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For example, when someone identified the
think/pair/share technique for the “fish” activity,
a teacher pointed out how helpful it had been for her to work
individually on the task first. Others corroborated this
observation, noting the value of getting personally engaged in
a task before interacting with others. In contrast, one
participant expressed his relief at knowing that this
individual stage would last only a few minutes, since he
initially believed he would never be able to compute the area
of the “fish” alone. This discordant opinion
invited some considerations about differences in
individuals’ preferences and learning styles. Other
people then commented on the power of the whole group
discussion and how it had helped them go well beyond what they
had achieved working with just one partner. The group agreed on
the value of being able to explain one’s strategies and
solutions to another person first, and all the participants
felt that this stage had been beneficial not only for gathering
courage to report their ideas to the whole group but also for
clarifying and expanding ideas by talking with a partner.
This reflective session also enabled the
participants to recognize and discuss the role of less evident
yet equally important pedagogical decisions, such as starting
the unit with the complex, open-ended task of finding the area
of the “fish.” Participants noted the marked
contrast between this decision and the traditional practice of
assigning complex problems only after students
have learned specific procedures that are presumed to be
prerequisites for solving problems efficiently. This insight
led to discussing the different assumptions about learning that
distinguish constructivist/inquiry-based mathematics from
traditional practices grounded in behaviorist learning
theories.
Illustration 2: Working alongside
mathematicians in a real-life setting
We adapted the illustration in this
section from the Growth
in Education through a Mathematical Mentorship Alliance Project
(GEMMA) (ENC, 2000; Farrell,
1994).
As part of the GEMMA project,
teachers participated in an eight-week summer internship in
local businesses heavily involved in the use of mathematics and
science, such as consumer marketing companies, scientific
consulting firms, and automobile and other manufacturing
companies. Each teacher was assigned a mentor in a company, and
they worked together solving authentic problems that confronted
the business. These projects included analyzing market surveys,
testing fan blades for engines, researching the operation of a
microwave that was being installed on a factory production
line, determining and graphically displaying the relationship
among molecules in a new material, and creating a computerized
model of transportation systems. The companies expected
teachers to be fully contributing members of the
problem-solving team. In doing so, teachers had to learn about
current industry practices for solving problems and to identify
where and how mathematics was used.
During the internship, teachers attended
a series of seminars where they discussed what they were doing,
what mathematical applications they were learning, and what new
instructional practices they were generating from their
experiences with industry. By the end of the summer internship,
teachers were expected to have designed some applied
mathematical problems that they would pilot in their own
classrooms. The project goal was to create a booklet of such
“applications problems” to share with the other
mathematics teachers.
The outcomes far surpassed the GEMMA project
directors’ expectations. They hoped the teachers would
discover applications for the kind of mathematics they taught,
which they did. However, the directors found that the
internship experiences also introduced and/or reinforced many
of the current reforms in pedagogy. In their final papers, for
example, teachers wrote that they teach with a greater purpose
and that they feel a need to integrate mathematics and science.
They also wished to create collaborative learning environments
in their classrooms and to give students much more
responsibility for their learning.
Main elements and variations
The previous illustrations highlight
several of the elements we believe need to be a part of any
high-quality experience-as-learners.
Some of these elements have to do with
the nature of the mathematical
learning experience for the
teachers. In order to be effective, we believe that these
mathematical experiences need to accomplish the following:
• Challenge
the participants intellectually, regardless
of their mathematical backgrounds or the grade levels they
teach. Only under these conditions can teachers be genuine
learners and benefit fully from participating in these
instructional experiences.
• Be
mathematically sound and address key concepts. In order to strengthen teachers’
knowledge of mathematics and invite them to rethink the goal of
school mathematics, these experiences must offer opportunities
to learn worthwhile and significant mathematics.
• Allow
for mathematical reflection and discussion in addition to
mathematical problem-solving. Doing
so is essential to ensure that teachers revise and enhance
their current understanding of key mathematical concepts and
procedures, and do not just engage in “activities for
activity sake.”
• Model
non-traditional ways of learning and/or teaching mathematics. Participants must experience alternatives
to traditional school mathematics in order to appreciate their
potential for student learning.
Another set of characterizing elements
involves the reflections that follow the mathematical learning
experience itself. As both illustrations show, these
reflections are critical to the success of any
experience-as-learners in initiating teachers’ rethinking
of their views of mathematics, teaching and learning. The
following list captures the characteristics of optimal
reflective activities:
• Reflective
activities should occur after the learning experience is over,
not during it. In this way,
participants may find it easier to abandon their teacher roles
as they engage in the mathematical learning experience and be
genuine learners in it.
• There
should be opportunities for individual reflections as well as
group discussion. Participants
need to make personal sense of the experience as well as hear
other people’s insights and perspectives.
Despite these common characteristics,
successful experiences-as-learners can also differ in
substantial ways, as reflected by our two illustrations.
Important variations can occur along any of the following
dimensions:
• Duration
and complexity of the mathematical experience. Both of our illustrations included intense
mathematical experiences – a 7-hour inquiry on area in
Illustration 1, and a summer-long project in Illustration 2. In
contrast, there are examples in the literature of shorter
mathematical experiences, involving the solution of a problem
or other isolated mathematical tasks.
• Diversity
of participants. Participants
may be a rather homogeneous groups of mathematics teachers
teaching at the same level of schooling or they may include
mathematics specialists and non-mathematics specialists at
different grade levels (as it was the case in Illustration 1).
• Facilitator’s
role. The facilitator may
purposefully model some innovative teaching practices (as in
the inquiry on area reported in Illustration 1) or simply work
alongside teachers in a joint task (as expert mathematicians
did in Illustration 2).
• Scope
and structure of follow-up reflections. Reflective activities may be open-ended or focused
explicitly on specific aspects of the learning experience. For
example, facilitators may ask teachers to reflect on the
teaching practices modeled, the reactions of different learners
to the experience, or their views of mathematics. Leaders may
also elicit individual reflections in different ways, such as
asking teachers to respond in writing to written prompts, to
write in journals or to brainstorm ideas with a partner before
having teachers share and discuss them.
Experiences-as-learners can also take
place in a variety of professional development formats. They
can be part of an after-school workshop, a summer institute, a
university course, an on-site study group, or even an immersion
situation in which teachers become mathematics-learners and
problem-solvers alongside mathematicians in real-world
settings. In many cases, part of the participants’
mathematical experience may require projects or other
assignments that are undertaken by each teacher independently.
Experiences-as-learners can be conducted
by facilitators with a variety of backgrounds. Although
mathematicians might seem to be ideal facilitators for this
type of professional development, they may need to work
collaboratively with experienced teachers or mathematics
educators who can complement their subject matter expertise
with experience in instructional innovation. Conversely,
experienced teachers playing the facilitator’s role may
benefit from coaching on the differences between teaching
adults and K-12 students and from readings about the “big
mathematical ideas” that form the core of any experience
as learners. Regardless of their affiliation, facilitators
leading experiences-as-learners need both a strong mathematical
background and the ability to model innovative teaching
practices.
Teacher learning needs addressed
Experiences-as-learners have the
potential to address many of the teacher learning needs we
identified in Chapter 1, yet the extent to which they do so
depends on how the activity is implemented. In this section, we
discuss what specific variations of experiences-as-learners can
best help meet the needs of teachers who are interested in
pursuing school mathematics reform and how.
• Developing
a vision and commitment to school mathematics reform. Mathematical experiences-as-learners can
be powerful to help teachers understand what school mathematics
reform really mean and why it should be promoted. When a
skilled mathematics teacher educator designs the activities to
demonstrate the kind of mathematics instruction promoted by the
reform movement, teachers can appreciate the vast difference
between traditional and constructivist-based practices. For
example, the inquiry on area reported in Illustration 1 allowed
the teachers themselves to learn about a traditional
mathematical topic by focusing on big mathematical ideas,
solving problems through inquiry and constructing knowledge
with others. It also illustrated concretely the new roles that
teachers and students must play when a constructivist view of
learning informs mathematics instruction.
The personal success and enjoyment that participants experience in novel mathematical activities are powerful motivators toward instructional innovation. Committed teachers want their students to experience the same positive emotions about mathematics. We have observed this happen, especially with teachers who have bad memories of being students in traditional mathematics classes. Even teachers who were successful students in traditional settings, however, can experience vicariously their colleagues’ delight when they share such thoughts as “I never knew I could do mathematics! If only I had been taught this way!” This kind of response is especially common when the group includes non-mathematics specialists.
Continued
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CHAPTER 4 continued