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This article is another in a series by Doug Clements and Ann-Marie DiBiase, State University of New York, Buffalo, resulting from the May 2000 Conference on Standards for Preschool and Kindergarten Mathematics Education. This article picks up where September’s left off. Many thanks to the authors. Ed.
Last issue we illustrated the
"developmental paths" that emerged from the Conference
on Standards. We also said that, more important than knowing at
what age most children acquire specific mathematical concepts, is
knowing how these mathematical concepts develop, and how we can
enrich and deepen children’s concepts. In this issue and the
next, we shall look at this learning process by considering a
specific developmental or learning continuum—one in each
issue. 
We shall attempt to show how such a
continuum can help facilitate developmentally appropriate
teaching and learning—that is, how it can help generate
activities that are attainable, but challenging, for each child.
This information, and much more, will be available in the book
that resulted from the conference, 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).
For this issue, let’s start with a continuum—with which most teachers will be familiar—dealing with one of the many components of addition and subtraction. A couple of the rows of the developmental guidelines from Engaging Young Children in Mathematics are shown in Figure 1. The first row (a.) involves nonverbal adding and subtracting. When one ball is put into a box and then another ball is put into the box, even toddlers expect that the box then contains two balls. They are surprised if the box is opened (after the teacher secretly adds another ball) and there are three balls. As a curriculum example, consider the "Double Trouble" activity from the Building Blocks project1 that we discussed in the previous issue. Recall that Mrs. Double is throwing a birthday party for her twins. The twins like their cookies to have the same number of chocolate chips. The teacher puts two pretend chips on a cookie (paper plate), covers it with a napkin, says "Watch!" slips another chip under the napkin onto the cookie, and asks her preschoolers to put the same number of chips on their cookies. Some preschoolers may count, but many solve the task nonverbally by creating "seeing" the "two and one more" chips in their "mind’s eye" and recreating that "image" on their own cookie. Later, the "Double Trouble" computer activity has Mrs. Double similarly asking children to make a "twin" cookie with the same number of chips as a cookie Mrs. Double made by putting two groups of chips on a cookie hidden by a napkin.
As shown on Figure 1, row (b.), the next developmental step is that children 4-5 years old can solve word problems involving small numbers by using concrete objects. Children who cannot yet count on usually follow three steps to solve a problem such as "How many chips will be on a cookie if we put on three chips and then two more chips": (1) counting out three chips onto a cookie; (2) counting out two more chips; and, (3) counting the five chips again, starting at one. Such problems are presented in Building Blocks as word problems (with small groups having chips and paper-plate cookies), as story problems the group acts out (rugs are cookies and children are "chips"), and on the computer (Mrs. Double shows a cookie with three chips and says, "This cookie should have five chips. Make it five! Then click on ‘GO.’").
| Figure 1.
Developmental Guidelines for Number and Operations-One
Small Part of Addition and Subtraction |
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Adding To/Taking Away A collection can be made larger by adding items to it and made smaller by taking some away from it. |
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Topic |
PreK2 |
Kindergarten |
1 |
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2-4 years |
4-5 years |
5-6 |
6-7 |
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| a. Nonverbal problem solving supports later adding and subtracting. | Nonverbal addition and subtraction… |
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one item + one item or two items – one item |
sums up to 4 and subtraction involving 1 to 4 items |
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| b. Solving problems using informal counting strategies is a critical step in learning adding and subtracting. | Solve and make verbal word problems; add and subtract using |
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concrete modeling (objects or fingers), totals to 5 |
counting-based strategies such as counting on, totals to 10 |
advanced counting strategies, e.g., counting -on or -up (for subtraction and unknown addends) to 18 |
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| 2Ages reflect those typically found in classes or groups of children; for example, 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. | ||||
Research and expert practice indicate that next children of ages 5-6 develop, and eventually abbreviate, such solution methods. For example, when chips are hidden from view, children may put up fingers sequentially while saying, "1, 2, 3" and then continue on, putting up two more fingers, "4, 5. Five!" As children continue to develop, they abbreviate their counting methods even further. Rather than putting up fingers sequentially to count the three hidden items, children who can count on simply say, "Threeeee…4, 5. Five!" Such counting on is a landmark in children’s numerical development. It is not a rote step. It requires conceptually embedding the 3 inside the total, 5.
Counting on when increasing collections and the corresponding counting-back-from when decreasing collections are powerful numerical strategies for children. However, these are only beginning strategies. In the case where the amount of increase is unknown, children 6 years and older count-up-to to find the unknown amount. If six items are increased so that there are now nine items, children may find the amount of increase by counting, "Six; 7, 8, 9. Three!" And if nine items are decreased so that six remain, children may count from nine down to six as follows: "Nine; 8, 7, 6. Three!"
There are other learning continua for different aspects of addition and subtraction (e.g., composing and decomposing numbers), and many more for number and arithmetic. Next time, we’ll take a look at a learning continuum for another topic altogether—geometry!
1Building Blocks is a National Science Foundation-funded project developing and evaluating an innovative curriculum for early childhood education, preschool to grade 2. The Building Blocks program incorporates both old and new technologies, from blocks and puzzles to multimedia computer programs. Preliminary evaluations show that the program’s approach of finding the mathematics in, and developing mathematics from, children’s everyday activity, allows children to learn and do more mathematics than previously assumed.
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.
As you may recall, the November/December issue featured reflections of DMI participants about their summer experiences at Mount Holyoke College. So many reflections were received that space did not allow all of them to be published last time. Please find the rest below. Thanks to all contributors! Ed.
During July 2001, the Fort Bend Math Specialists were given the opportunity to attend Developing Mathematical Ideas (DMI), an intense investigation into the development of mathematical thinking in students. Because we were all math specialists, we hypothesized that the two weeks would be a tweaked rendition of previous math trainings we had experienced. However, the results of the experience far exceeded our expectations. Composing, decomposing and recomposing numbers had never taken on such a captivated audience.
With our bags packed and our mathematical confidence in check, we boarded our plane and migrated north to the beautiful Mount Holyoke College campus in South Hadley, Massachusetts. On arrival, we were warmly greeted by the DMI staff, issued our bed linens and box fans, and escorted to the rooms that were to be our accommodations for the next two weeks.
Those two weeks proved to be engaging, thought-provoking and enlightening. Each morning and afternoon we were given reading assignments that delved into the mathematical thinking of students. We analyzed case studies that illustrated interactions between teachers and their students. We observed questioning strategies and student responses by viewing taped sessions recorded in various classrooms of DMI-trained teachers. In addition to whole group roundtable discussions, we also participated in small group activities that fostered the sharing of personal experiences from our own classroom environments, as well as our own thoughts about mathematical ideas and thinking.
After analyzing and evaluating mathematical ideas for the duration of the institute, the participants left DMI with a new perspective on students’ mathematical thinking. We had many opportunities to reflect, revise and reconstruct our own philosophies. The facilitators of the DMI institute shared such valuable and powerful insights into the big ideas of mathematical thinking that our math specialists were truly compelled to transfer this excitement and information to the teachers in our own district.
The initial in-service to "kick off" the current school year afforded us the opportunity to provide limited experiences for our district teachers in composing, decomposing and recomposing numbers. The flow of information to our teachers is continuing throughout the year through sharing of our DMI experiences with the other district math specialists each month. The enthusiasm for learning and experiencing in more depth the ways in which students develop mathematical ideas has generated an overwhelming demand by our math specialists to attend the institutes during the palindrome summer of 2002. We look forward to the continued opportunity to participate in DMI programs and to further investigate ways in which students develop mathematical ideas.
Beth Williams, Paul Webb and Pam Phillips authored this reflection together. Attending DMI with them was Loren Pitt from the University of Virginia. His article follows this one. Ed.
Our two weeks at Mount Holyoke College was the most incredible professional development experience. The campus is beautiful! The buildings are gorgeous and well maintained. The grounds are full of lovely plantings and walking paths, and the lakes are covered with ducks and swans. The food was wonderful! Three meals a day and constant snacking opportunities were the norm.
Our dorm experience was interesting. Living with the other participants—studying, eating, and talking together—quickly turned us into a strong community. Sharing a room with a colleague was fun. The communal bathrooms and showers were something we hadn’t experienced since our own college days.
The end of July—ninety-plus degree days with no air-conditioning—was tough. We studied and hung out in the air-conditioned rooms. When we went to sleep on the top floor of the dorm, it felt like we were in an oven. Thank goodness the weather changed by the end of the week!
The surrounding community of South Hadley, Massachusetts has made sure that everything you might need can be found in the Commons shopping area one block from the main campus entrance. Early morning coffee at the Tailgate Picnic Coffee Shop helped us jump-start every day. We continued our studying or finished our journal writing beginning around 6 AM. The work was intense! Thinking through the big ideas of mathematics, reading through the case books, working through the children’s thinking, and making the mathematics shown fit the thinking expressed in numbers, was some of the hardest work we’ve ever done. Listening carefully to colleagues as they shared their thinking, making sense of what we thought, and writing about it daily both informally in journals and formally in papers was a powerful learning experience.
We met daily as the "Virginia team" to talk about what we were learning and how it would impact our work back home. We knew by the end of the first week that we needed to share this experience with others.
Since summer, we have been facilitating the DMI materials with all Bedford County mathematics lead leaders during our monthly meetings. The sharing that takes place as we study the casebook Building a System of Tens has been a wonderful community-building experience for our lead teachers. Most have never studied cases before, and none have looked at children’s work to determine how they come to understand things. Looking for correct thinking, even in wrong answers, has changed our focus in teaching. More classroom time is going to build student understanding.
This experience was wonderful for us all. We hope to return this next summer to take the Facilitators course on Measurement and Geometry. Thank you ExxonMobil Foundation for another incredible learning opportunity!
One of the more intriguing ideas I encountered at DMI was the notion of a classroom as a "learning community." This was first introduced in a series of goals for the seminar, one of which was for participants to experience being in a supportive learning community. This community was further fleshed out during a discussion of routines the group should adopt to support productive discussions, and was continuously enhanced and reinforced during follow-on activities and discussions.
Because of the demonstrated power of the learning community, I decided to use the approach with my middle school Algebra I classes. Several days into the school year, after I had done a series of activities involving hard thinking and discussions of the different methods behind them, I introduced this notion of the class as a learning community. I encouraged them to talk about the conditions needed in the class to enable everyone to feel safe about sharing ideas and discussing their thinking. Suggestions were listed and then a composite list from all four classes was prepared and posted. This list of norms has guided class behavior and aided discipline, acting as a touchstone when necessary to bring the class back to the business at hand. The learning community model has been an effective framework for the classes as we have worked at learning together by sharing ideas about the complex problems we encounter.
On a personal level, the DMI seminar challenged me to accept students with a variety of skills and abilities at their current level, to focus on what they currently understand, and to then move them forward. By pursuing the thinking behind their statements, even when the answer is incorrect, I can find what is right about their thinking and then work to move them forward. Allowing students to share their thinking has produced rich conversations where students have demonstrated an understanding of mathematics ideas and concepts well beyond their current level of study. I have truly been impressed at what my students are capable of when I don’t limit them to the narrow confines of a textbook curriculum. Many of their approaches to problems are considerably more elegant than my own!
The questioning techniques modeled by our course instructors have been so beneficial. I watched them lead us all to understanding by posing the next question. "What do you think about…?" "Do you agree with…?" "What would you do if…?" Our instructors asked someone else to restate what was shared in different words. They expected us to listen carefully for what was said, not for what we expected to hear. There was always wait time. This gave the more cautious responder extra time to process the question and come up with an answer.
These techniques have become new tools in my toolbox. They have helped me reach all of my kindergarten students, giving them the time they need to process and respond. They have given me new insight into my students’ understanding. They will continue to help me reinforce the learning for daily success in school.
Last July I joined a team of three mathematics lead teachers from Bedford County, VA attending the DMI Leadership Institute. I was the higher ed part of the team. In fact, I was the lone traditional mathematician from a large research department who was attending the institute. DMI was not, perhaps, designed for those with my background, but the institute opened windows for me that were totally unexpected on how children (people) learn mathematics. This is my attempt to describe the experience from my perspective. I will also focus a bit on how I came to attend a DMI institute and what I gained from the experience.
Some years ago, I served as the mathematics department chair at the University of Virginia. Like other first tier research universities, perhaps our strongest focus was on research. But as chairman I met and served the public in ways that were new to me. In particular, the problems facing our schools and the assistance that my department could provide to them captured my attention. Over time these issues drew me deeper into K-12 education and into the Virginia Mathematics and Science Coalition (VMSC).
I have worked with mathematics teachers and educators on numerous projects. In recent years, working with support from both NSF and the Federal Eisenhower program, I have concentrated on geometry. These efforts were a constant stimulus and source of inspiration. Along the way I learned about teaching and learning geometry and acquired strong opinions about how the subject can be taught successfully. I also became a champion of elementary and middle school mathematics specialists or lead teachers.
In the spring of 2000, Beth Williams, a third-grade mathematics teacher from Bedford, contacted me and asked me to join her and her colleagues in developing a mathematics lead teacher program. Bedford had worked with ExxonMobil Foundation support for several years to strengthen their elementary mathematics teaching, and Jean Moon had suggested that it was time to develop a comprehensive lead teacher program including, in particular, a lead teacher professional development component.
The content piece of this component was to be my particular contribution to their efforts. Because I had never engaged in the issues of how children learn and build their understanding of arithmetic, I felt ill prepared for the expert’s role, but I agreed to join their team. A year later I was attending the DMI institute to study the modules, "Building a System of Tens" and "Making Meaning of Operations," as the research mathematician in a team whose other members— Pam, Beth, and Paul—taught kindergarten, third-grade, and eighth-grade. The institute became one of the best experiences of my professional life.
Our work, or at least my attention, focused on the case studies in these modules, and I found them absolutely first rate. The teachers who wrote them succeeded—"far more than I could have imagined—in capturing the students’ thoughts and efforts as they built their understanding of mathematics. For the first time, I felt that I was allowed a meaningful glimpse of what is happening mentally as children assemble the pieces of their understanding.
The contributions of our community of learners (the institute participants) were also great, perhaps greater than that which I learned from the case studies. Much of the process in and interpretation of the case studies was invisible to me alone. To see the full picture, I needed not only the help and insights of the teachers, but the benefits that came from being part of a group that included individuals who were building their own understanding of the mathematical ideas. The institute staff was superb at the task of keeping us on target, asking profound questions, and getting strong contributions from all participants.
Having been part of this learning community was very powerful. I have many new friends and I am sure that my teaching will be changed forever. Virginia Bastable’s recurring questions such as, "What are the big mathematical ideas that you have been working on?" and "What does his work show us that he understands and where does he need more work?" will never be far from my thoughts.
I need to report that our DMI and lead teacher efforts are bearing new fruit. I hope to be attending DMI L 2 as part of the Bedford team this summer, and we have just written an Eisenhower proposal with eight school divisions—including Bedford and the VMSC—for a summer geometry institute for lead teachers. Several of these divisions will send teachers to DMI this summer, and the new DMI geometry and shape modules will make up a large piece of the project’s activities.
A year ago last August, our principal, Mrs. Kathy Barnett, and a co-teacher, Mrs. Ellen Knudson, shared at our staff meeting their experiences from a two-week DMI Leadership Seminar. As a result, ten of our staff members signed up for DMI training. We were allowed two half-days a month of professional leave to attend the training. Because of our district’s support, we not only felt that our time was valued but also our quest to improve our own understanding of how children learn mathematics. The two of us joined Kathy and Ellen the next summer to continue our training.
We were excited about our commitment to attend the DMI Leadership Seminar from July 17 to August 3. We looked at it as such an opportunity to grow. We were impacted in so many ways that we really weren’t ready for, but we adjusted. Our own DMI 1 had been with building peers, and at Mount Holyoke, we were sitting shoulder-to-shoulder with fellow teachers from across the country. This was powerful. To be able to have math conversations with teachers from Texas, New York, Florida, Michigan, etc. was rewarding in itself. The study was intense. It took us to places we had never been. The teachers were persistent, knowledgeable and focused.
We gained the insight that our ability to have a constructivist math classroom would happen because we believed in what we were seeing, experiencing, reading about, and studying. We knew we could give children learning experiences that would create the understanding they needed to become mathematicians.
Support is essential to us as we proceed to build a constructivist classroom. It is important to us that our district, administrators and building principal understand and show their support of DMI. We are fortunate that eight of our fellow teachers are continuing with DMI 2 and that twelve more are beginning the DMI 1 training. This year we also have ten other schools involved in DMI 1 training with eight principals and over one hundred teachers.
Within our building we study how children learn. We are able to support and seek knowledge from each other. We help each other with questions, "What am I missing here?" "Where do I go next?" "What questions could I ask to lead the student to the next level of understanding?" Being able to assess student learning across grades 2-6 is "eye opening".
Our first attempt at informing parents about standards-based mathematics at our nine-week parent conferences was successful. We shared student interviews and work that displayed understanding through the children’s diagrams, mathematical sentences, and written explanations. Parents could see what their children understood. It was more effective than pages of skill and drill showing a percentage of accuracy.
We know parent education is so important because standard-based mathematics is not how they were taught. We are aware that we need to send home newsletters about math and host parent math nights. In Bismarck, we are excited about honoring our students as mathematical thinkers and providing them with math instruction that supports this belief. We are finding ourselves gaining confidence in our ability to lead a constructivist classroom. We delight in watching our students make sense of math.
Cathy Kinzer is an elementary teacher, doctoral candidate and preservice instructor for mathematical methods in Las Cruces, NM. She was NM’s Presidential Awardee for Excellence in Mathematics and Science Teaching last spring. Many thanks, Cathy! Ed.
Reviewed by Cathy Kinzer
Thinking Like Mathematicians, Putting the NCTM Standards into Practice, Updated for Standards 2000 (Heinemann, 2001), by Thomas Rowan and Barbara Bourne is a wonderful book for inservice teachers, preservice teachers, and those involved in making decisions that affect mathematics instruction. As I read this book, it made me wish I had a copy for all the preservice teachers I work with because the book delivers a practical vision of implementing NCTM Standards, provides contexts through vignettes, and brings together research behind standards-based teaching. Some of the chapters have discussions, guidelines, planning, and implementation considerations for classrooms built around the NCTM Standards.
The book begins with a brief history of major efforts to reform mathematics and the reasons for the needed changes. The NCTM Standards documents provide the framework for Rowan and Bourne to build their case that students can become confident, successful constructors of knowledge in developing an understanding of mathematics. The authors provide numerous examples of learning opportunities where students develop, modify and reflect on their mathematical thinking and reasoning. Children are actively involved in an ongoing process of thinking, communicating and constructing meaning—all of which are goals of the Standards.
The book focuses on the elementary levels and shares very important teaching and learning strategies that are essential in developing mathematical thinking and empowering students. There are activities, student work, and classroom vignettes that bring meaning to the ideas and discussions of the Standards 2000. One, in particular, that stood out for me involved honoring and understanding the cultural diversity of students. Cultural factors are especially important in regards to language. An example was given of a second grade class where a student was using a counting process that confused his understanding of the value of money and his understanding of the number system in the U.S. Through questioning, the teacher realized that the student’s confusion was based on cultural factors in expressing monetary values.
Thinking Like Mathematicians serves as a great review for practitioners or an essential read for novice teachers. The suggestions provided to put the NCTM Standards into practice are very useful and accessible. The book concludes with questions and answers that weave together the recommended classroom practices.
Thanks to Heinemann for providing this title. Ed.
We’re in the dead of winter. Add some color to your life by making plans now to enroll in one of the spring or summer professional development sessions described below. If you don’t, you might be blue later! Ed.
Thanks to Pat Baggett for providing this information. Ed.
You are cordially invited to our fifth annual Mathematics Education Institute, to be held from 1 PM Saturday, March 16 through noon Tuesday, March 19, 2002, at New Mexico State University in Las Cruces. The Institute is for those interested in university mathematics courses for prospective and practicing K-8 (and higher) teachers. Practicing K-8 teachers are also invited to attend.
Attendees are invited to make presentations at a mini-conference on innovations occurring in K-8 (and higher) mathematics teacher preparation and development in their home institutions, and to see a "partnership" program for prospective and practicing teachers in action. We will visit schools and see lessons studied in two university math classes being taught to children.
The Institute is a part of the New Mexico Collaborative for Excellence in Teacher Preparation, sponsored by NSF. To help defray expenses, we have up to $200 in funds for those attending from inside New Mexico, and up to $300 for those coming from outside the state.
Sunny Las Cruces is a great place to visit in March. For details and on-line registration including submitting papers for the mini-conference, visit the "Breaking Away" web page or e-mail Pat at baggett@nmsu.edu.
This information was sent by Sheryl Russell. Many thanks! Ed.
Mathematical Perspectives will host a Math Leadership Institute for K-6 Teachers and Administrators in Bellingham, WA between July 9 – 12, 2002. Led by Kathy Richardson, the workshop will allow participants to examine how children learn mathematics; learn how to create an environment that supports and encourages each child’s development of mathematical concepts; confront political realities in mathematics education; and discuss common concerns regarding current issues in mathematics with teachers and leaders from districts across the country.
The registration fee is $350/person if payment is received on or before March 1; after that, it is $375/person. All registrations must be received by May 1. Since space is limited, early registration is recommended. A special hotel rate of $69/room/night has been arranged at the Best Western Heritage Inn in Bellingham. It’s located approximately 85 miles north of Seattle and 55 miles south of Vancouver, B.C.
For more information, please visit www.mathperspectives.com. You may also contact Sheryl by phone at (360) 715-2782 or via e-mail at sheryl@mathperspectives.com.
Thanks to Virginia Bastable for forwarding this item. Ed.
For summer 2002, there will be two DMI Leadership programs: DMI L 1 and DMI L 2. All DMI Leadership programs are designed for teams of staff developers, teacher-educators, teacher-leaders, and others who support the professional development of teachers in mathematics grades K-7. The DMI L Institutes are based on the Developing Mathematical Ideas curriculum published by Dale Seymour and are partially funded by the ExxonMobil Foundation. They will be held at Mount Holyoke College.
DMI L I is a two-week program designed for educators who are new to the DMI professional development model. The number and operations modules —Building a System of Tens and Making Meaning for Operations—are the core curriculum of the institute.
DMI L 2 is a two-week program designed for educators who are already familiar with the number and operations modules. A prerequisite is participation in a DMI seminar based on these materials or experience facilitating such a seminar. DMI L 2 is designed to help facilitators dig into both the mathematical goals of DMI and into facilitation issues.
Educators participating in DMI L 2 will choose among four mathematical foci for their work: Working with Data; Measuring Space in 1, 2, and 3 Dimensions; Examining Features of Shape; and Rational Numbers.
DMI L 1 and 2 will be offered during the same time block so teams may include participants in both programs. Programs will open Sunday evening July 7 and close after lunch on Friday July 19. (There are no scheduled activities on the weekend of July 13 - 14). The program fee of $1800 includes room and board and 4 graduate credits in mathematics education. Plan to register early.
To learn more, visit DMI's web page, call (413) 538-2063, or send inquiries via e-mail to smt-dmiinfo@mtholyoke.edu.
It’s almost February, and Cupid’s afoot. Why not get into the spirit of things, and send other readers something even better than flowers or candy—an article from you!
Many thanks to this month’s contributors. Next month’s deadline is Friday, February 15. Please send articles to Jean Ehnebuske, 105 Hideaway Cove, Georgetown, TX 78628; phone, (512) 869-1580; fax, (512) 869-8477; e-mail, jean@intersectionlive.org. Thank you.
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