers come to consensus and make generalizations.
This model suggests to us that good mathematical learning
experiences for teachers need to invite active engagement,
provoke cognitive dissonance, and encourage social as well as
individual construction of meaning. Simon’s model further
claims that what is learned in one cycle can be used to
stimulate another cycle of learning. We suggest that reflecting
on these mathematical learning experiences can become the
catalyst for teachers to begin yet another “learning
cycle,” this time focusing on the nature of mathematics
as a discipline, how people learn and what can best support
such learning.
Research corroborates the benefits of
teachers experiencing mathematics as learners articulated
above. This type of professional development experience plays a
central role in several professional development programs with
documented success (Simon & Schifter, 1991; Schifter &
Fosnot, 1993; Borasi, Fonzi, Smith & Rose, 1999). A
systematic study conducted by Simon and Schifter (1991) in the
context of one of these programs has specifically shown changes
in teachers’ beliefs and practices toward a more
constructivist approach to teaching mathematics. Since
mathematical experiences-as-learners were not the only kind
of professional development experience employed in these
professional development programs, the results may not be
considered conclusive. However, case studies and anecdotal
evidence (Schifter & Fosnot, 1993; Borasi, Fonzi, Smith,
& Rose, 1999) further confirm that experiences-as-learners
were a critical element in changing the beliefs and practices
of several participants in these programs.
Illustration 1: A facilitated inquiry on
area for teachers
We derive the illustration in this
section from one of the Introductory Summer Institutes in the Making
Mathematics Reform a Reality in Middle School (MMRR) project
described in Chapter 2. This experience-as-learners was
designed to help teachers analyze how an inquiry approach to
teaching mathematics involves a radical rethinking of both
mathematical content and pedagogical practices. It was also
intended to introduce teachers to an “illustrative
inquiry unit” they might be teaching in their own classes
later -- a unit on area formulas designed for middle school
students (the same unit featured in the classroom vignette
included in Chapter 1). This experience-as-learners thus
engaged participants in an inquiry similar to one they might be
using with students.
The participants in the implementation
described here included elementary teachers, secondary
mathematics teachers, and special education teachers at the
middle school level. It took about seven hours over three
consecutive days to complete.
The instructor began by asking
participants to take off their “teachers’
hats” and become learners in a series of activities about
the concept of area. The instructor warned participants that
this was not going to be a simulation in which they
should pretend to be elementary or secondary students. Rather,
the content would challenge everyone at their own level of
expertise, so they should participate as genuine learners and
use all they knew to deal with the tasks presented to them.
The first task was to find the area of a
“fish” similar to the one middle school students
worked with in the classroom vignette (see Figure 8).
Figure 8
The “fish”
Each teacher worked on this task first
individually, then with a partner. The pairs then shared their
results with the whole class. Most secondary mathematics
teachers broke the fish into simpler figures, computed their
areas using formulas they knew, and then added up those areas.
A special education teacher had used a similar approach, yet
made more efficient by using the symmetry of the fish and
folding the figure in half. An elementary teacher showed
instead how she had “boxed” the fish and then
subtracted the area of the “extra pieces.” Another
elementary teacher “admitted” that she had simply
“counted the squares,” matching partial squares as
best as she could to form whole squares.
Everybody was surprised by the variety of
these approaches and by the fact that non-mathematics
specialists had proposed the most creative solutions. A lively
discussion surrounded this sharing, and participants came to
appreciate the value of alternative strategies for finding the
area of complex figures and the role that area formulas played
in some of these strategies.
Next, the instructor challenged the
participants to develop some area formulas on their own. First,
she modeled this novel process by creating, together with the
participants, an area formula for “diamonds.” Later
in the activity, she defined a diamond as “a
quadrilateral with perpendicular diagonals.” This task,
and the reflection that followed it, highlighted important
elements in the process of developing area formulas.
Participants then worked independently in
small groups to develop area formulas for “regular”
stars. The next day, they shared the area formulas they had
created and explained the process they had used to derive them.
Once again, everyone was amazed by the variety of area formulas
thus created and by the creativity shown by several class
members who had little mathematical background.
To help participants further appreciate
the complexity of the mathematical concept of area, the
instructor asked them to grapple with some thought-provoking
questions for homework:
• Why are squares chosen as the
“unit” to measure areas? Could other shapes be
used? Why or why not?
• How do we choose the “size” of the
squares to be used as units? Can this choice affect the value
of the area of a given figure?
• Area formulas essentially enable us to compute
the area of a two-dimensional figure by taking only linear
measures (i.e., the length of the height, base, radius, etc.).
How is this possible? Does this mean that you can measure area
with a ruler?
• Can we ever find the area of a curved figure
EXACTLY? For example, does A=π r2 give us the exact
value for the area of a circle or just a good approximation?
The difficulty they encountered
responding to these apparently simple questions astounded the
teachers. In all of their years as students of mathematics, not
even the secondary mathematics teachers had been asked to think
about questions like these, because learning about area had
been reduced to memorizing and applying area formulas.
In the next session, the group discussed
these questions in depth. At the end of this discussion, the
facilitator handed out a mathematical essay on area as a
follow-up reading assignment, to both validate some of the
conclusions the group had reached and expand them further.
A number of follow-up activities
encouraged the participants to reflect on this unusual learning
experience and to analyze it from different perspectives.
Participants listed “what they had learned” about
area from this experience. This list was quite detailed and
complex. Interestingly, although the teachers included a few
technical facts, such as learning area formulas for diamonds
and stars, they primarily identified elements related to
mathematical processes and the nature of mathematics. For
example, they highlighted the importance of learning to develop
area formulas on their own, of understanding the role played by
the choice of unit in measuring area, and of recognizing that
mathematical problems could have more than one acceptable
solution. Several participants also mentioned gaining increased
confidence as learners of mathematics as a result of this
experience.
The facilitator then began a discussion
on the instructional goals that should inform a unit on area for their students. Not surprisingly, the group established quite
different goals for their students than it is traditionally the
case, such as: Students should understand the concept of area
(how it is useful, what is actually measured); students should
understand the concept of scale; students should discover that
there is more than one formula for a given figure; students
should be able to derive formulas.
A day later, as a culminating experience
for the whole Summer Institute, the facilitator asked
participants to reflect on this experience-as-learners on area
and another experience-as-learners on tessellations they had
engaged in a few days earlier. This time, the participants were
asked to identify the teaching
practices that the institute
instructors had modeled in these experiences. As individuals
shared their reflections with the whole group, the facilitator
probed responses to elicit the rationale for, and potential
effects of, these practices in their own classroom instruction.
Continued
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CHAPTER 4