Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


VIDEO CLIPS: Internet access via modem may mean very long download times for video clips. If you are not on a fast line, you may want to read this paper without viewing the clips.

 
Authors' Biographies



Jon Basden    Susan Boone    Annie Fetter    Judith Koenig
Cynthia Lanius    Art Mabbott    John McKinstry    Ann Renninger
Roya Salehi    Susan Stein    Jody Underwood    Stephen Weimar



 Jon Basden       [Project Reflections]
School type Public middle school
Student population 99% White, 1% other, rural, mostly middle class
Grade level 7th
Subjects Math (4 Sections of Regular Math & 1 Pre-Algebra Section)
Student ability range We have full inclusion of special education students, but most of the top students are in Pre-Algebra, so the gap is pretty wide but not as wide as it could be.
Class sizes Regular Math: 24, Pre-Algebra: 32
Administrative support Better than most
Curricular reforms The Illinois State Goals and testing are starting to emphasize open-ended problem solving and writing in the math classroom. As a result, there is a large priority placed on those parts of the curriculum.
Technology used in class Computers (web, spreadsheets, & software like Sketchpad for demonstrations) & calculators (non-graphing)
No. years teaching experience 2 full years plus this one
Internet experience email, web page creation, teach web page design classes to district and area educators
Professional goals I would like to be a technology coordinator for a school district.


 Susan Boone       [Project Reflections]

School type Private, College Preparatory, All Girls Catholic (Dominican) High School
Student population 11% Asian or Pacific Islander
3% African-American
10% Hispanic
76% All others
Grade level 9-12
Subjects Algebra 1 and Algebra 2
Student ability range Students are admitted partly based on test scores, so most students are middle- to high-ability. In addition, students are grouped into honors and regular sections for mathematics. The Algebra 2 class in the video is middle- to high-ability.
Class sizes 20
Administrative support Very supportive.
Curricular reforms Support the NCTM Standards.
Texts: Algebra I & II: Integration, Application, and Connections; Glencoe
Technology used in class Computers (web, spreadsheets, & software like Sketchpad for demonstrations),
TI-83 Graphing Calculator required,
Laptop Program to be implemented 2000-2001 school year for all incoming freshwomen.
No. years teaching experience 14 years in urban middle schools, 8 years at Saint Agnes Academy
Internet experience Master Teacher for GirlTECH '96, '97, '99;
email, web page creation, teach web page design classes to area educators;
Technology Team at Saint Agnes Academy
e-mail
On-Line Algebra Lessons
Teach web page design classes to area educators;
Web-Site Team at Saint Agnes Academy
Professional goals Be the best teacher I can personally be.
Other mentions Susan has been recognized for the following:
  • On-Line Algebra Lessons
  • her work with girls in the book by Roberta Furger in Does Jane Compute.
  • participates in the Park City/Institute for Advanced Studies Mathematics Institute,
  • presented at SuperComputing '96
  • ENC's 1996 Hard Drive Cafe.
  • She was selected in '96 to be a Teacher Advisor at the AAUW Teacher Institute.
  • 1990-1991 Eleanor Roosevelt Fellow awarded by the American Association of University Women(AAUW)
  • 1986-1987 GTE GIFT (Growth Initiative for Teachers) Fellow. This fellowship is awarded to teams of one math andone science teacher to develop and implement an innovative integrated math and science curriculum.

  •  Annie Fetter
    Annie Fetter has a passion for math and education. She also has the good fortune to have a career that allows her to pursue that passion daily.

    After helping launch the Geometry Forum, the precursor to today's Math Forum, Annie started the Geometry Problem of the Week in 1993, which has, through the worldwide scope of the Internet, grown into a global classroom where students share ideas on geometry and its applications, develop analytical thinking skills, and learn to express themselves clearly in written form. It is believed to be the longest-running interactive student project on the Internet, and even predates the World Wide Web (yes, there was an Internet before the Web).

    In addition to handling the administrative tasks of the Problem of the Week, Annie often travels and visits schools, putting faces to the names of her Internet students, running workshops on technology in the classroom, guest-teaching, and just simply sharing her infectious energy about math and education with everyone she meets. She's also closely involved in the Forum's professional development efforts.

    After graduating from Swarthmore College in 1988 with degrees in math and music (though she earned more varsity letters than she did credits in either of those subjects) and a secondary math teaching certificate, Annie started down the trail of math education working with Professor Gene Klotz as part of the Visual Geometry Project, producing instructional workbooks and video tapes. At the same time, a college classmate of hers, Nick Jackiw, was developing software called the Geometer's Sketchpad, and Annie got interested in the project. Today she does teacher workshops on Sketchpad for Key Curriculum Press, which affords even more opportunity to travel to schools and generate enthusiasm for math and technology in education.

    She's an author of three workbooks published by Key Curriculum Press (The Stella Octangula, The Platonic Solids, and Three-Dimensional Symmetry), has written a number of Sketchpad resources, and once received an acknowledgement for technical assistance in a published physics paper for providing the three-dimensional coordinates of the vertices of a soccer ball.


     Judith Koenig       [Project Reflections]
    Judith Koenig teaches grade 8 in Boulder, Colorado. After many years teaching grade 7, she has moved ahead with her students. She says it is very interesting to teach the same students for two years, especially since she uses The Connected Mathematics Project and gets to see student progress and retention.

    Judy has almost 20 years experience, starting at the elementary level and moving to middle school in the mid 1980's. Before moving to Boulder in 1991, she taught for many years in Los Angeles. She has always taught in public schools. Students in the middle school where she teaches are mostly Caucasian, due to the fact that Boulder does not have a very diverse population. Classes have between 25 to 30 students and her entire school uses the same math program. This contributes to collegiality among peers and good administrative support for Connected Mathematics.

    The 8th grade classes have a choice of Regular, Honors, or Algebra I Honors. Both the Regular and Honors classes use Connected Mathematics but with different emphases and requirements. The Algebra class uses a more traditional text. Connected Mathematics is one of the NSF-funded programs designed to encourage reform in the middle grades. Students excel in this program and demonstrate a good understanding of concepts. The graphing calculator is incorporated in all classes, as are appropriate computer programs.

    Judith received the 1996 Presidential Award from Colorado for Secondary Mathematics and the Impact on Learning Award from the Boulder Valley School District. She supports the NCTM Standards and has been active in implementing them with teachers and students in Colorado.


      Cynthia Lanius       [Project Reflections]
    Before Cynthia Lanius came to Rice University in March 1998, she taught mathematics at Milby High School, the largest school in Houston ISD, for 8 years. She currently serves as Program Manager for the National Education Outreach and Training Partnership for Advanced Computational Infrastructure (EOT-PACI) and co-leads the EOT-PACI Access and Inclusion Team.

    In this position she co-directs GirlTECH and The Mathematics and Computational Sciences Awareness Workshop (MCSA). She serves as Director of Research and Evaluation for Rice University's Diversity Graduate Program for Science and Engineering.

    Ms. Lanius is a member of the Executive Committee, Rice University School Mathematics Project (RUSMP), National Council of Teachers of Mathematics (NCTM), the Texas Computer Education Association and the American Association of University Women (AAUW). She has received the following awards:

    Ms. Lanius has become a leader in publishing and using mathematics on the web. An index of her lessons can be found at http://math.rice.edu/~lanius/Lessons . She has given numerous presentations on related topics. Ms. Lanius is especially interested in increasing the participation of women and underrepresented minorities in the computational sciences. She has written essays entitled:


     Art Mabbott       [Project Reflections]
    Art Mabbott has been teaching kids mathematics at the secondary level for most of his nearly 30 years in the profession. While he has spent a great deal of his time working with high school aged students in both public and private schools, Art has made a couple of forays to work with younger and older students. During the last two years, he taught "severely gifted" elementary and middle school students in the Bellevue, Washington, Prism Program, in which younger students explore high school mathematics. He has also taught at the university level while completing his master's degree. However, Art finds that he always migrates back to the 9-12th grade classes.

    Art is currently teaching Geometry and Integrated Algebra/Geometry at Sammamish High School in Bellevue, Washington. His great love is geometry and he is working very hard to make certain that as the transition is made to an integrated curriculum, geometry is not forced into the role of the poor step-child to algebra.

    Art continues to explore new topics in mathematics at the University of Washington and is always looking for ways to bring them back into his classroom. He is a lead instructor at the summer Northwest Mathematics Interactions at the University of Washington, teaching teachers how to incorporate more hands-on experiences as part of real mathematics. NWMI is the northwest outreach of the Park City Mathematics Institute where Art assists in the classroom and the computer lab. Art also presents at local and national mathematics conferences.

    School type Public high school
    Student population Diverse -- mostly white with a large percentage of Latino (a lot are ESL) and African American and SouthEast Asian (VN, Thai, and Chinese - both from mainland and Taiwan also ESL). I have about six students who speak another language other than English at home.
    Grade level Mostly 10th and 11th graders
    Subjects Geometry (video) and Integrated Algebra/Geometry
    Student ability range I have a small number of mathematically talented students. But most of my kids are in the middle to lower range of mathematical ability. I have two students who are identified as "special needs" who get support (a study/learning skills class) from the building.
    Class sizes There are 24 kids in the class that was video taped.
    Administrative support The district has been very supportive in this project and "in stretching the envelope" in math curriculum. The superintendent and the math curriculum developer are encouraging us to be on the front line of the reform movement. The building has a more traditional outlook, and is having a harder time implementing and supporting the reform movement.
    Curricular reforms We have just moved to a 90-minute block period meeting daily and we now complete a year's work in 90 days of class. As a district, we are changing to an integrated mathematics curriculum using the Core Plus Mathematics Project.
    Technology used in class I have access to the web in my classroom, in a Mac lab, and at home. I encourage my students to do the geometry Problem of the Week from the Math Forum. We use my computers and graphing calculators as tools whenever necessary and appropriate - usually daily.
    No. years teaching experience Almost 30 years
    Professional goals Always looking for ways to bring new topics in mathematics back into my classroom.


     John McKinstry      [Project Reflections]
    School type a Quaker middle school
    Student population mostly white; upper, upper-middle and middle class
    Grade level 7th and 8th
    Subjects 7th and 8th grade math
    Student ability range The range of student abilities in the school is fairly narrow, since all are fairly strong, and in my class the range is even narrower, with their abilities very strong.
    Class sizes My class size is 9 in my 7th grade class, and 12 in my 8th grade class.
    Administrative support My principal (i.e. the head of the middle school) is outstanding and incredibly supportive, as is the Head of the School.
    Curricular reforms We are presently re-examining the entire curriculum (all subject areas) to see how to make our approach more comprehensive (id est so that subjects can tie into each other) so that our curriculum is more experiential and more service oriented. We are just beginning this process.
    Technology used in class The technology I use in my classroom includes the Geometer's sketchpad, plus the Math Forum's Middle School Problem of the Week. I also encourage my class to use Ask Dr. Math.
    No. years teaching experience 7 (1 year at 1st grade, 2 years at 4th grade, 2 years at 5th grade, and 2 years teaching 7th and 8th.)
    Internet experience My class's internet experience is use of the MF POW.



     Ann Renninger

    K. Ann Renninger is a professor of education at Swarthmore College and director of research for the Math Forum. She also worked as a consultant and evaluator on projects that preceded the development of the Math Forum, including the Visual Geometry Project (origin of Geometer's Sketchpad) and the Geometry Forum. She has been a National Academy of Education Spencer Fellow and an Educational Testing Service Postdoctoral Fellow. Her research focuses on the role of interest in students' learning and development, and the implications of research on student learning for classroom practice and project design. A former classroom teacher, she received her Ph.D. in Education and Child Development from Bryn Mawr College. Her contributions to the present paper include consultation on the design and development for the BRAP project, research liaison, and paper revision and development.

     Roya Salehi      [Project Reflections]

    Before Roya Salehi joined the Math Forum team in 1998, she taught mathematics to middle and high school students for five years at Muslim Community School, a K-10 private school in Potomac, Maryland. She facilitated communication between the administration and the upper grades' teachers; she also served as the math and computer curriculum advisor and resource person for the school. Before MCS, she taught computer classes at the Islamic School of Seattle, a private elementary school in Seattle, Washington, for three years.

    Prior to her professional career as a teacher, Roya worked at Lawrence Berkeley National Laboratory, doing research on molecular models using computer simulation. She received her undergraduate degree from the University of California at Berkeley in pure mathematics, minoring in computer science.

    Roya is interested in identifying effective ways of implementing on-line teacher professional development and the generative facilitation of this process. She is also interested in the use of the Internet in the classroom to enhance student mathematical thinking. She is pursuing her interests in math education and artificial intelligence, particularly machine learning, at George Mason University.

    Currently, Roya is leading the Bridging Research and Practice (BRAP) project and Teacher2Teacher at the Math Forum.


     Susan Stein       [Project Reflections]

    School type independent Quaker school
    Student population mostly white; upper middle- and upper-SES.
    Grade level I teach seventh grade and one section of eighth grade.
    Subjects Math 7 and Eighth Grade Algebra 1 (This was the class video taped.)
    Student ability range Students are admitted partly based on test scores, so most students are above the 50th percentile. In addition, students are grouped into advanced and regular sections for mathematics. The Algebra 1 class was a "regular" section.
    Class sizes 20 students
    Administrative support The administration has been consistently interested in learning more about mathematics education and how it has changed in the last 10 or 15 years. As most of them are former History or English teachers, they are particularly excited when they recognize that mathematics education is becoming more interested in developing a student's ability to make sense of mathematical ideas and connections between ideas by explaining their reasoning orally and in writing. What seems to constrain us is that a need to communicate to our wider community how changes in curriculum and pedagogy will still provide our students with superior skills and knowledge of procedures in addition to any problem-solving and higher order reasoning skills they may acquire.
    Curricular reforms I use CMP (Connected Math Project) in seventh grade & we're using a combination of CMP, CORE PLUS and a traditional text (McDougal Littell) in the Algebra 1 course. We may be moving to greater use of CORE PLUS or CMP next year, but those decisions have not been made yet.
    Technology used in class TI 83 calculators are introduced in 8th grade. There is just about no computer use in class. Students are expected to work on several Math Forum PoWs per year as out-of-class assignments.
    No. years teaching
    experience
    I began teaching in 1968, and teaching mathematics in 1984. (Before that I taught English and Social Studies, grades 9 - 12). I've taught Middle & Upper School math (one year) and I was a Lower School math coordinator for two years. I am currently Mathematics Department Head at Wilmington Friends.
    Internet experience I use email and have been known to use the Internet for information searches, but I don't like to surf. I get lost in and frustrated by its non-linearity.
    Professional goals I would like to help the wider community - not just the school community, but others -- understand that demanding that our students make sense of mathematical ideas does not replace the basics as we were taught them when we were children. Rather, it expands those basics to include more than facts and procedures, to include an ability to solve complex problems based in real contexts, make connections between mathematical ideas, and communicate mathematical reasoning using words, symbols and other mathematical models.

    On a personal-professional level, I always want to improve my teaching. One goal I have for my classes is to find ways to support my students as they make a bridge from mathematics as the discipline of the "give me the rule and I'll give you the right answer," to the "problem-solving, reasoning, and explaining how I make sense" course. In addition, I would love to be able to work with pre-service teachers more than I do now.


     Jody Underwood      [Project Reflections]

    Jody Underwood joined the Math Forum in February, 1999, where she is an Evaluator and Project Developer. Prior to coming to the Forum, Jody did her post-doctoral work at Vanderbilt University's Learning Technology Center. Her main focus there was working with middle school teachers to figure out how to use technology for social studies and reading as part of the LTC's Schools For Thought program. She did her Ph.D. in Science and Technology Education at the Technion-Israel Institute of Technology, and has an M.S. and B.S. in computer science. Many of Jody's computer programming years were spent developing software for education applications.

    Jody is interested in helping people learn in general, and specifically in using technology in appropriate ways to facilitate learning. In particular, she would like to determine appropriate ways to use technology in the classroom, and to disseminate that information to teachers. This interest extends to professional development and helping teachers become comfortable using the Internet and other technologies. She is addressing these issues on both research and development projects at the Math Forum.


     Stephen Weimar      [Project Reflections]

    Steve was brought on board in 1993 by Gene Klotz, Principal Investigator of what was then the Geometry Forum, to help figure out how to make the Internet useful for mathematics teachers and students. He worked with Annie Fetter and others to bring together a group of leading Internet-savvy educators for a summer institute to design promising projects. This led to many of the Math Forum services available today. Steve has been involved ever since and in 1996 he joined Gene as Co-Director of the Math Forum. Currently he straddles his jobs as President of mathforum.org, through which he attempts to give the Math Forum a life after NSF funding, and director of project development and research.

    Steve began his professional work in the high school and middle school math classroom, from which he migrated in 1983 to join a group of educators establishing Educators for Social Responsibility, a national organization, known best for its work in conflict resolution. He then supervised student teachers for Swarthmore College and served as a consultant to schools, supporting professional development programs. At Swarthmore College, he coordinated a variety of mathematics and technology programs including the Kids Network Teacher Education Pilot Project at Swarthmore College and Diverse Perspectives in Education, a telecommunication simulation on issues of diversity in the classroom, linking student teachers at Swarthmore College and the University of Michigan.

    Steve is committed to finding ways for the Internet to support learning environments for teachers that enable us to learn by doing, just as we seek to do with our students -- practicing our strategies, exploring math problems, and building a trusted community together.

    Steve is an author along with Ann Renninger and Gene Klotz of "Teachers And Students Investigating And Communicating About Geometry: The Math Forum," a chapter in Designing Learning Environments for Developing Understanding of Geometry and Space, Richard Lehrer and Daniel Chazan, editors, Mahwah, NJ: Lawrence Erlbaum Associates, 1998.


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