HE MATHEMATICS EDUCATORmath.coe.uga.edu/tme/issues/v16n1/tmev16n1_final.pdf · mathematics preparation of in-service middle school mathematics teachers, and 2) incorporating the use

____ THE ______MATHEMATICS___

_________EDUCATOR _____ Volume 16 Number 1

Spring 2006 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff Editor Ginger Rhodes

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

A Note from the Editor 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. Ginger A. Rhodes 105 Aderhold Hall [email protected] The University of Georgia www.coe.uga.edu/tme Athens, GA 30602-7124

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

2 Project InterMath

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

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 web-enhanced 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 in-service 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

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.

James W. Wilson 3

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 pre-proposal 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

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.


4 What Learner-Centered Professional Development Looks Like

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

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 content-focused 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).

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).

Chandra Hawley Orrill 5

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

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


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.

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


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 over-reliance 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

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


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

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 pre-algebra 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


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

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.


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

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


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 open-ended 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

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


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.

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


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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What don't we know about teacher learning — but would make good working hypotheses? Paper presented at the Teacher Enhancement in Mathematics K–6, Arlington, VA.

Brown, S., Erbas, A. K., Glazer, E., Orrill, C. H., & Umberger, S. (2001). Learner-Centered Professional Development Environments in Mathematics: The InterMath Experience. In Simonson, M. (Ed.) 2001 annual proceedings: Selected research and development papers presented at the national convention of the Association for Educational Communications and Technology (AECT) (pp. 329–335). Bloomington, IN: AECT.

Cochrane, D. (1999). A wake-up call for U.S. educators: The Third International Mathematics and Science Study. Policy Forum, 2(1), 1, 6.

Coffey, A. & Atkinson, P. (1996). Making sense of qualitative data: Complementary research strategies. Thousand Oaks, CA: Sage Publications.

Cohen, D., & Ball, D. (1990). Policy and practice: An overview. Educational Evaluation and Policy Analysis, 12(3), 347–353.

Erbas, A. K., Umberger, S., Glazer, E. M., & Orrill, C. H. (2002). InterMath: Technology-enhanced, learner-centered professional development. In Mewborn, D. S., Sztajn, P., White, D. Y., Wiegel, H. G., Bryant, R. L., Nooney, K. (Eds.), Proceedings of the 24th annual meeting of the North American Chapter of the international group for the Psychology of Mathematics Education (pp. 1608–1612). Columbus, OH: ERIC Clearinghouse on Science, mathematics, and Environmental Education.

Hawley, W. D., & Valli, L. (1999). The essentials of effective professional development: A new consensus. In L. Darling-Hammond & G. Sykes (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 127–150). San Francisco: Jossey-Bass.

Jackiw, N. (1990). The Geometer's Sketchpad [Computer software]. Berkeley: Key Curriculum Press.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping students learn mathematics. Washington, DC: National Academy Press.

Krajcik, J. S., Blumenfeld, P. C., Marx, R. W., & Soloway, E. (1994). A collaborative model for helping middle grade science teachers learn project-based instruction. The Elementary School Journal, 94(5), 483–497.

National Council on Mathematics and Science Teaching. (2000). Before it's too late: A report to the nation from The National Commission on Mathematics and Science Teaching for the 21st century. Jessup, MD: National Commission on Mathematics and Science Teaching.

National Partnership for Education and Accountability in Teaching. (2000). Revisioning professional development: What learner-centered professional development looks like. Oxford, OH: National Assessment of Educational Progress.

National Commission on Teaching and America's Future. (1996). What matters most: Teaching for America's future. New York: Teachers College.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Renyi, J. (1996). Teachers take charge of their learning: Transforming professional development for student success. Washington, DC: National Foundation for the Improvement of Education.

Southern Regional Education Board. (1998). Education's weak link: Student performance in the middle grades. Atlanta: Author.

Sparks, D., & Hirsch, S. (1999). A national plan for improving professional development. Oxford, OH: National Staff Development Council.

U.S. Department of Educational Statistics - OERI. (2001). Mathematics highlights 2000: The nation's report card. Washington, DC: U.S. Department of Education.


14 Participants’ Focus in a Professional Development Program

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 learner-centered 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 (Loucks-Horsley 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).

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, hands-on 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

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.

Drew Polly 15

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.

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 post-hoc case studies of three InterMath participants. This study was driven by the following questions:

What do the participants report learning during an InterMath course?

What participant characteristics influence what they report learning?

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.

Table 1

Description of Participants Name InterMath

Course Position during course

Mathematics background

Teaching Experience

Kendra Number Sense

5th Grade teacher assistant

No college mathematics

courses

1st Year

Lauren Algebra and

Number Sense

Career Exploration

teacher

Numerous college

mathematics courses

5th Year

Sheila Algebra Middle grades

mathematics teacher

A few college

mathematics courses

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 Interviews and open-ended survey data were used

in this study.

Pre-Course Survey 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.


Interviews Participants were interviewed twice using a semi-

structured 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 Content Knowledge (MCK)

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 Content Knowledge (TCK)

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 to make graphs from a table in Microsoft Excel.”

Pedagogical Content Knowledge 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.”

Drew Polly 17

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

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 pre-course 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.


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,

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.

Drew Polly 19

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.”

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


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.

Implications for Research 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.,

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 learner-centered 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.

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22 An Analysis of Using Graphical Representations

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

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.

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.

Laurel Bleich, Sarah Ledford, Chandra Hawley Orrill & Drew Polly 23

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 reform-oriented 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

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


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.

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 Mathematics (CM)

Incorrect Mathematics (~CM)

Correct Answer (CA) 170 22

Incorrect Answer (~CA) 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.


Table 2 Descriptions of Representation Categories and Coding Strategies.

Type of Graphical Representation

Definition (Adapted from Monk, 2003)

Coding

Communicate Graphic is used to convey a meaning, or express one’s ideas

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”

Make sense Graphic is used in the understanding of the problem or in the process of finding a solution

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

Cosmetically enhanced Graphic is neither appropriate nor relevant to the investigation or solution

Randomly placed; graphic without explanation or reference; graphic not appropriate/relevant; used for organization purposes only; used as filler

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 write-ups 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

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!IHF+m!HFI+m!FIH = 180°

m!HFA = 63°

m!AHF = 81°

m!FAH = 36° Line j is parallel to LIne qm

q

543

12

A

H

F

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.

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 write-up, 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).

Investigation: …How does the combined area of all of the shaded circles relate to the area of the entire circle?

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:

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

Small Circles Comb. Area

Small/Lrg Circle Ratio

3.14 0.78 2 1.56 0.496815287 3.14 0.35 3 1.05 0.334394904 3.14 0.2 4 0.8 0.25477707 3.14 0.13 5 0.65 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.

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 write-up depicts a cosmetically enhanced write-up (see Appendix C for the entire write-up).

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.


G

F

E

H

B

A D

C

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.

G

F

E

H

DC

BA

The image above seems to show that if the original quadrilateral is a trapezoid, the smaller quadrilateral will be a rectangle.

G

F

E

H

B

A D

C

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

sense of the mathematics or as a means to communicate a point, the write-up was coded as cosmetically enhanced.

Findings and Discussion 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.

Table 3

Distribution of Types of Incorrect Write-Ups and Uses of Graphical Representations

Communication Making Sense

Cosmetically Enhanced

Total

Correct Mathematics Incorrect Solution

6 8 9 23

Incorrect Mathematics Correct Solution

6 7 6 19

Incorrect Mathematics Incorrect Solution

2 7 11 20

Total 14 22 26 62

Communication 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


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” write-ups, 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 write-ups provide evidence that participants without a sense

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


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.

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: Jossey-Bass.

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 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.


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!IHF+m!HFI+m!FIH = 180°

m!HFA = 63°

m!AHF = 81°

m!FAH = 36° Line j is parallel to LIne qm

q

543

12

A

H

F

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.


Click here for interactive sketch (shown below).

m!EBA+m!AEB+m!EAB( )+ m!ECB+m!CEB+m!CBE( ) = 360.00°

m!EBA+m!AEB+m!EAB = 180.00°

m!ECB+m!CEB+m!CBE = 180.00°

m!EBA = 45°

m!AEB = 45°

m!EAB = 90°

m!ECB = 90°

m!CEB = 45°

m!CBE = 45°

E C

BA

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 Sum of Angles Vertex Angles 3 180 60 4 360 90 5 540 108 6 720 120 7 900 128.5714286 8 1080 135 9 1260 140 10 1440 144 11 1620 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.


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 Small Circles Comb. Area

Small/Lrg Circle Ratio

3.14 0.78 2 1.56 0.496815287 3.14 0.35 3 1.05 0.334394904 3.14 0.2 4 0.8 0.25477707 3.14 0.13 5 0.65 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 Small Circle

Circumference # of Small Circles Small Circles Total

Circum. Small/Lrg Circle

Ratio 6.28 3.14 2 6.28 1 6.28 2.1 3 6.3 1.003184 6.28 1.57 4 6.28 1 6.28 1.26 5 6.3 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.


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.

G

F

E

H

B

A D

C

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.

G

F

E

H

DC

BA

The image above seems to show that if the original quadrilateral is a trapezoid, the smaller quadrilateral will be a

rectangle.

G

F

E

H

B

A D

C

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.


Area D F E( )+ Area FCH( )+ Area EGB( )+ Area GAH( ) = 19.91 cm2

Area GAH = 4.07 cm2

Area EGB = 4.11 cm2

Area FCH = 5.84 cm2

Area DFE = 5.88 cm2

Area FEGH = 19.91 cm2

H

G

F

E

AB

CD

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.


Ayhan Kursat Erbas, Erdinc Cakiroglu, Utkun Aydin & Semsettin Beser 35

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)

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

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 [email protected]. 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 [email protected] 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 [email protected]. 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 [email protected].

36 T-Math Project

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).

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).


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

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 technology-based 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. T-Math 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

38 T-Math Project

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

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

EXPERIENCE in problem solving through technology

REFLECTION upon the personal and

collaborative experience

COLLABORATION for a new culture of

mathematics in schools

TECHNOLOGY


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 open-

ended 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?)

a. What are the dimensions of an acceptable box-shaped 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 box-shaped 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 C) 24

D) 16 E) 8

This problem was adapted into an open-ended problem as “You are making a rectangular flower

40 T-Math Project

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. T-Math 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

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. T-Math 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


(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 5. Ivory Carving Belt Piece in Topkapi Museum in Istanbul, Dating from 1500s

Figure 6. Panel from the Muradiye at Bursa, Dating from 1426

Figure 7. Tile from an Arched Panel in Iznik, Dating from Mid 16th Century

42 T-Math Project

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

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.


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 in-service 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 in-service 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

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 open-ended 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 in-service 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 T-Math 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

44 T-Math Project

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 face-to-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.

Author’s Note 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: [email protected]

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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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 sub-competencies 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.


Sarah Ledford 47

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!

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

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.

48 Teaching InterMath

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

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. .


Laura Grimwade 49

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 non-traditional 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.

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 standards-based 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

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.

50 The InterMath Experience

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

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.

.

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CONFERENCES 2006, 2007…

AMESA Twelfth Annual National Congress http://www.amesa.org.za/AMESA2006/

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 http://pme30.cz

Prague, Czech Republic July 16–21, 2006

JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings/jsm/2006/

Seattle, WA August 6–10, 2006

Thailand International Conference on 21st Century Information Technology in Mathematics Education http://www.cmru.ac.th/conference/page.php

Chiang Mai, Thailand September 17–20, 2006

GCTM Georgia Council of Teachers of Mathematics Annual Conference http://www.gctm.org/

Rock Eagle, GA October 19–21, 2006

SSMA School Science and Mathematics Association http://www.ssma.org

Missoula, MT October 26–28, 2006

International Symposium: Policy and Practice in Mathematics and Science Teaching and Learning in the Elementary Grades http://www.aub.edu.lb/~websmec/

Beirut, Lebanon November 8–10, 2006

PME-NA North American Chapter: International Group for the Psychology of Mathematics Education http://pmena.org

Mérida, Yucatán, Mexico

November, 9–12, 2006

MAA-AMS Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org

New Orleans, LA January 5–8, 2007

AMTE Association of Mathematics Teacher Educators http://amte.net

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 http://www.ncsonline.org/

Atlanta, GA March 19–21, 2007

NCTM National Council of Teachers of Mathematics http://www.nctm.org

Atlanta, GA March 21–24, 2007

AERA American Education Research Association

Chicago, IL April 7–11, 2007

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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

Documents

HE MATHEMATICS EDUCATORmath.coe.uga.edu/tme/issues/v16n1/tmev16n1_final.pdf · mathematics preparation of in-service middle school mathematics teachers, and 2) incorporating the use