February / March 2002


A Learning Continuum in Geometry

More from the Conference on Standards

Thanks to Julie Sarama, Doug Clements and Ann-Marie DiBiase, State University of New York, Buffalo, for yet another in the series of articles resulting from the May 2000 Conference on Standards for Preschool and Kindergarten Mathematics Education. This one continues where January’s left off. Ed.

In the last issue, we illustrated teaching with a "learning continuum" in number. We wanted to start with an area with which most teachers are familiar. In this issue, we’ll try to show a similar learning continuum in geometry—one that many may not have thought a lot about. Please remember that all this information, and much more, will be available in the book Engaging Young Children in Mathematics: Findings of the 2000 National Conference on Standards for Preschool and Kindergarten Mathematics Education by D.H. Clements, J. Sarama, and A-M DiBiase, Eds., (Mahwah, NJ: Lawrence Erlbaum Associates, Inc., in press).

The continuum is summarized in one row of the developmental guidelines from Engaging Young Children in Mathematics (Fig. 1).

PreK* Kindergarten
2-4 years 4-5 years 5-6 years
Compose (put together) 2-D shapes to make new shapes
Level A: Use shapes in isolation to make a picture.

Level B: Cover an outline with shapes without leaving gaps, first with trial-and-error, then with foresight.

Makes a picture by combining shapes.

1st Grade 2nd Grade  
6-7 years 7-8 years  
Compose (put together) 2-D shapes to make new shapes  
Level C: Compose, combining shapes into new shapes

Level D: Compose, combining shapes into new shapes, substituting a combination of smaller shapes for a larger shape * Ages reflect those typically found in classes or groups of children. For example, in the first category, a typical classroom of “3-year-olds” may begin the year with some 2-year-olds and end the year with some children just turning 4 years of age.
Figure 1. Developmental Guidelines for Geometry—One Small Part of “Putting Together” and “Taking Apart” Shapes. Shapes can be decomposed into other shapes or into their component parts; shapes can be composed into other shapes and structures, such as tilings.

Let’s just examine one activity that, with research-based planning, covers the range of this learning continuum. That activity is called "Shape Puzzles" and, as the name implies, children complete puzzles by filling a region with shapes. Of course we do this off- and on-computer, but will illustrate it with our computer version (Fig. 2). Children find this version very motivating, and they not only learn about putting together shapes but, as they use the "tools" along the bottom edge (which allow children to "act on" shapes by sliding, turning, flipping, and so forth) they learn about geometric transformations. When the child completes a shape, it "comes alive" (see Fig. 3).

How does "Shape Puzzles" fit the learning continuum? By automatically adjusting the type of puzzle each individual child works on. Children who are at level A in Figure 1 are offered puzzles such as Figure 4A. All they have to do is match shapes. Children at level B are offered puzzles such as Figure 4B, or the crab in Figure 2, which is at the same level. They have to combine several shapes to make the top of the flower, but the outline still suggests what shapes to use. Children at level C solve puzzles such as Figure 4C that are very difficult to fill. They have to think about how to combine shapes to make new shapes that cover a larger region! At this level, children have started to achieve real competence in composing shapes.

Author Note

Time to prepare this material was partially provided by three grants, two from the National Science Foundation (NSF), ESI-98-17540: "Conference on Standards for Preschool and Kindergarten Mathematics Education" and ESI-9730804, "Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development" and one from the ExxonMobil Foundation, also entitled, "Conference on Standards for Preschool and Kindergarten Mathematics Education." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of either foundation.


Permission to reproduce images in figures accompanying this article has been granted by the publisher, DLM Early Childhood Express, SRA/McGraw-Hill.

Mathematical Education of Teachers

Focus of National Summit

Many thanks to Jean Moon, Advisor to the ExxonMobil Foundation, for contributing this article. Ed.

In early November, almost three-hundred participants and speakers gathered near Washington, D.C. for the National Summit on the Mathematical Education of Teachers co-sponsored by ExxonMobil Foundation and the National Science Foundation. The Summit was an intensive two-day event, organized by the Conference Board of the Mathematical Sciences (CBMS). The CBMS is an umbrella organization consisting of sixteen professional societies all of which have as one of their primary objectives the increase or diffusion of knowledge in one or more of the mathematical sciences. The purpose of the Summit was to launch the document, The Mathematical Education of Teachers (MET), and to stimulate the mathematics community into making the mathematical education of teachers a priority for this decade. Instructions for ordering copies of this report can be found at www.cbmsweb.org.

Together, participants formed a diverse group—not only in terms of where they came from, but also by their specialties and by their affiliations with higher education. Attendees came from thirty-five states, the District of Columbia and the Virgin Islands. There were participants from the areas of mathematics, mathematics education, and education representing universities, four-year colleges and two-year colleges. Together, attendees heard recommendations of MET and the challenges of implementing them and learned of efforts underway at several institutions to improve the mathematics education of teachers.

The Summit consisted of plenary sessions, addresses, and working sessions. The plenary sessions and addresses were designed to frame the issues associated with the mathematical education of teachers. Working sessions were led by people from institutions that have had success improving that education. Topics included elementary, middle and high school teacher preparation, and partnerships between higher education mathematics departments and school districts, schools of education and two-year colleges. Several K-5 Mathematics Specialist projects were part of the working session program: Donna Little-Kaumo (Albuquerque), Deborah Schifter (Developing Mathematical Ideas), and Bill Fisher and Margaret Owen (California State University at Chico).

Mathematical Education of Teachers (MET)

This document, released in August (2001), is directed primarily at mathematicians and math educators—those who actually teach mathematics to future teachers. It urges mathematicians to recognize their role and responsibility in the education of future teachers and it focuses on issues of mathematical content and teaching.

Two general themes guide MET: the intellectual substance in school mathematics, and the special nature of mathematical knowledge needed for teaching.

Mathematics education research over the past decade has highlighted the substantial mathematical understanding that is needed to teach well even basic topics like whole number arithmetic. This work has helped persuade mathematicians that teaching and learning basic mathematics involves very complex cognitive demands for students and for teachers. MET provides a practical guide for mathematics departments and faculty in developing courses that will give future teachers a deep understanding of the mathematics they will teach.

The report offers recommendations on mathematics curriculum and instruction for future teachers, on the need for cooperation among different parties involved in educating teachers, and on advocay for policies that support high quality mathematics. Three overview chapters give a brief introduction to the mathematics needed by teachers at the elementary, middle, and high school levels. Three longer content chapters give detailed discussions of the mathematics needed by teachers at each level. These chapters provide a substantial resource for faculty members who teach courses for future teachers or who assume leadership roles in designing and offering their department’s courses.

ExxonMobil Innovation Grants

Building on a long-standing tradition of supporting planning initiatives, five ExxonMobil Innovation grants were made at the Summit, and additional grant recipients were named in December. Prior to awarding the initial round of grants, ExxonMobil Foundation President Ed Ahnert made the following remarks:

"In support of the belief that the mathematics community can lead the way in providing good models for building the kinds of cooperative efforts needed for long term improvement of teacher education, the ExxonMobil Foundation will offer planning grants of $3,000 each to assist you, the National Summit participants, in developing plans for partnerships or other innovative cooperative efforts among groups involved in the mathematics education of teachers. Teacher education in this context includes both the preparation of future teachers and the on-going education of practicing teachers. These grants are intended to provide the participant teams with the resources needed to engage in thoughtful planning with appropriate partners to develop an action plan which will then be supported by the local institutions involved or which can be used to obtain external federal or private foundation funding."

Recipients of the ExxonMobil Innovation Grants were: East Tennessee State University; Humboldt State University; Northeastern State University (Oklahoma); University of Illinois at Chicago; University of Southern Colorado; American Statistical Association in collaboration with the University of Georgia; Montgomery College; Towson State University; Minnesota State-Mankato; Ohio University; Rutgers University; University of Virginia–Wise; Jacksonville State University; Minnesota State-Moorhead; and Columbus State University.

Reviews by Readers

Many thanks to Pat Hess and David Chia for these reviews of two titles in a new series, Teaching Arithmetic, from Marilyn Burns’ Math Solutions Publications. Thanks, also, to the folks at Math Solutions who provided these titles gratis. You may read reviews of other titles by Intersection readers at www.mathsolutions.com. Ed.

Teaching Arithmetic: Lessons for Extending Multiplication

Reviewed by Pat Hess

Pat Hess is the past program facilitator for the Foundation’s K-5 Math Specialist Program. She is retired and lives in Albuquerque, NM. Many thanks, Pat! Ed.

I’ve been waiting for this book by Maryann Wickett and Marilyn Burns. It meets all my expectations. It is a collection of lessons extending the understanding of multiplication presented in the previous book, Lessons for Introducing Multiplication, Grade 3. All the lessons detail how students express and reflect on their mathematical thinking. Each lesson contains a script of what the teacher says and the expectations of the student’s responses.

The content of the book covers understanding of multiplication, linking multiplication and geometry, investigating multiplication number sense and explorating real-world problems. Additional activities include assessments, parent communication and blackline masters. The blackline masters are given as aids to the exploration of mathematics—aids such as centimeter grid paper, game boards, and a 100 chart.

This book is a winner and it engenders in me "what if" feelings. What if school systems supported study groups so thatteachers could discuss lessons such as these, note expected responses from students, and, after teaching the lesson, discuss the observed mathematical understanding and learning progressing in their classes? This type of education requires support from the administration, but Lessons for Extending Multiplication gives teachers support of well-thought-out lessons in an easy-to-follow format.

Teaching Arithmetic: Lessons for Introducing Fractions

Reviewed by David Chia

Many thanks to David Chia, staff development teacher at Broad Acres Elementary School, Montgomery County Public Schools, MD, for this review. Ed.

If I’m at work for nine hours today and I spend one-third of half of my workday co-teaching with a new teacher, how many minutes did I spend with her? It’s questions like this that sometimes cause students (and teachers) to scream "I hate fractions!" For students, fractions can be a tough concept and fraction applications are not always easily mastered. For teachers, teaching fractions can be a challenge and sometimes a frustration. In this book, author Marilyn Burns notes "learning about fractions in the upper elementary grades is hard. Really hard. Fractions are hard not only for children to learn but for teachers to teach." With this in mind, the author sets out on a reflective journey as she teaches fractions to fourth- and fifth-grade students.

There are fifteen lessons organized into individual chapters beginning with introducing fractions, developing a fraction kit, and exploring fractions with various manipulatives. In the middle chapters, lesson topics focus on using one-half (1/2) as a benchmark fraction, comparing fractions, and finding fractional parts. The final chapters conclude with comparing and combining fractions. Each chapter and its lesson is organized with an overview, a materials list, suggested timeline for lessons, teaching directions, teaching notes, a vignette of the lesson with reflective thinking, sample student work, extensions, assessments, and questions and discussions which address issues that other teachers have asked. Several assessments and numerous blackline masters are included along with an index.

While the author recommends beginning with the introductory lessons and then chunking the remaining lessons throughout the course of a year, I don’t believe that is always necessary. Each chapter is so well outlined that depending on the outcomes and indicators you are teaching, you could find and use the chapter and its lesson for the area of "greatest need." The "greatest need" could be your students’ struggle in attaining a concept or your need as a teacher to expertly deliver a lesson. The most intriguing parts of each chapter are the shared vignette and the question and discussion pages. I find each vignette interesting because it shares a situation I can relate to as a math teacher or it broadens my perspective of how other students learn and how other teachers teach. Additionally, the question and discussion pages allow me to review issues and concerns that other teachers might have when teaching a similar lesson. The quality in reflective thinking about how students learn is certainly one of the highlights of the book.

If you are an upper elementary teacher who would like to broaden your thinking and expertise in teaching fractions, this book will be an added resource to you. If you are a new teacher wondering about how students really learn fractions or asking yourself how you can begin to help your students learn fractions, this is an excellent resource that will guide you in introducing fractions to upper elementary students.

Learning Opportunity

Math Solutions Summer Institutes

Many thanks to Tracey Schuyler for this information. Ed.

Marilyn Burns Education Associates will present five-day mathematics institutes for math coaches, teacher leaders, and teachers of kindergarten through grade eight in partnership with more than 30 school districts and education agencies nationwide. The institutes present practical and proven ways to understand and implement standards-based mathematics instruction, and make use of effective instructional strategies. Using classroom-tested activities, teachers learn how to develop students’ ability to think and reason, build students’ number sense and computation skills, help students learn to solve problems, use manipulative materials, and organize instruction for cooperative and individual learning. Special grade-level sessions help teachers focus on the skills and concepts appropriate for their students.

The registration fee is $420, which covers the cost of the five-day institute, sample manipulative materials, and the course book About Teaching Mathematics, by Marilyn Burns. Discounts are available for teachers from the hosting district or agency.

For information and a complete summer schedule, visit www.mathsolutions.com, or call 800-868-9092. Send questions to courseinfo@mathsolutions.com.

Saying Goodbye

This news and these comments come from Bernie White, Senior Program Officer for ExxonMobil Foundation, and Jean Moon, Advisor to the Foundation. Many thanks for their kind words and warm wishes. Ed.

With this issue of Intersection, we bid farewell to our editor of many years, Jean Ehnebuske. Jean has been the editor since 1997 when she assumed this responsibility from Pat Hess, retired program facilitator for the K-5 Math Specialist Program. During Jean’s tenure, Intersection has grown to fully reflect the vibrant community of mathematics educators associated with the Foundation’s K-5 Mathematics Specialist Program. We thank Jean for her dedication to Intersection, and wish her the very best in the years ahead.

We are now in the process of identifying a new editor. You will note that this issue is for February/March. We expect to have a new editor in place by April 1, 2002. During this transition, please continue to send newsletter articles to Jean.

Farewell, Friends

After five years, I’ve decided it’s time for me to exit as editor of Intersection—a decision not lightly nor easily made. That’s because of you, the readers. If the publication has been meaningful and useful, it’s because of your contributions. Over the years, you have sent a wealth of material that has allowed me to document time after time that knowledgeable, thoughtful, committed educators are making a positive and profound difference in the way mathematics is taught and learned. The way I see it, I’ve been getting presents every month for five years. I can’t thank you enough.

Since I’m an editor—and a former English teacher—I’ve got to leave you with these words, "please write." I don’t mean to me (although I’d also like that very much), but just write. Break into print, tell your stories, put down in words the wonderful things you and your colleagues and your students are doing. Overwhelm the new editor with material. In my experience, teachers are a modest bunch. But you’re changing the world, each one of you. And the world should know more about it.

As Bernie mentioned, I assumed this position from Pat Hess in 1997. A "tough act to follow," Pat was my role model and remained my mentor until she retired in December 1999. Every editor should be so lucky! Thanks, Pat.

I’d like to thank Bob Witte— senior program officer, now retired—for placing his trust in me when he made me Pat’s successor. I am grateful to Bob for his sage advice, sustained support, and frequent contributions to the newsls.etter. I also enjoyed working with Joe Gonzales, who succeeded Bob, and most recently, with Bernie White. I’ve known Jean Moon for nearly ten years and value all I’ve learned from her. I would like to thank Ed Ahnert, ExxonMobil Foundation’s president, for his long-term support of the newsletter since its inception in 1989.

It has been my good fortune and great pleasure to work with Nihad Ziad and her staff at NCTM all this time. Even though the mailing list has grown from 150 to 850 recipients and the newsletter itself has grown in length, Nihad and her staff have never missed a beat. Every month copies are placed in envelopes and mailed out all over the country—quickly and efficiently. Thanks, Nihad!

My thanks go also to my husband, David, for his invaluable technical help and his sincere and abiding interest in this endeavor.

I wish the new editor every success. Until that person is in place, please send contributions to me, Jean Ehnebuske, at 105 Hideaway Cove, Georgetown, TX 78628; phone, (512) 869-1580; fax, (512) 869-8477; e-mail, jean@intersectionlive.org. So long and best wishes!

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