____ THE
______ MATHEMATICS ___
_________ EDUCATOR _____
Volume 16 Number 1
Spring 2006
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Ginger Rhodes
Dear TME Readers,
Along with a team of editors, I am proud to present the first issue of Volume 16 of The Mathematics
Educator. Even though our field is filled with various debates that range in topics from reform to
accountability, we are all united in our goal to improve mathematics teaching for all students. As we
consider the ways to improve mathematics teaching we must consider a central player – the classroom
teacher. There is a growing body of literature documenting effective professional development experiences
for teachers. This TME issue contributes to the professional development literature by presenting a collection
of articles related to InterMath (http://intermath.coe.uga.edu/), a professional development project created in
adherence to recommendations for high quality professional development.
The stated focus of InterMath website is to build teachers’ mathematical content knowledge through
mathematical investigations that are supported by technology. The articles presented in this issue focus on
the courses provided by InterMath, but I want to acknowledge that InterMath is much more than the courses
it has developed. With a quick look at the website, one would notice that it provides an array of resources to
teachers, such as a dictionary and lesson plans. It is documented that over 1.4 million users have benefited
from the website to date. It is also notable that InterMath has provided learning opportunities for
mathematics educators with varying professional titles, such as classroom mathematics teachers, district
mathematics coordinators, and doctoral students. The influence of InterMath can be traced from Georgia
throughout the United States and to other countries, such as Turkey.
This special issue on InterMath opens with an editorial piece from an innovative and renowned
mathematics educator who has been fundamental in InterMath’s development and implementation. James
W. Wilson – a faculty member at the University of Georgia (UGA) and a recipient of the Lifetime
Achievement Award for Distinguished Service to Mathematics Education from the National Council of
Teachers of Mathematics – has played a vital role in the InterMath project. In his editorial piece he
highlights his journey with InterMath and offers both optimism and pessimism about the future.
Chandra Hawley Orrill, a Research Scientist at the UGA, continues the InterMath discussion by
examining findings from two pilot studies on InterMath and the subsequent project modifications based on
those findings. She raises challenges and questions about professional development designed for teachers’
own learning. Following Orrill’s piece is a research study by Drew Polly that examines three participants’
focus during InterMath courses by providing insights into their background and reported learning. Next,
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill, and Drew Polly take a closer look at the graphical
representations in the write-ups of InterMath students, discussing some implications for professional
development and research.
A recent InterMath evaluation report states, “The impact of the InterMath project has been greatest in
its diffusion to other mathematics education programs at the state and national levels.” In this issue, Ayhan
Kursat Erbas, Erdinc Cakiroglu, Utkum Aydin, and Semsettin Beser describe the development of their
project T-Math, which is an adaptation of InterMath, in Turkey. They provide examples of mathematical
investigations adapted for use in that country as well as anticipated challenges.
The final two articles are personal accounts from a former InterMath instructor, Sarah Ledford, and a
student, Laura Grimwade. Each narrative piece documents its author’s InterMath experience and
professional growth.
It is our goal with TME to provide a range of articles that are thought provoking and insightful for
readers. Within this issue we have presented research studies and personal reflections about the InterMath
project. It is my hope that with these brief descriptions of the articles that you will be enticed to read further.
In my final remarks as TME editor I wish to recognize the many people who contribute to the success of
TME, including reviewers, authors, and faculty. In addition, I would like to offer a special thanks to the
editorial team. It is because of their countless hours of work, dedication, and support that TME has grown
and will continue to grow in the future.
Associate Editors
Rachel Brown
Erin Horst
Na Young Kwon
Kyle T. Schultz
Margaret Sloan
Catherine Ulrich
Publication
Stephen Bismarck
Advisors
Denise S. Mewborn
Dorothy Y. White
MESA Officers
2005-2006
President
D. Natasha Brewley
Corbin
Vice-President
Bob Allen
Secretary
Erin Horst
Treasurer
Na Young Kwon
NCTM
Representative
Sarah Ledford
Undergraduate
Representative
Erin Cain
Jessie Rieber
Ginger A. Rhodes
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@uga.edu
www.coe.uga.edu/tme
I wish to acknowledge a mistake in the previous publication of TME 2005, Vol. 15, No. 2. On page 2 in the biographical information about the
author it should read Andrew Izsák is Assistant Professor of Mathematics Education at the University of Georgia.
This publication is supported by the College of Education at The University of Georgia
____________ THE ________________
___________ MATHEMATICS ________
______________ EDUCATOR ____________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Spring 2006
Volume 16 Number 1
Table of Contents
2 Guest Editorial… Project InterMath
JAMES W. WILSON
4 What Learner-Centered Professional Development Looks Like: The Pilot Studies
of the InterMath Professional Development Project
CHANDRA HAWLEY ORRILL & THE INTERMATH TEAM
14 Participants’ Focus in a Learner-Centered Technology-Rich Mathematics
Professional Development Program
DREW POLLY
22 An Analysis of the Use of Graphical Representation in Participants’ Solutions
LAUREL BLEICH, SARAH LEDFORD, CHANDRA HAWLEY ORRILL, & DREW
POLLY
35 Professional Development Through Technology-Integrated Problem Solving:
From InterMath to T-Math
AYHAN KURSAT ERBAS, ERDINC CAKIROGLU, UTKUM AYDIN, &
SEMSETTIN BESER
47 In Focus… Teaching InterMath: An Instructor’s Success
SARAH LEDFORD
49 In Focus… The InterMath Experience: A Student’s Perspective
LAURA GRIMWADE
51 Upcoming conferences
52 Subscription form
53 Submissions information
© 2006 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2006, Vol. 16, No. 1, 2–3
Guest Editorial…
Project InterMath
James W. Wilson
My recent e-mail had a message that read, “The
current InterMath team will be having cake to say
goodbye to a few project personnel and to the end of
InterMath as a formal project.” Hmm. Am I doing an
editorial about something that is over? I do not think
so . . .
In fact, a primary point I wish to make in this
editorial is that we have a very RARE situation with
Project InterMath. Most National Science Foundation
(NSF) funded projects operate only as long as the
funds last, and then the activities cease. InterMath is
alive and well and continuing. More formally,
University of Georgia’s Learning Performance and
Support Laboratory (LPSL) continues to coordinate
45-hour workshops for in-service teachers in the areas
of Number, Geometry, Algebra, and Statistics, using
materials and syllabi from the site. Recently, the web
site has been moved to http://intermath.coe.uga.edu,
and the site will continue to be available for use.
Instructors at various sites use material from this web
site in their workshops and courses. Additionally,
InterMath materials are used in other projects.
For
example,
the
Interactive
Dictionary
(http://intermath.coe.uga.edu/dictnary/homepg.asp) has
been adopted by the Georgia Department of Education
for its official use. InterMath has been adapted to allow
it to continue to be a viable project while assuring the
work will continue after NSF funding has gone.
The NSF funded Project InterMath is a
collaboration among the Department of Mathematics
Education (EMAT), the LPSL at the University of
Georgia, and the Center for Education Integrating
Science, Mathematics, and Computing (CEISMC) at
the Georgia Institute of Technology. NSF funding
began in April 1999 and was for a five-year project.
So, Project InterMath was a formal project as
designated by the NSF funding that ended in 2004, and
we have had no-cost extensions.
In my view, the time of NSF support was only a
midpoint of this journey. Elements and ideas for
James W. Wilson is Professor of Mathematics Education at the
University of Georgia. His research interest includes
mathematics problem solving processes, teacher education in
mathematics, and the use of technology in mathematics teaching
and learning.
2
Project InterMath grew out of my early involvement
with LPSL, the Georgia Research Alliance (GRA), and
the Georgia Center for Advanced Telecommunications
Technology (GCATT). Support from GRA through
GCATT helped with the early development of webenhanced courses such as my courses, Technology in
Secondary School Mathematics (EMAT 6680) and
Problem Solving in Mathematics (EMAT 6680). These
efforts combined the use of open-ended mathematics
explorations with the use of technology tools.
By the mid-point of the 1990s there were various
efforts in EMAT to address the mathematics
preparation of middle school teachers. In the LPSL
meetings, in which I continued as an adjunct staff
member, we explored the ways we could incorporate
technology into the professional development of inservice middle school teachers. We began to develop
ideas for a project that would address: 1) improving the
mathematics preparation of in-service middle school
mathematics teachers, and 2) incorporating the use of
technology tools into these teachers’ explorations with
mathematics.
Several partnerships for these efforts were sought.
Using grants from the Eisenhower Plan Prototype
Development, we spent two years putting together a
proposal for NSF, a team of players from EMAT,
LPSL, and CEISMC, and prototype materials for what
was to become the InterMath Web Site. The Georgia
Department of Education pledged its support and
participation through the INTECH Centers, with
Valdosta State University and Kennesaw State
University as test sites.
More importantly, we had a talented team of
graduate students, teachers, and staff working with us.
We found assistantship support from a variety of
sources in EMAT, LPSL, and Eisenhower Grants. Our
team included experienced mathematics teachers at the
middle school and secondary school levels. As the
project has continued, that talent pool has been
replenished each year as graduate students completed
their degrees or moved on to other opportunities and as
the project moved from the development phase to the
field phase.
Leaving out lots of details, essentially the initial
years of NSF funding had an emphasis on building the
Project InterMath
InterMath web site (http://intermath.coe.uga.edu). The
content and explorations were built with a team in
EMAT, the structure of the web site was built by
CEISMC, and the coordination and management of the
project were provided by LPSL. Then as we were
ready to use the InterMath materials with middle
school teacher workshops, most of the operation was
housed in LPSL.
The Dictionary was not a part of the NSF proposal.
Rather, it grew out of an early identification by the preproposal staff for having something readily available
for definitions and elementary descriptions. It was
funded initially by an Eisenhower Plan grant and then
incorporated into the Web Site as it was developed at
CEISMC. Our focus was on a set of definitions
appropriate to the middle school and the writing was
targeted for middle school students, middle school
teachers, and the parents of middle school students.
One pre-dictionary e-mail I received said “I am trying
to help my child with his homework but I do not know
what an acute angle may be.” She needed a dictionary.
To our knowledge, this is the only mathematics
dictionary specifically targeted to the middle school
level.
We opted to put the dictionary on the InterMath
web site and eventually arrived at some compromise in
precision to be offset by the use of examples and
descriptions. Many of my e-mails from throughout the
world mention the InterMath mathematics dictionary
and how useful it is.
Did the project go as we planned? Of course not.
For example, the state of Georgia demanded that all
teachers complete INTECH training, and suddenly the
INTECH centers were no longer available to us. When
we were ready for field-testing of materials, Valdosta
State University and Kennesaw State University had
other agendas with higher priority. Our plans for
workshops for principals never materialized—their
attention span is too short for even a scaled down
workshop on mathematics. It took us almost a year to
get acceptance by the Professional Standards
Commission for approval of staff development credits
for the course syllabi we produced in Number,
Geometry, Algebra, and Statistics. In our planning, we
envisioned that college credit would be a primary
direction for 45-hour courses; however, the option of
staff development credit for which the school system
hires an instructor proved to be a more viable option.
Much more remains to be done to provide
professional development for middle school in-service
James W. Wilson
mathematics teachers. InterMath can continue to be a
vehicle to incorporate efforts to impact on and learn
from teachers’ practices in mathematics teaching.
I am optimistic that the new standards for
mathematics instruction will bring about improvement
in teaching practices. It will, however, be a slow
process. The new Georgia Performance Standards
(GPS) will demand enhanced content knowledge, new
pedagogical practices, and a deeper understanding of
both. The InterMath materials have been indexed with
the new GPS as well as with the NCTM standards.
I am pessimistic, however, about whether teachers,
as a profession, can get beyond the search for the
magic band-aid. One of the hard lessons of InterMath
is that many teachers just could not devote the time
needed for a 45-hour workshop. Yet, we feel strongly
that extended professional development activities of
that extent are going to be needed. Furthermore, as the
mandate goes out to implement the new GPS
standards, I worry that studying (the jargon is
‘unpacking’) the new standards in mathematics will
become the goal of professional development rather
than attending to the demands for deeper understanding
of mathematics.
InterMath has been a good journey for me. I have
developed a greater appreciation for the challenges
faced by middle school mathematics teachers. I have a
better respect for the expertise they bring to my classes
and workshops. I have been rewarded by seeing many
of our graduate students develop expertise in
developing materials, managing web page, organizing
workshops, writing syllabi, teaching middle school
teachers, and teaching school system instructors to run
InterMath workshops.
Moreover, I get a lot of e-mail from instructors and
students who are making use of the InterMath
materials. Some of them want help with problems.
Some of them have suggestions. Sometimes it is just a
“thanks for making this site available.”
I have chosen to avoid naming all of the students,
faculty, and staff members who have participated in the
InterMath experience over the years. It is appropriate
to close with a note of thanks to three colleagues. Mike
Hannafin, director of LPSL, and Paul Ohme, director
of CEISMC, were Co-Principal Investigators on the
NSF Project. Thanks to them for putting up with me
and being great team members. Chandra Orrill,
however, has been the LPSL Staff member who has
provided the leadership to make InterMath work and to
engineer its continuation in new directions.
3
The Mathematics Educator
2006, Vol. 16, No. 1, 4–13
What Learner-Centered Professional Development Looks Like:
The Pilot Studies of the InterMath Professional Development
Project
Chandra Hawley Orrill
& The InterMath Team
In recent years professional developers have reached a consensus about what constitutes effective professional
development, referred to in the literature as “learner-centered professional development.” InterMath is a
professional development project that was developed to address the recommendations for high quality
professional development for middle grades mathematics teachers. In this report, I will highlight two cases of
InterMath implementation. Then, I will offer a discussion of changes that have been made to InterMath in light
of the findings from the pilot studies and report preliminary analysis of the impact of these changes.
How can teachers teach a mathematics that they
have never learned, in ways that they never
experienced? (Cohen & Ball, 1990)
The above quotation is one of the underlying
conundrums of mathematics reform. Recent analyses
of mathematics assessments show American students’
failure to achieve even basic levels of proficiency on
national tests (U.S. Department of Educational
Statistics - OERI, 2001) and their low performances on
international tests (Cochran, 1999). These test results
have been accompanied by the National Council for
Teachers of Mathematics’ call for mathematics
teaching to embody the tenets of constructivism by
focusing more on hands-on engagement with
mathematics in the service of developing
understanding (NCTM: 1991, 1995, 2000). The NCTM
standards recommend that teachers pose meaningful,
complex tasks for their students, provide opportunities
Chandra Hawley Orrill is a Research Scientist in the Learning
and Performance Support Laboratory at the University of
Georgia. Her research interest is in how teachers make sense of
professional development and how the professional development
impacts learning opportunities for students.
Acknowledgements
The National Science Foundation under grant number
ESI9876611 has supported InterMath and the work reported
here. Opinions expressed are those of the researchers and do not
necessarily reflect the opinions of NSF. Pilot study data
collection, analysis, and writing previous versions of this report
was completed by the author and members of the InterMath team,
including Summer Brown, A. Kursat Erbas, Chad Galloway,
Evan Glazer, Brian Lawler, and Shannon Umberger Patton.
Ongoing analysis of post-Pilot Study InterMath data has been
supported by Laurel Bleich, A. Kursat Erbas, Sarah Ledford, and
Drew Polly. Previous versions of this paper have appeared in the
Proceedings of the Association for Educational Communications
and Technology (2001) and the proceedings from PME-NA XXIV
(2002).
4
for students to engage in real-world problems, and use
manipulatives and technology to support learners in the
construction of their own personal understanding of
mathematics concepts. While these recommendations
are very clear to the NCTM authors, they are
completely foreign to many classroom teachers. Given
the shifts called for by NCTM and the ongoing
problems with student performance, there is clearly a
rationale for rethinking both the role and the format of
professional development (e.g., National Council for
Science and Mathematics [NCSMT], 2000; National
Partnership for Education and Accountability in
Teaching [NPEAT], 2000; National Commission on
Teaching & America’s Future, 1996; Renyi, 1996;
Sparks & Hirsch, 1999).
Professional developers in recent years have
reached a consensus about what constitutes effective
professional development, referred to in the literature
as “learner-centered professional development” or
“research-based professional development” (NPEAT,
2000). These recommendations include extending
professional development beyond the "one-shot
workshop," promoting opportunities for teachers to
learn in the same ways they are expected to teach,
focusing on reflection, and pushing for more contentfocused teacher learning (e.g., Ball, 1994; Hawley &
Valli, 1999; Krajcik, Blumenfeld, Marx, & Soloway,
1994). As summarized by Kilpatrick, Swafford, and
Findell (2001):
Teachers’ professional development should be high
quality, sustained, and systematically designed and
deployed to help all students develop mathematical
proficiency. Schools should support, as a central
part of teachers’ work, engagement in sustained
efforts to improve their mathematics instruction.
This support requires the provision of time and
resources (p. 12).
What Learner-Centered Professional Development Looks Like
The National Partnership for Excellence and
Accountability in Teaching (NPEAT) has outlined the
aspects that should be included in this new kind of
professional development (NPEAT, 2000). Aligned
with other proposals for improving professional
development, the NPEAT Research-Based Principles
provide a guide for professional development. These
principles include:
The content of professional development (PD)
focuses on what students are to learn and how to
address the different problems students may have
in learning the material.
Professional development should involve teachers
in the identification of what they need to learn and
in the development of the learning experiences in
which they will be involved.
Most professional development should be
organized around collaborative problem solving.
Professional development should be continuous
and ongoing, involving follow-up and support for
further learning — including support from sources
external to the school that can provide necessary
resources and new perspectives.
Professional
development
should
provide
opportunities to gain an understanding of the
theory underlying the knowledge and skills being
learned (NPEAT, 2000).
In short, numerous researchers and policy-makers
now assert that teachers should take charge of their
learning, be provided with motivational and
challenging ways to learn, and should have the
opportunity to decide what is most relevant for their
students (Hawley & Valli, 1999).
InterMath
InterMath, a National Science Foundation-funded
initiative,
was
developed
to
address
the
recommendations for high quality professional
development for middle grades mathematics teachers.
Originally, InterMath was developed to be a 15-week
(45 seat hours) face-to-face workshop supported by a
variety of technologies including an extensive Web site
that provides over 500 open-ended investigations
(http://www.intermath-uga.gatech.edu).
InterMath’s
Web site also included an interactive dictionary of
common middle grades mathematics terms, a
discussion board, and a section designed to house
teachers’ electronic portfolios of work from their
InterMath courses. InterMath was specifically created
to help address a critical deficiency in teacher content
knowledge in the state of Georgia (Southern Regional
Chandra Hawley Orrill
Education Board [SREB], 1998). This problem was a
result of the number of middle grades teachers teaching
out of field or holding a “generalist” degree in
elementary education that did not provide the teachers
with a rich enough content background to develop
needed content and pedagogical knowledge.
InterMath’s initial goals included the improvement
of teachers’ mathematical skills and knowledge
through open-ended explorations; an understanding
and ability to use software to support the development
of mathematical thinking; and the creation of a
community of teachers who support each other in
implementing the explorations-based approach in their
classroom. In implementation, there is considerable
room for teachers to choose their own path to success –
they select which problem(s) they want to work on in
each of the critical content areas; they select the
approach they want to use to solve the problem; and,
ultimately, the teachers decide the depth of learning
they take from the class by choosing to explore more
challenging problems or add extensions to the
problems.
InterMath embodies many of the professional
development principles mentioned earlier. It provides
an extended opportunity for teachers to engage in
mathematics in the same ways they should engage their
own students in mathematics. Further, the format of
InterMath allows teachers to work with their peers,
select the problems on which to focus, and use a
variety of tools to support their own work. In fact, in
the pilot offerings and many of the current offerings of
InterMath courses, teachers have developed their own
calendars for completing assigned work. While many
of the teachers who have participated in InterMath
courses were not necessarily seeking an introduction to
reform-based approach to mathematics, all have
reported learning about aspects of the NCTM standards
that help define a quality mathematics experience.
In this report, I will highlight two cases of
InterMath implementation. Because these have been
discussed elsewhere (e.g., Brown et al., 2001; Erbas,
Umberger, Glazer, & Orrill, 2002), they will be brief
with particular emphasis on the findings. Then, I will
offer a discussion of changes that have been made to
InterMath in light of the findings from the pilot studies
and report preliminary analysis of the impact of these
changes.
The Pilot Studies
Two InterMath pilot studies were conducted
simultaneously in two different locations. One began
with seven teachers and ended with four, while the
5
other included 24 to 28 teachers at various points in the
semester. Both courses lasted an entire semester,
meeting three hours per week every week. Both pilots
used the original InterMath format, which engaged
learners in mathematics from across the four strands:
algebra,
geometry,
number
sense,
and
statistics/probability.
Both studies relied heavily on field notes taken by
graduate students who acted as participant observers
during each workshop. In both pilot studies, these
students supported the InterMath instructors and
recorded field notes for the research. Additionally, I
(Orrill) visited the larger workshop three times and the
smaller workshop one time, taking field notes as an
external observer. In those visits, the goal was to gain a
non-participant view of the learning environment. For
the purposes of the pilot study, three weeks of field
notes were selected from each class. They came from
early in the semester (week 2 or 3), mid-semester
(week 6), and later in the semester (week 12–14).
These weeks were chosen because they represented the
beginning, middle, and end experiences for the
courses.
In addition to the extensive field notes analyzed,
the data analyzed for this report included tape-recorded
interviews with several participants (eight in the larger
class and all four participants in the smaller class) and
both instructors. Interview participants in the smaller
workshop included all of the participants at the time of
the interview. In the larger group, the participants were
randomly selected. Pre- and post-workshop surveys
were administered, asking participants to rate the
importance of, and their comfort with, using
technology in mathematics and using open-ended
investigation approaches. We also considered the
written work of those participants who were
interviewed as part of the data analysis.
We relied on traditional qualitative data analysis
methods of coding and sorting to find emergent
categories (Coffey & Atkinson, 1996). Using this
approach, we identified several emergent categories
that appeared repeatedly and used those as a
framework for our thinking. Those included: Support,
Interaction, Barriers, Presentation, and Adoption. Once
we had defined the categories and made initial
assertions, we checked the data to find examples both
supporting and refuting those assertions and then
refined the assertion as appropriate. Each case is
briefly discussed below with a cross-case analysis
following.
6
Case 1
Description
One of the two InterMath Pilot workshops took
place near Atlanta, GA, and was taught by a University
of Georgia (UGA) mathematics education professor.
The participants included 24 to 28 full-time middle
school teachers who had enrolled in a UGA graduate
program. Even though the participants were all
certified to teach mathematics, some were teaching
subjects other than math. The teachers participated in
the InterMath workshop as their first experience in a
degree program established between their school
district and UGA’s mathematics education department.
Participants had chosen to join the degree cohort, but
had no choice in their coursework as part of the
program. Two InterMath team members offered
assistance in the class each week and participated in
the data collection effort.
The class met weekly in the evening for three
hours. During the first hour portion of each class, the
instructor demonstrated one or two problems, talking
through the mathematics and the technology used. For
the remainder of the class, participants explored the
investigations using software programs such as NuCalc
(http://www.nucalc.com/), Geometers’ SketchPad®
(GSP; Jackiw, 1990), and spreadsheet software and
completed reports of their problem solving processes to
include in their electronic portfolios. The instructor and
graduate assistants walked around the room to assist
the participants, when requested, with technological
and mathematical questions.
Trends
Over-reliance on the instructor. The participants
seemed to perceive the instructor and graduate
assistants as experts. They relied on the instructor
rather than each other for technological and content
area support. Moreover, they seemed to view the
instructor as the “owner” of the class. Even after
seeking help from the graduate assistants, the
participants often wanted the instructor’s approval. In
one instance, a participant was exploring an
investigation in which he needed to find the maximum
volume of a box. The participant asked one of the
graduate assistants how he could incorporate
technology into the investigation. More specifically, he
wanted to know what technology he could use. The
graduate assistant discussed some of his options.
Instead of exploring these routes on his own and
finding multiple representations of the problem, the
participant told the graduate assistant that he was going
What Learner-Centered Professional Development Looks Like
to ask the instructor which way he should explore the
investigation. The participant was seeking a “correct
process” for solving the investigation. He only wanted
to explore the problem the way the instructor/owner
would.
The instructor’s actions both encouraged and
discouraged this over-reliance on him. In our analysis,
the instructor’s actions that encouraged an overreliance included his positioning himself in an
ownership position in the way he directed the
workshop conversations and selected problems to
investigate. Further, he sought little input from the
participants about exploring the problems he had
chosen during the first hour of class, leaving the
participants in a passive role, which was characterized
by noticeable off-task behavior by some participants.
However, the instructor promoted participant
independence and ownership during the second portion
of each class period. During this portion of each
meeting, participants were given the freedom to select
which problems they worked, how they chose to work
those problems, and what technology they used.
Further, they were able to work with partners or alone.
In this workshop, participants chose to work
individually on their write-ups with little
communication with other participants even though
they were able to work with partners. This instructional
approach may have contributed to participants’
frustration with the level of support they received in
the workshop. The choices that the participants made
within the learning environment, because of the
freedom offered by the instructor, actually contributed
even more to the frustrations as the participants chose
not to rely on each other, instead preferring to rely on
the instructor, or, if the instructor was not available,
graduate students, for support. The class, in
observations during the second half of the workshop,
when the participants were engaged in their own
investigations, was described as being very quiet other
than the sound of mouses clicking and graduate
assistants talking to the participants.
View of InterMath. The data showed that the
participants’ views of the goals and purpose of
InterMath fell into one of three categories. In the first
category, participants saw InterMath as a “make and
take” activity to take into their middle school
mathematics classrooms. They selected investigations
based on their students’ level of mathematics
knowledge rather than their own levels. Because of
this, the participants did not appear to push themselves
to increase their own mathematical understandings. In
one class meeting, a participant voiced concern that the
Chandra Hawley Orrill
investigations seemed too difficult for middle school
students. The instructor explained, correctly, that the
investigations were meant for the teachers and that the
teachers would have to adapt them if they chose to use
them with middle school students. Despite this
explanation, some of the participants continued to cling
to the idea that the investigations were suitable for their
middle school students with little modification or
consideration of how to present such an activity to that
age level. The participants who treated InterMath in
this way likely did not benefit much from their
participation, given that InterMath is intended as a
personal growth activity for the teacher and does not
include the creation of materials suitable for classroom
use.
A second group viewed InterMath as a technology
course in which they wanted to learn how to use the
software tools but took little interest in using the tools
to develop their own mathematical understandings. In
the workshop observations, these participants became
excited when using the technology or learning
something new on the computer but seemed to focus
very little on learning new mathematical concepts and
making connections. For example, one of the graduate
assistants showed a participant which button to push to
display all the Excel functions she might have needed
to create a spreadsheet. The participant exclaimed,
“Woo-hoo! I’m finally excited about something in
here!” This participant apparently either wanted or
expected participation in InterMath to lead to more
effective technology use rather than to deepen her
understanding of mathematics.
The last group saw InterMath as an opportunity to
enhance their mathematical understandings. In the
interviews, these four participants stressed the learning
of mathematics over the learning of the technology as
the focal point of the course. Given that mathematical
development was one of the key goals of InterMath,
the low number of people in this group was
disappointing. One explanation for this might be the
paradigm shift represented by InterMath. Rather than
being focused on the development of classroom
materials or other products for student learning,
InterMath focuses on teacher learning, and this is a
different way of thinking about professional
development for the teachers. Another explanation for
the small number of participants in this group may be
the rather low mathematical knowledge base evident in
the participants. The participants particularly seemed to
experience difficulties in geometry and thus struggled
to make mathematical connections and develop
multiple representations that were crucial in the
7
investigations. However, these were the teachers who
seemed most interested in further exploration of the
mathematics and also the most reflective about their
own mathematical ability.
InterMath adoption to the classroom. Some of the
participants had already begun to use InterMath
investigations in their classrooms before the end of the
workshop. Surprisingly, in class discussions, teachers
reported little or no adaptation of the InterMath
investigations when they used investigations with their
middle school students. This is ironic, given the
teachers’ discussions about InterMath investigations
being too difficult for middle school students. Late in
the semester, a participant pulled one of the graduate
assistants to the side and shared with her what she had
been doing in her middle school classroom. She had
assigned her students to choose three InterMath
investigations directly from the Web site to work on
and to write-up over two weeks. This participant did
not make any modifications to the investigations, nor
did she offer any guidance to the students in selecting
their investigations. However, consistent with
underlying philosophies of InterMath, the teacher did
encourage her students to work together. In her
particular case, the students rose to the challenge of
successfully completing the investigations. However,
we saw this instance as an atypical occurrence.
Given the fact that a large number of the workshop
participants saw InterMath as a course designed to
provide them with materials that were suitable for use
in their own classrooms, it is not surprising that they
used the investigations in their classrooms. From the
interviews, we noticed that this wholesale transfer of
InterMath investigations from the web site to the
classroom was accompanied by teachers encouraging
their students to work alone – a mirror of how the
teachers chose to work in the workshop. Further,
students often did not receive guidance and were
attempting to work problems that were not appropriate
for them but were meant to be investigations for the
teachers.
Case 2
Overview
The second pilot of the InterMath workshop was
led by a mathematics professor and offered on the
UGA campus. The workshop met one evening per
week for an entire semester, as in Case 1. Two
graduate assistants, one from UGA's Mathematics
Education department and one from the Instructional
Technology department, regularly attended the class to
support the learners. The Instructional Technology
8
graduate student, in fact, served in the role of an
assistant instructor. A third graduate assistant attended
the first few meetings to help support the participants
in learning how to make and publish Web pages for
their electronic portfolios.
The class began with seven teachers; however, by
the end of the fifteen weeks, there were only four
participants in regular attendance. One of the teachers
who dropped the course did eventually complete it as
an independent study. Two of the participants who
completed the pilot course taught eighth grade prealgebra and algebra at a rural middle school. The other
two participants came from a private middle school—
one was a sixth grade mathematics teacher, and the
other was the school's technology support person who
also had a mathematics education background. In
general, this class was highly cooperative, with
teachers from the same schools working together both
on solving problems and creating their portfolios.
What the Participants Learned
There were some overarching successes in this
pilot class. First, the participants learned how to use
technology to create and post write-ups of their
mathematical investigations on the InterMath website.
Specifically, the participants learned how to use
computer software that included web page
development tools and FTP (file transfer protocol)
clients. On average, the participants posted seven
write-ups during the course. These write-ups often
included links to spreadsheets and/or GSP files.
Second, the participants learned to identify and
appreciate certain aspects of reform-based issues in
mathematics teaching and learning. As evidenced
through their final interviews, the participants noted
the value of problem solving, learning through
collaboration and communication, finding multiple
solutions and answers, and asking extension questions.
For example, when asked what students in an ideal
mathematics classroom would be doing, one
participant commented, “Well, after all this, problem
solving.” Another participant said that an ideal
classroom to her would be one in which the students
were “asking questions, and they’re showing their
classmates what's happening and sharing ideas and
thoughts and communicating with each other.” A third
participant mentioned that the most important things
she learned from the InterMath experience were “The
importance of thinking and not just computation. ...
And collaboration.” She also stated, “I've even told my
kids that there are lots of ways to find an answer, and
oftentimes the answer's not the important part.” It was
What Learner-Centered Professional Development Looks Like
clear that mathematics and mathematics education
pedagogy were key issues to these participants.
What the Participants did not Learn
There were also some critical areas in which
learning did not seem to occur as expected. First, the
participants did not seem to greatly expand their
mathematics content knowledge. Approximately 61%
of the write-ups posted were about investigations that
were taken from the Algebra or Number Concepts
units on the InterMath Web site. These units
correspond to the majority of the topics that are
covered in middle grades mathematics. Only 25% of
the write-ups focused on Geometry problems, and only
7% were Data Analysis problems. One participant
mentioned that after she and her partner struggled with
a problem that was hard, they would simply, “close
that one up, and we'd do another one.” Issues with
participants’ lack of perseverance and unwillingness to
try new areas, possibly relating to issues of low
mathematics efficacy or the perpetuated notion that
mathematics problems should be easily solved within a
short time, were prevalent.
Second, the participants did not become
comfortable with using a variety of mathematical
software in doing their investigations. Approximately
86% of write-ups indicated that the authors used
spreadsheets to help them with the investigations. Not
surprisingly, spreadsheets were the only software with
which the teachers had considerable experience when
they began the workshop. Only 18% of write-ups
illustrated use of geometry software, and only 4%
mentioned the use of graphing software. One
participant stated that she and her partner “felt more
comfortable using a spreadsheet. And it's just
because...that's what we could maneuver better with.”
Again, the teachers were not pushing themselves very
far in terms of the problems they chose to work and the
ways in which they chose to work them.
Finally, the participants did not develop a variety
of mathematical approaches to solving problems. Most
of the participants relied solely on numeric patterns or
measurements to justify their solutions to the
investigations. None of the write-ups offered
conceptual explanations or tried to rationalize why the
numeric patterns or measurements must have given the
correct answer. More disturbing, they also did not seek
to use extensions to push their thinking and/or their
students' thinking further, even though that was an
explicit focus of the instructor.
The instructor
commented that even when the participants wrote
extensions, they did not try to solve them. This fact
Chandra Hawley Orrill
may relate to the same issues that prevented attempts at
difficult problems —including seeing the extensions as
something their students would not be able to do or
worrying that they, themselves, could not adequately
answer the extensions they had written.
Cross-Case Analysis
Several findings spanned across both cases. There
were also some findings within each case that we were
unable to reconcile. For example, we are not sure why
our attrition level was high in the second pilot. For that
class, the three participants who dropped out were all
from a single school. This raised questions for us about
the nature of working with peers as well as whether it
is feasible to keep teachers from diverse districts
engaged in this professional development if they are
not working with others from their school district. We
can speculate about the role of peer participation in
keeping the teachers engaged or the need for more
accessible locations; however, it is difficult to know
how to address the attrition problem, which has
persisted since the pilot studies.
For our cross-case analysis, we adhered to the
categories introduced previously. Based on careful
analysis of the findings within the coded categories of
the two separate case studies, we were able to develop
assertions about the professional development that
were true for both cases.
Support and Interaction
We found that support and interaction became
intertwined in our cross-case analysis. This
intertwining was a direct result of the nature of
interactions in these courses. It seemed that nearly all
interactions, whether between participants or including
the instructor, were focused on addressing the
participants’ concerns about their activities at a given
moment. We noted that there were two distinct kinds
of interactions: affective (those aimed at providing
positive feedback or other information to keep the
teachers motivated) and intellectual (those interactions
that provided the information teachers needed in order
to make progress on the problem with which they were
working). Based on our data, the affective interactions
were particularly important between participants.
Several times the learners commented that they felt
unprepared for InterMath until they began talking with
the other participants or until they began to find out
from the support staff that others were having the same
kinds of problems. In more than one case, this “same
boat” effect prevented our participants from dropping
out of the workshop.
9
Another support and interaction issue that appeared
was the overwhelming number of procedural questions
that were asked by the participants. In both of the pilot
workshops, the participants’ questions often focused on
how to use particular pieces of software until about the
halfway point of the course. Later in the courses, there
was more focus on process-oriented thinking, but the
procedural questions never faded entirely. This finding
raises a number of questions about supporting the
teachers in learning what was intended in the
workshops and about who needs to provide support
and what that support should look like. In Case 1, we
had about 25 teachers with three support people (two
graduate assistants and one instructor). In Case 2, we
had one instructor and either one or two graduate
assistants in every class session, but ended with only
four participants. Despite, or because of, the presence
of so many knowledgeable others, the participants
resisted engaging with each other for problem solving,
instead turning to those perceived as owning
information. This phenomenon leaves an open question
about whether InterMath was successful in helping the
participants see mathematics as being about problem
solving and other processes. It seems likely that they
still held the traditional idea that math is about right
answers and that the teacher’s role is to have those
answers.
Finally, while we provided every opportunity for
collaborative learning, few teachers chose to engage in
it. Even in those instances where teachers worked as
pairs or trios, they tended to work individually on the
problems and relied on others only when they were
confused or unable to continue alone. We also found
that among the teachers who did work together, almost
every group included teachers from the same school.
These findings, taken together, lead to two insights:
first, teachers seem to work with people they already
know and with whom they feel "safe," and second,
teachers are not naturally predisposed to working in
groups. This second point may explain many teachers’
reluctance to include group work in their classrooms
—reinforcing the need for the professional
development environment to model the desired
classroom environment.
Barriers and Difficulties
There were two main barriers to InterMath's being
as successful as possible across the two cases:
technology and “goals.” The technology problems
were related to participants’ inexperience with the
specific mathematics tools (e.g., Geometer’s
SketchPad®) and the need for them to learn to use web
10
development tools to be successful in the workshops.
Hardware problems and firewall issues throughout the
workshops exacerbated this lack of knowledge and
comfort. These were particularly common in Case 1. In
both classes, the difficulties with technology were
worse during the first several weeks of class with the
first half of each 15-week workshop being spent with
participants struggling to make and publish web pages.
Given that the Web page development component of
the class was secondary to the mathematical goals, this
was particularly problematic.
The barrier due to “goals” was caused by a
mismatch between the participant goals and the
workshop goals. In our follow-up interviews and
surveys, for instance, a large number of participants
indicated that learning technology was their personal
goal for participating in InterMath. While this group
was satisfied with their InterMath experience, learning
technology was not the InterMath team’s primary goal
for the participants. The InterMath team had hoped to
promote a different vision of teaching and learning
mathematics — certainly technology was a part of that
vision, but not the central focus.
Another large group of teachers seemed to think
that the InterMath workshop provided an opportunity
to become familiar with the InterMath website as a tool
that could be used in middle-grades classrooms. While
there are investigations on the InterMath website that
could be useful for middle school students, the purpose
of the site is to enhance teacher mathematical
understanding. Because teachers saw the site as a tool
for use in their own classrooms, many completed only
problems they felt their students could complete. This
meant that many of the teachers did not challenge their
own mathematical abilities at all.
On one hand, because the participants were able to
define and follow their own goals, they were pleased
with the outcome. On the other hand, we have concerns
about the kind and quality of learning because many of
the participants did not seem concerned with their
mathematical development. This is a recurring theme
in our ongoing professional development work. It is
unclear how to balance the identified content needs of
teachers as a group with the need for each teacher to
buy into the goals of a course.
Adoption
Our final major finding in the cross-case analysis
was a disturbing trend among the teachers who
implemented the InterMath problems in their
classrooms to structure their students’ learning
experiences exactly as their InterMath workshop
What Learner-Centered Professional Development Looks Like
experience had been structured. This was alarming for
a number of reasons. First, it may have demonstrated
little reflection on the part of the teachers about their
students’ abilities in mathematics. Further, the teacher
participants in both of the pilot workshops had
complained that there was not enough structure
because there were no clear guidelines for assignments.
Yet, they reported implementing this same kind of
approach for students who did not have the maturity
upon which to draw to cope in this extremely openended environment. In short, it seemed that the
teachers borrowed InterMath rather than adapting it for
use in middle grades. It may be argued that this is the
first step of changing practices, but at the conclusion of
the workshop, there was no further support for the
teachers unless they specifically requested it.
Further, post-workshop surveys indicated that
participants were not yet comfortable with the
implementation of technology-enhanced problem
solving in their own classrooms. This was corroborated
by the teachers we interviewed who asserted that they
could use InterMath problems and technologies in their
classrooms by demonstrating them and by those
teachers who asserted that they needed more practice
themselves before they could implement InterMath in
their own classrooms. While this is, in a sense, the
opposite of the problem we saw with wholesale
adoption of InterMath for middle grades classrooms,
the teachers’ discomfort with technology-enhanced
problem solving likely prevented their students from
having successful experiences with mathematical
explorations. However, the InterMath team also
recognizes that technology access in many schools
precludes the use of technology in ways other than as
demonstration tools.
Follow-Up
In the three years since the original InterMath
pilots, we have been able to collect data on
approximately 12 more offerings of different versions
of InterMath. While none of these has been as
thoroughly observed and documented as the initial
workshops, we have collected survey data from 10
courses, interviews from participants in approximately
five courses, and other data, such as performance
assessments, in a handful of courses. During the course
of these workshops, we have moved to a different
implementation plan that involves a train-the-trainer
model in which UGA personnel train district-based
instructors to teach the courses. Because of the results
of the first workshop and our work with various school
districts since then, we have modified InterMath in
Chandra Hawley Orrill
some ways that have led to some different findings
from our initial study. Here, I report some of the
preliminary findings of these later studies.
One major change to InterMath that is pervasive in
these more recent courses is a change to teaching
InterMath courses that focus either on only one strand
of content (Algebra, Geometry, Number Sense, or Data
Analysis) or focus on the issues of using open-ended
problem solving to meet the state’s new mathematics
standards. While we cannot report on data related to
the latter point, our sense as a team is that having a
single mathematical strand on which to focus helps the
participants develop a broader understanding of each of
the strands of mathematics.
Support and Interaction
Consistent with our findings in the pilot studies,
we have found an ongoing theme that participants
perform best when they realize that they are all
struggling together. We refer to this theme as the
“same boat effect.” Participants have described a sense
of comfort in knowing that they are all going through
the same thing together. In fact, in many of the courses
since the pilot studies, and even in the smaller pilot
class, working together in some way was critical to the
success of the participants.
We assert, based on our more recent data from
three workshops in one county that were taught by
InterMath team members, that there seems to be a shift
in teacher attitude about who “owns” the knowledge.
The teachers in this district, unlike those in the pilot
studies, seem to recognize that participating in a
community is critical for their learning and they rely
more on each other. It should be noted that they also
rely on the instructor, but they feel more empowered
about their own mathematical understandings, as
evidenced by comments they have shared on their
surveys and with their instructors. Most important,
however, is that many teachers now report that they are
better able to empathize with their students’ struggles
because they have gone through similar struggles
themselves while participating in InterMath.
Technology
We have taken a number of steps to alleviate some
of the technology problems associated with the initial
offerings of InterMath. Three of these steps have
significantly impacted the amount of technology
participants have to learn in the course of a workshop.
First, we have switched from having an HTML-based
electronic portfolio to having participants create
documents using a word processor that they simply
11
link to from their main web page. This has
significantly decreased the anxiety level of teachers as
they work on their write-ups. Second, because of
security problems on the InterMath server, we have
moved away from using an FTP client to move files to
the web site. In some cases, this has lowered anxiety
levels, while in other cases it has caused tremendous
problems because of incompatibilities between the new
technologies and the school districts’ technology
infrastructure. While we cannot be certain of the effect
this will have on how teachers use technology beyond
the workshop, we feel confident that this kind of
barrier is a significant factor in whether a teacher
chooses to use technology in his or her own classroom.
Finally, because we have moved, largely, to single
content strand courses (e.g., Algebra or Geometry), the
number of mathematics applications has decreased
from the three covered in the pilot workshops to one or
two depending on the specific course in which a
participant is enrolled. We do not have any evidence
that the number of mathematical applications were
problematic for participants before; however, we have
noticed that there are still a considerable number of
questions in the workshops focused on the technology.
Mismatched Goals
Finally, the pervasive misunderstanding of the
purpose of the InterMath courses and web site has
continued. Teachers regularly comment that the
problems on InterMath are too difficult for their
students. More pleasantly, some have noted that they
are surprised that their own mathematics knowledge
has been pushed beyond the point at which they began.
In four years of offering courses, we have concluded
that teachers simply have a different mindset about the
purpose and goals of professional development than
those on which InterMath was developed. This is a
problem that we continue to address.
Conclusions
In four years of successful InterMath
implementation, we have seen the classes take many
forms and we have seen a variety of participants
ranging from elementary-certified teachers to those not
certified in mathematics at all. We have enrolled high
school, middle school, and elementary teachers. All of
the InterMath participants have reported that they
learned from the InterMath course and many of them
say that they would recommend InterMath to a
colleague. Considering the paradigm shift InterMath
represents in professional development, we see this
high level of satisfaction as a success.
12
Returning to the learner-centered professional
development framework upon which InterMath was
built, there are some interesting trends and questions
that remain. The most important is the question of how
we support teachers in understanding that professional
development is about their own learning rather than
about supporting their students’ learning. This mindset,
we believe, is largely responsible for the mismatched
goals of the project and the participants. Teachers are
accustomed to participating in workshops focused on
either “make and take” philosophies or focused on
pedagogical strategies. The teachers’ mental models
for professional development are often challenged by
participation in a workshop that is focused on their
own content knowledge development rather than how
to teach content to children.
Second, we note that the learner-centered
professional development frameworks recommend that
teachers need to own their own learning. This has been
a challenge for InterMath participants as they struggle
with a number of issues that are largely related to their
own efficacy as mathematics learners and teachers and
their view of the role of an instructor. Our participants
struggled with (a) the notion that they could determine
what “adequate” levels of work were, (b) that they
could help each other, and (c) that their ideas about
how to solve mathematical investigations were worthy
of consideration. Because of these mindsets, the
participants in earlier InterMath courses often
complained that they needed additional feedback or
that they did not have as much support as they would
like. In response to these concerns, in later courses,
instructors provided more structure and feedback,
including intermediate due dates and providing
feedback on early write-ups, to alleviate these
complaints; but in doing so, they limited the level of
ownership the participants had.
Finally, InterMath participants have reported that
they learn mathematics and they enjoy the course once
they are past the technology problems. However, we
lack the data necessary to understand how the
participants use InterMath ideas (e.g., using
technology-enhanced investigations or open-ended
problems) in their own classrooms once the workshop
has ended. The ultimate goal of professional
development is to positively impact student
performance and we simply do not know enough to
know whether InterMath is doing that. It could be
argued, in fact, that even the data upon which the
learner-centered professional development principles
are based have this same shortcoming. Clearly more
work needs to be done by the professional
What Learner-Centered Professional Development Looks Like
development community to help develop an
understanding of how professional development can
positively impact student learning.
Author’s Note
InterMath and the work reported here have been
supported by the National Science Foundation (NSF).
Opinions expressed are those of the researchers and do
not necessarily reflect the opinions of NSF.
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Chandra Hawley Orrill
13
The Mathematics Educator
2006, Vol. 16, No. 1, 14–21
Participants’ Focus in a Learner-Centered Technology-Rich
Mathematics Professional Development Program
Drew Polly
Leaders in professional development have called for more learner-centered professional development opportunities for
teachers. These approaches allow teachers to have some choice about the content and pedagogies on which they focus
during professional development courses. This paper shares case studies of three participants from InterMath, a
learner-centered professional development program for middle grades mathematics teachers. The findings indicate that
participants’ backgrounds in both mathematics and technology as well as their goals for the course significantly
impacted what they reported learning. The paper concludes with implications for the design and research of learnercentered professional development programs.
Professional development programs and opportunities
for teacher learning are an essential component for
improving student learning (Joyce & Showers, 2002;
Loucks-Horsley, Love, Stiles, Mundry, & Hewson,
2003; No Child Left Behind, 2002; National
Partnership for Excellence and Accountability in
Teaching, 2000a). In the past decade, leaders in
professional
development
have
offered
recommendations
for
designing
professional
development programs (e.g. Guskey, 2003; Hawley &
Valli, 1999; Loucks-Horsley et al., 2003; National
Partnership for Excellence and Accountability in
Teaching [NPEAT], 2000a, 2000b) and theoretical
perspectives about how teachers learn (e.g. Cohen &
Ball, 1999; Putnam & Borko, 2000; Richardson, 1996).
Effective professional development focuses on
improving student learning (Hawley & Valli, 1999;
Joyce & Showers, 2002), is based on teachers’ practice
(Cohen & Ball, 1999; Putnam & Borko, 2000), and is
designed to give teachers ownership of their learning
(Hawley & Valli, 1999; Loucks-Horsley et al., 2003).
Furthermore, professional development should allow
teachers to collaborate with colleagues (LoucksHorsley et al., 2003), be carried out over a long period
of time (Garet, Porter, Desimone, Briman, & Yoon,
2001; Richardson, 1990), and be closely aligned with
goals for comprehensive change and reform (Fullan,
1995). These characteristics embody the description of
learner-centered professional development (LCPD)
programs developed by the NPEAT (2000b).
Drew Polly is currently a doctoral candidate in the Department
of Educational Psychology and Instructional Technology at the
University of Georgia, and beginning August 2006, he will be an
Assistant Professor in Elementary Education at the University of
North Carolina at Charlotte. His research focuses on examining
the influence of professional development on teachers'
instructional practices.
14
In
mathematics
education,
professional
development programs have been cited as an essential
part of current reform efforts (National Council of
Teachers of Mathematics, 2000). Research indicates
that student learning is positively influenced by four
teacher characteristics: teachers’ content knowledge
(Ball, Lubienski, & Mewborn, 2001; Hill, Rowan, &
Ball, 2004), pedagogical content knowledge (Marzano,
Pickering & Pollock, 2001), teachers’ understanding of
student thinking (Fennema, Carpenter, Franke, Levi,
Jacobs, & Empson, 1996) and teachers’ use of specific
instructional practices such as using technology, handson activities, or mathematical manipulatives (National
Center for Educational Statistics [NCES], 2001;
Wenglinsky, 1998).
Intuitively, mathematics
professional development programs should focus on
these characteristics.
InterMath: Learner-Centered Professional
Development
The InterMath project is an example of LCPD
designed to impact middle grades mathematics
teachers’ content knowledge, comfort with technology,
and experience with an investigative-based approach to
teaching and learning mathematics. Participants have
been surprised by the fact that InterMath differs from
traditional professional development programs in that
it focuses on teachers’ content knowledge rather than
providing activities that they can take directly into their
classrooms. The InterMath research team has also
found that teachers tend to get frustrated by the use of
technology, especially in the first few class meetings of
an InterMath course. From the first meeting,
participants actively engage in using technology as a
tool to explore mathematical concepts. After a few
class meetings, technology remains the primary focus
of the class, but many participants realize there is more
to InterMath than just learning to use technology. In
Participants’ Focus in a Professional Development Program
our early research, interviews indicated that
participants focus to various degrees on the
mathematics content, the technology, and the ways
they can use the InterMath content in their classrooms.
Table 1
Description of Participants
Name
InterMath
Course
Kendra
Number
Sense
Lauren
Algebra
and
Number
Sense
Algebra
Courses Discussed in this Paper
This paper presents case studies of three InterMath
participants from two InterMath courses: an Algebra
course and a Number Sense course. While the courses
were taught by different instructors, both featured the
same course components. Both courses involved 45
hours of face-to-face classes which involved three
major components: discussing investigations that were
modeled and led by the instructors, working
individually or with a partner on investigations and
completing write-ups of solutions, and designing
technology-rich investigations to be used in the
classroom.
Due to InterMath’s learner-centered nature,
participants took ownership of the content and
investigations. While instructors guided participants
through investigations in the respective content areas,
participants were able to select investigations from any
content area. Despite this freedom, participants in the
Number Sense course chose only number sense
investigations, and participants in the Algebra course
selected only algebra investigations.
Research Design and Methods
In order to more closely examine the teachers’
focus during the InterMath course, I conducted posthoc case studies of three InterMath participants. This
study was driven by the following questions:
Sheila
Position
during
course
5th Grade
teacher
assistant
Career
Exploration
teacher
Middle
grades
mathematics
teacher
Mathematics
background
Teaching
Experience
No college
mathematics
courses
Numerous
college
mathematics
courses
A few
college
mathematics
courses
1st Year
5th Year
8th year
* all names are Pseudonyms
As indicated in the table, Sheila was the only
participant that was teaching middle grades
mathematics, and she was selected due to the relevance
that InterMath had to her job as a classroom teacher.
Kendra was selected because she reported having
limited mathematics content knowledge and was
working with elementary school children while she
was taking the course. Lauren was not teaching middle
grades mathematics while taking the course but
reported having a high level of mathematics content
knowledge and comfort with technology. It was my
hope that selecting three such different participants
would provide insight into how participants’
backgrounds, jobs, and goals for the course influenced
how they focused on their learning.
Data Sources
What do the participants report learning during an
InterMath course?
Interviews and open-ended survey data were used
in this study.
What participant characteristics influence what
they report learning?
Pre-Course Survey
Participant Selection
The three participants were purposefully selected
for this study (Patton, 2002). These participants were
chosen because: 1) they all took InterMath during the
Fall 2002 semester; 2) the instructors were part of the
InterMath research team, ensuring there was high
fidelity between the implementation of the course and
the syllabus; and 3) the three participants had diverse
backgrounds and different reasons for taking the
courses. Table 1 describes the demographic
information for each participant.
Drew Polly
Participants filled out the pre-course survey during
the first class meeting. The instrument included 26
Likert-scale items and four open-ended items about
what the participants hoped to learn in the InterMath
course. For this study, only the open-ended questions
were examined because they were deemed relevant to
the research questions. On the pre-course survey,
participants were asked to explain their uses of
instructional technology in their teaching, why they
signed up for InterMath, and what they hoped to learn
during the course. On the post-course survey,
participants completed the same Likert questions as the
pre-survey as well as open-ended questions about what
they had learned during the course.
15
Interviews
Participants were interviewed twice using a semistructured interview protocol. The research team
interviewed participants approximately halfway
through the course and during the last course meeting.
The interviewers asked participants what they were
learning in InterMath and how they felt InterMath had
influenced their mathematics content knowledge,
views about how to teach mathematics, and views
about technology’s role in a mathematics classroom.
Analysis
Qualitative analysis methodologies guided by
principles of interpretive inquiry (Miles & Huberman,
1994) were used to analyze the interview data and the
open-ended survey questions. I examined instances in
the interviews during which participants discussed
what they hoped to learn, what they had learned, and
how they felt this experience would impact their
classroom practice. The data were analyzed using each
individual as a separate unit of analysis.
I then analyzed each individual interview transcript
and open-ended survey response, coding the data. The
first set of codes I used originated from my previous
experiences with InterMath participants (Table 2). In a
spreadsheet, I pasted the coded data along with labels
with codes and sub-codes. Preliminary analyses of data
from other InterMath participants suggest that
InterMath’s three-pronged approach of enhancing
participants’ mathematical content knowledge,
proficiency with technology, and learning of
mathematics through technology-rich mathematical
investigations typically results in the participants'
focusing their learning on various parts of the course
(Erbas, Umberger, Glazer, & Orrill, 2002; Brown,
Erbas, Glazer, Orrill, & Umberger, 2001). Based on
those observations and related literature, I constructed
preliminary codes about how participants might focus
their learning, began to analyze data, and revised the
codes according to the initial analysis. The preliminary
codes used at the beginning of analysis are in Table 2.
For each participant, I coded and sorted the data
and then created sub-codes. I then used the coded data
to generate themes for each participant. The themes
addressing each participant’s experience in InterMath
are reported below for the three individual cases.
Table 2
Initial Codes: InterMath Participants Areas of Learning
Category
Description
Citation
Example
Content Knowledge
Mathematical
Knowledge (MCK)
Content
The participant discusses learning
specific mathematical content or
mathematical processes.
Ball, 1994; Ma, 1999
“I learned that the graphs of
two linear equations will
intersect at only one point
unless they are the same line.”
Technological
Knowledge (TCK)
Content
The participant discusses learning
specific technology content, such
as how to use a piece of software.
Ertmer, 1999; National
Research Council, 2002
“I learned how
graphs from a
Microsoft Excel.”
Mathematical
Pedagogical
Content Knowledge
(MPCK)
The participant discusses learning
either how to teach mathematics
more effectively or how to better
understand students’ learning of
mathematics.
Schulman, 1987; Marks,
1990
“I learned that I can teach
linear equations by giving my
students an investigation to
solve and letting them
discover the mathematics that
is embedded.”
Knowledge about Teaching
with Technology
The participant discusses learning
how to integrate technology into a
classroom of K-12 students.
Ertmer, 2003; NCES, 1999;
NCES, 2002
“I learned how I can use
Microsoft Excel with my
students to help me teach
patterns.”
to make
table in
Pedagogical Content Knowledge
16
Participants’ Focus in a Professional Development Program
The participants represent a diverse range in terms
of their backgrounds: a current middle grades
mathematics teacher who wanted to learn how to
integrate more technology into her teaching, a teacher
who began the course with high comfort with
technology and high mathematical knowledge, and a
teaching assistant who began the course with low
mathematical knowledge and some comfort with
technology.
Findings: Three Case Studies
Sheila: A Middle Grades Mathematics Teacher
Background. Sheila was one of the few people
teaching mathematics while taking InterMath during
that session. On the pre-course survey, Sheila said that
she was taking InterMath so she could learn “new
concepts and ways to improve my math understanding,
so I can better teach my students.” In terms of
technology, Sheila reported, “I am not afraid to try new
things but do not feel as accomplished as many peers
in the field of technology.” In her mid-point interview,
Sheila also reported that she lacked confidence in her
knowledge of mathematics because she had not taken a
mathematics course in more than a decade. She hoped
that InterMath would give her a deeper understanding
of mathematics, which would, in turn, make her a more
effective teacher.
Learning about technology. Sheila reported that
her comfort level with, and views about, technology’s
role in her mathematics teaching changed during the
course. Although Sheila’s students did use technology
prior to the InterMath course, technology was only
used as an add-on or enrichment activity after the
mathematics content had been taught. Sheila had
experience
using
spreadsheets,
Geometer’s
SketchPad® (Jackiw, 1990), and other computer-based
technologies prior to InterMath, but still she reported a
lack of confidence that limited her use of those
technologies with her students.
At the end of the course, Sheila reported that she
viewed technology as a more powerful tool during
those moments of instruction when students discuss
specific concepts and struggle to understand
information. In her post-course survey, Sheila said:
I have learned to integrate technology into the unit
instead of making it a separate activity. I have
more confidence in trying to use the different
technologies when the opportunity presents itself
as a ‘teaching moment.’ Now I see technology as
being integrated, which is better than how it was
before. There are a couple of situations where I had
kids that I’ll say, ‘run back there and open this and
Drew Polly
try this.’ This year I have a computer that is hooked
up all the time to the presenter box.
She talked about the difficulty in getting access to
her school’s computer lab and that the only way to
bring technology into her teaching was to use a
computer and a projector. Although this lack of access
limits the activities that her students can do with
technology, the projector allows Sheila to use
technology in ways that enrich the mathematics content
she is teaching.
Learning about mathematics. On her pre-course
survey and during both interviews, Sheila reported that
she wanted to become a more effective mathematics
teacher by learning more mathematics. On the precourse survey she wrote, “[I want to learn] new
concepts and ways to improve my math understanding,
so I can better teach my students.” While she
considered herself to be an accomplished middle
grades mathematics teacher, Sheila reported she had
forgotten a lot of mathematics that she had in college.
Furthermore, she felt that she lacked a thorough
understanding of some of the mathematical concepts
that she taught. Her feeling that she lacked
mathematical knowledge and her belief that
contemporary teaching practices had changed since she
was a student motivated her to learn more mathematics
and new ways to teach mathematics.
Completing investigations in InterMath. In her
post-course interview, Sheila said that to successfully
use new teaching strategies (e.g. mathematical
investigations), she would have to not only experience
learning in this new way but also be more comfortable
with the content in order to help her students when
they had struggles and questions about mathematics.
She reported in an interview, “I was taught
[mathematics] in a different way. I was one of those
who were taught math by memorizing, and I wanted to
[teach] in a more contemporary style that would
benefit the students.”
At the end of the course, Sheila reported that she
was “comfortable enough to get in there and try
investigations.” Completing the write-ups gave Sheila
a better appreciation of her students’ struggles with
problem solving. She said in her post-course interview:
I have a better appreciation of my students’
struggles. I can better empathize with, oh, they’ve
heard this concept or they’ve heard or seen this,
that, or the other … but when I put it in writing …
I can see where it has been hard to grasp. And at
the same time, I now know better how to say well,
go for it. Work this out. Where do you think this
is going to go? Well, try this.
17
By working on the investigations, Sheila not only
has a deeper understanding of mathematics, but also
has a better idea of how to guide her students through
the problem solving process.
Lauren: High Mathematics Content Knowledge, High
Comfort with Technology
Background. Lauren was taking both the Algebra
and the Number Sense courses because she was
working with a provisional teacher certification. She
needed to earn ten professional learning units (PLUs),
which she could do by completing two InterMath
courses. Lauren reported that she already had a strong
mathematics background and a high comfort level
using technology. Lauren’s secondary motive for
taking the InterMath courses was to learn how to use
technology more effectively in her teaching. At the
time of the course, she was teaching Spanish and
Career Explorations, a course in which students apply
mathematical concepts in real-world activities, such as
setting up budgets, calculating interest on credit cards,
and planning their own businesses.
Integrating technology into her teaching. Prior
to InterMath, Lauren had extensive experience
creating web pages and using spreadsheets for
budgets. She felt that her next step was to carry
her technology skills into her classroom, which
she did a few times while taking InterMath. In her
mid-point interview, Lauren explained:
One day we talked about credit card risk. We tried
to figure out how long it would take to pay back
credit card debt if you only paid the minimum
payment. Luckily, I had a computer right there so I
threw it on an Excel spreadsheet. They thought it
was great, and they were doing the same thing in
their business education class. They were just
learning how to do that, so they were excited to see
it elsewhere, too…
Lauren extended this activity during the next class
period by having her students apply the concept of
interest rates using both calculators and spreadsheets.
Her students used both technologies and then discussed
which technology was more useful in solving the
problem that she posed. She reported:
[The credit card activity] was initially set up using
calculators. So what I did was — there were two
separate charts. I had them first use the calculator,
then showed them how much easier it was using a
spreadsheet. They then did the second chart
completely on the spreadsheet.
In this activity, Lauren was able to integrate not
only technology but also multiple forms of technology,
18
which is emphasized in InterMath. The InterMath
investigations allow participants to use multiple
technologies to explore the mathematics, and Lauren
was able to extend this idea into her classroom as her
students used both spreadsheets and calculators to
explore the idea of credit card interest.
Lauren reported that she uses technology “as often
as I can in my teaching.” She feels that using
technology in schools is essential since the students
have access to it at home and they will be required to
use computers when they enter the workforce. From an
instructional perspective, Lauren sees technology as a
"tool in a teacher's repertoire" that provides more
avenues for learning.
Beliefs about mathematics. While Lauren learned a
great deal about integrating technology into her
classroom, she reported that her biggest takeaway from
InterMath was a shift in her views about mathematics.
The investigations that she completed allowed her to
explore and continually unpack mathematics and see
connections between various mathematical concepts.
I was doing an investigation the other day, and it
was just a pattern, and it turned out to be this
investigation that had to do with relatively prime
numbers. And I would have never thought of that
as — it could have been just a fun little thing —
find the pattern, but as I went through it more, I
was, like, wait a minute, this happens here and it
was a mathematical relationship that just came
about because of this pattern in this problem…one
investigation can contain a number of different
math concepts across various content areas that can
continue to unfold as the learner digs deeper and
deeper into each problem.
InterMath convinced Lauren that mathematics
classes should get away from the traditional approach
in which facts are accepted as stated and enable her
students to explore and figure out why certain
mathematical concepts are true. Lauren stated that
mathematics teachers should “teach students a way to
think, rather than simply a way to do or solve
problems.”
Lauren repeatedly mentioned in interviews that
technology can help students learn mathematics, but it
must be used appropriately:
Kids think it’s neat. They think the computer is
solving problems for them. They think that a
computer will just answer. They don’t realize that
they need to know the math in order to put the
correct formula in the computer … they don’t
know that they are doing math and things like
algebra, but they are.
Participants’ Focus in a Professional Development Program
Kendra: Low Mathematics Content Knowledge, Some
Comfort with Technology
Background. During the InterMath course, Kendra
was a paraprofessional in a 5th grade classroom, and
she hoped to gain her teaching certificate and teach
elementary school the following year. Kendra came
into the course with very limited mathematics content
knowledge. She had not taken a mathematics course
since she graduated from high school more than a
decade before her participation in InterMath. In terms
of technology, Kendra described herself as, “Not very
comfortable but very open to learning.” Kendra had
seen students use computer-based drills and practice
software in mathematics, but had no experience using
any of the InterMath technologies. On the pre-course
survey, Kendra said that she hoped to learn “how we
can use the computers more effectively to supplement
teaching.”
Learning mathematics. On the pre-course survey,
Kendra saw InterMath as a technology course, and did
not report any intentions of learning mathematics.
Throughout the course, Kendra recognized that
InterMath was also a mathematics course, as she
experienced learning mathematics in a way that was
different. Kendra reported:
It has been a different classroom environment from
what I have seen in the past. I have never been in a
math class that we discussed so much, immensely
… really improved my level of confidence with
my mathematical ability. I’m surprised how much I
have been capable of now, especially in problem
solving.
This new experience shifted Kendra’s perspective
about how mathematics should be taught towards a
more hands-on approach that gives students the chance
to discuss the problems they are solving. Kendra’s
experience in previous mathematics was, “this is how
you do it, do these problems, and we will see you
tomorrow.” Kendra reported being amazed at how
much mathematics she learned by completing the
investigations.
Learning technology. Kendra had no experience
with any of the InterMath technologies prior to the
course but left with what she reported as “substantial
knowledge” in regards to solving problems by using
formulas, functions, and the graphing tools in Excel.
When asked if she could use Excel to help her go
through an investigation, she said, “I am pretty
confident. I’d like to practice even more but I certainly
feel more confident now in working with them on my
own.”
Drew Polly
Despite being in an elementary classroom that has
three computers, Kendra had not seen computers used
in mathematics lessons other than situations in which
students played skills-based games. While she thinks
that the potential is there for technology to enhance
student learning, Kendra offered numerous ideas about
why technology may not be appropriate in elementary
schools. She cited problems with technology access,
finding time to use technology, and having to manage a
classroom when the students were using the
technology. While Kendra is convinced that
technology can help teachers, like herself, learn
mathematics, she is still skeptical that technology is
appropriate for helping elementary students learn
mathematics.
Discussion
In each of the three case studies, participants
reported leaving InterMath with more knowledge about
mathematics, approaches to teaching mathematics,
ways to use technology, and strategies for integrating
technology into mathematics classrooms.
Learning Mathematics
All three participants’ learning related to
mathematics centered on the process of completing
InterMath investigations. During the interviews, Sheila
and Kendra both shared that their K-12 experiences of
learning mathematics were drastically different from
those they had in InterMath. Further, Sheila believed
that, in order to be effective, she needed to experience
learning in the manner in which she was expected to
teach. Lauren had a strong mathematics background
prior to InterMath but reported learning about
connecting mathematical ideas while exploring an
investigation. Specifically, Lauren contended that
mathematics instruction needs to focus on “a way to
think rather than a way to do.”
Despite the participants’ diverse mathematical
backgrounds, each reported an increased comfort in
learning mathematics through an investigation-based
approach. Each participant reported seeing the value of
completing mathematical investigations. Sheila and
Kendra explicitly recognized the importance of being
comfortable solving investigations prior to using them
with their students. This finding supports Cohen and
Ball’s (1990) sentiment that teachers must experience
learning mathematics in the same manner as they are
expected to use it in their teaching.
Learning Related to Technology
Prior to the course, Lauren and Sheila reported
being comfortable with technology, while Kendra had
19
never used any of the InterMath technologies. Lauren’s
strong background with technology enabled her to
focus on integrating technology and InterMath-like
investigations into her classroom. As seen in her credit
card lesson, Lauren reported being able to use multiple
technologies to allow students to gain a deeper
understanding of interest rates. Sheila reported a shift
in her views about technology, such that technology
was now a tool that could be woven into her
mathematics classroom rather than being used as an
add-on. This affordance of being able to focus on how
technology could be used in her classroom was not
available to Kendra, since her lack of experience
required her to focus her learning on mastering the
basics of each technology. This finding supports a
variety of technology integration models that contend
that teachers must first develop basic technology skills
before considering how to integrate them in their
classroom (Dwyer, Ringstaff, & Sandholtz, 1991;
Hooper & Rieber, 1995; Mandinach & Cline, 1992). In
Kendra’s case, more extensive time with the
technology would allow her to master the basic skills
so she could more closely attend to how these
technologies might be used with her future students.
middle grades mathematics teachers, participants that
report having high comfort with technology and
mathematics, etc.). Finally, the current emphasis in
professional development is on making a link between
teacher learning to both their classroom practices and
their students’ learning (Guskey, 2000; NCLB, 2002),
and in order to address these issues, longitudinal
studies are needed to examine participants prior to
InterMath, during the course(s), and then examine
teachers’ instructional practices and their students’
mathematical learning in their classrooms.
This study indicates that InterMath, a learnercentered professional development program, enhanced
participants’ mathematical content knowledge,
technological skills, and comfort with mathematical
investigations. Future studies will provide further
evidence about how, and the extent to which, teachers
learn during these experiences.
Implications for Research
Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research
on teaching mathematics: The unsolved problem of teachers'
mathematical knowledge. In V. Richardson (Ed.), Handbook
of research on teaching (4th ed.). Washington, DC: American
Educational Research Association.
This paper presents the cases of three participants
with different backgrounds and different reasons for
enrolling in an InterMath course. The findings indicate
there are individualized benefits for participants who
are learning in a learner-centered professional
development program that is aimed at developing
participants’ mathematical content knowledge,
technology skills, and comfort with mathematical
investigations. Professional development programs that
allow teachers to take ownership of their learning and
give teachers choices about the content and the
activities in which they engage have been highly
regarded (Hawley & Valli, 1999; Loucks-Horsley et
al., 2003; NPEAT, 2000a). However, these programs
can be problematic. In this paper, InterMath
participants focused, to varying degrees, on the
mathematics, the technology, and the process of doing
investigations. Further research is needed to more
closely examine the ways in which participants decide
how to focus their learning.
While these case studies begin to examine
participants’ focus in a learner-centered professional
development program with multiple foci, further
studies in this area are needed to generalize the
findings presented here. Future studies should examine
more participants that have similar backgrounds (e.g.,
20
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115. Stat. 1425.
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(3rd ed.). Thousand Oaks, CA, Sage.
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and thinking have to say about research on teaching?
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Loucks-Horsley, S., Love, N., Stiles, K. E., Mundry, S. & Hewson,
P. W. (2003). Designing professional development for
teachers of science and mathematics (2nd ed.). Thousand Oaks,
CA: Corwin Press.
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to teach. In J. Sikula, T. Buttery, & E. Guyton (Eds.),
Handbook of research on teacher education (pp. 102–119).
New York: Simon & Schuster Macmillan.
Ma, L. (1999). Knowing and teaching elementary school
mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.
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the new reform. Harvard Educational Review, 57(1), 1–22.
Mandinach, E., & Cline, H. (1992). The implementation of
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ftp://ftp.ets.org/pub/res/technolog.pdf.
Marks, R. (1990). Pedagogical content knowledge: From a
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Teacher Education, 41, 3–11.
Drew Polly
21
The Mathematics Educator
2006, Vol. 16, No. 1, 22–34
An Analysis of the Use of Graphical Representation in
Participants’ Solutions
Laurel Bleich
Sarah Ledford
Chandra Hawley Orrill
Drew Polly
InterMath participants spend time in workshops exploring technology-rich mathematical investigations and
completing write-ups. These write-ups include a written explanation of their problem solving process, screen
captures of files that they generated while completing the investigation and links to these files. This paper
examines the use of graphical representations in write-ups that included incorrect mathematics and/or incorrect
solutions. Our findings indicate that a large number of incorrect write-ups included a graphical representation
for cosmetic purposes, meaning that it was not used to explain or justify participants’ solutions.
InterMath1 is a professional development effort
designed to strengthen teachers’ mathematical content
knowledge through the exploration of mathematical
investigations using various technologies. It was
created to address a critical problem encountered in
many middle schools: the mathematics teachers are
deficient in content knowledge or in content-specific
pedagogy because they do not have adequate
preparation (Wilson, Hannafin, & Ohme, 1998).
InterMath addresses this goal by engaging teachers in
open-ended explorations that simultaneously allow
them
to
develop
their own
mathematical
understandings and learn to use technologies to support
their mathematical thinking.
InterMath participants have the opportunity to
select which problem(s) they want to work; the
Laurel Bleich is a currently a Program Manager in the Learning
and Performance Support Laboratory at The University of
Georgia. She has worked on the InterMath project for the past
two years. She received her Masters in Mathematics Education in
2003 from The University of Georgia. Prior to her work in
education, she worked numerous years in the computer industry.
Sarah Ledford is a doctoral student in the Mathematics
Education program at the University of Georgia. She has been a
graduate assistant on the InterMath project for the last four
years.
Chandra Hawley Orrill is a Research Scientist in the Learning
and Performance Support Laboratory at the University of
Georgia. Her research interest is in how teachers make sense of
professional development and how the professional development
impacts learning opportunities for students.
Drew Polly is currently a doctoral candidate in the Department
of Educational Psychology and Instructional Technology at the
University of Georgia; and beginning August 2006, he will be an
Assistant Professor in Elementary Education at the University of
North Carolina at Charlotte.
22
approach they use to solve the problem; and,
ultimately, the depth of learning they take from the
class by choosing the appropriate difficulty level of the
problems they worked. Consistent with many current
professional development guidelines (Hawley, 1999;
Loucks-Horsley, Love, Stiles, Mundry, & Hewson,
2003; National Partnership for Excellence and
Accountability in Teaching [NPEAT], 2000),
InterMath allows the participating teachers to identify
their own needs and direct their own learning in a
supportive environment. In a sense, InterMath
participant teachers determine what they need to
succeed as mathematics teachers and learners. The
primary deliverable of this process is a series of
documents called “write-ups” in which participants
“communicate and synthesize investigations involving
exploration, solving a problem, or working with an
application” (Wilson et al, 1998, p. 18). The key
elements of a write-up include a restatement of the
problem, the writer’s initial plan for solving the
investigation, an explanation of how the investigation
was actually approached, and a statement of the
findings. Write-ups may also include justifications of
solution processes, answers, extensions, or ideas of
how the problem might be modified for classroom use.
Most InterMath write-ups include screen captures of
relevant technology-enhanced work the participants
engaged in, and links to files that they created, for
example, Microsoft Excel or Geometer’s SketchPad
(GSP; Jackiw, 1993) files, as they solved the problems.
For each InterMath course in which we have collected
data, the participants were asked to complete
approximately 10 of these write-ups.
An Analysis of Using Graphical Representations
In our current analysis, we have focused on a
subset of these write-ups that include either incorrect
mathematics and/or incorrect answers. Our analyses
began with an effort to make sense of what participants
were learning, and quickly became focused on how
participants used graphical representations (defined
here as any kind of graphical representation including
graphs and GSP sketches) in their work.
Theoretical Framework
InterMath, as a professional development
experience, is grounded in theory and research that
suggests that teaching mathematics should be
something other than “chalk and talk” models in which
the teacher transmits his or her knowledge to the
students (e.g., NCTM, 2000). Specifically, InterMath
takes Cohen and Ball’s (1990) question to heart: “How
can teachers teach a mathematics that they never have
learned, in ways that they never experienced?” (p.
238). To this end, InterMath engages teachers in
learning to teach differently by engaging them in a
variety of mathematical activities as learners.
Consistent with the NCTM (2000) vision, the
InterMath workshops engage teachers in making
conjectures,
communicating
mathematically,
reasoning, and problem solving, while simultaneously
strengthening their content knowledge. By involving
participants in these activities as engaged and reflective
owners, InterMath seeks to foster a sense of comfort
and confidence for using similar teaching approaches
so that the teachers will begin to change their own
practices.
InterMath is also strongly based on the notion that
technology should be used in mathematics to support
problem solving and reasoning, as well as to reduce the
tedious aspects of certain calculations. We recognize
the potential of technology for promoting reformoriented approaches to mathematics, but also realize
that simply using technology is not reform in itself;
e.g., Kaput (1992). After all, mathematical learning is
not fundamentally different if students are using a drill
and practice program as opposed to a worksheet.
However, if the learners are engaging with a
technology to explore an aspect of mathematics or to
tie together different understandings, then that
technology offers an innovative learning opportunity.
Building from this vision of mathematics and the
belief that, through InterMath, teachers are
experiencing mathematics in a different way and
exploring some new understandings for the role of
technology in their classrooms, the research team has
set out to understand what the impact of this learning
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
is. To this end, we have begun considering the
mathematics the participants in the workshops seem to
be learning and/or applying to the investigations
presented in InterMath. In evaluating a subset of the
write-ups from a set of courses, we recognized a trend
that a high number of write-ups that included incorrect
mathematics and/or incorrect solutions2 also featured
the use of a graphic element such as a graph or a
geometric construction. Inspired to understand why
there were a high number of incorrect write-ups that
included a graphical representation, we turned to the
literature to determine the different ways graphic
elements might be used.
According to the NCTM Standards (2000),
mathematical representations are useful tools for
building understanding and for communicating both
information
and
understanding.
Given
this,
representations are critical elements in write-ups as
they relate to mathematical communications. But, what
is the role of the graphic? Do the teachers participating
in InterMath understand the power of representations
for both solving problems and communicating their
understandings?
Graphical Representations
There has been a shift in recent years in
mathematics educators’ views on the role of drawn
representations. As presented in Monk (2003), graphs
can be viewed in two distinct ways. First, and more
traditionally, a graph is a tool for communication. That
is, graphs describe a set of data or a solution of a
problem to the reader. However, Monk introduces the
notion that there is a second way to use graphs – as
tools for generating meaning. Monk elaborates saying,
“Whereas a graph had earlier been seen exclusively as
a conduit, a carrier of information, for example, about
the motion of a car, it can now also be seen as a lens
through which to explore that motion.” (p. 251,
emphasis in the original). Monk continues to point out
that these are not opposites, nor is one preferable,
rather that they are two different approaches to using
tools that look the same. Consistent with InterMath’s
goals and vision, it was expected that participants
would use graphs (and other visual representations) in
both of these ways. Further, it had been assumed that
the participants were using the representations as
problem-solving tools because that was the approach
modeled for them in the course.
More specifically, by using visual representations
as problem-solving tools, participants would be able to
see some benefits – particularly in their abilities to
solve the kinds of complex problems they were often
23
faced with in InterMath. Consistent with Monk’s
views, the InterMath team considered a number of
benefits to using graphs and graphic elements in this
way. These included
- Using graphics to explore aspects of a context
that might otherwise not be apparent;
- Developing a deeper understanding of a context
through the use of graphics that elicit particular
questions about those contexts; and
- Developing a deeper understanding of the kinds
of information that can be conveyed through
graphics (Monk, 2003).
Additionally, building on Gagatsis and Shiakalli
(2004), we assert that it is most important for teachers
to be able to work with these representations in both
ways – as communicating and problem solving. While
Gagatsis and Shiakalli were more concerned with
moving between representations, their point applies to
InterMath teacher participants. That is, translating
between representations and within representation
systems is a vital aspect of teaching. If a teacher is
unable to interpret a graphic representation that has
been developed by her students, she or he has lost one
way of making sense of (a) whether the student
understands a concept and (b) where the student may
still need additional support in refining his or her
understanding. In their assertion that students often
need nonstandard representations in order to support
their mathematical problem solving, Greeno and Hall
(1997) highlighted this need for teacher development
even more. If teachers are to fully support their
students, they need to be able to understand how
students are using graphical elements to not only
explain their answers but also to solve problems.
We believe that InterMath provides participants
with opportunities to develop these kinds of
dispositions toward graphical representations as well as
to refine their ability to interpret a wide range of
representations. While the investigations and
technologies used in InterMath do inherently support
more traditional forms of representation, they also do
promote multiple forms of representation. In classes,
participants are encouraged, but not required, to use
one or more technologies for their investigations;
InterMath instructors often demonstrated two or three
different approaches to solving the investigations, each
with their own use of representations. Participants
experienced the same kinds of teaching and learning
opportunities we hope they will develop for their
students.
24
It is our view that the use of graphical elements
should greatly enhance the problem solver’s ability to
successfully complete an investigation. Yet in our
sample, this was not necessarily true. This study,
therefore, considers why teachers who were using one
or more visual representations in their write-ups used
mathematically inappropriate approaches and/or got
wrong answers. For the purposes of this study, we
consider the following questions: How did participants
use graphical representations in their problem-solving
processes? How did the graphical representations allow
the participants to stray from correct or appropriate
mathematical approaches and/or fail to reach correct or
appropriate solutions?
Methods
This post-hoc study examined InterMath
participants from five InterMath courses taught
between 2001 and 2004. The workshops lasted
between 1 and 15 weeks and included 3 to 24
participants each. The content in each course varied; in
some cases all four strands (Number Sense, Algebra,
Geometry, Data Analysis) were included, whereas in
others only one strand was emphasized.
We examined all of the participants’ write-ups in
the smaller classes (n < 10). In the larger classes, 25–
30% of the participants were randomly selected. In all,
236 write-ups from 27 participants were coded into 4
categories: correct math/correct answer (CM/CA),
correct math/incorrect answer (CM/~CA), incorrect
math/correct answer (~CM/CA), and incorrect
math/incorrect answer (~CM/~CA). See Table 1 for the
breakdown of the write-ups in terms of mathematics
and answers.
Table 1
Distribution of Write-Ups in Terms of Correctness of
Mathematics and Answers
Correct Answer
(CA)
Incorrect Answer
(~CA)
Correct Mathematics
(CM)
Incorrect Mathematics
(~CM)
170
22
23
21
Each write-up was examined by two researchers
independently and coded based on the elements above.
The two analyses were then compared and a consensus
was reached when there was a disagreement. In all
cases, there was 100% inter-rater agreement before the
analysis proceeded.
An Analysis of Using Graphical Representations
Table 2
Descriptions of Representation Categories and Coding Strategies.
Type
of
Graphical
Representation
Communicate
Make sense
Cosmetically enhanced
Definition (Adapted from
Monk, 2003)
Graphic is used to
convey a meaning, or
express one’s ideas
Graphic is used in the
understanding of the
problem or in the process
of finding a solution
Graphic
is
neither
appropriate nor relevant
to the investigation or
solution
Of the 236 write-ups, over one-fourth of them, 66,
had incorrect mathematics and/or an incorrect answer.
Further study showed that of these 66 write-ups, 62
(94.93%) used some sort of graphical representation
(i.e. any kind of graph or diagram). By contrast only
48 (28.24%) of the remaining 170 write-ups (CM/CA)
used graphical representation. Therefore, we focused
our attention on these 62 write-ups that included some
level of incorrectness.
Identifying Types of Graphical Representations
Aside from Monk’s (2003) two roles of graphical
representations as mentioned above, we recognized a
third role from our analysis, one in which the graph
was used to cosmetically enhance their write-ups. Each
write-up was coded into only one category:
communication, make sense, or cosmetically enhanced.
Table 2 displays the definitions of each category and
describes how the representations were coded. All four
members of the research team were trained on a subset
of the write-ups (approximately 15) to reach consensus
on definitions of categories. Then one member of the
team coded each remaining write-up. A sampling of
the codes underwent inter-rater reliability and in all
cases where there were initial inconsistencies a 100%
agreement was reached.
The following examples chosen from the 62 writeups provide further details about the meanings of the
categories. These write-ups were taken verbatim from
the participants’ portfolios, including any misspelled
words, grammatical errors, and inconsistencies with
the graphic and the discussion of the graphic. They are
taken from 2 of the 5 courses and vary in their level of
correctness (one shown with CM/~CA, one shown
with ~CM/CA, and one shown with ~CM/~CA).
Communication example. The following example
by participant G83 was coded as having incorrect
mathematics and an incorrect answer resulting from the
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
Coding
Write-up could have been done without graphic; discussion of
graphic occurs before graphic; graphic is referred to, i.e. “as you
can see from the diagram”
Reference to “I” or “we” implying a collaborative effort; graphic
referred to throughout investigation; discussion of graphic occurs
after graphic; graphic used to find solution
Randomly placed; graphic without explanation or reference;
graphic not appropriate/relevant; used for organization purposes
only; used as filler
author’s assumption that the given polygons are
regular. The write-up illustrates the communication
category due to the discussion of the graphic coming
after the graphic. The graphic is referred to as the
participant is trying to communicate to the reader what
can be seen in the graphic (see Appendix A for the
entire write-up).
Investigation: What is the sum of the angles of a
triangle? Of a quadrilateral? Of a pentagon? Of a
hexagon? What is the sum of the angles in convex
polygons in terms of the number of sides?
Write-up: When using Geometer Sketchpad to
create a triangle that is formed by having two
transversals intersect a set of parallel lines, students
can then use the properties that they have learned
about angles to determine the sum of the angles of
a triangle.
m!AHF = 81°
m!HFA = 63°
m!FAH = 36°
m!IHF+m!HFI+m!FIH = 180°
m
5
2
A
3
Line j is parallel to LIne q
H
4
1
F
q
Because lines j and q are parallel to one another,
then lines x and m are transversals to these parallel
lines. Thus, angle 1 and angle 4 are congruent
because they are alternate interior angles.
Similarly, angle 2 and angle 5 are congruent.
Therefore, the following angles are congruent:
25
Angle 1 = Angle 4, Angle 2 = Angle 5, Angle 3 =
Angle 3
Conclusion: We can conclude that Angle 5 +
Angle 3 + Angle 4 = 180 degrees since these three
angles form a straight angle. From the above
conclusion, we see that: Angle 1 + Angle 2 +
Angle 3 = Angle 4 + Angle 5 + Angle 3 = 180
Thus, we can conclude that the sum of the
measures of the three vertex angles in a triangle is
180 degrees.
G8 uses the graph of two parallel lines cut by two
transversals as a means to illustrate to the reader that
the angles are congruent due to alternate interior
angles. Since the graphic is not used as a means to
make sense of the mathematics, and it appears that it is
appropriate for the discussion that follows in the writeup, it was coded as communication.
Making sense example. The following example by
participant G7 was coded as having incorrect
mathematics and a correct answer because the answer
is written as a ratio of 1:2 and is said to be equivalent
to 0.496, showing that there is lack of understanding of
rounding and truncating numbers. The write-up
illustrates making sense because the participant uses
the first person to reason through the investigation by
creating a representation and then “talking” through it
(see Appendix B for the entire write-up).
I used Excel to record the data about the areas of
each circle and its tangent circles. That table is
below and is colored to correspond to the
constructions above:
Investigation: …How does the combined area of all
of the shaded circles relate to the area of the entire
circle?
Cosmetically enhanced example. The following
example by participant C6 was coded as having correct
mathematics and an incorrect answer. Despite the fact
that the participant refers to the graphs in such a way
that might imply that the graphical representations are
being used as a tool to communicate, further analysis
revealed that what the participant wrote and what was
displayed were not in alignment (i.e. no measures were
taken to show that the quadrilateral referred to in the
graphic was actually a rectangle). Therefore, the writeup depicts a cosmetically enhanced write-up (see
Appendix C for the entire write-up).
Write-Up: I constructed circles using 2, 3, 4, and 5
smaller tangent circles along the diameters in
Geometer's Sketch Pad. Those constructions are
shown below:
Area of
Large
Circle
Area of
Small
Circle
3.14
3.14
3.14
3.14
0.78
0.35
0.2
0.13
# of
Small
Circles
2
3
4
5
Small
Circles
Comb.
Area
1.56
1.05
0.8
0.65
Small/Lrg
Circle Ratio
0.496815287
0.334394904
0.25477707
0.207006369
In each situation the relationship formed between
the circle and its tangent circles along its diameter
could be closely described in terms of the number
of tangent circles. When there were two tangent
circles, the area relationship was 1:2 or .496. When
three, the relationship was 1:3 or .33, etc.
Investigation: A number of investigations can be
done involving quadrilaterals. One investigation
that can be explored deals with drawing an original
quadrilateral. Then by marking the midpoints of
the sides and connecting the midpoints with line
segments to create an inscribed quadrilateral.
Write-up: This can be done using GSP.
example can be seen in the sketch below.
26
An
An Analysis of Using Graphical Representations
B
sense of the mathematics or as a means to
communicate a point, the write-up was coded as
cosmetically enhanced.
C
E
Findings and Discussion
H
Table 3 shows the distribution among the types of
incorrect write-ups as well as the uses of graphical
representations in the incorrect write-ups.
F
Table 3
G
A
Distribution of Types of Incorrect Write-Ups and Uses
of Graphical Representations
D
One idea that can be explored is if the shape of the
new quadrilateral depends on the shape of the
original quadrilateral. One could determine if the
two quadrilaterals will have the same shape or
different shapes. This could be explored using
GSP. In GSP, one could drag the different vertices
of the original quadrilateral to form different
shapes.
Examples of this using GSP are shown below.
A
E
B
Correct
Mathematics
Incorrect
Solution
Incorrect
Mathematics
Correct
Solution
Incorrect
Mathematics
Incorrect
Solution
Total
Cosmetically
Enhanced
9
Total
6
Making
Sense
8
6
7
6
19
2
7
11
20
14
22
26
62
23
Communication
H
F
G
D
C
The image above seems to show that if the original
quadrilateral is a trapezoid, the smaller
quadrilateral will be a rectangle.
E
B
C
H
A
Communication
F
G
D
The image above seems to show that is the original
figure is a rhombus; the second quadrilateral will
again be a rectangle.
It appears in the above write-up by C6 that in the
construction of the various quadrilaterals little
discussion was presented for proving the internal
quadrilateral is a rectangle. Since C6 showed little
indication of using the graphics as a means for making
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
Fourteen of the 62 incorrect write-ups (22.6%)
included graphical representations for the purpose of
communicating
mathematics.
These
graphical
representations were more likely than the other types
of representations to result in either correct
mathematics or a correct solution. Only 2 out of the 14
write-ups using graphics for communication (14.3%)
had both incorrect mathematics and an incorrect
answer. Participants that used these representations
included an explanation to either the investigation or
the solution prior to the graphical representation in
their write-up. The representation in these instances did
not help them complete the investigation, but rather it
served as an additional way of representing the
investigation. However, in these write-ups either the
discussion of the mathematics or the solution was
incorrect.
Making Sense
Twenty-two of the 62 incorrect write-ups (35.5%)
included representations that were used to make sense
of the mathematics and generate a solution to the
investigation. Fifteen of the write-ups in this category
(68.2%) had either incorrect mathematics or an
incorrect answer only. In these 15 write-ups,
participants portrayed an understanding that the
27
representation was going to be used as a tool to help
them reach a solution. In most of these write-ups,
participants correctly explained the mathematics in the
investigation or the process of finding a solution, but
used the representation erroneously or did not correctly
interpret the representation. In the seven write-ups that
had both incorrect mathematics and an incorrect
answer, participants explained that the representation
would lead them to a solution, but their discussion of
mathematical concepts and their solution were
incorrect or incomplete.
Cosmetically Enhanced
Twenty-six of the 62 incorrect write-ups (41.9%)
included
cosmetically
enhanced
graphical
representations. Eleven of those 26 write-ups (42.3%)
had both incorrect mathematics and an incorrect
solution. In these cases, participants’ write-ups were
brief, as they did not explain their process or discuss
their solution, and the write-ups included a graphical
representation that did not seem to enhance their work
with the investigation. We speculate that participants
felt obligated to use technology to create graphical
representations of the investigations (e.g. a graph in
Excel, a Geometer’s SketchPad sketch), and so
technology was being used just for the sake of using
technology.
Teachers’ Use of Representations
One of InterMath’s goals is to shift participants’
thinking to a more constructivist view of mathematics
by using technology to generate representations that
will help participants make sense and communicate the
mathematics embedded in the investigations. We
speculate that in the “cosmetically enhanced” writeups, the participants felt compelled to include
technology, and therefore a representation, because of
the emphasis on technology in InterMath courses. We
assert that in these cases, the participants created a
representational graphic without a clear sense of the
type of representation that should be created or how it
should be used to reach a solution.
The idea that students think that technology (e.g.
calculators and computers) will provide them with
answers is a concern many educators share. It is our
experience that teachers are concerned that technology
does the work for the student and/or that the student
accepts the answer without question because the
technology generated it. We believe that this tendency
occurred for the InterMath participants in the form of
cosmetically enhanced representations. These writeups provide evidence that participants without a sense
28
of the mathematics or a way of finding a solution used
technology to generate a representation in an effort to
miraculously come to one.
Write-ups that used representations for the
purposes of communicating and making sense of
mathematics were not flawless either. Participants who
used representations for communication did not always
have an accurate grasp of the mathematical concepts,
made careless errors while reaching a solution, or
wrote a solution based on something not visible in the
graphic. Representations for making sense also led to
incorrect
write-ups.
Participants
used
these
representations to lead them towards a solution, but the
representations were often incorrect (e.g. dimensions
of a geometric figure, pattern in an Excel spreadsheet).
Further, the interpretation of these representations led
to incorrect solutions.
Implications for Professional Development
The data discussed in this article illuminates
dilemmas concerning professional development for
mathematics teachers. First, professional developers
need to be more explicit in guiding teachers through
the use of graphical representations. InterMath is
designed to be very learner-centered. Participants have
the freedom to select investigations, their approach to
completing investigations, the technology that they
employ and how they write up their solution. As seen
in this paper, this approach can be problematic. While
instructors serve as a model and guide students through
a few investigations, our findings suggest that more
guidance and explicit attention should be given to the
use of representations. In our view, this is not a
paradox, but a very real part of learning how to
structure a learner-centered professional development
program. That is, the learners need to own aspects of
their learning, however, the instructor needs to be
sensitive to the scaffolds the teachers need in order to
be successful.
Second, participants need opportunities to engage
in the process of effectively using mathematical
representations. This process extends from choosing
the representation that will be created to interpreting
the representation to find a solution. This builds on
Greeno and Hall’s (1997) contention that teachers need
professional development that prepares them to support
their students’ use of representations to solve
mathematical problems. Greeno and Hall focus on the
use of non-standard representations, which do not
include Geometer’s SketchPad sketches and Excel
spreadsheets. Still, we posit that teachers need to know
how to effectively generate and use these
An Analysis of Using Graphical Representations
representations in order to support their students’ use
of them.
Our findings indicate that, while teachers did use
technology to generate representations, their use of the
representations was not always what the InterMath
team had hoped. Specifically, representations were
used for the sake of using them or were included as an
add-on at the end of their write-up. Further, efforts are
needed to help teachers understand the value of
representations as a tool for communicating and
making sense of mathematical concepts.
Implications for Research
While this study highlights the use of graphical
representations in incorrect mathematical write-ups,
the findings only show half of the picture. An
examination is needed of correct write-ups to see how
graphical representations were used in those write-ups.
While we hypothesize that the correct write-ups
included mostly representations for the purpose of
sense making, we have not conducted the necessary
analysis.
Further analysis is also needed to examine the
incorrect write-ups included in this paper. More
information is needed about whether and how
participants justified the use of a representation in their
write-up. Our hypothesis is that the participants were
attempting to fit the use of representations into their
belief structures about mathematics. For example,
many write-ups that had cosmetically enhanced
representations
or
used
representations
for
communication included algebraic work and an
explanation of how to use paper and pencil to solve the
problem. This suggests that the participants may have
believed that they had to include a graphic even though
they were not using it in a way that promoted
understanding. Another examination of the write-ups
might help us better understand this trend.
References
Cohen, D., & Ball, D. (1990). Policy and practice: An overview.
Educational Evaluation and Policy Analysis, 12(3), 347–353.
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one
representation of the concept of function to another and
mathematical problem solving. Educational Psychology,
24(5), 645–657.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation:
Learning with and about representational forms. Phi Delta
Kappan, 78(6), pp.361–367.
Hawley, W. D. & Valli, L. (1999). The Essentials of Effective
Professional Development, In L. Darling-Hammond, & G.
Sykes (Eds.), Teaching as the learning profession: Handbook
of policy and practice (pp. 127–150). San Francisco: JosseyBass.
Kaput, J. J. (1992). Technology and Mathematics Education. In D.
A. Grouws (Ed.), Handbook of research on mathematics
teaching and learning (pp.515–556). New York: Simon and
Shuster.
Jackiw, N. (1993). The Geometer's Sketchpad [Computer
software]. Berkeley, CA: Key Curriculum Press.
Loucks-Horsley, S., Love, N., Stiles, K. E., Mundry, S. & Hewson,
P. W. (2003). Designing professional development for
teachers of science and mathematics (2nd ed.). Thousand Oaks,
CA: Corwin Press.
Monk, S. (2003). Representation in school mathematics: Learning
to graph and graphing to learn. In J. Kilpatrick (Eds.), A
research companion to Principles and Standards for School
Mathematics. Reston, VA: National Council for Teachers of
Mathematics.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
National Partnership for Excellence and Accountability in
Teaching. (2000). Improving Professional Development:
Research Based Standards. Washington, DC: Author.
Wilson, J. W., Hannafin, M. J., & Ohme, P. (1998). Inter-Math:
Technology and the teaching and learning of middle grades
mathematics. Grant proposal submitted to and accepted by the
National Science Foundation–Teacher Enhancement Program.
1
For a more detailed description of InterMath, refer to the InterMath
website at http://intermath.coe.uga.edu.
2
By incorrect mathematics, we mean that the participant’s write-up
contains some mathematical errors, which may or may not lead to the
correct answer. By incorrect answer, we mean that part or all of the
participant’s answer is incorrect.
3
Each participant has been assigned a unique identifier. The letter
refers to the course and the number refers to which person this is within that
course.
29
Appendix A
Investigation: What is the sum of the angles of a triangle? Of a quadrilateral? Of a pentagon? Of a hexagon?
What is the sum of the angles in convex polygons in terms of the number of sides?
Write-up: When using Geometer Sketchpad to create a triangle that is formed by having two transversals intersect
a set of parallel lines, students can then use the properties that they have learned about angles to determine the sum of
the angles of a triangle.
Click here for InterMath dictionary if unfamiliar with terminology.
m!AHF = 81°
m!HFA = 63°
m!FAH = 36°
m!IHF+m!HFI+m!FIH = 180°
m
5
2
A
3
Line j is parallel to LIne q
H
4
1
F
q
Because lines j and q are parallel to one another, then lines x and m are transversals to these parallel lines. Thus,
angle 1 and angle 4 are congruent because they are alternate interior angles. Similarly, angle 2 and angle 5 are
congruent. Therefore, the following angles are congruent:
Angle 1 = Angle 4
Angle 2 = Angle 5
Angle 3 = Angle 3
Conclusion: We can conclude that Angle 5 + Angle 3 + Angle 4 = 180 degrees since these three angles form a
straight angle. From the above conclusion, we see that: Angle 1 + Angle 2 + Angle 3 = Angle 4 + Angle 5 + Angle 3
= 180
Thus, we can conclude that the sum of the measures of the three vertex angles in a triangle is 180 degrees.
We can use the angle sum in a triangle property to find the measure of the vertex in a regular n-gon. By forming
triangles within a polygon, we can find the sum of the vertex angles in any polygon.
In a quadrilateral, we formed two triangles. Since each triangle has 180 degrees, and there were two triangles
formed in the quadrilateral, we can conclude that the sum of the measures of the vertex angles is 2 x 180 = 360.
Furthermore, the individual angle measures of each vertex angle within the quadrilateral would be 360/4 = 90 degrees.
30
An Analysis of Using Graphical Representations
Click here for interactive sketch (shown below).
m!EBA = 45°
E
C
A
B
m!AEB = 45°
m!EAB = 90°
m!EBA+m!AEB+m!EAB = 180.00°
m!ECB = 90°
m!CEB = 45°
m!CBE = 45°
m!ECB+m!CEB+m!CBE = 180.00°
(m!EBA+m!AEB+m!EAB)+(m!ECB+m!CEB+m!CBE) = 360.00°
In a pentagon, we formed three triangles. Since each triangle has 180 degrees and there were three triangles
formed in the pentagon, we can conclude that the sum of the measures of the vertex angles is 3 x 180 = 540 degrees.
Furthermore, the individual angle measure of each vertex angle within the pentagon would be 540/5 = 108 degrees.
In a hexagon, four triangles can be formed. Since each triangle has 180 degrees and there were four triangles
formed in the hexagon, we can conclude that the sum of the measures of the vertex angles is 4 x 180 = 720 degrees.
Furthermore, the individual angle measures of each vertex angle within the hexagon would be 720/6 = 120 degrees.
Using the relationship shown above, we created a spreadsheet to show how to calculate the sum of the angles in
convex polygons in terms of the number of sides.
Click here to view spreadsheet (shown below).
n
3
4
Sum of Angles
180
360
Vertex Angles
60
90
5
6
540
720
108
120
7
8
900
1080
128.5714286
135
9
10
11
1260
1440
1620
140
144
147.2727273
12
1800
150
In each instance, the number of triangles formed in each polygon is equal to the number of sides in the polygon
minus two. Therefore, to find the sum of the angles in any convex polygon in terms of the number of sides, the
formula 180(n – 2) can be used.
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
31
Appendix B
Problem: Along the diameter of a circle you can construct circles with equal radii that are tangent to each other.
The outermost circles in the string of circles will be tangent to the large circle. (Tangent means that the circles touch
each other but do not cross each other, nor do they leave gaps.) How does the combined area of all of the shaded
circles relate to the area of the entire circle?
Solution: I constructed circles using 2, 3, 4, and 5 smaller tangent circles along the diameters in Geometer's Sketch
Pad. Those constructions are shown below:
I used Excel to record the data about the areas of each circle and its tangent circles. That table is below and is
colored to correspond to the constructions above:
Area of Large Circle
Area of Small Circle
# of Small Circles
3.14
3.14
3.14
3.14
0.78
0.35
0.2
0.13
2
3
4
5
Small Circles Comb.
Area
1.56
1.05
0.8
0.65
Small/Lrg Circle
Ratio
0.496815287
0.334394904
0.25477707
0.207006369
In each situation the relationship formed between the circle and its tangent circles along its diameter could be
closely described in terms of the number of tangent circles. When there were two tangent circles, the area relationship
was 1:2 or .496. When three, the relationship was 1:3 or .33, etc.
Extension: If you were to walk along the circumferences of all the small circles, there is almost a 1:1 relationship
to the circumference of the original circle. Examine the table below. Again it is color-coordinated with the
constructions above.
Large Circle
Circumference
6.28
6.28
6.28
6.28
Small Circle
Circumference
3.14
2.1
1.57
1.26
# of Small Circles
2
3
4
5
Small Circles Total
Circum.
6.28
6.3
6.28
6.3
Small/Lrg Circle
Ratio
1
1.003184
1
1.003184
It is interesting that when there are an even number of tangent circles along the diameter, the relationship is
precisely1:1. However, when there are an odd number of tangent circles the relationship is not precisely one to one.
The sum of the circumferences of the tangent circles is slightly more than the original circle, but appears to always be
the same difference.
32
An Analysis of Using Graphical Representations
Appendix C
Investigation: A number of investigations can be done involving quadrilaterals. One investigation that can be
explored deals with drawing an original quadrilateral. Then by marking the midpoints of the sides and connecting the
midpoints with line segments to create an inscribed quadrilateral.
Write-up: This can be done using GSP. An example can be seen in the sketch below.
B
C
E
H
F
G
A
D
One idea that can be explored is if the shape of the new quadrilateral depends on the shape of the original
quadrilateral. One could determine if the two quadrilaterals will have the same shape or different shapes. This could
be explored using GSP. In GSP, one could drag the different vertices of the original quadrilateral to form different
shapes.
Examples of this using GSP are shown below.
A
E
B
H
F
G
D
C
The image above seems to show that if the original quadrilateral is a trapezoid, the smaller quadrilateral will be a
rectangle.
E
B
C
H
A
F
G
D
The image above seems to show that is the original figure is a rhombus; the second quadrilateral will again be a
rectangle.
Another idea that could be explored is if the four smaller triangles created near the vertices of the original
quadrilateral have the same area. This can be explored using the measure and calculate features of GSP. An example
of this can be seen below.
Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly
33
D
Area FEGH = 19.91 cm2
Area
DFE = 5.88 cm2
Area
FCH = 5.84 cm2
Area
EGB = 4.11 cm2
Area
GAH = 4.07 cm2
F
E
H
B
( Area
C
D F E) + ( Area
A
G
F C H) + ( Area
E G B) + ( Area
GAH) = 19.91 cm2
The calculations show that the four small triangles do not have the same area. However, the calculations do show
that the sum of the areas of the four triangles is equal to the area of the smaller quadrilateral.
GSP could be used to explore a number of different aspects of this problem. Examples would include, is there a
constant ratio between the perimeter of the smaller quadrilateral and the larger quadrilateral. Another investigation
could be done comparing the areas of the two quadrilaterals.
34
An Analysis of Using Graphical Representations
The Mathematics Educator
2006, Vol. 16, No. 2, 35–46
Professional Development Through Technology-Integrated
Problem Solving: From InterMath to T-Math
Ayhan Kursat Erbas
Erdinc Cakiroglu
Utkun Aydin
Semsettin Beser
The ability to integrate technology into instruction is among the characteristics of a competent mathematics
teacher. Research indicates that the vast majority of teachers in Turkey believe the use of computers in
education is important, but have limited knowledge and experience on how to use technology in their
instruction. This paper describes the T-Math project (http://www.t-math.org), which adapted the InterMath
(http://intermath.coe.uga.edu) knowledge base for mathematics teachers in the United States and developed
relevant resources for professional development of Turkish mathematics teachers to guide them in constructing
useful strategies for their students while developing as expert mathematics teachers. Examples of mathematical
investigations adopted and developed in the T-Math project are presented as well as the anticipated challenges
and subsequent strategies for integration.
Schools throughout the world recognize the need,
but still struggle, to integrate technology into
mathematics education. The development of teachers
who can flexibly adapt technology into their teaching
of mathematics is crucial for technology to have a
positive impact on student performance. In order to
develop teachers’ flexibility in selecting instructional
alternatives, technology should be integrated as a
central aspect of teacher education programs (Sudzina,
1993).
The InterMath project promotes such an approach
with an Internet-based (http://intermath.coe.uga.edu)
Ayhan Kursat Erbas is an Assistant Professor in the Department
of Secondary Science and Mathematics Education at the Middle
East Technical University, Ankara, Turkey. His research interests
include teaching and learning of algebra, integrating technology
into mathematics education, and teacher knowledge and beliefs.
His e-mail is erbas@metu.edu.tr.
Erdinc Cakiroglu is an assistant professor of mathematics
education in the Department of Elementary Education at Middle
East Technical University, Ankara, Turkey. His research interest
includes curriculum development and mathematics teacher
education. His e-mail is erdinc@metu.edu.tr
Şemsettin Beşer is a doctoral student and research assistant in
the Department of Secondary Science and Mathematics
Education at the Middle East Technical University, Ankara,
Turkey. His research interests include web-based adaptive
learning and computerized adaptive testing in mathematics. His
e-mail is sbeser@metu.edu.tr.
Utkun Aydın is a graduate student and research assistant in the
Department of Secondary Science and Mathematics Education at
the Middle East Technical University, Ankara, Turkey. Her
interests include metacognition, preservice teacher education,
and technology at secondary level. Her e-mail is
utkun@metu.edu.tr.
Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser
professional development effort with the goal of
designing and implementing a series of workshops and
ongoing support programs that feature contemporary
applications of technology and mathematics pedagogy
in the middle-grades. It focuses on building teachers'
mathematical content knowledge through mathematical
investigations that are supported by technology. As a
result of working on the InterMath project and seeing
firsthand how teachers become better mathematics
educators
through
completing
technology-rich
mathematical investigations, the first author of this
paper sought to adapt InterMath for professional
development of mathematics teachers in Turkey. An
international extension of the InterMath project was a
natural consequence, because e-mail communications
and web hits suggested that InterMath’s knowledge
base and resources were already being used widely,
both in the United States of America (USA) and
internationally.
Before delving further into T-Math, we will look at
some of the most evident similarities and differences
between the educational systems in the USA and
Turkey to better understand the adaptation of T-Math.
Unlike the USA, the Turkish school system and
curricula are centralized. All educational institutions
are under the control of the Turkish Ministry of
National Education (MNE). All important policy and
administrative decisions, including the appointment of
teachers and administrators, the selection of textbooks,
the selection of subjects for the curriculum, and the
management of in-service teacher education, are made
by the MNE. A national mathematics curriculum is
35
followed in every school and supervisors assigned by
the MNE control all educational activities in schools.
In both Turkey and the USA, pre-service
mathematics teachers are required to have an
undergraduate degree. Unlike the United States,
Turkey has a unified system of higher education under
the umbrella of the Higher Education Council of
Turkey, which is responsible for the planning,
coordination, and supervision of higher education.
Teacher education programs in different universities
usually require the coursework suggested by the
Higher Education Council of Turkey (Yükseköğretim
Kurumu, 1998).
As for similarities between the educational
systems, high stakes tests are an important issue in
both Turkey and the USA. In Turkey, nationwide
examinations for university and high school entrance
are very important factors that influence what
mathematics teachers do in the classrooms. The
pressure these exams put on students, parents, and
teachers easily changes the perception of “good
teaching” in schools. Teaching to the test and solving
as many multiple-choice questions as possible are
highly valued teaching behaviors by most of the
stakeholders. This results in appreciation of such
student behaviors in mathematics classes as solving
mathematics questions as quickly as possible, or
remembering the rules that will help them reach quick
solutions. In Turkey, due to the centralized education
system, such tests have an extensive nationwide impact
in almost all schools.
Similar to the impacts of Principles and Standards
for School Mathematics (National Council of Teachers
of Mathematics, 2000) in reforming mathematics
education in the USA, the development of new
elementary and secondary school mathematics
curricula in Turkey supported the idea of adopting
InterMath in a Turkish context. The new Turkish
curriculum deviates from its precursor, and includes a
larger emphasis on learner-centered instruction,
problem solving, open-ended explorations, modeling
real-life situations, and the use of technology as a tool
to support mathematics learning (MNE, 2005a, 2005b).
In Turkey, most teachers neither have experienced
such instructional approaches as learners nor used them
in their teaching. T-Math, like InterMath, aims to
address the concern of, “How can teachers teach a
mathematics that they never learned, in ways that they
never experienced?” (Cohen & Ball, 1990, p. 238).
36
The Pebbly Road to Technology Integration in
Teaching and Learning of Mathematics
Integrating technology into mathematics education
is not easy or straightforward, and many barriers exist.
Such barriers include the lack of a unified meaning of
integration of technology (Willis & Mehlinger, 1996);
common teacher perception that technology and its
integration would not have a positive impact on student
learning (Coffland, 2000; Ertmer, Addison, Lane,
Ross, & Woods, 1999; Ertmer & Hruskocy, 1999;
Slough & Chamblee, 2000); lack of access to
technology and related resources (Hadley & Sheingold,
1993; Manouchehri, 1999; Parr, 1999); lack of training
and support in both pre-service and in-service teacher
education programs (Ertmer & Hruskocy, 1999;
Wetzel, Zambo, & Buss, 1996); and discouraging
school environments, curriculum requirements, and
heavy
teacher
course-load
(Coffland,
2000;
Manouchehri, 1999). In addition, research has shown
that teachers teach in the same manner in which they
have been taught, making the integration of technology
quite difficult, since most teachers have never used
technology as a tool for meaningful learning (Ball,
1990; Frank, 1990; Quinn, 1998; Trueblood, 1986;
Vannatta & Fordham, 2004).
In Turkey, the integration of technology into
school mathematics is moving at a very slow pace
compared to other countries in the Organization for
Economic Co-operation and Development (2005), and
barriers to integration are similar to the ones in the
USA. For example, strict curriculum requirements,
heavy content of mathematics lessons, and a lack of
time to integrate technology into teaching are some of
the obstacles that teachers in Turkey have to overcome
(Cakiroglu, Cagiltay, Cakiroglu, & Cagiltay, 2001).
Further, tests given nationwide at the end of primary
education (i.e., High School Entrance Examination)
and secondary education (i.e., Student Selection Exam
for University Programs) may result in teachers
focusing mainly on test preparation, which makes the
implementation
of
technologically-oriented
applications and problem solving even more
challenging (Kellecioglu, 2002). Many teachers think
that using calculators or computers in a mathematics
course, before students have mastered basic concepts
and skills may limit their cognitive abilities and hinder
their
computational
skills
(Fleener,
1995).
Nevertheless, other research shows that some teachers
do see technology as a tool to develop their students’
critical thinking processes (Aloff, 1999; Hembree &
Dessart, 1992; Yoder, 2000).
T-Math Project
Similar to research in the USA, Turkish studies
indicate that a majority of teachers believe the use of
computers in education is important, but they have
limited knowledge and experience on how to use this
technology in their teaching (e.g. Cakiroglu et al.,
2001; Cakiroglu & Haser, 2002). Two more major
obstacles to the use of technology were the lack of
hardware and the lack of teachers’ knowledge about
using computers (Cakiroglu et al., 2001). Teachers
expressed concern about classroom management,
including issues such as keeping track of student
progress and maintaining control of the lesson
(Cakiroglu & Haser, 2002). Further, teachers felt that
they had a more “passive” role in lessons when
computers were involved and that students were less
“serious” when using computers (Cakiroglu & Haser,
2002).
Further, these negative perceptions of computer
use (or other technologies such as graphing calculators)
in mathematics influence whether technology is
integrated into their teaching (Norton, McRobbie, &
Cooper, 2000). Other studies on technology use
suggest that, even if the computers are available and
accessible, mathematics teachers tend not to use
computers in their classrooms (Rosen & Weil, 1995).
Need for Professional Development
Providing professional development activities for
teachers who do not feel prepared to integrate
technology into their instructional practices is crucial
for supporting technology integration into mathematics
classrooms (Liu, 2001). As part of their education
reforms, the MNE has attempted to improve the
technological infrastructure (e.g. hardware and Internet
access) in Turkish schools and has mandated that
teachers must learn how to use technology and
integrate it into their teaching (MNE, 2005c).
Technology-related teacher competencies defined by
the MNE can be seen in the Appendix. Despite these
visions, Turkish teachers, like American teachers,
typically only learn about the basic uses of technology
(e.g., how to operate a computer, how to use Microsoft
Office programs, and how to do basic computer
programming), rather than learning how to use these
technologies to enhance their teaching. External
factors, such as poor administrative support, lack of
access, limited or no budget, inadequate training on the
use of hardware and software, additional work and
preparation time that technology may demand from
teachers, curriculum requirements, and teachers’
insufficient pedagogical content knowledge, also
Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser
inhibit the implementation of technology-rich activities
(Halpin & Kossegi, 1996; Hanks, 2002; Mouza, 2003;
Tozoglu & Varank, 2001).
A new vision of school mathematics requires a
new vision of teacher education. For a successful
implementation of recent curriculum changes in
Turkey, there is a need for professional development
efforts aiming to influence teachers’ beliefs about,
improve knowledge of, and increase comfort with
technologies that are likely to enhance student
learning. As Ball (1990) suggested, professional
development can be achieved by having teachers fully
immersed in a context of professional development
where they can experience the use of technology, first
as a learner in investigating problems for improving
their understanding of mathematics, and then as a
teacher in their actual instructional practices. This
would be an effective way of having teachers improve
themselves for better implementation of the recent
curriculum changes and serve to mediate between
reform dictations and classroom implementations. The
existing in-service teacher education programs in
Turkey are far from addressing such expectations
(Cakiroglu et al., 2001). Thus, we are in the process of
adapting InterMath into T-Math, which aims to utilize
the principles explained above to come up with
effective professional development activities for
Turkish mathematics teachers.
Goals of the T-Math Project
Derived from InterMath, the overall aim of T-Math
is to provide a professional development environment
for mathematics teachers. To attain this goal, three
principles were considered as a basis for the main
activities of T-Math (Figure 1). First, T-Math aims to
help teachers experience the use of technology as
learners in a problem-solving environment. Second, it
facilitates teachers’ reflections on their technologybased problem-solving experiences. Third, T-Math
provides environments for teachers to collaborate with
each other to establish a shared understanding of a
technology-rich mathematics learning environment. TMath aims to address all of these goals through the use
of interactive and dynamic technologies. The
expectation of T-Math is that teachers will construct
their own understanding in a context where
collaboration and problem-solving activities engage
them in debating ideas, communicating with each
other, transferring knowledge, making predictions, and
deriving new questions (Cobb, 1994). T-Math employs
a mixed approach that combines on-site workshops and
37
EXPERIENCE
in problem solving
through technology
TECHNOLOGY
COLLABORATION
for a new culture of
mathematics in schools
REFLECTION
upon the personal and
collaborative experience
Figure 1. The professional development principles promoted in the T-Math project.
online help systems to facilitate the various activities.
These activities are detailed below.
T-Math Investigations
Similar to InterMath, the face-to-face workshops
of T-Math provide the opportunity for teachers to
explore
technology-rich
investigations.
These
investigations allow teachers to develop their
mathematics content knowledge, hone their technology
skills in the context of doing mathematics, and
experience learning mathematics in an investigative,
learner-centered manner.
Integrating Technology-Rich Investigations
In addition to the goals described above, the
workshops are expected to provide participant teachers
with meaningful learning experiences and motivate
them to adapt and use T-Math investigations, or
integrate technology in general, into their instructions.
Teachers in the InterMath workshops made progress in
learning how to use technology and provided evidence
that they saw technology as being important in their
own learning of mathematics (Orrill, Polly, Ledford,
Bleich, & Erbas, 2005). However, the majority of the
participants believed that their students could not
benefit from this use of technology because of
logistical barriers or because the students had not yet
developed an understanding that watching a
demonstration provided a fundamentally different
learning experience than engaging with the technology.
Considering that InterMath courses had little focus on
how a teacher can use the technology in their own
classroom, these results were reasonable. In T-Math
workshops, we intend to give more emphasis on
38
classroom integration and give support to individual
teachers in transferring what they learn into their
classrooms. We anticipate that in this way teachers will
be more willing to adapt T-Math and use technology
for and with their students.
Addressing Beliefs about Technology Use and
Integration
There is a growing body of research literature
indicating that the beliefs teachers hold directly affect
both their perceptions and strategies of teaching and
learning interactions in the classroom, and that these,
in turn, affect their teaching behaviors (Clark &
Peterson, 1986; Clark & Yinger, 1987). Trumbull
(1987) has shown, for instance, how teachers’ beliefs
limit their ability to find solutions to pedagogical
problems. While teachers are trying to adopt
innovations
related
to
technology-integrated
mathematical applications into their classrooms,
negative attitudes towards technology impede both
their teaching and their students’ learning (Hazzan,
2002; Margerum-Leys & Marx, 1999). Teachers’
negative beliefs and attitudes towards technology and
its integration deriving from their lack of experience
and knowledge would be addressed in T-Math
workshops, on-line and collaborative colleague support
systems, and by providing first-hand experiences
related to the learning and teaching of mathematics. As
it was found in the Apple Classrooms of Tomorrow
research project (Sandholtz, Ringstaff, & Dwyer,
1997), only after teachers had learned the fundamentals
of using the technology and had become more
comfortable would they drop their negative beliefs,
attitudes, and concerns about using technology in the
T-Math Project
classroom. Without establishing this level of comfort,
we cannot expect teachers to adopt or begin to think
about how they could use the technology as part of
their instruction.
T-Math Resources
The major components of T-Math include openended problems and investigations, materials and plans
for workshops, and a mathematics dictionary.
Technologies such as spreadsheet applications,
dynamic geometry software, graphing tools, and
graphing calculators are suitable to investigate the
open-ended problems in T-Math. In the initial phase,
the problems in T-Math were translated and adapted
from InterMath, making problems more culturally
relevant when necessary.
The T-Math project aims to organize its knowledge
base within a user-friendly web-based system so that
teachers and other users can easily access organized
information without any frustration. To help teachers
better organize and select problems, T-Math organizes
and presents problems based on their mathematical
content. For this purpose, the new mathematics
curriculum in Turkish schools serves as a foundation.
The curriculum is divided into five domains of
mathematics: numbers, geometry, algebra, probability
and statistics, and measurement. T-Math problems
were categorized according to these five categories and
also identified based on (a) the technological tools that
may be used to investigate them, (b) the grade level(s)
in which these problems could be used, and (c) the
objectives of the new mathematics curricula that they
correspond to. Investigations in the T-Math project
consist mainly of the following four types:
1. Direct translations of the InterMath’s
investigations into Turkish. In doing these translations,
we have considered Turkish educational and cultural
contexts so that problem contents match Turkish
school mathematics curricula, and problem statements
and wording are appropriate for culture and curricula.
The following case illustrates how we translated and
adopted an InterMath investigation to a T-Math
investigation:
InterMath version: The U.S. Postal Service will
only mail packages that meet certain size
requirements. For cylinder-shaped packages (or
"rolls"), the minimum length is 4 inches and the
maximum length is 36 inches. There is also a
restriction that the length plus two diameters can be
no more than 42 inches. (Why do you think they
have this restriction?)
Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser
a. What are the dimensions of an acceptable boxshaped package that will have the greatest volume?
b. When there must be only two opposite faces that
are square, what are the dimensions of an
acceptable box-shaped package that will have the
greatest volume? The smallest volume?
c. What are the dimensions of an acceptable boxshaped package that will have the greatest volume
if each dimension is different?
T-Math version: The Turkish Postal Service will
only mail packages as letter post that meets certain
size requirements. For box-shaped packages, it
should have a side width at least 14 cm by 9 cm
dimensions. Also, the longest side of the package
cannot be longer than 60 cm. There is also a
restriction that the sum of the width, length, and
depth of the package cannot exceed 90 cm. What
are the dimensions of an acceptable box-shaped
package that will have the greatest volume?
Extension: For cylinder-shaped packages (or
"rolls"), the minimum length is 10 cm and the
maximum length is 90 cm. There is also a
restriction that the length plus two diameters can be
no more than 104 cm and no less than 17 cm (why
do you think they have this restriction?). What are
the dimensions of an acceptable cylinder-shaped
package that will have the greatest volume?
In translating the InterMath version to T-Math, the
Turkish Postal Service restriction values were obtained
to provide a cultural context for students. Also, unlike
the InterMath version, the T-Math version extends the
problem by adding a second case (i.e., box-shaped
package) and by including an additional restriction (the
minimum for the length plus two diameters) for the
cylinder-shaped packages.
2. Adapted multiple-choice items used in previous
Turkish standardized tests such as the Students
Selection Exam (OSS), the Student Placement Exam
and the High School Entrance Exam and converted to
open-ended investigations. This adaptation was meant
to eliminate students’, teachers’, and parents’ concerns
for learning, teaching, and preparation towards tests.
As an example, the following item was used in OSS in
1999:
If a is a positive real number, at most how many
cm2 can the area of the rectangle with dimensions a
cm and (8 – 2a) cm be?
A) 64
B) 32
D) 16
E) 8
C) 24
This problem was adapted into an open-ended
problem as “You are making a rectangular flower
39
garden. What is the largest area of the garden whose
dimensions are a meters by 8 – 2a meters? Extension:
What is the largest area of a rectangular garden that
you can enclosure by using 16 – 2a meters of fencing?
If one side of the rectangle can use a barn wall, what
are the dimensions of the enclosure with the largest
area?” Students can use a spreadsheet, graphing
calculator, and dynamic geometry software to
investigate and solve the problem. This allows students
to use multiple approaches or representations to
conceptually investigate and understand such
problems. Investigating the area of a rectangular region
with a fixed perimeter can also extend the problem.
Such problems dealing with optimization are covered
in the InterMath project as well.
3. Investigations added by the T-Math team. TMath also extended the InterMath investigations by
adding new investigations for middle and high school
that are not drawn from standardized tests. A sample
investigation of this type is given below.
A Pythagorean triple is an ordered triple (a, b, c) of
positive integers satisfying
a2 + b2 = c2. Find as many Pythagorean triples as
you can. Can you come up with an easier way to
find Pythagorean triples? Your friend claims that
for any positive integer m, the triple (2m, m2 – 1,
m2 + 1) is a Pythagorean triple. Does this work?
Why?
The Pythagorean triples mentioned in the problem
can be investigated through calculators, spreadsheet
applications, and dynamic geometry applications
(Figure 2 and Figure 3). Learners may use a
Figure 2. Investigating Pythagorean triples in a
spreadsheet application
40
spreadsheet application to investigate Pythagorean
triples mentioned in the problem (Figure 2). They may
assign positive integer values to the first two variables
and calculate the third one using the first two values
and observe if it is integer or not. In this way they can
determine whether a triple (a, b, c) is a Pythagorean
triple or not.
Similarly, learners may use a dynamic geometry
application, such as Geometer’s Sketchpad (GSP)
(Jackiw, 2001), to investigate the same problem. A
right triangle may be constructed whose vertices are
snapped to grid points so that the side lengths of the
triangle are integer values (Figure 3). Learners may
play with the vertices to obtain right triangles with side
lengths that satisfy the Pythagorean triples rule. In this
way, a connection between the algebraic and geometric
representation of the same problem can be made. This
investigation can also be extended by allowing more
advanced learners to use a graphing application that
has 3-D graphing capabilities to explore the graph of x2
+ y2 = z2 to determine the integer values that satisfy this
equation by intersecting the graph with various x and y
values such that x = n and/or y = m where m, n ∈ Z+.
4. Investigations added by the T-Math team that
make use of local cultural elements. Anatolian land has
been a crossroad for many cultures and civilizations
such as Greek, Roman, Islamic, and many others. For
this reason, Turkish history and culture offer culturally
rich contexts to explore many mathematical topics. TMath utilizes this opportunity to provide an
ethnomathematical perspective in investigations while
integrating technology. For example, traditional
Turkish handicrafts – carpets and rugs, marbling
Figure 3. Investigating Pythagorean triples in a
dynamic geometry application
T-Math Project
(ebru), stone carvings, wood carvings, ivory carvings,
tiles, calligraphy, embroidery, quilts, knitted socks,
felts, fabrics and textiles, yazma (hand printed textiles),
etc. – and historical structures offer superb
opportunities to investigate symmetry, asymmetry,
grids and tessellations, and other geometrical content.
For this purpose, students were asked to visit websites
(e.g., http://www.turkishculture.org) containing digital
examples of traditional Turkish art styles mentioned
above or to take their own digital pictures of the
traditional art styles and historical structures around
them. Students were asked to copy and paste the
images into GSP to explore and reproduce reflection
and rotation symmetry, asymmetry and tessellations
such as the ones shown in Figures 4, 5, 6 and 7.
Figure 4. Wood Carving Shutter Panel in the Kilic Ali
Pasa Mosque in Istanbul
Figure 6. Panel from the Muradiye at Bursa, Dating
from 1426
Figure 5. Ivory Carving Belt Piece in Topkapi
Museum in Istanbul, Dating from 1500s
Figure 7. Tile from an Arched Panel in Iznik, Dating
from Mid 16th Century
Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser
41
Role of T-Math Instructors
T-Math, like InterMath, is founded on the premise
of providing teachers time during workshops to
collaborate and work on problems. In workshops,
instructors will act as a facilitator while teachers use
technology to explore the investigations. Further,
instructors attempt to help participants connect
mathematical concepts to real-life and use a variety of
activities to see how technology can be integrated in
meaningful ways. The collaborative workshops are
designed to assist in the establishment of a
mathematical culture in which technology helps
teachers develop their mathematics content knowledge.
Mathematical understanding and communication is
built on modeling and problem solving with interactive
technological tools. These workshops are meant to help
teachers reflect on their experiences and use them to
develop content knowledge, as well as to help teachers
become comfortable exploring mathematical problems.
Future Endeavors
The T-Math project staff is also adapting
InterMath’s mathematics dictionary and the
Constructionary for the Turkish elementary and
secondary school mathematics curriculum. The
purpose of the T-Math dictionary will be to present
explanations of mathematical terms for students,
teachers, parents and other potential users so that they
can study mathematical terminology, terms, and
concepts in an interactive environment. The T-Math
dictionary will help us meet a significant need for a
mathematics dictionary in Turkish for elementary and
secondary students. Existing printed mathematics
dictionaries in Turkish only cover upper level
mathematics, and the language used in defining and
explaining terms is more suitable for advanced levels.
The T-Math dictionary will use clear language that is
both age and content level appropriate. It will be
equipped with pedagogical elements such as links to
related terms, real life examples and applications, and a
forum where users can express and discuss their
opinions about each component of the on-line
mathematics dictionary. Considering that there are
certain debates and disagreements about some of the
mathematical terms in Turkish, this dictionary could be
a platform for dialogue on Turkish mathematics
terminology. We anticipate that the dictionary will be
useful to teachers who are about to implement the new
national curriculum for primary schools, since many
teachers in elementary schools are not familiar with
concepts such as patterns, tessellations, and
42
transformations that are new in the curriculum and new
to the Turkish school mathematics terminology.
Formerly, geometric constructions using Greek
construction rules were covered only in the 10th grade
geometry curricula in Turkey. However, with recent
changes in the elementary school mathematics
curriculum, geometric constructions are now covered
in sixth, seventh, and eighth grades as well. Therefore,
T-Math is adopting InterMath’s Constructionary,
which is an online tool designed to help users create
geometric constructions using Geometer's Sketchpad,
and include more constructions that will address the
new curriculum covered in the 6-8 grades. This will be
highly valuable for middle school mathematics
teachers, as they may lack such content knowledge.
Challenges in the Adaptation Process
The issues around mathematics education in the
USA and Turkey have many similarities and
differences. School systems, classroom cultures,
curriculum climates, and the teacher education systems
in both countries should be carefully examined before
adopting any educational innovation. Similarities
between the two systems encouraged the T-Math team
to benefit from the InterMath content and strategies in
mathematics teachers’ professional development. The
differences, on the other hand, compelled us to come
up with additional strategies for developing relevant
professional development resources and for gaining
acceptance of the teachers and the local mathematics
education community.
Adopting InterMath principles and content into a
different educational system is a challenge in many
senses, especially considering the unique issues
surrounding mathematics education in the USA and
Turkey. A project aiming to place open-ended
mathematical investigations into the heart of
mathematics instruction will have to confront
traditional attitudes towards mathematical tasks in both
contexts. In the adaptation process of the InterMath
principles, the T-Math project team has been
developing and implementing the following strategies
to deal with such unique challenges:
Making T-Math Content Culturally Relevant
As explained earlier, the investigations that were
translated and adopted from InterMath have been
revised and additional problems surrounding Turkish
culture have been developed to make T-Math content
more relevant for Turkish mathematics teachers’
professional development.
T-Math Project
Working with Private Schools.
There is a competitive environment among private
schools in Turkey about the innovative educational
initiatives. In its initial phases, T-Math aims to work
with private schools that already have an agenda of
integrating technology into instruction in order to
create samples of exemplary T-Math implementation.
Emphasizing T-Math’s Potential Contributions to the
Implementation of the New Curriculum and to the
Change Efforts.
On-going curriculum reform efforts in Turkey are
hoped to trigger a culture of change in teachers’
perceptions of school mathematics. Turkey’s progress
toward joining the European Union (EU) and the
process of accession negotiations with the EU is an
important motivation for Turkish institutions to
change. In this sense, the MNE is open to innovative
teacher education programs, which provides an
important opportunity for T-Math to contribute to the
change efforts. This possibility will be used to
persuade teachers, private school administrators, and
the MNE authorities to support mathematics teachers’
participation in T-Math.
Convincing the Central Authority
Reaching the teachers of public schools in Turkey
requires not only convincing them to participate but
also getting the approval of the MNE. After the recent
curriculum changes, the MNE has been searching for
ways of collaborating with universities on the inservice training of teachers. By convincing the
administrators in the MNE about the possible
contributions of T-Math to teachers’ professional
development, we hope to establish an important
channel for reaching a large number of schools and
teachers throughout Turkey. Once the T-Math
knowledge base (i.e., open-ended investigations, a
mathematics dictionary, a dictionary of geometric
constructions, etc.) is established, and pilot workshops
are conducted, the T-Math team plans to submit a
proposal to the MNE to take an integral part in their inservice teacher education agenda. We believe that data
from the pilot implementations of T-Math is important
in demonstrating its potential in teachers’ professional
development about integrating technology in
mathematics education.
Learn from the InterMath Experience
Although there are considerable differences
between Turkish and American education systems,
there are many things that we can learn from the
Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser
InterMath experience, especially on how to deal with
challenges in changing mathematics teachers’
conceptions of technology integration in mathematics
education. In this sense, as explained earlier in this
paper, T-Math team is making use of the feedback and
the knowledge base shared by the InterMath project in
planning workshops, selecting and developing openended mathematical investigations, and developing
other components of T-Math project.
Conclusion
The revolution of technology in education,
according to the curriculum reform in Turkey, requires
mathematically sophisticated teachers that can
integrate technology in meaningful ways. Regarding
teachers’ essential role in their classrooms, meaningful
reform is more likely to succeed if teachers are
adequately prepared to use mathematics-related
technologies in ways that develop students’ conceptual
understanding and problem solving skills.
The studies and efforts by the MNE mentioned in
this paper highlight the significance of technology for
the future of education in Turkey. George Cantor (1845
- 1918) once said that, “The essence of mathematics is
freedom” and we believe that technology can free
teachers and students in their teaching and learning
efforts. As White and Frederikson (1998) indicated,
with the aid of technology teachers and students should
and can question, “why it is they believe what they
believe, and whether there is sufficient evidence for
their beliefs” (p. 7). With the use of open-ended and
interactive technologies, learners and instructors can
model most mathematical situations as problems and
investigate them interactively. One who is familiar
with such technologies should recognize that what can
be done with open-ended environments is usually
limited by the computing knowledge of the user. With
more and various technology usage, one can model
mathematical problems in various ways with various
technologies. In technology supported mathematics
education, integrating technology into pre- and inservice teacher education in terms of not only how to
teach with technology but also how to learn with it
gains importance. Teachers will need to practice with
relevant technology resources before they implement
them in their classroom environments. Through the TMath project site and its sophisticated publishing
environment, teachers will be able to reach and use
several technologies through a web interface.
Additions and changes have been made to InterMath
investigations, dictionary, Constructionary, and other
tools to make them more culturally relevant and match
43
curricular issues so that teachers feel more familiar
with the overall content within the existing culture and
educational systems. Thus, all the resources and
applications that are developed for workshops or faceto-face implementations will also be available for
teachers everywhere and all the time. Accessibility to
resources anytime may motivate teachers who are not
able to participate in workshops and other project
activities because of the location, time, and cost
problems.
In conclusion, we can and should utilize
functionalities of computer technologies in learning
and teaching for understanding in mathematics
education. As Gottfried Wilhelm Leibniz (1646-1716)
said long before the invention of computers, “It is
unworthy of excellent men to lose hours like slaves in
the labor of calculation which could safely be relegated
to anyone else if machines were used.” With this vision
in mind, T-Math project, as an extension of InterMath,
aims to contribute to the effective and meaningful use
of technology in exploring and learning mathematics
by providing professional development opportunities
for teachers.
Cakiroglu, E., Cagiltay, K, Cakiroglu, J., & Cagiltay, N. (2001,
April). Elementary and secondary teachers' perspectives
about the computer use in education. Paper presented at the
annual meeting of American Educational Research
Association. Seattle, WA.
Author’s Note
Cohen, D. K., & Ball, D. L. (1990). Policy and practice: An
overview. Educational Evaluation and Policy Analysis, 12,
347–353.
The research reported here paper was supported by
the Middle East Technical University (METU) under
Grant No. BAP-2005-07-02-08. The opinions
expressed in this paper are those of the authors and do
not necessarily reflect the views of the METU.
Correspondence concerning this article should be
addressed to Ayhan Kursat Erbas, The Department of
Secondary Science and Mathematics Education,
Middle East Technical University, 06531, Ankara,
Turkey. E-mail: erbas@metu.edu.tr
We are most grateful for the critical comments and
suggestions on earlier versions of the manuscript from
Chandra Orrill, Drew Polly and other members of the
InterMath project team, the TME editorial staff and the
reviewers
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45
Appendix
Technology-Related Teacher Competencies as Defined by the Turkish Ministry of National Education
In supporting the Basic Education Project, the Turkish Ministry of National Education initiated a study to redefine
the qualifications for teachers (Turkish Ministry of National Education, 2005c). Six main competency areas, together
with 39 sub-competencies and 244 performance indicators, are determined for in-service teachers teaching in Turkey.
The six main competencies are (a) personal and professional values – professional development, (b) acknowledging
students, (c) the teaching and learning process, (d) tracking and evaluating learning and development, (e) school-parent
and school-society relations, (f) curriculum and content knowledge. Knowledge of technology and integrating that into
the teaching and learning process is highly emphasized in this vision of a competent teacher. Some of the subcompetencies related to technology knowledge and its utilization emphasized for a professionally competent teacher
are given below:
A5.12. Technologically literate (has knowledge and skills of concepts and applications related to technology).
A5.13. Follows developments in information and communication technologies.
A6.2. Uses information and communication technologies in order to support his/her professional development and
increase his/her productivity.
A6.8. Utilizes information and communication technologies (on-line journals, software, e-mail, etc.) to share knowledge.
B2.3. Utilizes information and communication technologies to prepare suitable learning environments for students with
different experiences, characteristics, and talents.
C1.8. Gives place to ways of using information and communication technologies in lesson plans.
C3.8. Sets an example for effective usage of technological resources and teaches them.
C5.8. Takes various needs of students into consideration and utilize technologies to promote and support student-centered
strategies.
C7.8. Develops and uses strategies for behavior management in technology-rich learning environments.
D3.2. Analyzes data by using information and communication technologies.
46
T-Math Project
The Mathematics Educator
2006, Vol. 16, No. 2, 47–48
In Focus…
Teaching InterMath: An Instructor’s Success
Sarah Ledford
This paper discusses the InterMath courses from the perspective of an instructor. The instructor writing this
paper was teaching her sixth InterMath course in the same school system at the time this was written. This paper
describes a typical InterMath class and the success stories of many of the teachers participating in the courses.
The instructor also reflects upon her growth as a teacher during her experience with InterMath.
When I was a classroom teacher, I often found
myself thinking about how I could be a better teacher.
When I decided to pursue my doctorate, my thinking
shifted to how I could help others be better teachers. I
have always had an innate need to help others and I
found my vessel in teaching InterMath professional
development courses.
InterMath is a five-course series including Number
Sense, Algebra, Geometry, Data Analysis, and an
alignment course for the Georgia Performance
Standards (GPS). The first four courses allow middle
grades teachers to gain a mathematics concentration,
which is a recently added requirement for Georgia
middle school mathematics teachers. The GPS
alignment course was added to assist teachers in
dealing with the new state-mandated standards. In the
InterMath GPS course, teachers build their content
knowledge and gain a better understanding of the GPS
through exploration of problems.
Taking an InterMath course is a big commitment.
The participating teachers attend a weekly 4-hour class
after they have been teaching all day. Each of the
courses that I have instructed required 50 seat hours
from the teachers, which allows them to earn 5
Professional Learning Units. When I was teaching high
school, I am pretty sure that I would not have been as
committed as these teachers have been.
In the InterMath courses, teachers are allowed to
explore mathematics with technology. A typical
InterMath session involves the whole class looking at
the syllabus on-line, reading the topics to be covered in
the lesson, choosing a problem (or two or three) to
examine together, and then choosing a problem to
investigate and write up. We have been known to
spend over an hour on one problem!
Sarah Ledford is a doctoral student in the Mathematics Education
program at the University of Georgia. She has been a graduate
assistant on the InterMath project for the last four years. Her
dissertation topic is also directly related to the InterMath project.
Sarah Ledford
Three goals of the InterMath course are for
participants to learn mathematical content, to use
different technologies (spreadsheets, Geometer’s
Sketchpad (GSP) (Jackiw, 1993), graphing
calculators, etc.) to explore the mathematics, and to
think about their teaching and their students’ learning. I
have no doubt that the first two goals are always
accomplished. While all of the participants have a
varying degree of mathematical understanding and
technological savvy, they all learn something about
mathematics and technology. The third goal is a little
harder to recognize and poses a greater challenge for
the instructor. That is what I love about teaching
InterMath – getting the teachers to think about their
practice. To really think about it.
The abilities of my students in InterMath courses
have been as diverse as their experiences as teachers. I
have taught primary, elementary, middle, and high
school teachers. I have taught mathematics, science,
social studies, special education, and home economics
teachers. My participants have ranged from highly
confident in technology and/or mathematics to having
very low confidence in these areas. All of the
participants start and finish the course in different
places, but they all learn from each other and get to
think!
I know that I have had success getting the teachers
to think deeply about mathematics. Several teachers
have expressed that they did not know that there were
so many ways to solve the same problem and that no
single approach seemed to be better than others. Often
the approach depended on where the students were
mathematically when they were trying to solve the
problem. Teachers expressing this thought, including a
kindergarten teacher, were proud to learn so much
mathematics while never being made to feel inferior to
more proficient mathematics teachers.
I consider anyone who completes an InterMath
course to be successful because it is a large
commitment of time and effort. Of course, there have
been exceptional cases where teachers went beyond my
47
expectations for them in the course. For example, a
special education teacher often took technologies that
she mastered into her classroom. Her students often
relied on technology to communicate so she had access
to it in her classroom and wanted to use it as much as
possible with her students. Before the InterMath class,
she had never used Geometer’s Sketchpad or a
spreadsheet program to explore and solve a
mathematics problem. During that semester, she
allowed her students to use GSP to construct geometric
shapes and created a lesson for students to explore the
real-life situations of payroll and budget using a
spreadsheet.
For many participants, success meant mastering
mathematical content. Another special education
teacher, who is currently enrolled in her third
InterMath course, had to take content exams in order to
continue teaching the different subjects to her students.
She was most worried about the mathematics test but
was overcome with relief when she looked over the
test. She reported recognizing a lot of the mathematics
from our InterMath classes. She passed with a 92% and
was amazed at her inner mathematician. I would be
unaware of many of my students’ successes if they had
not decided to share them with me. I always get so
excited when I find a teacher who is thinking on the
next level – one who is thinking about students and
learning instead of only focusing on himself as the
teacher. It always makes me proud to think that I may
have contributed to that.
As the instructor, I also had a major success of my
own. Coming from a more traditional background, I
taught my first InterMath course in a more directed
manner. These classes had a different procedure than
the InterMath courses previously discussed. Once the
participants had read what topics would be covered
during the class, I went to the board and methodically
talked about each topic. I had a nagging suspicion that
this was not the way that InterMath was meant to be
taught. Sometime during my third course, I
experienced a paradigm shift. I realized that the content
was still addressed in the exploration of the problems
even if I did not try to pour it into the minds of the
participants beforehand. This idea of teaching was not
new to me, as I had read research, listened to
discussions, and thought about it often. However, I was
unable to simply pick up the teaching ideas and
implement them. I had to first re-work my philosophy
of what constitutes mathematics, how it should be
taught and how students learn. This kind of change in
48
thinking is a difficult one for many teachers, myself
included.
The biggest part of my paradigm shift probably
came from my newfound ability to say, “I don’t
know.” As a student, I had always thought that the
teacher was the knower of all things mathematical. As
a teacher, I had to realize that I do not always have the
answer and that there may not even be one. In my
classes, we explore the problems and learn together
from each other. Of course, there are many instances
where the participants in the class ask me for the
answer placing me in the position of the knower of all
things mathematical. Instead of telling them my
answer, I push them more to rely on each other and
themselves. I think that this realization has made me a
more honest instructor, which seems to be appreciated
by the participants in the classes.
My teaching philosophy has changed dramatically.
Now we discuss the mathematics as it comes up during
an exploration rather than me just trying to pass on
what I know about the mathematics to the participants.
We may not cover all of the pre-determined topics for
the night and we may cover other topics not included
in the syllabus, but the content inevitably gets covered
during the semester. We have richer discussions
because I have learned to embrace the fact that
everyone in the class has different abilities and
interests. Those who may think they know all of the
mathematics are sure to learn something from someone
who is a lot less confident. Those who lack
mathematical confidence are sure to learn from those
who have already made mathematical connections.
I have had success as an instructor in helping the
participants think about mathematics, students,
teaching practices, and even life in general. Also, of
major importance is the success that I have
experienced in my growth as a teacher. I, too, have
learned a lot from the InterMath courses and I consider
that a great success.
Author’s Note
The National Science Foundation under Grant No.
9876611 supported the professional development
reported here. The opinions expressed in this paper are
those of the author and do not necessarily reflect the
views of the NSF.
References
Jackiw, N. (1993). The Geometer's Sketchpad [Computer
software]. Berkeley, CA: Key Curriculum Press.
.
Teaching InterMath
The Mathematics Educator
2006, Vol. 16, No. 2, 49–50
In Focus…
The InterMath Experience: A Student’s Perspective
Laura Grimwade
In this paper I give a personal account of my experience as a student in an InterMath course. InterMath is a nontraditional course that focuses on mathematical content and technology. In the discussion I highlight particular
experiences that stood out for me as well as what I learned through the experience.
As a student taking math courses in college, I
wished for an opportunity to participate in classes that
would allow me to grow conceptually but would be
non-threatening to my GPA and eliminate the fear of
failing the course. I wanted a depth of knowledge but I
did not know where I could get it without facing the
apprehension of getting in over my head, or feeling
inadequate in front of other students. I experienced
similar feelings during professional development
workshops after becoming a teacher. Finally, along
came InterMath.
The timing was perfect. The area of mathematics
that I consider to be my weakest is Probability and
Statistics, which happened to be the InterMath course
offered. After reading through the course information, I
thought it sounded too good to be true. I looked up
InterMath on the Internet and read through the web
page. Again, it sounded too good to be true. In addition
to this course claiming to be exactly what I had always
hoped for, they were going to pay me for meeting the
course requirements.
During the day of the first class I wondered what I
had gotten myself into. I would work all day from 7:30
am to 4:30 pm and then voluntarily agree to sit in a
class from 4:30–8:30 one night a week for an entire
semester. However, that class went very well and was
over before I knew it. What most impressed me was
that I had learned a great bit about probability without
actually being taught in a traditional way. The teacher
did not stand up in front of the class, demonstrate and
lecture on how to do the work, and then assign a bunch
of problems for homework. There was not a huge,
thick textbook that I had to purchase. All of the
resources were web based and we used a lot of
technology. The entire course was very non-traditional
and extremely performance based.
Laura Grimwade is the K-12 mathematics coordinator in the
Rockdale County School System. She previously taught grade 7
mathematics for 9 years. She took the InterMath course in the fall
of 2004, and has been instrumental in her county continuing to
offer the InterMath professional development courses to their
teachers.
Laura Grimwade
As student participants, we sat together and
discussed our mathematical and teaching weaknesses.
We looked at problems together and then worked
through them as a collaborative effort. We posed
questions to our instructor, who acted as a facilitator to
help us work through our inquiries. When we asked
questions, we were directed with questions from the
instructor that prompted our thinking. She never gave
us answers but guided our thoughts so that we worked
through finding the answers together as a group. Our
thoughts and ideas generated more thinking and
working together allowed us to solve the problems and
develop our own conceptual knowledge. We were not
told formulas and procedures that we had to memorize.
Instead, we worked through the problems and
developed our own thought processes, which gave us
ownership of the problems. It became a part of us
through our experiences.
I have previous experience with professional
developments that attempted to implement standardsbased reform. With the new Georgia Performance
Standards (GPS), professional development of this sort
is more important now than ever because we want out
teachers to experience the teaching style that we would
like to see them implement in their own classrooms. I
immediately recognized how well InterMath supported
the GPS by modeling this style of teaching where
students experience learning by using concepts in a
range of situations and in complex problem solving,
representing concepts in multiple ways, and explaining
concepts to other students.
As a participant in the class I found that I was
gaining a deeper understanding by using mathematics,
representing mathematics, and explaining mathematics,
which provided evidence that performance-based
learning has a much deeper, conceptual meaning to
students as opposed to the traditional methods. I was
provided the opportunity to choose which problems I
wanted to solve. If there were problems that seemed to
overwhelm me, we worked those out together as a
class under the guidance of our facilitating instructor.
By offering her support and guidance, I gained
49
confidence in my weaker math areas and came to
recognize how well coordinating actions with others
assists the learning process.
InterMath provides the opportunity for teachers to
participate as students in learning important
mathematical
concepts
and
processes
with
understanding. Research shows that one of the most
important indicators of student achievement is teacher
quality (Hill & Ball, 2004). InterMath promotes
professional growth and development to strengthen
teacher content knowledge in mathematics and models
performance-based teaching and learning, supporting
the new Georgia Performance Standards (GPS). In
turn, taking an InterMath course should lead to student
achievement.
After completing this course I became very
interested in the next course being offered, a GPS
alignment course. While a conflict kept me from
enrolling in the GPS alignment course, I often sat in on
the course just for the fun of it. If I am able to fit in
other courses being offered, I plan to attend.
Otherwise, I will again sit in on as many of the classes
as possible to learn from the same instructor and offer
my assistance to all of the system teachers to whom I
have expressed this to be one of the best opportunities
50
they will ever have to take an in-depth, non-threatening
course that will provide them with professional growth
and development. I firmly believe InterMath will
provide teachers with a stronger understanding of math
and technology, model performance-based teaching,
and prepare the teachers to promote an increase in
student achievement by providing students with
opportunities to demonstrate their conceptual
understanding of math concepts that go beyond recall
because this ideology matches my experience in the
InterMath course.
Author’s Note
The opinions expressed in this paper are those of
the author and do not necessarily reflect the views of
the Rockdale County Public School System.
Reference
Hill, H., & Ball, D. (2004) Learning mathematics for teaching:
Results from California’s mathematics professional
development institutes. Journal for Research in Mathematics
Education, 35(5), 330–351.
.
The InterMath Experience
CONFERENCES 2006, 2007…
AMESA
Twelfth Annual National Congress
Polokwane, South
Africa
July 3–7, 2006
International Workshop on Research in Secondary and Tertiary Mathematics Education
http://www.mathed.baskent.edu.tr/
Ankara, Turkey
July 7–11, 2006
PME-30
International Group for the Psychology of Mathematics Education
Prague, Czech Republic
July 16–21,
2006
Seattle, WA
August 6–10,
2006
Chiang Mai, Thailand
September 17–
20, 2006
Rock Eagle, GA
October 19–21,
2006
Missoula, MT
October 26–28,
2006
Beirut, Lebanon
November 8–10,
2006
Mérida, Yucatán,
Mexico
November, 9–
12, 2006
New Orleans, LA
January 5–8,
2007
Irvine, CA
January 25–27,
2007
RCML
Research Council on Mathematics Learning
http://www.unlv.edu/RCML/conference2007.html
Cleveland, OH
March 1–3,
2007
NCSM
National Council of Supervisors of Mathematics
Atlanta, GA
March 19–21,
2007
Atlanta, GA
March 21–24,
2007
Chicago, IL
April 7–11,
2007
http://www.amesa.org.za/AMESA2006/
http://pme30.cz
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings/jsm/2006/
Thailand International Conference on 21st Century Information Technology in
Mathematics Education
http://www.cmru.ac.th/conference/page.php
GCTM
Georgia Council of Teachers of Mathematics Annual Conference
http://www.gctm.org/
SSMA
School Science and Mathematics Association
http://www.ssma.org
International Symposium: Policy and Practice in Mathematics and Science Teaching and
Learning in the Elementary Grades
http://www.aub.edu.lb/~websmec/
PME-NA
North American Chapter: International Group for the Psychology of Mathematics
Education
http://pmena.org
MAA-AMS
Joint Meeting of the Mathematical Association of America and the American
Mathematical Society
http://www.ams.org
AMTE
Association of Mathematics Teacher Educators
http://amte.net
http://www.ncsonline.org/
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
AERA
American Education Research Association
51
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53
In this Issue,
Guest Editorial… Project InterMath
JAMES W. WILSON
What Learner-Centered Professional Development Looks Like: The Pilot Studies of
the InterMath Professional Development Project
CHANDRA HAWLEY ORRILL & THE INTERMATH TEAM
Participants’ Focus in a Learner-Centered Technology-Rich Mathematics
Professional Development Program
DREW POLLY
An Analysis of the Use of Graphical Representation in Participants’ Solutions
LAUREL BLEICH, SARAH LEDFORD, CHANDRA ORRILL, & DREW POLLY
Professional Development Through Technology-Integrated Problem Solving: From
InterMath to T-Math
AYHAN KURSAT ERBAS, ERDINC CAKIROGLU, UTKUM AYDIN, &
SEMSETTIN BESER
In Focus… Teaching InterMath: An Instructor’s Success
SARAH LEDFORD
In Focus… The InterMath Experience: A Student’s Perspective
LAURA GRIMWADE