v
Introducing Bernie WhiteOn July 2, Bernie H. White became the new Senior Program Officer for the ExxonMobil Foundation. He follows Joe Gonzales who is pursuing a new opportunity in Houston as Staff Public Affairs Representative at the corporation’s Upstream Public Affairs, Operations, Contributions and Community Relations Office. (Please see Intersection, May 2001.)
Prior to taking his new position, Bernie was the vice president of ExxonMobil Catalyst Services, Inc. in Houston. Hired by Exxon in 1997 in Baytown, TX, Bernie has held a variety of technical, operations, and managerial assignments in Exxon Research and Engineering and in Exxon Company USA.
Bernie holds undergraduate degrees in physics, mathematics, and chemical engineering. His Ph.D. is in fast-reaction (nanosecond) kinetics—a specialty within chemical engineering and physical chemistry. In the course of working on his doctorate, Bernie taught a variety of undergraduate classes in thermodynamics, physical chemistry, reactor design, transport phenomena, and differential equations.
Welcome, Bernie! Ed.
Temperatures in the nineties notwithstanding, fall will be here soon. With fall comes the Annual Meeting of the ExxonMobil Foundation/NCTM K-5 Mathematics Specialist Program. Celebrating this year the thirteenth such gathering, representatives from projects supported by the Foundation will convene on the evening of Thursday, September 20 and adjourn at noon on Sunday, September 23. The conference will be held at ExxonMobil’s headquarters in Irving, Texas, a city between Dallas and Fort Worth.
Each year the meeting presents unparalleled opportunities for participants to share and compare experiences and to learn from presenters and colleagues in both formal and informal settings. At its best, the conference truly becomes a "harvest of ideas." This year, as in the past, the conference planning committee has put together an agenda that offers attendees the chance to hear presentations by recognized practitioners in the field; attend meaningful, interactive sessions; network with others from across the country; and enjoy the hospitality extended by the Foundation.
The planning committee is delighted to have as presenters the individuals introduced below. Each one has something to share from a unique perspective. In addition, the committee is honored to announce that NCTM president-elect Johnny Lott and NCTM Board Member Cindy Chapman will be coming to the meeting. While Johnny’s schedule will not allow him to attend the entire conference, Cindy will stay for the duration.
Another highlight is that on Friday afternoon, attendees will have an opportunity to hear a presentation by ExxonMobil Foundation President Ed Ahnert. An informal "question and answer" session will follow. Ed will also address guests and welcome keynote speaker Jeremy Kilpatrick at the traditional banquet on Saturday night.
Members of the planning committee are Bill Fisher, Chico, CA: Vandi Hodges, Ashland, VA; Donna Little-Kaumo, Albuquerque, NM; Charlotte Stadler, New Rochelle, NY; and Bonnie Tank, San Francisco, CA. Many thanks to committee members for planning what promised to be a stellar conference!
The individuals introduced below will be leading sessions at this year’s conference. In this column, readers will become better acquainted with them through the biographical sketches that they have graciously provided. Readers will also learn a little here about the topics of their respective sessions.
Information is presented according to the schedule of presentations over the course of the conference.
Thanks to all who were able to forward information and photos for this column. Ed.
While the conference will not "officially" begin until the welcome buffet on Thursday evening, some participants will arrive a little early to attend a three-hour presession on the issue of project evaluation. Rose-Ann McKernan, director of research and evaluation for the Albuquerque Public Schools, NM, will lead that workshop (please see Intersection, June 2001).
Susan Jo Russell, Traci Higgins and Cornelia Tierney from the Education Research Collaborative at TERC, Cambridge, MA, will kick off the annual meeting with a session entitled, "Thinking About Algebra." Susan Jo provided these remarks: "During the session, we’ll be exploring some of the mathematics of algebra ourselves, then thinking about how these ideas relate to the ‘story of algebra’ for the elementary grades."
Susan Jo is a principal scientist at the Education Research Collaborative at TERC. After ten years of classroom teaching and staff development in elementary schools, Dr. Russell became involved in research and development—first in the Laboratory for Computer Science at M.I.T., then at TERC, a nonprofit group that works to improve mathematics and science education. She directed the development of the NSF-funded K–5 curriculum, Investigations in Number, Data, and Space, and co-directs, with Deborah Schifter and Virginia Bastable, a project that is creating professional development materials for elementary grade teachers, Developing Mathematical Ideas. She was a member of the grades 3–5 writing group for the NCTM’s (2000) Principles and Standards for School Mathematics. Her current work focuses on issues of professional development and leadership development in elementary mathematics and on understanding how practicing teachers can learn more about mathematics and about children’s mathematical thinking.
In addition to co-authoring Investigations and DMI materials, Susan Jo’s recent publications include: "Developing Computational Fluency with Whole Numbers," Teaching Children Mathematics, NCTM, November 2000; "Mathematical Reasoning in the Elementary Grades," in Lee V. Stiff and Frances R. Curcio (Eds.), Developing Mathematical Reasoning in Grades K-12, 1999 Yearbook, (NCTM, 1999); Relearning to Teach Arithmetic: A Professional Development Package, (Dale Seymour Publications, 1999, co-authored with David A. Smith, Judy Storeygard and Megan Murray).
Also at TERC, Traci is a senior research and development specialist. While completing a doctorate in cognitive psychology, she became involved in research on conceptual change in science education and subsequently, the development of mathematical thinking in children. As a postdoctoral fellow, she studied the impact of the Number Worlds curriculum on both students and teachers in K-2 classrooms. She has co-authored several research review essays on children’s understanding of data with Cliff Konold. Her current work focuses on the evaluation and analysis of school and district-wide elementary mathematics reform efforts. She is also studying algebraic thinking in grades K–8 and collaborating on the development of a conceptual review paper that will highlight student thinking around some of the big ideas central to this domain.
Cornelia is a senior scientist at ERC at TERC. Before coming to TERC, she taught mathematics in grades 5–9 and to pre-service elementary and middle school teachers. Her interest focuses on working with teachers to better understand students’ mathematical thinking and to engage students in their own learning. She was a principle author of the NSF-funded K–5 math curriculum, Investigations in Number, Data, and Space and wrote units on the mathematics of change based on research with Ricardo Nemirovsky. Working with Sheila Sconiers, she leads TERC’s participation in the development of Bridges to Classroom Mathematics staff development materials in mathematics for elementary school teachers who are implementing NSF-funded curricula. She and Judy Storeygard, also at TERC, are starting a new NSF-funded project, Accessible Math, to work with teachers to find ways to include students with disabilities in learning mathematics with NCTM Standards-based mathematics curriculum.
Among Cornelia’s recent publications are: "Children Creating Ways to Represent Changing Situations," Educational Studies in Mathematics, (in press, co-authored with Ricardo Nemirovsky); Ten-Minute Math, (Dale Seymour Publications, 2001, co-authored with Susan Jo Russell); Investigations in Number, Data, and Space, (Scott Foresman, 1998, co-authored with Susan Jo Russell, Karen Economopoulos and Jan Mokros).
"Designing Professional Development to Invite and Sustain Diversity among Teachers" is the subject for Friday afternoon’s session. It will be led by Carne Barnett-Clarke, Alma Ramirez and Henry Phillips. Carne and Alma are from WestEd in Oakland, CA and Henry is a fourth-grade teacher in Stockton, CA.
Carne forwarded these notes: "Case discussions offer special opportunities for teacher leaders and their peers to have sincere and provocative conversations about mathematics teaching and learning. Yet, in the beginning, many voices were missing from those conversations.
"In this interactive session, we will discuss how we continually adjusted the design of the Mathematics Case Methods Project to meet our commitment to diversity. We will share strategies for recruiting and mentoring a diverse group of teacher leaders and openly discuss what worked and what didn’t and speculate why. Participants will experience part of a case discussion, see a videotape of a case discussion among teachers, and discuss critical issues for broadening the participation of teachers. Although this session is oriented toward the process of case discussions, the principles that we share for facilitating discussions and making them more inclusive may be adapted to fit different professional development experiences."
Carne
Barnett-ClarkeDr. Carne Barnett-Clarke is a Senior Research Associate at WestEd in Oakland, California, where she directs the Mathematics Case Methods Project. Teachers in this professional development project discuss cases about mathematics teaching dilemmas. Dr. Barnett’s own teaching experiences led to her interest in this pioneering work, which is patterned after methods used in other professions such as business and health. She has written numerous books for teachers and students and has published in research journals. She was formerly a teacher educator at the University of California, Berkeley and has also conducted professional development across the United States and internationally.
Recent publications by Carne include: "Case Design and Use: Opportunities and Limitations," Research in Science Education, 2001; "Cases," Journal of Staff Development, 1999, 20(3), 26-27; "Case Methods and Teacher Change: Shifting Authority to Build Autonomy," in M. Lundeberg, B. Levin and H. Harrington (Eds.), Who Learns What from Cases: The Research Base for Teaching with Cases (Lawrence Erlbaum Associates, 1999, co-authored with P. Tyson); "Mathematics Teaching Cases as a Catalyst for Informed Strategic Inquiry," Teaching and Teacher Education, 1998, 14, 81-93, co-authored with P. Tyson.
Alma Ramirez Alma coordinates the Mathematics Case Methods Project under the direction of Dr. Carne Barnett-Clarke at WestEd. She works with K–8 teachers using case discussions as a catalyst for enhancing pedagogical content knowledge, co-develops case-like materials for students at grades K–8, and focuses on increasing diversity in the leadership of mathematics education. Ms. Ramirez was an elementary school teacher for nine years, and served as a consultant and teacher for a variety of mathematics projects, including EQUALS, Family Math, and the Head Start Math Readiness Project.
Ms. Ramirez has co-edited a book for teachers of K–3 students focusing on number sense, place value, and operations (in press); co-authored a chapter on cases as professional development tools; and is co-authoring a chapter on increasing diversity in mathematics professional development for the National Institute of Science Education (in press). She is a K–6 mathematics author for Scott Foresman Publishing Company. She is also mom to a 5-year old and a 3-year old.
Currently teaching fourth grade at Taylor Elementary School in the Stockton Unified School District, CA, Henry has been an elementary teacher for the past five years. He has also been a teacher and an aide in alternative education for over eight years. For the last three years, Henry has been involved with the Mathematics Case Methods Project. Over the time that he has been involved, Henry has participated in a variety of roles.
Saturday morning will begin bright and early with the session, "Professional Development of Math Coaches," presented by Amy Morse from the Education Development Center in Boston, MA.
Amy provided the following information: "The focus of the session will be to explore both the role of the math coach and the ways coaches are coming to identify and analyze the components of their work. We will ground our discussion in an examination of several cases written by elementary school math coaches. In one case, a coach re-examines her own understanding of division after a perplexing discussion during a classroom visit. In another case, a coach sifts through the elements of building relationships. She explores the relational nature of her work and what these, sometimes fragile, partnerships afford. We will have an opportunity to discuss the writing of cases as a way of examining the intricacies of the coach/support/specialist role and how writing and analyzing cases might enrich the professional development of coaches. In addition, we will reflect on the complex confidentiality issues inherent in this type of work." Liz Sweeney and Carol Walker will be facilitating small group discussions afterwards.
Amy MorseAmy Morse is a Senior Research Associate in the Center for the Development of Teaching at the Education Development Center. She works primarily as a consultant to the Elementary Math Department of the Boston Public Schools. Amy’s work involves both supporting the implementation of a citywide restructuring of math education in the elementary schools and facilitating professional development for the cadre of K–5 math coaches. Amy is also collaborating with several coaches to design seminars in mathematical thinking (based on the Developing Mathematical Ideas seminars) for parents in the Boston community.
Cathy Fosnot and Maarten Dolk have been collaborating for over ten years on the designing, implementing and researching of inservice projects in mathematics education. Important aspects of their work include the role of context in learning and the role of the teacher in facilitating math workshops. A focus has been helping teachers understand the development of children’s mathematical ideas, strategies and models (the landscape of learning). Drawing from the work of the Dutch mathematician, Hans Freudenthal, they define mathematics as "mathematizing"—the activity of structuring, modeling, and interpreting one’s "lived world" mathematically. In their writing and work with teachers, they move beyond the current debate on the teaching of algorithms to argue for deep number sense and the development of a repertoire of strategies based on landmark numbers and operations.
Cathy wrote: "We approach inservice from a constructivist view of learning. We do not tell teachers what to see in children’s work, or what to do. We do not attempt to transmit theory. Instead we engage participants in investigating children’s work, in lesson study, and in designing contexts. We construct with them landscapes of learning, and we encourage them to think deeply about their practice. In our institutes and workshops, participants learn to see themselves as mathematicians, to work on the edge, to inquire.
"Funded by NSF, we are currently engaged in the development of a collection of inservice materials in the form of interactive CD-ROM-based learning environments, books, teacher educator casebooks, and hypertext versions of the NCTM Standards 2000 on number and operation. The materials and inservice activities we are designing are interactive and reflective. Users of the materials are asked to look at lessons, to discuss what they expect children to do, to reflect on the teacher’s use of context, and to design follow-up lessons given what they saw children do. They are presented with children’s work, and let in on the teacher’s decision-making. Via video and CD-ROMs, users will actually have the context of the classroom as a lab to explore. In our session we will use these new materials and focus on the role of context in the development of subtraction."
Cathy FosnotCathy Fosnot is Professor of Education at the City College of New York and Director of Mathematics in the City, an in-service project for mathematics education reform, PK–8, offering summer institutes, year-long courses , inservice workshops, and on-site staff development.
She has twice received the "best writing" award from AERA’s Constructivist SIG as well as the "young scholar" award (1984) from the Educational Communication and Technology Journal. Cathy was one of the original founding members of the Association for Constructivist Teaching and served as editor of that association’s journal, The Constructivist, for over ten years, recently resigning to focus her time on the development of inservice materials for mathematics education reform.
Cathy is the author of numerous books and articles, among them: Young Mathematicians at Work (Heinemann, 2001, co-authored with Maarten Dolk); Enquiring Teachers, Enquiring Learners (Teachers College Press, 1989); Constructivism: Theory, Perspectives and Practice (Teachers College Press, 1996); Reconstructing Mathematics Education (Teachers College Press, 1993, co-authored with Deborah Schifter).
Maarten DolkMaarten Dolk is a researcher and developer of mathematics education at the Freudenthal Institute in the Netherlands, where he has been involved in the development of inservice materials for teachers and of multimedia learning environments for student teachers (MILE). He has also directed an inservice project in the Netherlands for teacher educators and staff developers.
Please see "Reviews by Readers" below for reviews by three individuals of Cathy and Maarten’s most recent title, Young Mathematicians at Work. Ed.
The Foundation is pleased and honored that Jeremy Kilpatrick will not only deliver the keynote address at the traditional banquet on Saturday night, but also lead the "town hall" meeting on Sunday.
Thanks to Jean Moon for writing the article that follows. Ed.
Jeremy Kilpatrick is currently chair of the National Research Council’s Mathematics Learning Study Committee that produced the recently released report, "Adding It Up: Helping Children Learn Mathematics." He has been on the faculty at the University of Georgia, Athens, since 1975 where he is presently Regents Professor of mathematics education. Prior to coming to Georgia University, he taught at Teachers College, Columbia University. His doctorate in mathematics education is from Stanford University where he worked with George Polya. Professor Kilpatrick also has an honorary doctorate from the University of Gothenburg in Sweden. In addition, he has had an opportunity to work in New Zealand, Spain, Colombia and Sweden as a Fulbright scholar.
His contributions to the mathematics education community have been many. He was editor of the Journal for Research in Mathematics Education from 1982 to 1988. He co-edited the series Soviet Studies in the Psychology of Learning and Teaching Mathematics from 1969 to 1975. Professor Kilpatrick was a member of the executive committee (1987-1998) and was vice-president (1991-1998) for the International Commission on Mathematical Instruction. He has been a board member for the National Board for Professional Teaching Standards. Presently, Dr. Kilpatrick is a member of NCTM’s Committee on the Future of the Standards. His publications in the arena of mathematics education have been numerous and influential.
"Adding It up: Helping Children Learn Mathematics" is the product of an 18-month study by a 16-member committee composed of mathematicians, mathematics educators, cognitive psychologists, teachers, and a representative of business. It focuses on mathematics learning from pre-kindergarten through grade eight and was sponsored jointly by the Department of Education and the National Science Foundation. The pre-publication report, released in January 2001, maintains that all students can and should acquire a deep knowledge and understanding of mathematics. Diverse perspectives around the teaching and learning of mathematics were represented among committee members. Everyone connected with the report hopes it will serve as a unifying document in the midst of continuing rancor—often termed the "math wars"—concerning how students best learn mathematics.
The report summary focuses on the mathematical knowledge children bring to school with them; the need for both procedural and conceptual understanding; and the importance of how teachers frame and present work to students, engage in conversations with them, and convey their expectations. It also addresses the kind of mathematical knowledge teachers need. A primary focus of the report is on children’s developing proficiency with whole numbers and rational numbers.
The study will not be available in its final publication format until later in the year. A pre-publications copy of the study is being sent by the Center for Education of the National Research Council to those attending the K-5 Mathematics Specialist annual meeting in anticipation of Professor Kilpatrick’s talk at this meeting. As mentioned above, a discussion in a "town meeting" format will be held Sunday morning.
Joining Professor Kilpatrick will be Casilda Pardo and Ed Robinson. Casilda is an ExxonMobil Foundation teacher-leader who has been working for three years with K–5 teachers Valle Vista Elementary School in Albuquerque, NM. Now in her twenty-third year of teaching, she has been a resource math teacher for the district for the past seven years. Casilda has presented at both national and regional NCTM conferences, and is a national trainer for the Investigations curriculum. Ed Robinson is a recent retiree from the ExxonMobil Corporation. Both served on the 16-member committee. ExxonMobil Foundation is supporting a version of "Adding It Up" intended for the public. It will be a much shorter version, written to make the report easily understandable and useable to parents, civic organizations, parent organizations, school boards and other local policy setting groups.
Susan Ohanian posted to the listserv the news that a 29-page Executive Summary of the report may be read and printed out. Prepublications (uncorrected proofs) may be ordered there as well. Thanks, Susan! Ed.
Many thanks to Ellen Knudson, project director for a newly-funded site in Bismarck, North Dakota, for the introduction that follows. Welcome! Ed.
Four years ago, our district contracted with Marilyn Burns Math Solutions to do a weeklong summer workshop. Bismarck Public Schools, ND, was interested in offering teachers development beyond their undergraduate experiences.
During the course of the seminar, K–12 teachers reflected on their understanding of mathematics. They solved problems in various ways and discovered a variety of ways to approach a problem. For a few of us who have been in active pursuit of a constructivist way of teaching mathematics, this form of staff development fueled our own inquiry. This was not the case for the majority of the participants. As I looked around the room, teachers with varying rings of experience seemed to be caught up in the wisdom of the instructors and the pursuit of mathematics in a problem-based approach. This was a new cultural experience for many. It did not remotely resemble what they grew up with nor the college prep classes that trained them to teach. Teachers became aware that knowledge of the facts was not a guarantee that students understood the operations. The formulas and equations that teachers professed often masked the misconceptions that were concealed in the minds of the learners! Wow! For the first time, teachers were confronted with the notion that a teacher’s role was to understand children’s thinking about mathematics. Break-time conversations revealed that teachers were intrigued with this paradigm and were eager to try out what they were learning in the fall.
Yet, after two consecutive summers of weeklong institutes, I failed to see the momentum build. Though many embraced the philosophy, gaps and questions remained. Familiar concerns seem to echo throughout the district in the months following the sessions. "How do we fit the ‘fun activities’ into a math class already brimming with textbook expectations?" "Too much time spent with the manipulatives resulted in a decline of the proficiency of facts. Facts are important. I am returning to what I have always done." The seeds were planted but we forgot to hire a gardener to tend to them! The weeds stifled the growth of the delicate seedlings. We were a textbook case. Without ongoing staff development and mentoring, the obstacles became too much for teachers to handle in isolation. It became clear to me and to others that a systemic plan for meaningful reform must include more than an occasional encounter with a new concept.
In the second summer of the Math Solutions workshop, I had the privilege to meet one of the trainers, Lucy West (Mahon) who is also the staff development director for mathematics in District 2 in New York. Not only is Lucy a stellar educator, but she has become a dedicated friend. She has willingly been there to lend an ear to my inquiries or her knowledge to help me to grow to be a better teacher. Through our long distance phone calls and e-mails, she has selflessly acted as a precious resource in my quest to create a quality staff development program in math for the Bismarck Public Schools. It was Lucy who suggested that I seek out information regarding DMI (Developing Mathematical Ideas)—one of the most incredible professional development experiences of my life.
After receiving a Target Teacher Scholarship to attend the DMI Leadership Seminar last summer, I approached our administration about funding another individual. The DMI literature stressed that district teams would be trained during the two weeks. Having the number sense to know two is better than one, I suggested that my principal, Kathy Barnett, be selected. My background had been exclusively in early childhood, and Kathy was a former intermediate teacher. A balanced equation! My rationale was that separately we could work as advocates as we built awareness and understanding regarding the importance of this work in our peer groups as teacher and principal.
The two-week DMI training at Mount Holyoke provided time to totally immerse ourselves in mathematics. No meals to cook, no grocery shopping, no housework. Nothing but deepening our own mathematical knowledge and appreciating the logic of learners as they wrestle with making sense of math. The two weeks also allowed my principal and I to get to know each other on a deeper level—void of daily distractions. We built a trusting relationship as we took giant steps revealing, on occasion, our naked number sense. It was awesome to see a principal as a passionate learner. I was respectful of her before our arrival at Mount Holyoke but our bonding experience has now made me revere her. Kathy is a true example of a life-long learner!
Near the end of the institute, participants were asked to discuss and write down what they intended to do when they returned to their districts. At that point, Kathy and I decided to pilot DMI and invite the teachers at our elementary school to join us in bimonthly professional development sessions. We met with our assistant superintendent, John Salwei, shortly after our return to discuss our plans. John, being in total support, provided funds for teachers to be released during the school day twice a month to attend the DMI sessions.
During our first staff meeting, Kathy and I reported "What We Did On Our Summer Vacation" to the group. Openly we shared where we were mathematically before attending, our humbling experiences while we were there, what we had gained, and our intentions for the nine-month staff development program. We invited them to think about our proposal and to let us know if they were interested. Within two days, half the staff had stepped forward to sign on for our pioneer program. We felt that a key to its success was participation grounded in teachers’ own recognition of a need to grow—not as an administrative initiative.
Through our shared experiences, we witnessed an epiphany of many veteran teachers. This uncommon professional development experience was met with passionate dedication and gratitude. Teachers grew not only in what they knew about the content but also in how to approach students in the journey of learning. It was not a ride void of bumps but each individual knew that our scheduled time and guidance from the facilitators and peers would support them through it. Eleanor Duckworth, author of The Having Of Wonderful Ideas, writes about shifting our overactive focus to the passive virtue of knowing the answer to exploring ideas and making sense of them. Exploring ideas and making sense of them is not just the exclusive privilege of young learners but is just as appropriate as teachers mount the daily hurdles that occur as we move from theory to practice.
Over neighborhood fences and in hallway passings, our pilot group of teachers began to recount what was happening in their classrooms as a result of the DMI sessions. Soon teachers not enrolled in our building and from other schools, began to inquire whether this form of staff development would continue. In our desire to build capacity, our team has multiplied. Bismarck Public Schools will send 13 educators (5 principals and 8 teachers) to receive DMI Leadership training in July of 2001.
From our pilot experience, the teachers revealed in their evaluation that they appreciated having the course led by a teacher/principal team. The consensus was that we brought different but essential perspectives into the program. Teachers also felt that having the principal as one of the instructors demonstrated support as they incorporated this research based approach into their classrooms.
As the new kids on the block, we are truly fortunate to join the ExxonMobil Foundation Math Specialist program. We are eager to learn from the experiences of others involved in the program as we begin to study the conditions that need to exist in a successful Standards-based mathematics program.
Last year, I shifted from solely working with six-year olds in all disciplines to facilitating a multi-age group of adult learners in mathematics. One similarity, common to the two groups, is that learners of any age have difficulty accepting new information that goes against firmly held beliefs. Eleanor Duckworth (1997) acknowledges this by pointing out that simply telling learners the truth about something cannot make them understand it. With this in mind, our future plans will expand the math specialist duties to include demonstration lessons and coaching in the classroom of the participants. In addition, we will have teachers from nine elementary schools involved in DMI staff development sessions facilitated by trained principals teaming with classroom teachers. We feel that development of the teacher-principal teams is the cornerstone in the change process. Finally, our planning committee will continue to examine the role and responsibilities of a math specialist in supporting and sustaining systematic mathematics reform in Bismarck.
Thanks to Beth Williams, project director in Bedford County, VA, and to Vandi Hodges, project director in neighboring Hanover County for contributing the article below. Ed.
The first of June was an exciting day for the administrators in Bedford and Hanover counties. They had the opportunity through our ExxonMobil Foundation networking grant to spend an entire day with Virginia Bastable. Virginia, from the DMI (Developing Mathematical Ideas) Institute at Mount Holyoke College, MA, came to the University of Virginia to lead an engaging day of learning and collaboration. Dr. Loren Pitt, Professor of Mathematics at the University, welcomed everyone to Charlottesville. He spoke of the work of the Virginia Mathematics and Science Coalition to support school systems across our state. He shared his excitement for this meeting in having two school systems present that were actively promoting lead teachers, and encouraged all in both systems to continue this work.
Jean Moon, Advisor to the ExxonMobil Foundation, welcomed participants to the meeting. Her comments about the work of the Foundation broadened the participants’ understanding of the current happenings in mathematics reform. She spoke to the critical role that administrators play, and encouraged them in their work to support good mathematics instruction. Patrick Dexter, from the Finance office at ExxonMobil in Fairfax was also able to be present. He shared a little about the work of ExxonMobil as a global operation, and shared Jean’s enthusiasm for the Foundation’s work.
Vandi Hodges, Mathematics Supervisor from Hanover County, then introduced Virginia Bastable. Vandi spoke of the research-based work that Virginia has been involved with and Vandi shared some of her own experiences as a DMI summer institute participant. Virginia then led workshop attendants through problem solving-activities, small group discussions, and the viewing of actual classroom videos. Participants had fun while they wrestled with the complexities of learning mathematics! Throughout the day, Virginia had participants reflect on the next steps—first in writing, then in small group discussions, and then—if they were willing—to the whole group. Feedol tback from this day of thinking and reflecting was encouraging. Administrators spoke of the need to allow teachers time to collaborate. Flexible scheduling, creative duty assignments and other ways to allow this to occur were discussed. Participants also recognized the potential for allowing lead teachers to observe in classrooms and model lessons. Finally, the need was voiced for more professional development for all involved in helping children understand mathematics.
As a consequence of our meeting, many exciting events are being planned. Jackie Getgood, President-elect for the Virginia Council of Teachers of Mathematics, has invited Virginia Bastable to speak at our March meeting, and share her research findings with the Mathematics Supervisors meeting the day before. This will benefit all teachers across our state. Dr. Pitt has also invited mathematics supervisors to consider working with the Virginia Mathematics and Science Coalition to develop a model for training elementary lead teachers that could be implemented across the state.
The Bedford and Hanover County administrators have asked for a follow-up session to continue to look at mathematics. Using the rest of our networking funds, we plan to bring administrators from both counties back together again this fall. We hope to have Dr. Tom Rowan, Consultant and Research Associate in Mathematics Education at the University of Maryland, conduct this session. It will focus on the ways that administrators can be pro-active in helping teachers use Cognitively Guided Instruction in teaching mathematics. Thank you ExxonMobil Foundation for all of these exciting opportunities!
Three individuals offered to review this recent title by Cathy Fosnot and Maarten. Many thanks to each one. Ed.
Jennifer teaches at Parkdale Elementary School in East Aurora, New York. Ed.
Catherine Fosnot’s and Maarten Dolk’s Young Mathematicians At Work: Constructing Number Sense, Addition and Subtraction, (Portsmouth, NH: Heinemann, 2001) is an excellent resource for educators involved in developing mathematics education. It is the first in a series of books and inservice materials (pre-k to grade 8) on the number strand.
In the Preface, the authors point out that it is important for mathematics teachers to "base their practice on how people learn mathematics, how they come to see the world through a mathematical lens—how they come to mathematize their world." Throughout the book, Fosnot and Dolk provide a wide variety of skills, strategies, tools, and classroom scenarios to assist teachers in learning and teaching mathematics. Classiroom conversations and descriptions enhance investigations and illustrate students constructing their own ideas about number, addition and subtraction.
In Chapter 1, Mathematics or Mathematizing, Fosnot and Dolk use classroom math explorations to show what it means to do and learn mathematics. The authors focus in on strategies as schemes, big ideas as structures, and models as tools for thought.
Next, Fosnot and Dolk explain what the "landscape of learning" is. The second chapter takes a look at the understandings that teachers must have so that they are able to recognize the strategies, big ideas, and models that children construct. Also in this chapter, the authors discuss the best contexts for learning. They contrast word problems and truly problematic situations. They stress that truly problematic situations give students the context they need to "generate and explore mathematical ideas."
Chapter 3, Number Sense on the Horizon, shifts the focus to preschool children as they begin to explore and develop number sense. A variety of games, activities, explorations, and investigations are offered for teachers to use to facilitate this learning.
In Chapter 4, Place Value on the Horizon, Fosnot and Dolk describe the development of the number system in our culture and other cultures around the world. The authors go further to use classroom situations to show how teachers can help children develop mathematical notation and facilitate place value development. Once again, the authors illustrate the value of real-life contexts in making learning meaningful.
In their discussions of mathematizing, Fosnot and Dolk say that it’s "impossible to discuss mathematizing without simultaneously discussing models." What are mathematical models? One could say that models are "mental maps used by mathematicians as they organize their activity, solve problems, or explore relationships." The children in our classrooms use many models as they work as "mathematicians" to solve problems that are presented. For one problem, children will come up with different models because they organize their thoughts in different ways. Models are "tools to think with." Even further, models can be used to communicate one’s thoughts about mathematics. Children can learn from discussing and interpreting other students’ models. In their presentations of many models, the authors focus on open number lines and the ways children use them to explore problems.
In Chapters 6, 7, and 8, addition and subtraction are explored. Chapter 6 begins with the question: "Basic facts: To Memorize, or Not to Memorize?" "Understanding what it means to add or subtract is necessary before facts can be automatic, but understanding does not necessarily transfer to automaticity." In teaching children relationships between facts, it is helpful to facilitate their discovery of strategies such as double plus or minus one, working with the structure of five, making tens, using compensation, and using known facts. Each of these strategies is explored in these chapters. The authors integrate literature, manipulatives, and a unique tool called the rekenrek into their lessons. Lastly, they offer ideas and examples of how to offer these math experiences through meaningful mini-lessons, full class explorations, and small group problem-solving activities.
Regardless of the manner in which math instruction is delivered, all math programs need to have appropriate means of assessment. In Chapter 9, the authors discuss performance-based assessment, portfolio-based documentation, assessing mathematizing, and standardized test results. Fosnot and Dolk encourage teachers to use types of assessment that match the teaching and learning that is taking place in the classrooms.
In their last chapter, Fosnot and Dolk switch their focus from young mathematicians to teachers as mathematicians. They offer some wonderful ideas for assisting teachers in learning to mathematize and in realizing opportunities for real-life problem solving in our mathematical world. This book is a great resource for teachers and school mathematics leaders. The focus on preschool children, primary school students, and on teachers as mathematicians will assist any school in taking a closer look at math instruction. I would recommend this guide for administrators, curriculum coordinators, teachers and teacher discussion groups interested in exploring how children construct number sense, addition, and subtraction.
Linda is the K–5 Mathematics Coordinator in Columbia, Missouri. Ed.
Young Mathematicians at Work is a must-have book for all elementary teachers and those who work with elementary teachers or young children. It is the first of three books that Cathy and Maarten are scheduled to write. The next book will continue the focus on number with an extension to multiplication and division. The third book will focus on rational numbers with an aim toward middle school. All three books are part of a professional development package including CD-ROM’s, videotapes and facilitator manuals that is being developed through a grant from NSF for which Cathy and Maarten are co-project directors.
My favorite chapter is Chapter Two: Landscapes of Learning. If you are familiar with Cathy’s and Maarten’s work, then you have heard them refer to "Learning Lines" which contain the strategies and models that help children frame or construct the "big ideas" of mathematics. The authors have changed the description of children’s mathematical journey from the "Learning Line" to the "Landscape of Learning." To this reader, the metaphor is more comfortable. My favorite passage from the text describes the process of learning—at least my learning.
When we are moving across a landscape toward a horizon, the horizon seems clear. Yet we never actually reach it. New objects—new landmarks—come into view. So, too, with learning. One question seemingly answered raises otrahers. Children seem to resolve one struggle only to grapple with another. It helps to have the horizon in mind when we plan activities, when we interact, question, and facilitate discussions. But horizons are not fixed points in the landscape, they are constantly shifting. (p. 18)
As I read teachers’ journals from within our district, the above quote appropriately describes their struggle with making sense of the learning process. Several teachers and I have decided to just accept our fate of living in a state of constant disequilibrium.
The chapter goes on to discuss the important role of contexts and constraints in setting situations that allow students to journey toward the horizon and yet also allow them to take the "paths" that are appropriate for them. The instructional examples that are used in this section provide great insight into the difference between word problems and "truly problematic situations." The final section of the chapter discusses turning classrooms into mathematical communities. Cathy and Maarten discuss the edge between the individual and the community, facilitating dialogue and structuring math workshops. The discussion is straightforward, important and contains great questions for teachers to use in developing community and facilitating a math congress.
Although the discussion about landscapes is my favorite, chapters one and two are full of important information for teachers of young children. In chapter one, the authors lay the groundwork for the rest of the chapters outlining their beliefs about what it means to do and learn mathematics. These beliefs and the process of "mathematizing"—as Cathy and Maarten have coined it—are masterfully revealed in visits in Madeline Chang’s K–1 classroom.
Teachers will find the other eight chapters wonderful resources for understanding and designing opportunities for their students to grapple with the big ideas of mathematics. The classroom vignettes help the reader visualize the "mathematizing" the students and teachers are doing in the chapters. Chapters three through six highlight the big ideas of number sense, place value, addition and subtraction and the development of mathematical models. Teachers who are struggling with the algorithms versus number sense issue will find chapter seven interesting and informative. Chapter eight follows the discussion of algorithms with a discussion of using mini-lessons to develop efficient computation skills. Chapter nine discusses the role and context of assessment in a reform classroom. Knowing what to look for is only half the "problem"; knowing what to do next is the other half. Knowing and understanding the landscape of mathematics is critical to helping young children take a worthwhile significant mathematical journey. Chapter ten sums up the journey for teachers as mathematicians.
For teachers to be able to teach in the ways illustrated in these chapters, they need to walk the edge between the structure of mathematics and child development, between the community and the individual. They need to be willing to live on the edge. They need to be willing to challenge themselves mathematically and to be willing to journey with their children. There is no one path, one line, one map for the journey. The landscape of learning has many paths, and the horizons shift as we approach them. Knowing the landscape, having a sense of the landmarks—the big ideas, the strategies, and the models—helps us plan the journey. We need to structure the e 9nvironment to bring children closer to the landmarks, to the horizon—to enable them to act on their world mathematically. (p. 181)
As I said at the beginning of this review, this is a must-have book for all those interested in young children’s construction of mathematics. I can’t wait to use Young Mathematicians at Work as the topic of a book study with elementary teachers in our district. The text, rich vignettes and examples offer a wealth of opportunities for a group of teachers to take a mathematical journey traversing the landscape heading for the horizon of knowledge and understanding of how mathematics is truly learned.
Gregg is a Mathematics Consultant with the Oakland Schools in Waterford, Michigan. He works with 28 school districts and the state of Michigan on a variety of curriculum, professional development, and assessment projects. Ed.
I was looking forward to reading this book based on my experience with the previous work of Cathy Fosnot with Deborah Schifter, Reconstructing Mathematics Education. Their book has been so useful in my work that I expected to find an equally powerful experience with Young Mathematicians At Work. I was not mistaken! The second author, Maarten Dolk, comes from the Netherlands. The influence of mathematics teaching there pervades both the project and the book. It makes for a fascinating read.
This book covers so much material within its pages that it is hard to do a thorough review. The authors describe their experiences working with teachers through a grant in the New York City Schools. During this project, they spent countless hours working to develop the mathematical power of classroom teachers. In doing this, they also spent extensive time in the teacher’s classrooms and working with children. In a series of sequential chapters, the authors describe these experiences in intricate and powerful detail. The rich scenarios and interesting ideas begin on the first page and continue through the book. The chapters proceed from the large concepts central to the project and to the book, then move through the development of number sense in the early grades and end with a look at assessment. The final chapter looks at teachers and how they learn and teach mathematics.
One example
of the influence of Dutch research in mathematics education is
the introduction of the rekenrek to the participants in
the project. The rekenrek is a manipulative similar to an
abacus but without place value columns (see illustration left.)
Translated, rekenrek means "calculating frame,"
or "arithmetic rack."
The rekenrek is a
powerful manipulative that can help children learn to subitize
numbers. By seeing numbers as groups rather than the result of
counting singly or counting on, children are able to
conceptualize groups of numbers and how they can be combined to
make new numbers. Since the beads on the rekenrek are
grouped in fives by color, children are able to see number
combinations easily. The illustration to the right, below shows 6
+ 7 on the rekenrek.
In a classroom scenario, the reader sees how children conceptualize this equation. Some children see it as 6 + 6 + 1, others as 5 + 5 + 3, and another as 7 + 7 - 1.
A central theme of the book, and the mathematics project, was to develop the ability to "mathematize". This means to "…see, organize, and interpret the world through and with mathematical models." In order to develop the ability to mathematize. both teachers and children—indeed all learners—need to have mathematical experiences that are based in the real world. The authors make the point that traditionally mathematics has been taught as" the presentation of ideas that others have already figured out. Therefore, in traditional instruction, to do math we just need to learn the procedures that someone else passes on to us. Of course, this has not been successful. This lack of success is demonstrated by the math phobia of American adults, low scores on the TIMSS, few students taking higher level math courses, and many other indicators. By "mathematizing," or organizing and interpreting reality mathematically, learners create their own understanding of mathematical concepts and thus make the ideas their own.
Another major thrust of the book is how to make mathematizing happen in the classroom. The authors believe this is done with strategies, big ideas, and models. These strategies, or organized patterns of behavior, begin to develop at birth and continue throughout our lives. The authors also make the important point that it is easy to confuse tools with strategies. Teaching a child to follow a routine with a manipulative, such as how to add with base ten blocks, is not a strategy.
Big ideas are the "central, organizing ideas of mathematics…" (Schifter and Fosnot, 1993, 35.) One example of a big idea is unitizing, or the ability to count both single items and groups simultaneously. This is absolutely essential for understanding place value, then addition and subtraction, then the other operations, and eventually higher mathematics. So a big idea is one which flows across the curriculum K–12 and beyond.
Models are the representations of mathematical ideas. In one scenario, children are designing necklaces using a particular pattern. Some students use Unifix cubes to model their ideas. Others use drawings. Either way, the model helps them think about and understand a real world problem they are working on.
As the authors present their ideas, they continually illustrate the concepts with vignettes and scenarios from classrooms. This discussion of ideas woven with children and teachers working and thinking together makes the ideas easy to visualize and understand. The authors present many strategies and lessons for teachers that will help primary children develop number sense and addition and subtraction concepts. A teacher could easily plan an entire year of teaching number sense and operation from the ideas in this book.
This book would be useful to many different readers. Whether in the classroom, working in professional development, teacher education, or designing learning experiences for any learner of mathematics, there are many ideas and strategies to be found. I highly recommend this book.
To order this title, please visit Heinemann's web site. Ed.
Five hundred sheets of shrink-wrapped loose-leaf paper, an unopened box of 48 Crayolas, a brightly-colored Flintstones lunchbox—what memories do you have of beginning a new school year? It’s getting to be that time again.
As the school year opens, please consider sharing with readers your meaningful summertime staff development experiences, your reviews of favorite poolside professional "reads, and your plans, hopes and dreams for the coming year. It’s one way we can all keep up with one another.
Many thanks to all those who contributed such wonderful articles this time. The deadline for the September issue is Monday, August 27. Please send items to Jean Ehnebuske, 105 Hideaway Cove, Georgetown, TX 78628; e-mail, jean@intersectionlive.org; phone, (512) 869-1580; fax, (512) 869-8477. Thanks!
 |
 |
 |