v
Thinking About Number and OperationsMany thanks to Francis (Skip) Fennell, professor of education at Western Maryland College, for this article. Ed.
Arguably the Number and Operations strand is the backbone of elementary school mathematics—curriculum-wise and expectation-wise. Activities involving counting, place value, and computation have historically been considered important topics within this strand—appropriately so, I might add. However, the Principles and Standards for School Mathematics (NCTM, 2000) broadens these historic expectations to also emphasize composing and decomposing numbers and computational fluency. These PSSM points of emphasis add flexibility and proficiency to number use—an updated spin, if you will, to the number sense emphasis of the Curriculum and Evaluation Standards (NCTM, 1989). PSSM further emphasizes the bridge between the original and revised standards by indicating that "central to this standard (number and operations) is the goal of developing number sense…"(p. 32, PSSM).
This brief manuscript will attempt a response to some frequently asked questions about number and operations, and particularly those elements that embody number sense.
People with number sense are able to use numbers and operations with flexibility. Computational fluency is a key ingredient to such flexibility. People (students or adults) with computational fluency have efficient and accurate methods for obtaining solutions. They understand when to estimate, use mental mathematics or stop and compute—using paper and pencil or calculator. They see relationships. Their methods exemplify conceptual understanding and computational proficiency. In sum, I liken number sense to common sense, involving number and operations. There are lots of examples of such behavior. Here are a few:
v How many different ways can we think about 24? Some typical responses include 6x4; 25-1; 3x8; Christmas Eve, a day, 2 feet, and many more. Such number "stories" are wonderful ways for children to develop a reservoir of meanings for numbers. Teachers can have students extend such stories through the day. This notion of generating equivalent representations by decomposing and composing numbers is a central idea of the grade 3-5 PSSM.
v Try an adult example. What do you do when it is time to approximate a tip at a restaurant? Some use mental math to determine 10% of the bill and take half more for 15%. Others, believing good service should be worth 18-20% of the total bill may take 10% of the bill and double it and then "shave a bit off" based on the quality of the service. In Maryland, a state with a 5% tax rate, people tell me they just triple the tax. A particular technique, of course, is not the issue. Thinking about deriving the amount and using techniques which are flexible and accessible are the points worth discussing.
v Another money-related example would include a problem like this: If you had $20, could you make all of the following purchases—$2.37; $5.67; $9.50; and $5.34? Such examples involve estimation and a level of "operation sense" as we determine that something would have to go back to the shelf.
v Think about this middle grade example. A friend of mine, a very prominent mathematics educator— Rick Billstein—thinks of % off sales in the following manner. Rick thinks of 25% "off of" say $80, as 75% "on," thus reducing the solution to one step, i.e., 3/4 or 75% of $80 is $60. My reaction to "Rick’s technique" several years ago was that of a child in your classroom. Wow, that’s pretty neat. I can do that! In fact, why didn’t I think of that?
v Finally, students need to develop a sense of flexibility and comfort with big (and small) numbers, particularly landmark numbers like 10, 100 and 1,000. A first-grade teacher friend has her students write about 100. She has them respond to the following queries: When is 100 big? When is it small? Students need experience not only with landmark numbers, but with both big and small numbers. Have them ponder situations like: What do you have 1000 of? What does one million look like? What is smaller than 1/2? Than 1/4?
Well this one is tricky, because you actually teach about number and operations— everything from and including counting, basic number combinations, and computation. But, you teach for number sense. To help clarify this point, I turn to an another old friend—Paul Trafton. Paul once used a description I like very much, relative to number sense. He indicated that number sense is "nurtured." I have not only come to learn, but firmly believe that number sense develops over time. Students who receive a rich background in number meanings (counting, subitizing), number relationships (patterns, principles), magnitude (relative and absolute) and understanding operations become flexible and comfortable with number and operations in a variety of contexts. However, I have this concern that while most people acknowledge that perhaps the ultimate goal of elementary school mathematics is for students to leave the elementary grades with a well understood and robust sense of number, people want a quick solution or fix to this multi-layered outcome. Number sense is not intended to be the newest or capstone chapter in a book or curriculum; it is to be the culmination of expectations involving number and operations, and therefore it emerges—but, gradually.
Ah, well here’s an issue to think about. One of the things we all need to do is consider what is important within number and operations—across the grades. If we really want to provide experiences where children can thinatk about and create equivalent representations of numbers, discuss and determine the reasonableness of computational results, and develop computational fluency, something has to give. We need to consider how much instructional time to provide for topics as well as for linking topics within number and operations. Consider this:
v Basic number combinations —while important building blocks for computational fluency—should not dominate a seemingly limited mathematics hour (far less time in many places). Find ways to work with strategies for learnin5g these combinations (note that PSSM has purposely referred to these as number combinations rather than as facts) and provide systematic opportunities for their acquisition.
v Topics within number and operation are not distinct. As children explore counting, they will naturally add and subtract. Multiplication and division may also be linked to early counting and pattern experiences. Such natural links between counting, place value and computation should be exploited.
v Children must also have the opportunity to develop and acquire a sense of understanding relative to fractions and negative integers.
v Finally, we need to be able to say enough is enough with regard to particular topics. Can’t we assume that students have an acceptable level of proficiency in addition and subtraction by the fourth or fifth grade? What determines the amount of time we spend on, say, place value at each grade level?
The issues above are but a sampling of the considerations we must make in order for teachers and children to develop computational fluency and that elusive sense of number so important for all mathematics learners. These words are an attempt to have us begin the process of thinking about issues along such a journey.
Many thanks to Susan Jo Russell of the Education Research Collaborative, TERC, for contributing the article below about the newly-funded project in which she is engaged. Welcome! Ed.
In a third-grade classroom, Katie is working on finding factor pairs for 120 and representing them by cutting arrays out of squared paper. She knows from a previous day’s work that a 6 by 20 array has 120 squares, so that 6 x 20 = 120. She has been experimenting to see if there is an array with one dimension of 3 that has 120 squares. At first, as she and her partner count, they end up with 42 rows of 3 squares and think they have an array of 120 squares, but Katie finds this result puzzling. She rechecks her work and realizes that the array should be 3 by 40 because, as she explains later, "at first we thought it was three by forty-two, but then on another day … it was six by twenty, and we thought, well, if we split six in half and we get three, then we would have to double twenty and we got forty, not forty-two (Russell, 1999)."
Katie could have found the 3 by 40 array simply by counting and recounting, but she was building on an important mathematical idea—that there is a relationship between the factors in equivalent factor pairs. Specifically, she has noticed that if you halve one factor, you must double the other factor to maintain equivalence. At this point in the development of her thinking, Katie may be basing her idea on the physical model she is using: If you cut a 6 by 20 array down the middle lengthwise, you end up with two 3 by 20 pieces. By joining these pieces, you can create a 3 by 40 array, and you have halved the width and doubled the length of the original array.
Katie is beginning to delve into a set of important mathematical ideas that elementary school students encounter at the juncture of number and algebra. While for her and her teacher, the entry into this work is the multiplication and division of whole numbers, the mathematical underpinnings of her argument have to do with the properties of multiplication—properties that hold whether we are dealing with whole numbers, rational numbers, integers, or variables. Katie’s reasoning opens up an opportunity for these students to think about important algebraic relationships: Does Katie’s method work for any factor pair in which at least one factor is even? Why? Can this idea be extended to work with factor pairs in which both factors are odd?
Our ExxonMobil-funded effort—Developing the Story of Algebra in Grades K-5—focuses on laying groundwork for curriculum and professional development work in elementary algebra being planned by our group in connection with the revision of the Investigations in Number, Data, and Space curriculum. The ExxonMobil grant will support a conceptual review and exploratory classroom work so that we will better understand what opportunities exist for significant work in algebra in the K-5 curriculum and what teachers need to know in order to take advantage of these opportunities.
In previous work we have identified two key areas of algebra content in the elementary grades. One focuses on the idea of a function. How can the relationship between two variables be represented and described? What can elementary grade students understand about such relationships? What contexts provide connections to these ideas for students at these ages? Ideas about functions were developed in grades 4 and 5 of our curriculum work (see, for example, the Investigations units, Changes Over Time in grade 4 and Patterns of Change in grade 5). Questions that emerged from this work include: What experiences in grades K-3 lead to the development of ideas about function in later elementary grades? In what ways does work with patterns, especially in grades K-2, lead (or not) to understandings in algebra?
The second area of algebra content focuses on generalized arithmetic. What are underlying structures and generalizations in number and operations? What can elementary grade students understand about these structures and generalizations? This area includes ideas about properties of numbers and operations (for example, Katie’s ideas described above), work with integers, and understanding equality. How can beginning understandings such as Katie’s be developed and used to solve problems? How do students generalize this understanding as they use larger numbers or rational numbers? Is it possible to begin working on some of these ideas in lower grades? What can students in these grades learn about the symbolic representation of these ideas?
In the next months, we will be gathering information from research, curriculum projects, and professional development projects to develop a conceptual landscape of the territory of algebra in the elementary grades. In addition, we will be collaborating with experienced local teachers to take a look together at the possibilities for algebraic thinking that arise in the context of number. Out of this work, we can develop beginning professional development experiences for our local teachers who will then work with us to expand these ideas in the following years. In the long term, we expect to develop:
v a story of the development of algebra, with examples of classroom episodes and student work in grades K-5 that can be used by teachers as a tool to become familiar with what algebra is and can be;
v readings for teachers, culled from our own work and the field; and
v approaches / structures / experiences in professional development for elementary teachers that include solving mathematical problems for themselves and analyzing student work.
Although it’s a challenge in February, your imagination may allow you to find the promise of April in those mounds of snow outside. Wrap yourself in a vision of Orlando, Florida … sun-splashed patios, playful tropical breezes, graceful palm trees, lush liberated houseplants, soaring seagulls, skyblue-pink sunsets. And if that’s not enough, NCTM has scheduled its Annual Meeting in Orlando this year, April 4-7.
As usual, the ExxonMobil Foundation will be hosting a reception in Orlando. It will be held on Thursday evening, April 5. Look for details next month.
Editor’s note: If you’re presenting a session, please let me know. I’d like to publish a list. Thanks.
In the section that follows, you will find reflective articles by Sherry Rosenberg, Charlotte Stadler and Vandi Hodges. Each contributor shares her experiences in DMI (Developing Mathematical Ideas) professional development and what that participation has meant. Many thanks to each of you! Ed.
Approximately ten years ago, Sheri Willebrand approached me and asked me to join a group of teachers who were getting together to experience mathematics as learners as well as teachers. I was in the process of completing my reading credential, and thought it would be one more thing to add to an already overflowing plate. However, when she mentioned that we would also be looking at student work, and processing the mathematical activities in order to understand the concept of the mathematics behind the procedure, I jumped at the chance. I was one of those students who was able to do all the algorithms correctly, but never understood why. When I took trigonometry in high school, I hit the proverbial wall at full speed. Therefore, I eagerly became a part of Four Square, financially and professionally supported by the Foundation with consistent aid from Jean Moon.
My enthusiasm could not be curtailed. My classroom teaching of mathematics started to include supplements of materials from Four Square, BAM, NCTM, Marilyn Burns, as well as my own creations. Other teachers commented about the noise and excitement during mathematics in my classroom. When the assistant-superintendent came across my room, he told the principal he had never heard so much noise in a classroom with actual discovering, discussing and learning going on. He set up an appointment to observe my classroom for a longer period of time.
When I shared with teachers the activities and student work, they merely smiled and thought I was very creative. They commented that when they collected data the first week of fifth-grade, most of the students I had taught as fourth-graders listed mathematics as their favorite subject. The teachers were puzzled that art or PE— the previous favorites—were not listed first. It was not until my colleagues had difficulty teaching a mathematical concept that they would come to me for suggestions. Although most teachers wrote it off as "Sherry being creative," and "It only works for Sherry," eventually one teacher at a time started talking to me about my mathematics program. Slowly but surely, a group at my site began to discuss the mathematics and student work in their classes.
I was excited, but very frustrated, because I felt that although the activities in Four Square were challenging, interesting and conceptually based, the program jumped from piece to piece with no continuum. There had to be something out there that would really help my colleagues and me understand how to be better teachers. The answer was a professional development program supported by the Foundation: Developing Mathematical Ideas (DMI). I found that DMI did not merely present an engaging activity that you try once and never revisit. Instead, DMI is on-going professional development in which you read and discuss student thinking; look at how students learn; acquire deeper mathematics for yourself; share and discuss your own students’ thinking; and always leave with ideas that must be revisited for your own clarification.
After experiencing the DMI training in the summer of 1997, I knew that I needed to change the direction of Four Square. At the time, I was the Teacher on Leave for Elementary Mathematics. I knew that the teachers and students in my district could benefit from this program.
I know that I am singing to the choir, but I want to share my aria.
The first year of DML through DMI (Developing Mathematics Leaders through Developing Mathematical Ideas) was a learning experience for all four trainers. The process continues to be a learning experience. I returned to Mount Holyoke College for the second training session (which I advocate for all trainers) with Jenine Willsrud, and learned how to be a better facilitator. The first year in my district, we had approximately 36 teachers taking Module I. Module II had about the same number, about half of whom were returning. This third year, with 5 trained facilitators, we have 40 participants piloting one of the new modules. Not only have I increased my mathematical knowledge, but I am constantly honing my facilitator’s skills. Among the responses participants have shared:
"I always leave with more questions than answers, and that is O.K."
"This workshop has given me the confidence to teach using Investigations."
I got a chuckle from a participant who commented, "She loves to make us think until our heads hurt."
My most recent journey began the day Virginia Bastable asked me to become part of the facilitator’s training. Last summer, I spent three weeks at Mount Holyoke and worked with Virginia Bastable, Deborah Schifter and Deborah O’Brien as co-facilitator trainers for DMI. I continue to develop and learn through the generosity of ExxonMobil. I especially enjoyed the time Jean Moon and Joe Gonzales spent with us during the training. I appreciated talking with Joe and expressing what DMI has allowed me to accomplish. This was a leap in my professional growth. Being very confident within my own district was not the same situation that occurred for me there working with my mentors, Virginia and Deborah. However, I was pleased to be asked to return the summer of 2001 to again co-facilitate the training of Modules I and II.
Besides growing professionally, I have met many wonderful teachers with various years of experience. They believe, as we do, that mathematics must be the window of the future for all students. In order for that to be accomplished, we, as teachers, must be willing to teach mathematics differently than we were taught. Working in California where the stress on raising scores is the foremost goal in education, I especially cherish the contact with NCTM and with other ExxonMobil sites that encourages me to do what is right for our students.
So, I continue to thank the ExxonMobil family for the many journeys it has enabled me to take—the physical ones that have enriched my geography of other states; the informational ones that have introduced me to diverse 5ways of thinking; the personal ones that have acquainted me with new friends; and the professional ones that have enabled me to expand my knowledge of mathematics and facilitation.
Last July, a team of teachers from New Rochelle who had recently attended a ten-day Math in the City Institute led by Cathy Fosnot from City College and Maarten Dolk from Holland’s Freudenthal Institute traveled to Mount Holyoke College to take part in two additional weeks of professional development in DMI.
Once at DMI, our group of eight broke into two groups. Five of us—three elementary teachers, a middle school math teacher, and the chair of middle school mathematics—participated in DMII. This session helps participan˙˙Pts explore the ideas which underlie elementary and middle school mathematics through analyzing cases and videos of children and by reflecting on one’s self as a learner. Three other participants— Dr. Jeff Korostoff, District Superintendent for Elementary Education, and two math leaders who had had previous experience with "DMII—attended the DMIII session. DMIII also focuses on the cases and videos in the units "Building a System of Tens" and "Making Meaning of Operations" but its emphasis is on helping participants feel comfortable as DMI leaders back in their home communities.
Though all of us came away from DMI with somewhat different perceptions, we all left with the common feeling that this was perhaps the most significant staff development experience we had ever had. This remarkable consensus is particularly impressive since our group consisted of some people who have been teaching less than five years and others who have been in education for more than thirty.
The chance to be living for a while in a college dorm gave us the wonderful freedom to spend blocks of time together, away from the day-to-day demands of job or family. I think we all reverted a bit to our "college days"—staying up late, writing papers, playing logic games, etc. This kind of quality time helped us bond together as a team and gave us a chance to talk about our common goals and problems.
However, this was not primarily what made our DMI experience so profoundly valuable. The authors of DMI—Deborah Schifter and Virginia Bastable—have put together a beautifully crafted program. The videos and cases perfectly complement the math work that we did as participants. Additionally, there is another thread that runs through DMI. This consists of journal entries by Maxine, an "everywoman" teacher-leader who is running a DMI course in her own district. Through Maxine’s eyes you reflect on yourself as a teacher-leader, and think about the situations you will face in using DMI to help teachers grapple with the big ideas involved in learning and teaching mathematics.
DMI is a remarkable course. Next summer we all want to go back to experience the new modules, "Measuring Space in One, Two and Three Dimensions," "Examining Features of Shape," and "Working with Data." Also, several on our math leaders team who were unable to attend in 2000 are planing to take DMII this coming summer. We are immensely lucky that we received a Goals 2000 grant to pay for this training.
In addition, three of us are currently leading a DMI seminar back in our own district. The seminar has been very well received, and the three of us are having a great time. It really seems that the more time you spend with the cases and materials in this program, the more you see and the deeper your own understanding becomes.
Since our district is involved with both Math and the City and with DMI, people have asked whether they have conflicted in any way. On the contrary, I think we all feel that the two programs compliment each other.
In my view, Math in the City focuses more on looking at the "landscape of learning" or the developmental way that children develop the big ideas underlying mathematics. It also helps us focus on ourselves as teachers and on what we can do to facilitate and contextualize learning so that the mathematics flows from the children’s own experiences and ideas.
DMI centers on looking intensively at student work and at student dialogue to help us tease out in what ways a child is making sense of mathematical situations. Both programs help deepen the participants’ own understanding of mathematics and attempt to help teachers see the immense complexity that exists in the development of mathematical ideas.
Those of us in New Rochelle who participated in both programs last summer really immersed ourselves in a lot of math. We loved it, though, and we could see ourselves grow both as teachers and learners.
Like Alice in Wonderland, my experience at the DMI Institute last summer was an incredible adventure. "Curiouser and curiouser" are appropriate words to describe the effect of this adventure on my thinking. It caused me to step back and examine my world and my teaching, and it heightened my curiosity about student thinking and learning.
Before I discuss the adventure, I need to give you a little background on what led to my involvement. About four years ago, Jean Moon visited the Hanover County Schools and encouraged us to focus our next Foundation grant on opportunities for teachers to develop a deeper understanding of mathematics and student learning. We hired Tom Rowan and Anna Suarez as consultants and they began sharing their knowledge and experiences gained in part from their work on the Cognitively Guided Instruction project. Two years later, as a complement to this work, Jean suggested that we send some teachers to the DMI Institute. The two teachers who attended returned enthusiastic about what they had learned. They shared information with the other Exxon leaders and with teachers at their schools but it became apparent that they would need more support if the message was to spread. This past summer two more classroom teachers attended DMII, and I attended DMIII.
I have always enjoyed traveling to new places, meeting new people and learning new things, so I had anticipated that the week at Mount Holyoke would be interesting. I was anxious to see a different part of Massachusetts, meet participants from all over the country, and learn more about reflective teaching. Still, I never expected to be so thoroughly captivated and inspired. After a delayed flight out of Richmond, VA and a dubious standby connection, I arrived late the first evening. Confused and tired, I felt somewhat disoriented, like I had fallen through the rabbit hole. Yet when I arrived at the dormitory, lights were bright, the conversations between old and new friends were animated, and Virginia Bastable greeted me with open arms. The adventure had begun.
Though I had taught and supervised mathematics for twenty years at all grade levels, I recognized that I still had much to learn both about mathematics and about how children learn mathematics. I was, however, over-confident. In my cockiness, I had asked to attend for only one week and to "audit" the DMIIi course. Dr. Bastable graciously accepted me, but due to high enrollment in DMII, she asked me to enroll in DMIII for teachers whndo already had experience in DMII and who would be learning to facilitate sessions. Since this was what I needed to do in my district, I had agreed. To further illustrate my cockiness, I must confess that I waited until I was on the airplane to begin reading the prerequisite articles—adding to my blurriness when I arrived. As I read, I quickly realized the enormity of my ignorance as I began to recognize the sophistication of the research that is the basis for DMI. I was about to embark on an academic journey that was going to stretch my mind. I began to get nervous. Was I prepared? Only time would tell.
One calming element was the idyllic setting for my adventure. Mount Holyoke is a beautiful campus with two lakes and a tremendous waterfall cascading right outside the dormitory. Another reassuring factor was my fellow participants. We bonded the first morning as we walked down the hall to the common bathroom and showers! After the first of many delicious meals, we were ready to get to work. My three leaders—my white rabbits— were mast tier teachers and experienced facilitators who tactfully began to guide and encourage us. They set the tone for everyone’s success. They challenged our thinking and patiently waited for responses. Each day they pushed a little more but they were careful to always be supportive. Our work focused on two DMI modules of 25-30 cases each. Most of the participants in DMIII were familiar with the two casebooks, "Buialding a System of Tens" (BST) and "Making Meaning for Operations" (MMO).
For me, everything was new so I read and studied like I hadn’t done since graduate school! The good news was that I was hooked from the first case study. The reading was fascinating. Like a good novel, it was hard to put down. Each module was about real children and what and how real children think about mathematics. The variety of approaches the children used to solve problems was amazing. At times, a child’s strategy might not appear to be mathematically sound, but on careful examination and through skillful probing by the teacher, the connections became clear.
Homework was expected, so at night we read and analyzed the case studies and did related problem-solving activities. Doing the activities allowed us to analyze our own strategies for solving problems and relate our approaches to those of the children in the studies. In class, we saw video clips related to the case studies and we worked individually, in pairs, and in small groups to share our thoughts. Listening to the insights and experiences of other participants helped me stretch my thinking and gain a deeper understanding of the issues.
Learning to facilitate a workshop is a goal in DMIII, so each of us was grouped with two or three other participants and assigned a lesson to present from the MMO module. Five or six other class members were our audience. They participated in the lesson and then critiqued our presentation. My group presented a lesson on division and fractional amounts. The participants had to develop their own word problems involving operations with fractions using some basic information. They could use chart paper, cubes or other manipulatives to help with their work. After working in small groups to write their problems, we shared responses. The participants were asked questions such as these: "What did you need to know about division and fractions to complete these problems?" "What issues did you discuss when you were creating the problems?" "What mathematics would a child need to know to solve these problems?"
Facilitating this discussion and formulating key questions was a challenging task that was essential to our learning and a critical component of the lesson. Though it was intimidating to present a lesson to our peers, the satisfaction and the rewards of successfully accomplishing this assignment helped all of us gain confidence in our ability to lead a group. We left with newfound friendships and a renewed love of learning.
I would recommend the DMI Institute to anyone teaching mathematics. I would encourage administrators and supervisors to attend along with their teachers. With administrators’ support, study groups can be established in a school or across a district so that the benefits of this program can be shared. We want to spark the curiosity of teachers so they will reflect upon their students’ work and we want students to develop a curiosity that will help them explore the world of mathematics and beyond! The DMI adventure can open the mind and imagination. This is a rabbit hole you don’t want to miss!
The Lamp, an ExxonMobil publication for shareholders, features a story in its most recent issue about the approach and strategies Donna and Brian Kaumo use to effectively teach mathematics to learning-disabled fifth-graders at Whittier Elementary School in Albuquerque, NM. The article will be reproduced in next month’s Intersection, but if you’d like to read it now, please visit the ExxonMobil web site.
Many thanks to new contributor, Gail Gibson, principal of Mapleton Elementary School in Mapleton, ME for sending the first review below. Thanks, also, to Jennifer Kibler, Parkdale Elementary School, East Aurora, NY, for her review of the second title. Ed.
Reviewed by Gail Gibson
The consortium our district is a part of was recently awarded an ExxonMobil Mathematics Grant. During the past summer, four teachers from Mapleton Elementary School, the building of which I am principal, attended a weeklong math institute. The timing was perfect since we are in the process of creating a new math curriculum. I was not able to attend the institute but because I am anxious to support the teachers as they work to improve their math instruction, I spent some time perusing articles and tradebooks the teachers shared.
Martha LaPointe introduced me to Leading the Way: Principals and Superintendents Look at Math Instruction (Math Solutions, 1999). Edited by Marilyn Burns, this book contains only seven chapters, but they are filled with many helpful hints. The first is an introduction by Marilyn where she states: "Just as there’s no one way for teachers to teach math effectively, there isn’t one right way for principals to be instructional leaders in this area [math]."
The other six chapters, written by administrators from across the United States, tell the unique stories of how they worked to improve mathematics instruction in their districts. Each district represented has been involved with Math Solutions inservice. Each district approached staff development in a unique way that met the needs of its teachers. Because I have spent the majority of my career teaching language arts, I was especially interested in Bob Wortman’s chapter entitled "Building Community through Mathematics: The Principal’s Role." Bob used the framework from Brian Cambourne’s model for how children learn language as the basis for reflecting on the teacher as learner and on learning in general. He told the story of his school’s staff development practices using Cambourne’s conditions for learning as a framework. Since I am very familiar with this model, it was easy for me to identify with this chapter. I felt very comfortable using these ideas to help me analyze our school’s needs.
Although I enjoyed the other chapters, most of my reflection was based on this chapter. Interestingly, when I reread the book to complete this review, I found that the other chapters also contained helpful and meaningful information. So, in short, this is a useful reference book that I want to keep on my desk. Each chapter stands independently. The introduction reminds administrators of the importance and the complexity of effective sitaff development in improving math instruction, as well as in other curricular areas. This is definitely a book that I would recommend to other administrators.
To order, please visit Math Solutions Online. Ed.
Reviewed by Jennifer Kibler
Sensible Mathematics, A Guide for School Leaders (Portsmouth, NH: Heinemann, 2000) is an excellent resource for educators involved in exploring changes in mathematics education. In the introduction, author Steven Leinwand points out that as we enter the twenty-first century, changes are being made to meet the national standards for school mathematics. He states that school leaders "help people envision the possible, show people what can be done, broaden people’s understanding of why change is being made, and garner support for these changes within the broader community." Throughout the book, Leinwand provides a wide variety of skills, strategies, tools, and plans to support the efforts of school leaders, administrators and teachers.
In Chapter 1, Mediating, Forestalling, and Even Winning the Math Wars, Leinwand discusses some changes that are being advocated and focuses on some common questions being raised about these changes. Questions from teachers, parents, and community members are all included in the discussion. The author highlights some school success stories and is quick to point out that in each case, "the principal or other school leader made the difference. He or she carefully analyzes the situation and strategically plans a course of action."
Next, Leinwand makes the case for change and offers strategies and compelling examples. The second chapter takes an in-depth look at four domains of change: Changes in Our Society, Changes in Expectations, Changes in Students, and What the Traditional Program Hath Wrought, and provides support for one justifying the reform of a school mathematics program. The discussions incorporate classroom strategies and experiments which can help to build the case for reform.
Once change is justified, it makes sense to focus on curricular shifts, instructional shifts, and assessment shifts. Leinwand outlines each of these topics and goes further to show many specific and concrete examples of "Where We’ve Been" and "Where We’re Moving To." The author concludes the chapter by taking the reader through a sample lesson using the traditional approach (before the three shifts mentioned above) and then using the alternative approach (after successfully carrying out all three shifts).
In Chapter 4, Building Sensible, Sense-Making Mathematics: What to Encourage and Implement, Leinwand explains that the "end result" of the three shifts described in the previous chapter is "sensible, sense-making mathematics." He states, "Sense-making mathematics is the mathematics of rules, procedures, techniques, and concepts that make sense to students." He outlines eight critical characteristics of sensible, sense-making mathematics. Leinwand does a fine job of showing what each of these characteristics looks like in practice so that principals and school leaders can both recognize and encourage the incorporation of these characteristics in their school’s program.
What should we see when all of these aspects of a high-quality math program are pulled together and put into practice? What should an observer expect to see in the classroom? Leinwand answers these questions by describing what the classroom should look like and feel like. Classroom environment, the role of the teacher, the roles of the students, homework, tests, quizzes, and grades are all mentioned and described. The author states, "Classrooms are active environments—often noisy, but clearly productive and purposeful—as students converse mathematically and wrestle with ideas in the course of solving interesting problems."
In Chapter 6, Leinwand offers insights and practical strategies for assisting individual teachers and/or teams in recognizing and overcoming obstacles. Some possible obstacles include professional isolation, lack of confidence and fear of change, fear of failure, lack of support, and insufficient time. "The deep, dark secret in schools is that all teachers and all schools are not the same." Since all teachers and all schools are not the same, the author presents many possible obstacles which may be encountered and a variety of strategies to overcome them. Different strategies will be appropriate for different teachers and classrooms.
In Leinwand’s final chapter, he offers one last tool for use on the road to reform. This extremely practical tool is a list of fifteen program components that should be considered throughout the process of changing a school’s mathematics program. The list begins with curriculum and ends with administrative understanding and support. Leinwand explains that "any and all discussions and planning relating to the improvement of mathematics programs must attend to each of the [fifteen] components." This list of components can also be used when reforming other content areas in the curriculum. Readers will find that this list provides guidance for a group of administrators or teachers to confidently and thoroughly reform any area of curriculum.
In his conclusion, Leinwand mentions that not everything requires change. "There is so much that is good in America’s elementary, middle, and high schools. There are thousands of pockets of extraordinary accomplishments occurring daily in America’s classrooms. "Every school and every district should engage in commensurate promotion of the positive."
Lastly, readers will find that the appendix contains eight wonderful articles to be used as discussion starters as schools and teams of teachers begin to change programs. The articles express many issues and obstacles that may be encountered throughout this reform process.
This book is a great resource for those considering program reform. I would recommend this guide for administrators, curriculum coordinators, and school leaders, as well as for individual teachers and teacher discussion groups exploring changes in school mathematics programs.
Please visit Heinemann's web site to order. Ed.
If you’d like a really fine read that will help you grow professionally, volunteer to review the new Heinemann publication, Teacher Leadership in Mathematics and Science: Casebook and Facilitator’s Guide. Jean Moon is one of the authors along with Barbara Miller and Susan Elko. Here are some comments from Lance Menster with the Annenberg Challenge, Houston:
"We have been using the cases from Jean Moon’s Teacher Leadership book in Houston. In the fall, our math initiative began with five new math specialists, each of whom received a copy. The specialists were captivated by the cases, and desired time to discuss the content of them. The teacher-leaders found that they are part of an even larger picture outside of Houston as people like themselves are grappling with real, complex and dynamic situations in creating and sustaining change in mathematics.
The cases are multi-layered with leadership themes from negotiating relationships, defining roles and sharing expertise. A note of additional praise—the facilitator’s section after each case isoci superbly written with strategies and options for creating an extensive conversation around the cases. We continue to look forward to working with the cases extensively as our work in Houston evolves."
There are three copies available, and there will be space in an upcoming issue for three reviews from three perspectives. One copy can be yours if you’ll review it. What do you say?
Your fellow readers would love to hear about your project, colleagues and students. So, have a heart—send something for the February newsletter!
The deadline for the next issue is Monday, February 12. Please send contributions to Jean Ehnebuske, 105 Hideaway Cove, Georgetown, TX 78628; e-mail, jean@intersectionlive.org; phone, (512) 869-1580; fax, (512) 869-8477. Many thanks!
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