• Strengthening one’s knowledge of
mathematics.
Experiences-as-learners are ideal for strengthening
teachers’ knowledge of mathematics. However, the nature
and extent of this learning depends on the duration and design
of the learning experience. For example, immersion experiences
(as shown in Illustration 2) expose teachers to mathematical
tools and applications used in business, not in the traditional
school curriculum. By seeing what mathematical knowledge and
skills are really needed to solve real-life problems, teachers
may begin to question what their students should learn.
Consequently, they may rethink the goals of the mathematics
courses they teach.
Teachers can also learn something new
about topics that are currently in the K-12 curriculum, as
shown in the area inquiry in Illustration 1. There are many
benefits to doing so, since teachers – even those who
have taken several college-level mathematics courses –
often lack the deep conceptual understanding of mathematical
topics in the K-12 curriculum that are necessary to implement
reform lessons. As reported earlier, several teachers in the
inquiry on area had never questioned the significance of using
squares as units when measuring area, nor had they really
understood what area formulas are or where they come from.
However, the mathematical insights these teachers gained might
not have been achieved at the same level without the reflection
and discussions that followed the learning experience itself.
Follow-up reflective discussions, such as the “What I
have learned” analysis that followed the inquiry on area,
are critical to challenge participants’ views of
mathematics as a discipline and their perceptions of themselves
(and their students) as learners of mathematics.
• Understanding
the pedagogical theories that underlie school mathematics
reform. Experiencing mathematics
as learners has also the potential to help teachers understand
better the pedagogical theories that inform current reform
efforts. As Simon’s (1994) model of learning cycles
suggests, this kind of professional development activity not
only provides an experiential basis for new learning approaches
but also stimulates teachers to reflect on, and inquire further
about, the theories of learning and teaching on which these
approaches are based. To ensure a thorough understanding of
learning theories, however, personal reflections need to be
augmented by specially designed follow-up readings and/or
presentations, something that was missing in our illustrations.
• Understanding
students’ mathematical thinking. Because experiences-as-learners focus on the teachers’
learning, they are not an ideal vehicle to pursue an
understanding of students’ learning and thinking processes.
However, these experiences do help teachers become aware of
their own – and other adults’ – mathematical
thinking and problem-solving strategies. This awareness can be
eye-opening for many teachers, and it can inspire them to
examine their students’ thinking in the future.
• Learning
to use effective teaching and assessment strategies. Experiences-as-learners are especially
appropriate for modeling effective teaching practices, at least
when the facilitator has the expertise to do so. As we argued
earlier, modeling is a critical part of learning complex tasks
(Collins, Brown, & Newman, 1989). To be most effective,
modeling should not stop with the expert performing the novel
task in front of the novice. Rather, it should be accompanied
by explicit reflection on the teaching practice that was demonstrated
so that participants can recognize and internalize its key
elements. We believe, therefore, that a focused follow-up
reflective session is necessary to help teachers identify the
teaching practices modeled and to analyze the implications for
mathematics instruction (as shown in Illustration 1).
• Becoming
familiar with exemplary instructional materials and resources. Depending on the content of the
mathematical learning experience, experiences-as-learners may
or may not help participants become familiar with exemplary
instructional materials and resources. Teacher educators who
want to introduce participants to an exemplary curriculum
series or to a replacement unit that teachers will be expected
to implement later in their classes need to select mathematical
tasks from these materials and adapt them for an adult
audience. This is what happened in the inquiry on area we
featured in Illustration 1, and it is a practice used in many
projects designed to support the implementation of NSF-funded
curricula.
• Understanding
equity issues and their implications for the classroom. By doing mathematics in a group, teachers are
inescapably confronted with the diversity in learning styles
and approaches that exist. This is especially the case, though,
when the mathematical task is open-ended and there are
opportunities to share different solution processes. The
experience can be especially powerful when the group is highly
diverse and the implications of the differences are addressed
explicitly. However, it is our experience that given an
appropriate mathematical task, any group of learners will
produce enough diversity in responses to begin a conversation.
Facilitated experiences-as-learners are also ideal for modeling
strategies for differentiated instruction based on diverse
learning needs and, then, discussing participants’
reactions to these strategies.
• Coping
with the emotional aspects of engaging in reform. Coping with the emotional aspects of engaging
in reform is not a central goal of engaging teachers in
experiences as learners of mathematics. Nevertheless, using this
kind of professional development experience early in a program can
be instrumental in creating a bond among participants and
engendering a “community of learners” that can
offer emotional support as the participants undertake
instructional innovation in their classrooms later on. It is
also important to recognize that for some elementary and
special education teachers just engaging as learners in a
mathematical task may evoke painful memories of failure and
raise anxiety levels. Acknowledging and addressing these
feelings within the context of an experience as learners may
help these teachers overcome their fears, thus mitigating
emotional obstacles to their individual efforts at
instructional innovation later on.
• Developing
an attitude of inquiry toward one’s instructional
practice. As teachers
critically analyze the experience they participated in as
learners, they begin to appreciate the power of reflecting on
instructional practice. These reflective sessions can also
model ways for teachers to structure their own reflections to
make the process more productive. Therefore,
experiences-as-learners can be valuable in addressing this
teacher learning need, provided that the follow-up reflective
sessions are designed to achieve that goal.
Summary
Our analysis shows that activities in
which teachers become learners of mathematics can be a powerful
way to accomplish multiple professional development goals,
especially when they are thoughtfully designed and led by a
capable facilitator. Any variation within this type of
professional development experience can promote the learning of
new mathematics and challenge teachers’ beliefs about
what students should learn and how. These experiences can also
help teachers develop a vision for school mathematics reform,
examine pedagogical theories and effective teaching practices
and become aware of diversity in approaches to problem-solving
and learning styles. However, we caution that these benefits
depend on whether a facilitator carefully models novel teaching
strategies and orchestrates focused reflections on these
experiences. The length of the activity, the complexity of the
tasks, the design of the format, and the structure of the
follow-up reflection may also determine the extent to which
this kind of professional development experience can meet
various kinds of teacher learning needs.
Suggested follow-up resources
If you are interested in learning more
about exemplary professional development materials that can
help teacher educators plan and facilitate mathematical
experiences-as-learners, we recommend the following resources:
Corwin, R.B., Price, S.L., and
Storeygard, J. (1996). Talking
mathematics: Resources for developing professionals. Portsmouth, NH: Heinemann.
This multi-media package is intended to
support teacher educators in planning professional development
for elementary teachers to help them promote and facilitate in
their classes the kind of mathematical discourse recommended by
the NCTM Standards. A main component of the proposed
professional development program are experiences-as-learners
where the teachers engage in a number of mathematical problems,
chosen because they are mathematically rich and
“engaging” yet accessible to elementary students.
The materials include a facilitator guide, videotapes providing
images of elementary classrooms engaged in mathematical
discourse and a book for the participants. The facilitator
guide provides considerable support for setting-up and
facilitating the suggested experiences-as-learners.
Friel, S.N., and Joyner, J.M. (Eds.).
(1997). Teach-Stat for
teachers: Professional development manual. Palo Alto, CA: Seymour.
This manual is intended to support
teacher educators interested in replicating the 3-week summer
institute developed and field-tested by the NSF-funded
Teach-Stat project. This program was designed to prepare
elementary teachers to teach statistics and at its core has a
carefully-designed series of experiences where the teachers
themselves learn statistics in the inquiry-oriented way they
are expected to encourage in their students. The manual
provides valuable directions and support about how to plan and
implement the summer institute.
Fonzi, J., and Borasi, R. (2000). Orchestrating math experiences for
teachers. (videotape +
facilitator’s guide) (available from the authors).
This 50-minute videotape features the
mathematical inquiry on area described in Illustration 1. The
accompanying guide provides additional information about and a
commentary on this mathematical learning experience and a rich
set of questions to help teacher educators use the illustration
to design similar mathematical learning experiences for
teachers.
Fonzi, J., and Borasi, R. (2000). Promoting focused reflections on
learning experiences. (videotape +
facilitator’s guide) (available from the authors).
This 40-minute videotape features
excerpts from three reflective sessions that followed the
inquiry on area featured in Orchestrating math experiences for teachers and another inquiry on the topic of
tessellations. Taken together, the three sessions illustrate
complementary ways to focus and structure follow-up
reflections, a critical component of effective experiences as
learners. The accompanying guide offers additional information
about and a commentary on the illustrations and questions to
help teacher educators analyze what it takes to successfully
design and facilitate this kind of reflective session.
Borasi, R., and Fonzi, J. (in
preparation). Introducing
math teachers to inquiry: A framework and supporting materials
for teacher educators.
(multi-media package) (available from the authors).
This multimedia package supports
mathematics teacher educators who want to implement a
professional development program to begin the process of school
reform. It shows teacher educators how to design experiences as
learners that introduce teachers to an inquiry approach to
teaching mathematics. The package contains two 2-hour-long
videos, each featuring an experience-as-learners. The CD-ROM
included in the package contains a detailed set of artifacts
from these experiences and suggestions for implementing similar
ones.
