virginia mathematics teacher · 2011. 9. 28. · volume 38, no. 1 fall, 2011 virginia mathematics...

VirginiaMathematicsTeacher

A Resource Journal for Mathematics Teachers at all Levels.

Volume 38, No. 1 Fall, 2011

THE MATHEMATICS SPECIALIST

Volume 38, No. 1 Fall, 2011

VirginiaMatheMatics teacher

The VIRGINIA MATHEMATICS TEACHER (VMT) is published twice yearly by the Virginia Council of Teach-

ers of Mathematics. Non-profit organizations are granted permission to reprint articles appearing in the VMT provided that one copy of the publication in which the material is reprinted is sent to the Editor and the VMT is cited as the original source.

EDITORIAL STAFFDavid Albig, Editor, e-mail: [email protected] Radford University

Editorial Panel Bobbye Hoffman Bartels, Christopher Newport University; David Fama, Germana Community College; Jackie Getgood, Spotsylvania County Mathematics Supervisor; Sherry Pugh, Southwest VA Governor’s School; Wendy Hageman-Smith, Longwood University; Ray Spaulding, Radford University Jonathan Schulz, Montgomery County Mathematics Supervisor

MANUSCRIPTS & CORRESPONDENCEFor manuscript, submit two copies, typed double spaced. We favor manuscripts on disk or presented electronically in Word. Drawings should be large, black line, camera ready, on separate sheets, referenced in the text. Omit author names from the text. Include a cover letter identifying author(s) with address, and professional affiliation(s).

Send correspondence to Dave Albig at: Box 6942 Radford University Radford, VA 24142

Virginia Council of Teachers of MathematicsPresident: Beth Williams, Bedford County SchoolsPresident-Elect: Ian Shenk, Hanover Public SchoolsPast-President: Carolyn Williamson, Retired from Hanover County Public SchoolsSecretary: Lisa HallNCTM Rep: Paul Webb, Forest Middle School (Bedford)Math Specialist Rep: Corinne MageeElected Board Members: Elem. Rep: Meghann Cope, Bedford County Schools Middle School Reps: Skip Tyler, Henrico County; Alfreda Jernigan, Norfolk Public Schools Secondary Reps: Lynn Foshee Reed, Maggie L. Walker Governor’s School 2 Yr. College Rep: Joseph Joyner, Tidewater Community College 4 Yr. College Rep: Lou Ann Lovin, James Madison University; Maria Timmerman, Longwood University Membership: Ruth Harbin-Miles

Publicity: Laura Scearce, Battlefield Park Elem and Cold Harbor Elem

Treasurer: Diane Leighty, Powhatan County Public Schools

Assistant to the Board: Ruth Harbin Miles

Webmaster: Jennifer Hackley, Charlottesville City Schools

Webpage: www.vctm.org

Membership: Annual dues for individual membership in the Council are $20.00 ($10.00 for students) and include a subscription to this journal. Registration is available online at the VCTM website.

Printed by Wordsprint Christiansburg225 Industrial Drive, Christiansburg, VA 24073

TABLE OF CONTENTS

Grade Levels Titles and Authors ................................................................. Turn to Page

General President’s Message ..........................................................................1 (BethWilliams)

General Year 1: Why am I here? ......................................................................2 (KathyDonovan)

General Keeping the Balance ..........................................................................4 (LizSinclair)

General Affiliates’ Corner .................................................................................5

General How Do Math Specialists Affect Change in School ............................6 (ChelyseMiller)

General Early Algebra and Mathematics Specialists ........................................8 (M.K.Murray)

General Approaching Mathematics Utopia? ...................................................12 (F.Morton)

General Resource Review..............................................................................16 (SusanO’Connell;MargaretS.Smith&MaryKayStein; JulieMcNamara&MeghanM.Shaughnessy;KariEverett)

General Problem Corner ................................................................................19 (RaySpaulding)

General Ten Strategies for Maximizing Opportunity with Instructional Coaching ......................................................................29 (TheresaWillsandMollyRothermel)

General Meet Billy ..........................................................................................32 (CandaceStandley)

General Math & What I Know for Sure: Reflections of a Soon-to-Be Math Teacher ....................................................................................34 (AntaresWinslow)

General Overcoming Resistance to Change: Why Isn’t It Working? ..............36 (TedH.Hull,RuthHarbinMilesandDonS.Balka)

General Mathematics Leaders’ Belief Statements .........................................39 (DianeKinch)

ABOUTTHECOVER:ThecovergraphicistheworkofJasonSchermofWestmorelandCountyPublicSchoolsandaparticipantintheNSFRuralMathematicsSpecialistGrantwithVCU,UVA,NSU,andLU.

Virginia Mathematics Teacher 1

GENERAL INTEREST

President’s MessageBeth Williams

Welcome to our exciting ‘Mathematics Specialist’ Edition of the Journal. Your colleagues across the state have gra-ciously shared stories and experiences from their practice. Our thanks go out to all of them as we enjoy this very special Journal. Many exciting things are happening across our Virginia Council. At our summer retreat, the VCTM Executive Board met to begin the planning process for our Annual Conference coming up in the spring of 2012. Make your plans now to attend our Annual Conference at the Hotel Roanoke March 9th and 10th. Our conference theme this year is “REACH FOR THE STARS: SOL and Beyond. “ We will highlight and define the NCTM Process Standards in the conference presentations as colleagues share ideas and strategies to improve mathematics instruction. Speaker proposals for our Annual Conference are posted on our website. Please consider sharing your expertise with your colleagues at this conference next spring. Sign up to present and register now at www.vctm.org Our own Virginia Department of Education Mathematics Team is hosting presentations this year in four locations across the state. The goal of these presentations is to improve mathematics instruction by providing district-level trainers with professional development resources that focus on facilitating mathematical communication and reasoning through problem solving. The trainers will bring back to their divisions these resources that focus on the process standards and the strands of mathematical proficiency. Both describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, represen-tation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up. These strands are adaptive reasoning, strategic competence, conceptual understanding comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and pro-ductive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). The content of the math-ematics standards is intended to support the following five goals for students: becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical

representations to model and interpret practical situations. More information about these workshop presentations and registration forms will be sent out via a Superintendent’s Memo in August. The dates and locations are: September 27 – Abingdon, September 28 – Roanoke, October 18 – Richmond, October 19 – Fredericksburg. Did you know that NCTM keeps an ongoing Capitol Report on the www.nctm.org website? Every two weeks an up-dated report keeps you informed about actions and decisions taking place in our nation’s capitol. In the July 6th report, Ellin J. Nolan, NCTM’s Government Relations Consultant, wrote about “Plan B” for the passage of the new Elementary and Secondary Education Act (ESEA). The July 20th article discusses a bill that has been put forward to work towards getting the No Child Left Behind Act (NCLB) replaced by updated legislation. This is a great site to bookmark to keep up with the mathematics educational happenings in Washington. As promised in my last message, this time, my question for you would be:

What do you know about student growth percentiles?

• A student growth percentile expresses how much progress a student has made relative to the prog-ress of students whose achievement was similar on previous assessments.

• A student growth percentile complements a stu-dent’s SOL scaled score and gives his or her teach-er, parents and principal a more complete picture of achievement and progress. A high growth percentile is an indicator of effective instruction, regardless of a student’s scaled score.

As the process of the evaluation of teachers’ changes across our state, these growth percentiles are part of the conversation. There is a lot of information available to you on the DOE website. Please read it at the following link and add your educated voice to the teacher evaluation conversations. http://www.doe.virginia.gov/testing/scoring/student_growth_percentiles/index.shtm

As this new school year begins, your VCTM organization will continue to support your learning, living and love of mathematics.

Best wishes to you all!Beth

2 Virginia Mathematics Teacher

GENERAL INTEREST

Year 1: Why am I here?Kathy Donovan

It was the end of the first 9-week marking period during my first week as a Math Coach. Up till this point, I really wondered what I was doing here. I wasn’t sure if I had con-nected with anyone (teachers or students) and was equally unsure of my role at this point. Report cards had just gone home and one afternoon, right after dismissal, I wandered into the office and a little girl said to me, “Ms. Math, now that you are here, I do good in math! I used to get Ds in math, but now I get Bs!” As they say, out of the mouths of babes. Here was a little girl whose name I couldn’t tell you, whose class or even grade level I didn’t know. But, thanks to her, it suddenly occurred to me that maybe I had made a mark! I came to teaching late in life – my children were both in college and my spouse was getting ready to retire from 30 years in the U.S. navy. I wanted to do something to help supplement our income and after spending time in my own children’s classrooms and being a fulltime substitute teacher, I decided to go back to school to get certified as a teacher. As of now, I have just finished my first year as a Math Coach in Norfolk, after spending the past 6 years in a 5th grade gifted cluster classroom. I truly love the teaching profession and have always felt like I was making a difference in someone’s life. When the opportunity to earn a Masters Degree and Math Specialist Certification was offered, I jumped at the chance, without really internalizing the fact that I would eventually have to leave my own classroom. When I was asked to interview for the position after two years of the Math Specialist Program, I did and was hired to become the ‘math lady’ at a school different than my own, where I knew no one. I wasn’t nervous because teachers are the friendliest and most welcoming people on earth. However, I hadn’t realized how sad I would feel on the first day of school when I didn’t have my own brood to guide and set up for success. Students were leaving saying good bye to their own teacher, but no one said goodbye to me! I felt like a nobody. During the first week of school the resource teachers went around to each classroom, and we introduced ourselves and gave the students a little insight into what we would be doing with them during the year. I teamed up with our new Instructional Specialist, Mr. Goodman. Between the two of us, the students were now confronted with two new people at the same time – Mr. Goodman and Mrs. Donovan. They couldn’t keep us straight! We heard all sorts of combinations – Mr. Goodvan, Mrs. Donoman. It was a riot! Finally, after continually reminding the students that I was the math lady, they hit on the solution of calling me Mrs. Math – so that is who I became. In most classrooms, at least with the younger students, I am now Mrs. Math. And I love it! I don’t hesitate to let them know that I love to read, talk science, and learn history, but my main message is that “Math Rocks – and we use it everywhere!” And thus began my connection with the students as a math coach. Generally speaking, the goal of the math coach is to im-prove the teaching of mathematics so that students will be challenged and pushed to maximize their learning experi-

ence. Basically, I want to make teachers so good that they don’t need me! Methods for doing this job vary with the school, other teachers, the individual math coach, the desires of the administration, and the needs of the students. Part of my job is creating grade level assessments every 6 weeks or so. I meet with teachers to review their class data from the assessments, and try to give advice as to what and how to teach and review various math topics. I also try to inspire them to pre-plan their own lessons and to incorporate many different styles of instruction with the hope of engaging all learning types. This is more effective with some teachers than with others. During the first few months of school, I tried to get into as many classrooms as possible, just to visit and see what math felt like at this new school. I was also still trying to sort out exactly what my role would be, and how I would serve each teacher’s needs. I know what I did in my own classroom, but what do other teachers do? I quickly learned that my presence in the classroom was not always welcomed by the corresponding teacher! I think many teachers viewed my visit as an observation, or a critique of sorts. After each visit I tried to send a note right away with at least two things I liked that I saw happening, and one or two things that could use improvement. I think word got around that I was coming into classrooms because some teachers asked if I was also coming see them. This still was not necessarily a welcome invitation to come in to their classrooms, but more so a question of curiosity – possibly to see if they needed to update their calendar math, or change anything they were doing. In the end, I was still unsure of my impact. In continuing to work with the teachers, my initial goal was to offer grade level professional development once every month, if possible. However, I found that often times I only got a portion of any one grade level team at a time. Some-one always had something more pressing to attend, or had another excuse to not be there. I struggled with convincing teachers to create their own daily math reviews, based on class needs, versus using a premade review from years past or created commercially. Because we had two long-term substitute teachers in our third grade, I began to make the weekly reviews for that grade. Luckily, the other teacher that taught math to the other two third grade classes used what I created too, so she saw how simple it was to do. As happens in many schools, there is heavy emphasis on the testing grades, so a lot of my time is spent in the upper grades (3-5) getting them ready for our high stakes tests. However, my office is located near the kindergarten classrooms and I had gotten to know some of those teach-ers quite well. I had tried to get in to visit with the students as often as possible. About half way through the school year, one of the kin-dergarten teachers asked me for some help with teaching shapes – she said that she had done everything she could think of and needed some help. I offered to come in and do a model lesson, using literature and some interactive


whole group strategies. She was thrilled. I did the lesson in her classroom and the kids responded very well – I read the book “The Greedy Triangle” by Marilyn Burns and had anchor charts of many of the basic shapes required for kin-dergarteners, with labels for the sides and corners that they placed on the shapes. That way I could leave a tangible record of my presence and the lesson – the teacher could refer back to it whenever necessary and, I hope, give the students some ownership of their new knowledge. About a week later one of the other kindergarten teachers mentioned in passing that she had heard that I had come to the first teacher’s classroom. Teacher number 2 felt slighted! I offered to come right away and, indeed, the very next day did the same lesson with that kindergarten class – with a few fine tunings that I learned from the previous lesson. A couple days later, another kindergarten teacher asked for the same thing and later the 4th, and final, kindergarten class asked me to come in. For the rest of the school year the kindergarteners would see me in the hallway or at lunch and show me their triangle, or another shape that they saw around them in the floor tiles, the ceiling panels, or bulletin boards nearby. This habit translated to counting by 5’s as we passed in the hall, as they learned that new skill. Each and every time they showed me a shape or counted to 30 by 5s it was as if they were as proud as the first time they did it! It was fabulous! The students were engaged in the topic and enthused about their learning. This is what I wanted! This was what we needed! This would make a difference! By the end of the third quarter I had spent a lot of time in two of our third grade classes assisting a couple of long term substitute teachers in those classrooms. I knew the students

quite well. Prior to the beginning of our SOL testing, at the request of the third grade team leader and the principal, I created four weeks worth of activities that would review the school year in ‘clumps’. One teacher and I would take those students who struggled with their quarterly testing and the SOL Practice test, and the others would be grouped with the other two third grade teachers with enrichment activities along the same SOL as what we were doing. The activities varied from drilling of the basics like place value to algebraic thinking in the third grade. We did menu-like activities after mini-lessons that focused on the skills that were most difficult for these students. By the end of the four weeks, prior to administration of the SOLs, we felt like we had made some headway with and for these students. Looking back over the year, I went from wondering why I had left the classroom to believing that I can make a differ-ence. I think with the show of support that I received last year, more teachers will be willing to let me in this year. I look forward to being a part of their planning and helping them to identify ways to improve instruction and thereby improve student ability and test performance. I hope that they will let me visit their classrooms and allow me to model and/or co-teach so that we can be sure that all students are engaged and acquiring the necessary tools to be successful. I still miss having students say “That’s my teacher!”, but am I becoming more and more satisfied with simply hearing, “That’s the math lady!” It’s a big transition, and I am taking baby-steps. Math is everywhere!!!

KATHY DONOVAN was a Math Coach with Norfolk Public Schools. Kathy Donovan died in May of this year.


GENERAL INTEREST

Keeping the BalanceLiz Sinclair

For the past six years I have served as a math special-ist/math coach in the Alexandria City Public School divi-sion. The theory behind a coach’s role is that everyone is involved in the learning cycle. Improving student learning is a group process and all are participants in life-long learn-ing. Students’ learning improves as teachers improve their practice. The goal of instructional coaching is to cultivate teachers’ academic habits of reasoning and discourse as-sociated with their particular discipline. It also helps them develop a specific skill set that will enable them to cultivate those habits in their students, habits which will promote student appreciation and understanding of the subject at hand (West, 2009). I am fully committed to this process. The most challenging part for me has been maintaining a balance between being nice to everyone vs. stepping out of my comfort zone to have crucial conversations, for suc-cess will not be had while remaining solely in one of these domains. When I first began this work, I felt I needed to befriend everyone. When teachers were resistant to work with me, I took things personally and was challenged by trying to find ways to improve teachers’ practice while establishing a trusting relationship with them. I’m not a quitter so I sought out advice and guidance from my supervisor, more expe-rienced coaches, and my mentor. In addition, I have had tremendously helpful professional development over the years – from Lucy West to Skillful Leader to Understanding by Design. All of this learning helped me adopt a laser-like focus on high expectations for ALL. This includes students, teachers, and most importantly – me. In my work with teachers my mantra is, “Everyone has something to learn; everyone has something to bring to the table.” Over the years I have learned as much from the teachers I coach as I hope they have learned from me. In order to collaborate around student achievement, it is imperative that coaches, teachers, principals, supervisors, and superintendents all see themselves as lifelong learn-ers. We adults must model this stance for the students so that they in turn will consider themselves lifelong learners. It is also incumbent upon us to model perseverance and struggling. Learning is about hard work, not passively ac-cepting a teacher “delivering” the curriculum. The Skillful Leader curriculum taught me much about teaching and learning. The most powerful lesson I learned was with regard to expectations. The bottom line of this area of performance is sending three key messages to all students: 1. This work is important 2. You can do it. 3. I won’t give up on you. (Saphier, 2008). We simply must have high goals for our students or they won’t have them for themselves. We are educating the next generation and it is our responsibility to engage them in 21st century skills so they will be successful in their endeavors. We also must have these same high expectations for the teachers we coach. Don’t we all want to be as good as we can be? We must continually model our commitment to be life

long learners. I have the utmost respect for the challeng-es facing the elementary classroom teacher. Don’t they have to teach math, science, social studies and language arts? There’s an inordinate amount of pressure on the classroom teacher. There is the content to master, which for many elementary teachers is a challenge. Then, there’s the pedagogy to be learned. Over the past several years, I have worked closely with teachers in guiding them in using instructional strategies to deepen students’ understanding of the mathematics. A focus on discourse across the disci-plines has proven to be a strategy that works. I have also found that engaging the teachers in professional learning communities provides time to plan, reflect, and analyze their lessons. What was the goal of the lesson? How was the goal maintained throughout the lesson? How were stu-dents engaged? What was the cognitive demand of the tasks? Were the problems worthwhile? What did the stu-dents do if they got stuck? How did they know if what they did was of good quality? “The Land of Nice” is where we don’t have the courage to move beyond pleasantries. It is a place where no one is challenged to push forward. It is a place where student achievement remains stagnant and expectations remain low. After reading Courageous Conversations by Kerry Patterson (Patterson et.al., 2005) I took on a whole new view of the importance of pushing forward with things that need to be said in order to improve relationships or pro-fessional growth. As a change agent, I have an important role to play in challenging others to do better in the name of student achievement. It’s not personal. It’s non-defen-sive, as I learned from Lucy West. It’s not about “us.” It’s simply about having open conversations about what drives student achievement higher. These conversations can be challenging but they are necessary and pivotal. I know and feel it when I’ve been pushed out of my comfort zone. It is discomforting but it often pushes me to improve my prac-tice. The disequilibrium is good because that is the place where we take the next step forward. Now, my personality is not to completely leave the “Land of Nice.” Some of that niceness is absolutely necessary. It’s important to get to know our colleagues’ interests, what drives them, and then ultimately to establish trusting relationships with them. I enjoy that part of my work – knowing them personally as well as professionally. Taking on the role of change agent as a math coach re-sults in planning our lessons in a differentiated manner, not just to reach those who may need some scaffolding, but also challenging those who need to reach a higher level in order to be engaged. It’s about allowing students to strug-gle and come up with strategies that work for them. It’s about perseverance. It allows for more student talk, less teacher talk. Formative assessment becomes the natu-ral norm. It is incumbent upon us as math specialists to meet the teachers where they are in order for them to be equipped to meet their students where they are. It is also


our job to learn as much as we can from the colleagues, teachers, and students who cross our path in order to keep getting better and moving forward. This important work doesn’t happen in “The Land of Nice,” where everyone remains in their comfort zone, unwilling to challenge the status quo. This past school year I was coached by Lucy West in front of 30 or so colleagues. It was challenging to say the least. I certainly had moments over the course of the day when I felt I was not where I wanted to be. I kept remind-ing myself that it was not about me. It was about all of us learning together. There was an “aha” moment for me, when critiqued by Lucy, where I reminded myself to be non-defensive and think about the learning experience for everyone that day. Once I stepped outside of myself, I real-

ized my hope was that each and every one of the coaches and specialists added something to their repertoire. That’s what it’s all about. I feel that I am maintaining a good balance at this point in my career. This, by no means, allows me to stop my drive for learning. I have books in my car, on my nightstand, and in my briefcase. I know I will never be where I’d like to be. But, that’s a good thing, for that’s when the learning stops. So, each year I look forward to the many challenges that I may face and keep reminding myself that I will only experi-ence success by keeping the balance between “The Land of Nice” and “Crucial Conversations.”

LIZ SINCLAIR is an Instructional Coach in the Alexandria City Public Schools.

GENERAL INTEREST

Affiliates’ CornerBATTLEFIELDS of NORTHERN VIRGINIA Fall Conference: Saturday, November 5 at Benton Middle Schooll, 8 AM to 1 PM President Kat Meints; [email protected]

BLUE RIDGE Fall Conference: Monday, October 3, 4 PM to 8 PM President Jonathan Schulz; [email protected]

GREATER RICHMOND Middle School Math Field Day: Tuesday, October 11

Fall Conference: Tuesday, October 18President Carrie Persing; [email protected]

NORTHERN VIRGINIA Fall Seminar: Tuesday, October 18 at NCTM Headquarters, Richmond President Gail Chmura; [email protected]

PIEDMONT Fall Conference: Tuesday, October 11, at Longwood University, 9 AM to 3 PM President Sharon Emerson-Stonnell; [email protected]

RAPPAHANNOCK Fall Conference: Tuesday October 4 at Gayle Middle School, 5 PM to 8:30 PM President Pam Bailey; [email protected]

SOUTHWEST VIRGINIA Fall Conference: Saturday September 17 at Abingdon Higher Education Center, 8 AM to 1 PM


GENERAL INTEREST

How Do Math Specialists Affect Change in SchoolsChelyse Miller

Mathematics specialists are in a unique position to affect change in schools since they serve as both teachers and as mathematics leaders who are in a position to develop teacher capacity within a building or district. Schools in America are in a period of transition. Some of the transition is technical in nature and some is more cultural. As 21st century educators, many teachers and administrators are working to transform into Collaborative Learning Communities where doors are being opened to colleagues to grow professionally. This is a shift in the way we have viewed education and taught historically. As a result many educators have to adjust be-liefs, attitudes about experiences, and habits as they move forward and specialists are instruments to support them in this shift. As transformation happens, specialists can use various strategies to affect change in instructional practices and belief structures that will ultimately help maintain healthy school cultures and allow collaborative learning communities to take root. Specialists need to understand and appreciate their role in regards to change. This quote by Carol Burnett sums it up, “Only I can change my life. No one can do it for me.” Specialists can support the teacher in trying to make changes. In building Collaborative Learning Communities, technical change occurs in schedules or policy decisions that make the collaboration possible. As a specialist the best way we can help our colleagues through this process is to always be a teacher advocate by asking for input from the teachers and sharing that with the administration. Once decisions are made, communicate with the teachers to help them un-derstand the change clearly. Communication is imperative. A lot of resistance can be avoided when communication is transparent and clear. Change, no matter how small or big, can be difficult be-cause it challenges beliefs, experiences, and habits. Some teachers may believe that “the way I was schooled by lec-ture was the best because after all I am successful.” Some teachers may believe students should sit still and be quiet to learn the most. These examples are educational beliefs about the role of the teacher, the student, and what learning means. As a specialist, being asked to work with teachers who may hold this belief can be very challenging especially when the belief structures may be opposing to the special-ist’s beliefs. Specialists must find common ground and not be afraid to express their own beliefs and why they hold them. Second, change challenges experiences. Maybe a teacher does not plan to teach any academics on Halloween because from experience, the students are always unruly on Halloween and they have never learned anything in the past on that day. As a specialist, that experience needs to be challenged maybe by teaching an engaging math les-son on Halloween. Lastly, change challenges habits. Ex-perienced teachers have habits that make their classroom work. These habits have become routine over the years and teachers may not want to change. For example, a teacher for years has been using an overhead projector to

write the morning work. Now the teacher does not have an overhead, but rather a Smartboard right where she used to project the overhead. The teacher is forced to change to using a new piece of equipment that she has not needed before and prepare slides with the morning work instead. As a specialist, we would have to assist this teacher make the technological transition in the mathematics classroom. Dr. Anthony Muhammad looked at school cultures in a va-riety of schools. He discovered four groups of teachers in every school he visited and in his book Transforming School Culture he identifies the four groups and then how to support each group.1 An awareness of this interplay of teachers is eye-opening for specialists who can affect the change more easily as a teacher and leader. The four groups are Tween-ers, Survivors, Fundamentalists, and Believers. The Tweeners are new teachers to the school system, to teaching, to the school, or to teaching math. By focusing on new teachers, we are building capacity within a group of teachers who will be in the field the longest. When working with Tweeners, specialists can provide best practices and resources, model specific teaching strategies, and be avail-able to support the teacher. Survivors are the next group. Survivors have given up, may be on probation, may not care, and may be counting the days until retirement. Sometimes a specialist may be asked to work with this group of teachers and in doing so, a specialist would value what the teacher does well, offer to share what the teacher does well with colleagues, boost the survivor’s self-esteem, and revive the teacher’s purpose for teaching. Fundamentalists are the next group and this is the most difficult group. Fundamentalists oppose change and maintain the status quo, usually openly and loudly. This group drums up support by having conversations openly in the lounge, workroom, or parking lot. Specialists need to stay positive about the change but keep their own emotion in check. If the person is going on negatively, acknowledge their view without adding opinion to it by saying, “What I hear you saying is…” After saying that, try to calm them. Lastly, try to redirect the person to something positive. To further explore this group Dr. Anthony Muhammad cites research entitled “Drop Your Tools” Research which comes from a book in 1992 written by Norman Maclean entitled Young Men and Fire. This book looked at two major Forest Fires in Colorodo. The research that Weick did using this book asked two questions, “Why is change so complex? And “Why would a person refuse to change at the risk of losing his or her own life?” Weick discovered four things. First, people will continue with the status quo when they have not been given a clear reason to change. If we want change within our schools in relationship to Math we must communicate clear reasons for change. Second, people will continue

1 Muhammad, A. (2009). Transforming School Culture: How to Overcome Staff Division. Bloomington, IN: Solution Tree Press.


with the status quo when they do not trust the person who tells them to change. As a specialist and a leader in math we have to build trust. Third, people will continue with the status quo when they view the alternative as more frighten-ing. Specialists need to be rocks for teachers. Specialists need to communicate with administrators when new things are attempted so administrators know they may look messy and observations should be avoided during this time. Lastly, people will continue with the status quo when changing may mean admitting failure. As a specialist, remind teachers that in the past, the way they taught was the best and it does not mean it was wrong. Show them research and discuss implications of the research to support best practices. Finally, the last group is the Believers. Believers embrace change to help promote student learning, Believers are our math teacher leaders, They want to learn as much as they can to help their students excel academically. They share

the strategies we relate with their colleagues. If a teacher is identified as a Believer, support them by do not hesitating to offer strategies to change. Build them up as much as you can. Run with their energy. Invite other teachers to observe various strategies in practice. In conclusion, change is difficult, but necessary. Clear communication with teachers and administrators is impera-tive in regards to technical changes. Clearly communicating why we are making the changes and voicing the opinions of the teachers is important as we make technical changes in the schools. Cultural change is more difficult but we can keep a positive attitude and use the strategies offered to affect the school culture in a healthy way. But the most important attitude to have is staying positive.

CHeLYSe V. MILLeR is a Mathematics Specialist at Lynnhaven Elementary School in Virginia Beach, VA.


GENERAL INTEREST

Early Algebra and Mathematics SpecialistsM. K. Murray

Abstract This paper discusses early algebra as it relates to the Mathematics Specialist program. Early algebra is described based on research and readings from the body of litera-ture focused on early algebra. Reasons why early algebra should be emphasized in elementary school mathematics are discussed, followed by a description of the role elemen-tary school Mathematics Specialists must play if schools are to begin to focus on early algebraic instruction. Finally, some suggestions are made for ways the Mathematics Specialist program might encourage more explicitly an ear-ly algebraic approach to elementary school mathematics.

Introduction I have been an instructor many times for the courses taught for Mathematic Specialists in Virginia. Each time I have taught one of these courses, I have deepened my own understanding of the mathematics that elementary children are capable of understanding, as well as of ways in which children come to express these understandings. However, it wasn’t until I had taught all three of the courses that make up what I think of as the “Numbers” sequence (Numbers and Operations; Rational Numbers and Proportional Rea-soning; and, Patterns, Functions and Algebra) that I be-gan to appreciate the connectedness and complexities of these courses. Furthermore, I had not really understood how these courses work together to support a curriculum focused on early algebraic reasoning. My work with these courses has led to my interest in early algebra and the re-search in the field. In this paper, I want to describe a little of what is meant by early algebra, based on research and readings from the body of literature focused on early al-gebra. I will discuss reasons why early algebra should be emphasized in elementary school mathematics. Next, I will look at the role elementary school Mathematics Specialists must play if schools are to begin to focus on early algebraic instruction. Finally, I will make some suggestions for ways I see the Mathematics Specialist program might encourage more explicitly an early algebraic approach to elementary school mathematics.

Early Algebra: What Is It? The Principles and Standards for School Mathematics describes six content standards for grades K-12 [1]. The Algebra Standard envisions students who:• Understand patterns, relations, and functions;• Represent and analyze mathematical situations and structures using algebraic symbols;

• Use mathematical models to represent and understand quantitative relationships; and,

• Analyze change in various contexts. It is important to realize that this Standard spans the el-ementary and secondary grades. Algebra is a body of knowledge that students learn over a long span of time, beginning in the early grades. Indeed, algebra is not sepa-

rate from the arithmetic studied in the elementary grades; rather, algebra and arithmetic are integrally connected. It is also important to understand that early algebra is not what we understand as high school algebra taught in earli-er grades. Most researchers echo Carpenter and Levi who claim the goal of early algebra is to develop algebraic think-ing [2]. They, like other researchers in the field, conceive of algebraic reasoning as the building, expression, and justification of generalizations, representing mathematical ideas with symbols, and using those symbols to represent and solve problems [3-8]. The algebraic reasoning most appropriate for elementary school that is the focus of these researchers’ work typically falls into one of two subcatego-ries: generalized arithmetic and functions. Generalized Arithmetic—This term refers to the reason-ing that occurs as students recognize patterns that emerge during their study of the four basic operations, and to the claims they make and later justify, and eventually express with symbolic notation. For example, a student solving the problem 37 + 28 may take 3 from the 28 and add it to 37; the resulting problem becomes 40 + 25. At first, the stu-dent may state a generalization of what he notices as with words: “When you take an amount from one addend and add the same amount to the other addend, you still get the same total when you add them together.” This serves as the basis for the symbolic expression of the relationship, (a+b) = (a+c) + (b-c). Functions—This term refers to the generalization of nu-meric patterns. Such patterns often arise from contextual situations, and may be represented with pictures, number lines, function tables, symbolic notation, and graphs. For example, six pennies are added to a jar every day and the children analyze the growth. An essential ingredient of early algebraic instruction is the focus on student reasoning and the discourse that al-lows students to identify connections among concepts, and then build on these connections to form generalizations. This discourse does not occur naturally, but rather is the result of a well articulated plan, developed by a teacher who herself understands the underlying algebraic aspects of the content. So early algebra is not just appropriate con-tent, but also requires effective pedagogy to bring the deep meaning of the content to the surface.

Why Emphasize Algebra in Elementary Grades? There are several reasons why an emphasis on early algebra in elementary grades is warranted. First, there is a call for early algebra on both national and state levels. Nationally, there is an emphasis on having all students complete at least one algebra course before graduating from high school. The NCTM released a position paper claiming all students should have an opportunity to learn algebra; furthermore, students need opportunities to en-counter algebraic ideas across the PreK-12 curriculum [9]. Statewide, Virginia students are required by the Virginia


Department of Education to pass at least three mathemat-ics courses at or above the level of Algebra I in order to obtain a Standard Diploma [10]. The Virginia Department of Education’s “Mathematics Standards of Learning” re-quire students to explore algebraic concepts in grades K-6 [11]. Some examples of algebra content in these grades include: the formal exploration before sixth grade of the commutative, associative, and distributive properties; an understanding of equality and inequality by second grade; and, the ability to recognize and “describe a variety of pat-terns formed using numbers, tables, and pictures, and ex-tend the patterns, using the same or different forms” by third grade [11]. Another reason to emphasize early algebra in the ele-mentary schools focuses on issues of equity. The Equity Principle states, “All students need access each year to a coherent, challenging mathematics curriculum taught by competent and well-supported mathematics teachers” [1]. Schifter, et al. report that a focus on algebraic representa-tions, generalizations, and connections supports students’ computational fluency [6]. Furthermore, in the same article they provide evidence that working on developing algebra-ic reasoning supports the range of learners in a classroom. Less capable students begin to find the mathematics more accessible as they are offered more entry points; more ca-pable students find the content associated with early al-gebra “challenging and stimulating.” Thus, a curriculum grounded in early algebra offers greater opportunities for differentiation practices that are focused on substantial mathematical thinking. A third argument for an emphasis on early algebra re-volves around improving overall elementary mathematics curriculum. A curriculum focused on early algebra, with a constant eye on helping children build on past experiences to form generalizations that can be justified, will be much more coherent than a curriculum that “covers the Stan-dards.” A curriculum tied together by algebraic concepts makes sense, and in fact might reduce what seems to be an overwhelming amount of material to learn by providing opportunities to teach more concepts simultaneously [12]. A simple case: understanding the commutative property reduces the number of basic facts one must learn by half. A less simple case: understanding how the distributive property is applied when multiplying whole numbers al-lows a student to apply the same process when multiplying mixed numbers. Another less simple case: approaching fact instruction through a functional lens creates opportu-nity for meaningful graphing experiences, tied to pattern exploration and tabular representations. One aspect of the work on early algebra that seems so promising is that it does not require an entire reworking of the current elementary curriculum. Rather, as Carraher, et al. state, “existing content needs to be subtly transformed to bring out its algebraic character” [7]. Kaput refers to this as “algebrafying” the elementary school curriculum [3]. This “algebrafication” requires “acknowledging the several different aspects of algebra and their roots in younger chil-dren’s mathematical activity.”

Enter the Mathematics Specialists Kaput and Blanton claim “elementary teachers are in the critical path to longitudinal algebra reform, yet they typically have little experience with the rich and connected activi-ties of generalizing and formalizing” [13]. One predictable result of this lack of experience may be a lack of depth of understanding achieved by students, even those who are successful with the Standards of Learning. For example, consider two students who are asked to decide if 37 + 52 > 38 +51. Student 1, taught by a teacher without a deep un-derstanding of algebraic concepts, will likely resort to sim-ply adding both sides of the equation, obtaining the same answer, and claiming the statement to be false. This is true, but an opportunity has been missed to use what Car-penter, Franke, and Levi refer to as relational thinking [14]. Also, this student has not been given an opportunity to solve this problem in ways that provide initial experiences with commutative and associative properties. Student 2, taught by a teacher with a deep understanding of the con-cepts and generalizations that can come from this problem, would likely solve this problem in a far different manner than Student 1. Student 2 might reason that 37 is one less than 38, but 52 is one more than 51, so the two sides are still even, using number sense and the relations between the numbers to arrive at a correct answer. If elementary teachers lack the content and pedagogical knowledge necessary for providing the type of instruction focused on early algebraic reasoning, then clearly this is an area for their professional development. Several groups have reported their efforts in working with teachers as they begin to approach instruction of the elementary mathemat-ics curriculum through an algebraic lens [15-17]. The ap-proaches of these groups reflect the “algebrafication” strat-egy described by Blanton and Kaput [15]. This strategy is focused on classroom teacher change, approached along three avenues: 1) the “algebrafication” of instructional ma-terials; 2) the support of students’ algebraic thinking; and, 3) the creation of a classroom culture and teaching prac-tices supportive of algebraic reasoning. Mathematics Specialists are in a critical position to pro-vide sustained professional development focused on alge-braic reasoning. In their daily work with teachers, Math-ematics Specialists regularly work with teachers to plan daily lessons and overall curriculum, work that includes modification of existing instructional resources. In schools with Mathematics Specialists, teachers are becoming bet-ter adept at listening to and exploring student reasoning, and helping students build on their own reasoning. As a re-sult of efforts on the part of Mathematics Specialists, more and more teachers afford students opportunities to explore and deeply engage in mathematical explorations, and classroom cultures are established that respect individual reasoning. So, the basic structures of “algebrafication” are in place as a result of Mathematics Specialists in schools. Yet for “algebrafication” to occur, early algebraic reason-ing needs to become a focus of the Mathematic Special-ists’ work. Specialists need to provide opportunities for the teachers in their school to explore algebraic concepts for


themselves in order to gain some depth of understanding of early algebra. As a Specialist works with teachers on lessons and curriculum, for example, the focus can be on underlying algebraic aspects of the concept in question, and how those aspects are brought to the forefront of dis-cussions and developed into generalizations. Mathemat-ics Specialists should work with teachers across the grade levels in their school to ensure that algebraic reasoning develops across concepts and from grade to grade, and that generalizations developed in one grade continue to be considered and reconceived or justified in the next. Math-ematics Specialists can help teachers recognize opportu-nities that arise to help children form generalizations, thus supporting students’ algebraic reasoning while creating a community where that reasoning is expected and valued.

Explicitly Focusing Mathematics Specialists on Early Algebra, During the Program and Beyond Much of the work done in the “Numbers” courses of the Mathematics Specialist program focuses on algebraic rea-soning. One of the first activities prospective Mathematics Specialists enrolled in the Numbers and Operations course engage in requires them to solve a problem like 57 + 36 using mental math. After a minute or so of reflection, par-ticipants share their strategies. Participants will propose a number of strategies, including: adding tens, then ones; changing 57 + 36 to 60 +33 or 53 +40, then completing the work with these easier, benchmark numbers; and, start-ing at 57 and counting on (57, 67, 77, 87, 88, 89, 90, 91, 92, 93). The language in the activity includes words like decomposing and recombining; the concepts being devel-oped underlie the commutative and associative properties of real numbers. Other work in this course continues to examine how children use number sense to develop mean-ingful approaches to the four operations; these approach-es often rely on (yet unstated or formulated) properties of equality. Algebraic reasoning is an essential component of the Rational Numbers and Proportional Reasoning course. Work with equivalent fractions, for instance, can be viewed through a functions lens. Examples of explorations teach-ers encounter include looking at similar rectangles, and ex-amining the ratio of height to width with tables and through graphing. An arrangement of nested similar rectangles on the coordinate grid reveals that the diagonals of similar rectangles fall on the same line, connecting the table to a linear function and a discussion of slope. Multiplication of fractions is analyzed through an area model, but also as the result of an operator acting on a quantity; again, a func-tion approach. The course Patterns, Functions, and Algebra, in its name and content, is the course most obviously focused on algebraic thinking. In the first half of this course, the focus is on the generalization of patterns, developing skills necessary to describe patterns with symbols. Participants develop fluency with algebraic notation as they learn how the symbols represent the physical quantities and actions. Conjectures (e.g., an odd plus an odd equals even) are justified and proven to hold over fields of numbers first

with models, then symbolically. Participants use models to justify laws of equality. In the second half of the course, activities explore various functions, with an emphasis on the connections between multiple representations. Work in this course includes developing an understanding of how young children can develop an understanding of functions. Clearly, opportunities to develop Mathematics Special-ists’ understanding of algebraic reasoning are available in the program courses. However, it is not clear that partici-pants in these courses are aware of the algebraic nature of this work until they enroll in Patterns, Functions, and Algebra. As instructors, we miss opportunities to explic-itly relate work in Numbers and Operations and Rational Numbers and Proportional Reasoning to algebra, and fail to explicitly highlight how algebra permeates the elementa-ry curriculum. Just as a focus on early algebraic reasoning ties together the elementary curriculum, creating opportu-nities to teach more concepts in a connected manner and with richer understanding, a focus on algebraic reasoning could also serve to tie together “Numbers” courses in a more cohesive program. How can the algebraic thread be made more explicit, in order to prepare Mathematics Specialists to think about early algebra in their own practice? First, some decision needs to be made as to the importance and relevance of algebraic reasoning as a unifying thread for these courses (and indeed, all content courses in the program.) If there is general agreement that algebraic reasoning should receive consistent, explicit focus, then instructional staff would benefit from professional development that highlights alge-braic reasoning in the courses, and how the courses are related in this regard. This seems especially important for instructors who have not had the opportunity to teach all three of these courses, and to experience these con-nections themselves. The present Mathematics Specialist curriculum implicitly encourages algebraic reasoning from the onset; would it be even more powerful to encourage algebraic reasoning with more intent? Mathematics Specialists also need support as they take on the work of implementing an early algebraic curriculum in their schools. This work should be focused on continu-ing to develop Mathematics Specialists’ understanding of early algebra. Some of this work already occurs through conference sessions, some through local efforts. While it is not (currently) in the scope of the Mathematics Specialist program, continuing professional development focused on increasing endorsed Mathematics Specialists’ knowledge of algebra could be considered in future initiatives. Finally, a focus on algebraic instruction in elementary school is fairly new in the arena of mathematics education. Teaching number facts through a functions approach will look and feel different to teachers, administrators, parents, and children; using patterns to learn these facts is also foreign to those who see this as a rote skill. Mathematics Specialists will need to advocate for this approach, and will need support in their advocacy. Mathematics educators involved in the Mathematics Specialist program need to work with administrations to develop an understanding and support for taking this approach to the elementary math-


ematics curriculum, because to be effective it will require time and effort in training staff and reworking curriculum. Early algebra and algebraic reasoning is a relatively new area of research in the mathematics education literature. There is still a lot of research that needs to be conducted to determine how children learn to reason algebraically, and what this means for instructional practices and resources. If this research is best conducted in school settings, it fol-lows that Mathematics Specialists should play a vital role, both as research subjects and researchers. To do so, they need to be prepared.

References[1] Principles and Standards for School Mathematics,

National Council of Teachers of Mathematics, Res-ton, VA, 2000.

[2] T. Carpenter and L. Levi, Developing Conceptions of Algebraic Reasoning in the Primary Grades, National Center for Improving Student Learning and Achieve-ment in Mathematics and Science, Madison, WI, 2000; Internet: http://www.wisc.wcer.edu/ncisla.

[3] J. Kaput, “Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by ‘Al-gebrafying’ the K–12 Curriculum,” in The Nature and Role of Algebra in the K–14 Curriculum: Proceedings of a National Symposium, Washington, DC, 1998.

[4] J. Kaput, “Teaching and Learning a New Algebra,” in E. Fennema and T. Romberg (eds.), Mathematics Classrooms that Promote Understanding, Lawrence Erlbaum Associates, Mahwah, NJ, 1993.

[5] J. Kaput, “What is Algebra? What is Algebraic Rea-soning?” in J. Kaput, D. Carraher, and M. Blanton (eds.), Algebra in the Early Grades, Lawrence Erl-baum Associates/Taylor Francis Group, Mahwah, NJ, 2008.

[6] D. Schifter, S.J. Russell, and V. Bastable, “Early Algebra to Reach the Range of Learners,” Teaching Children Mathematics, 16(4) (2009) 230-237.

[7] D. Carraher, A. Schliemann, B. Brizuela, and D. Ear-nest, “Arithmetic and Algebra in Early Mathematics Education,” Journal for Research in Mathematics Education, 37(2) (2006) 87–115.

[8] D. Carraher, A. Schliemann, and J. Schwartz “Early Algebra Is Not the Same as Algebra Early,” in J. Ka-put, D. Carraher, and M. Blanton (eds.), Algebra in

the Early Grades, Lawrence Erlbaum Associates/Tay-lor Francis Group, Mahwah, NJ, 2008.

[9] “Algebra: What, When, and for Whom: A Position Paper of the National Council of Teachers of Math-ematics,” National Council of Teachers of Mathemat-ics, 2008; Internet: http://www.nctm.org/about/con-tent.aspx?id=16229.

[10] “Graduation Requirements,” Virginia Department of Education; Internet: http://www.doe.virginia.gov/in-struction/graduation/index.shtml.

[11] “Mathematics Standards of Learning, 2009 Adoption,” Virginia Department of Education; Internet: http://www.doe.virginia.gov/instruction/graduation/index.shtml.

[12] J. Kaput, D.W. Carraher, and M.L. Blanton, “Skep-tics’s Guide to Algebra in the Early Grades,” in J. Kaput, D. Carraher, and M. Blanton (eds.), Algebra in the Early Grades, Lawrence Erlbaum Associates/Taylor Francis Group, Mahwah, NJ, 2008.

[13] J. Kaput and M. Blanton, “Algebrafying the Elemen-tary Mathematics Experience,” in H. Chick, K. Stacey, J. Vincent, and J. Vincent (eds.), Proceedings of the 12th ICMI Study Conference on the Future of the Teaching and Learning of Algebra, 1 (2001) 344–351.

[14] T. Carpenter, M. Franke, and L. Levi, Thinking Alge-braically: Integrating Arithmetic and Algebra in Ele-mentary School, Heinemann, Portsmouth, NH, 2003.

[15] M. Blanton and J. Kaput, “Developing Elementary Teachers’ ‘Algebra Eyes and Ears,’” Teaching Chil-dren Mathematics, 10 (2) (2003) 70-77.

[16] V. Jacobs, M. Franke, T. Carpenter, L. Levi, and D. Battey, “Professional Development Focused on Chil-dren’s Algebraic Reasoning in Elementary School,” Journal for Research in Mathematics Education, 18(3) (2007) 258-288.

[17] D. Schifter, V. Bastable, S.J. Russell, L. Seyforth, and M. Riddle, “Algebra in the K-5 Classroom: Learn-ing Opportunities for Students and Teachers” in C.E. Greene and R.Rubenstein (eds.), Algebra and Alge-braic Thinking in School Mathematics: Seventieth Yearbook, 2008.

This manuscript has been previously published by the Journal of Mathematics and Science: Collaborative Explorations and is re-printed with permission of the Virginia Mathematics and Science Coalition.


GENERAL INTEREST

Approaching Mathematics Utopia?F. Morton

Well, not really, but as Mathematics Specialists, we have an idea of what it looks like. The lessons learned during the first year of working with teachers will help us reach our ideal. That perfect world of mathematics instruction is best articulated in the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards for School Mathematics:

Imagine a classroom, a school, or a school district where all students have access to high-quality, en-gaging mathematics instruction. There are ambi-tious expectations for all, with accommodation for those who need it. Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals. The curriculum is mathematically rich, offering students opportunities to learn important mathematical concepts and proce-dures with understanding [1].

This is the “perfect world” Mathematics Specialists strive for as we set out to change the world—or the philosophy of mathematics instruction in our buildings, or the daily in-struction in a classroom. I have come to realize through the experiences of the last one and a half years that it is the small steps and the seeds that we plant in the actual classrooms that push us and those we work with in the direction of truly changing and reforming the mathematics we teach. Since instructional practices are a direct reflec-tion of a teacher’s belief system, one of the primary goals of the Mathematics Specialist is to promote the vision for mathematics education described above.

Stafford County’s Vision In Stafford County, there are currently nine Mathematics Specialists, each one working in a K-5 building. The Math-ematics Specialist Program is clearly defined in a five-year plan and was presented to administrators and staff before I started working in the building. The daily activities change as the needs in my building change, but the philosophy and intent of the Mathematics Specialist Program is focused on using best practices to improve student achievement in mathematics by improving instructional practices. Stafford County Mathematics Specialists established the following mission statement:

The Mathematics Specialist takes a leadership role to increase the awareness and value of mathematics for students, teachers, parents, and the community. They work with teachers and parents to increase their understanding of how children learn mathematics. In serving as a resource and a coach to teachers, they model best practices in teaching and learning math-ematics. They also collaborate with teachers and administrators to collect, analyze, and use data to in-form instruction in meeting the needs of all students.

The following Focus Goals have also been established:

1. Encourage and support teachers in modeling and developing students’ oral and written communication skills in mathematics to provide opportunities for stu-dent self-reflection and metacognition (through number talks, questioning, performance-based assessments, mental math activities, journals, etc.).

2. Support the divisionwide implementation of the Every-day Mathematics series throughout the year [2].

3. Revisit and refine current mathematics classes for parents and work to encourage more parental involve-ment.

4. Facilitate ongoing, monthly school-based professional development in content and pedagogy through grade-level meetings and at other times as needed.

5. Collaborate with teachers to examine formal and in-formal assessments in order to access student under-standing and to inform instruction.

6. Support teacher collaboration in developing common assessments and rubrics based on what they want children to know and be able to do, and to use those to measure student understanding and skills.

These statements are more than just words on paper; they guide our interactions with teachers, parents, and administrators and help us allocate our time. The Focus Goals and Mission Statement are designed to further our two primary initiatives: “Teaching and Learning for Under-standing” and “Every Child Mathematically Proficient.” In addition to the six Mathematics Specialists already in place, three Mathematics Specialists were placed in three schools participating in a National Science Foundation (NSF) grant in Fall 2006. It seemed like such a simple plan: a Mathematics Specialist would be placed in an el-ementary school as part of a NSF research project to work with teachers to enhance mathematics instruction, thereby improving students’ mathematical proficiency. Oh, and by the way, there would be a new mathematics curriculum tool to implement and support. Oh, and the report card would be changing for first through third grade in such a way that teachers would need to assess their students’ mathemati-cal understanding by evaluating their performance rather than scoring a worksheet. Oh, and one more thing—the county would be implementing an initiative to empower site-based decisions that would require teachers to meet continuously throughout the school year to complete and discuss research, develop action plans and strategies, and basically compromise any professional development time available. These were some of the more formidable issues


that arose during my first year—underscoring a situation in which a teacher leaves the classroom, moves to a new building, takes on a new role that is relatively new in the division and is still evolving, and is not clearly understood by her colleagues. Each of the aforementioned concerns could be the focus of discussion because they have truly impacted my abil-ity to fulfill the responsibilities of the Mathematics Special-ist. Instead, I have chosen to reflect more specifically on the coaching, co-teaching, and collaboration experiences during my first year and the lessons I learned from those experiences. On the surface, these three terms seem very similar. They all have the connotation of cooperation; they all seem to invoke a positive attitude with a common goal; and, they all even start with the letter “C”! As the lay-ers of each of these models of working with teachers are peeled away, these three aspects of being a mathematics education leader become very distinct, yet still very much entwined.

Collaboration I can remember clearly how excited I was to be starting this position. I had taken the math content courses that en-abled me to feel fairly confident in my teaching ability. I had first hand experience building a community of learners in my own classroom, and knew how powerful inquiry–based learning and teaching could be. In addition, I would be working in a new building with a group of teachers who re-ally wanted to be there. Most of them had competed for po-sitions at this school, and I thought that they would be more open to reform mathematics and problem-based teaching. In short, I felt fairly competent in using the new curriculum materials and had worked hard to be ready to support each grade level as we began the schoolyear. As Mathematics Specialists, we were asked to encour-age teachers to implement our new materials as holisti-cally as possible, keeping the spiral defined by the Every-day Mathematics curriculum intact [2]. At times, this went against the grain of my own notion of best practice; the material was more scripted than I was comfortable with, and many times I believed the concepts jumped around without enough time for students to fully grasp the ideas or practice application. To complicate the implementation of these materials, many teachers were just not willing to put in the time necessary to prepare for the daily lessons. At grade-level meetings, I would try to focus teachers’ at-tention on where the lessons were going or their purpose in laying the foundation for future lessons. Many times, this approach fell on deaf ears. The teachers were used to immediate results: teach a few lessons, check it off, and move on. Lesson Number One: Adult learners will only hear what they are ready to hear or what they perceive they need to hear. This was, and continues to be, one of the most frustrating aspects of being a mathematics education leader in my building. I know that through collaboration we can make great strides and effect change. In order for that collaboration to take place, all involved must realize the need for change and embrace the challenges that arise. I knew that supporting the implementation of the new

curriculum materials would give me an opportunity to work with more teachers; it would afford me an entry with some teachers who might not have been willing to work with me. Unfortunately, questions and strategies related to different aspects of the new materials consumed our precious time together. My grade-level meetings were spent disseminat-ing information or answering questions and encouraging teachers to “have faith in the spiral.” I was perceived as the “mouthpiece” for a curriculum with which I was occasionally uncomfortable. In order to model a more inquiry-based ap-proach to teaching with these materials, I offered to go into classrooms to demonstrate lessons. I would launch the lessons with number talks, while continually charting stu-dent responses, and then model questioning as much as possible. Many teachers asked me into their classrooms, to the point that I was having a hard time scheduling them all! Unfortunately, in my quest to get into more classrooms to model lessons, the collaborative planning and reflection on practice was lost. I know that I have to find a balance between modeling and collaboratively planning lessons, and realize that I can’t be everywhere all the time. Lesson Number Two: A little change in a teacher’s practice that will continue and sustain itself is better than one or two les-sons that will be forgotten the next day by both the teacher and students. By taking the time to debrief for just a few minutes after the lesson has been taught, the teacher can articulate some specific student learning behaviors they witnessed which could lead to a change in their approach the next time they teach a similar lesson.

Co-Teaching In Stafford County, the Mathematics Specialist co-teach-es with a teacher for the entire year at a grade level the Mathematics Specialist has not taught before. As I look back on my first few writings and synthesis of Lucy West’s Content-Focused Coaching, I am reminded again of my naiveté [3]. I was so enthralled in reading about this model of coaching. I really thought I could use these strategies of pre-conference planning, lesson observation, and post-conference debrief effectively to plan and teach lessons regularly with my co-teacher that would truly enhance stu-dent learning and our own pedagogy. I wrote about the coach and the “mathlete” with such high hopes. We were in this game together, and we would give 150% to reach our goal of student understanding. Thinking that my co-teach-er and I were on the same page, I was a little anxious to be teaching such a high-stakes testing grade for the first time, but I had had the opportunity to plan and discuss the first few weeks’ lessons with her and felt we were off to a good start. I remember vividly a day when my co-teacher and I had had an especially frustrating time. The students were confused, we were way behind in our pacing, the lessons in the teacher’s manual forged ahead, and I knew the stu-dents weren’t ready to move on. I suggested rearranging the sequence of the next few lessons so that there might be more continuity for the sake of student learning. My co-teacher was amazed when I suggested that we should al-ways make our instructional decisions based on the needs of our students. She actually said, “That’s the first time I’ve


heard you say that.” I had worked and planned with this teacher for two months, so I was surprised to learn that she didn’t know that student learning was the motivation and guiding principle for everything I had been doing. Lesson Number Three: Be explicit in your intentions and expecta-tions. This co-teaching is not as easy as it sounds. Looking back, I am sure her words were brought on by the same frustration I was feeling. To further complicate our co-teaching, we rarely had enough time for true col-laboration in order to plan our lessons. On occasion, we fell into the bad habit of “drive-by” planning wherein we would divide the lessons, but leave no time for reflection on pedagogy or student learning. In November, this finally reached a critical point: I backed away from instruction and insisted she do the teaching for a while. I wrote in my jour-nal that this allowed me the chance to observe her teaching and gave her the control of the classroom she needed. It was also an opportunity to appreciate what a truly gifted teacher she is—she is a master at making connections and promoting focused mathematical discourse among her students. In the day-to-day planning of instruction, I had focused too much on our new curriculum without giving her the freedom she was used to having in planning her own instruction. We had other incidences when one of us was not comfortable with the direction our math instruction was going, but we were able to work things out. Consequently, we reached a level of comfort and trust with each other that allowed us to trade off and build the other’s ideas during instruction. Many times, we knew exactly what the next question should be without having had the opportunity to plan for it. Lesson Number Four: The importance of plan-ning and reflection is critical to the success of a co-teach-ing experience. While it wasn’t a perfect situation all the time, I am happy to report that my co-teacher went against the grain of her grade level and did not insist on as much “paper proof” of student learning. She told me the other day that she felt that together, we had helped her students develop a much deeper conceptual understanding of the mathematics in fifth grade.

Coaching So I leave that which I have been able to do the least, for last. Again, I am reminded of my high hopes dashed by reality! It is extremely difficult to implement West’s model of coaching when there are so many other hats to wear and concerns to address. There have been a few occasions when I was able to use the “West protocol” with great suc-cess. In January, a third grade teacher asked me to help her understand and address her students’ confusion. She wanted them to develop a deeper understanding of multi-plication. We spent a lot of time discussing, planning, and anticipating student confusions with respect to the lessons that would build conceptual understanding and allow op-portunities for practice. Over a period of two weeks, I ob-served, co-taught, then observed as she taught. We were both pleased with the outcome. This leads to Lesson Num-ber Five: Observation and reflection of student learning enable the coach and teacher to work on the mathemati-cal understanding of students. Another short-term coach-

ing experience was an exciting opportunity to work with a teacher who had been resistant to concept of the Mathe-matics Specialist. She came to me asking for help in meet-ing the Standards of Learning (SOL), and I was thrilled to have the chance to work with her [4]. Together, we planned some solid lessons, but didn’t have as much time to actu-ally teach and observe together. It just worked out better to split the class and work with the smaller groups so in the end, we didn’t have an opportunity to share the nuances of our own practice and pedagogy. Once SOL testing started, she was feeling better about her students’ understanding, and we built a rapport that will continue to grow. West’s model of content-focused coaching is a frame-work for collaboration with teachers that I continue to study. I realize now that this collegiality is not something that can occur just because you know the questions to ask and the content knowledge to help develop student understanding. Lesson Number Six: True coaching must be a collabora-tion involving co-teaching wherein both parties believe in the process, with mutually agreed upon goals. It takes a skillful listener to determine what the teacher needs to fo-cus on, and the needs change with the teacher’s level of expertise, content knowledge, and philosophy of teaching. There is a delicate balance that must be achieved be-tween invitations for teacher contribution to lesson design and offering direct assistance in designing the lesson. It is much more difficult to finesse the coaching conversation so that the teacher and the coach have equal contribution and ownership in the goals and outcomes of a particular lesson. I have learned many lessons about myself and about the teachers.

Summary Collaboration, co-teaching, and coaching can be very different, yet each of these aspects of cooperative teaching involves similar layers of negotiating, planning, listening, and reflecting on the instruction and student understand-ing. I know that I must continue to work on being a good listener and ingrain those questions that will help a teacher reflect on the mathematics that will produce the collabora-tive teaching experience I envision. I have learned that it is critical to be explicit in my own intentions and expectations. Negotiating for time may be a constant frustration, but mak-ing the time for collaboration is crucial to the success of any and all aspects of teaching. Most importantly, the lesson to be learned is that the learning goes on—it doesn’t stop here—and that effect-ing change takes time. I’ve just come across something in my journal that I need to post on my wall at home and at school: “The true joy in improving things is the small, daily achievements along the way.” We must revel in the baby steps to appreciate the strides. Perhaps we will never reach “mathematical utopia.” There is always room for improvement, revealed through reflection on practice and pedagogy, and isn’t that the point? As stated in the Prin-ciples and Standards for School Mathematics:

Teacher Leaders can have a significant influence by assisting teachers in building their mathematical and


pedagogical knowledge. Leaders (especially Mathe-matics Specialists) face the challenge of changing the emphasis of the conversation among teachers from “activities that work” to the analysis of practice [1].

Enhancing mathematics instruction to facilitate math-ematical proficiency requires us to develop and design the best lessons possible, but we must continue to learn from our own lessons as well. References[1] Principles and Standards for School Mathematics,

National Council of Teachers of Mathematics, Res-ton, VA, 2000.

[2] Everyday Mathematics, University of Chicago School Mathematics Project, McGraw-Hill, New York, NY, 2004.

[3] L. West and F. Staub, Content-Focused Coaching: Transforming Mathematics Lessons, Heinemann, Portsmouth, NH, 2003.

[4] Standards of Learning for Virginia Public Schools, Board of Education, Commonwealth of Virginia, Rich-mond, VA, 1995.

Reprinted with permission from The Journal of Mathematics and Science: Collaborative Explorations, Vol. 9, Spring 2007, by the Virginia Mathematics and Science Coalition. All rights reserved.


GENERAL INTEREST

Resource ReviewThe resources in this section have been recently reviewed by VCTM members. Be sure to watch for the January VCTM newsletter which will provide detailed information on how you can submit resource reviews for consideration to the VCTM Teacher Resource Review Committee.

The Math Process Standards Series Grades Pre-K-2Susan O’Connell, ed,. 2007. Series Price $110 paper, Individual BookPrice $27.50 paper. Series ISBN 978-0-325-01305-3. Heinemann; www.heinemann.com.

The Math Process Standards series has five individual paperback books that each focus on a different NCTM process standard. The books in this series are a great source for teachers and are also appropriate as the focus of professional development.

Each chapter in Introduction to Problem Solving discusses different problem-solving strategies and how to develop the use of each strategy with Pre-K-2 students. Introduction to Communication focuses on fostering math talk, developing vocabulary, assess-ing communication, reading about mathematics, and writing about mathematics. Introduction to Reasoning and Proof provides grade-level appropriate suggestions for improving reasoning skills by making conjectures, making mathematical arguments, and assessing student reasoning. Introduction to Representation discusses how to use manipulatives, pictures, numbers, symbols, tables, and graphs to communicate mathematical thinking. Introduction to Connections offers suggestions on how to connect mathematics concepts to each other, other disciplines, and every-day experiences.

Each of the books in this series is filled with tasks which are also included on a CD in a format that can be easily edited for differentiation. These tasks are discussed in the text. Student work samples are also included that are appropriate for discussion by a professional development group, or, for use by an individual teacher to better understand a task and how it encourages the development of mathematics process skills. On the inside front cover of each book a table orga-nizes the tasks by content strand and book topic. The books in the series also provide valuable classroom tips for developing the process standards as well as questions for discussion appropri-ate for professional development.

I strongly recommend all of the books in this series for the teacher who already focuses on the development of the process standards and is looking for new ideas as well as the teacher who is just beginning to make the processes a focus of mathematics instruction. The activities and problems included in the books are appropriate for Pre-K-2 students and are ready to immediate-ly implement in the mathematics classroom. The books in this series are highly recommended for any teacher or leader of professional development interested in fostering the development of reasoning, communication, representation, connections, and problem solving among Pre-K-2 students.

Virginia V. LewisLongwood UniversityFarmville, VA 23909


Five Practices for Orchestrating Productive Mathematics DiscussionsMargaret S. Smith and Mary Kay Stein, 2011.$29.99. 978-0-87353-677-6. National Council of Teachers of Mathematics; www.nctm.org and Corwin Press; www.corwin.com

Five Practices for Orchestrating Productive Mathematics Discussions provides a framework for teachers, coaches and professional learning lead-ers to learn how to conduct classroom discussions using thoughtfully chosen

student work and talk moves to understand student thinking. This paperback book includes case studies and a Professional Development Guide.

Five Practices for Orchestrating Productive Mathematics Discussions guides readers through setting up goals and selecting tasks for meaningful mathematics lessons and into the pedagogy of using the five practices through case studies. Five chapters of the book are devoted to the prac-tices:

• Anticipating student responses and strategies used to solve the mathematical task

• Monitoring and asking probing questions of students as they work in pairs or small groups on the task

• Selecting student work that will move the whole class discussion of the task to reach the goal of the lesson

• Sequencing students’ work during the discussion to leverage higher potential for student understanding

• Connecting different strategies and solutions that were uncovered in the mathematical discussion.

There are also chapters devoted to asking good questions and holding students accountable through productive talk moves, developing lesson plans and making connections to the five practices using the “thinking through a lesson protocol” and overcoming obstacles to work with colleagues to develop these practices.

In my work as a mathematics content coach and staff developer, I have used Characterizing the Cognitive Demands of Mathematical Tasks, Professional Development Guidebook for Perspectives on Teaching Mathematics, NCTM Publications, 2004; Thinking through a Les-son: Successfully Implementing High-Level Tasks, Mathematics Teaching in the Middle School (October 2008) and Orchestrating Discussions, Mathematics Teaching in the Middle School, (May 2009) all written by Smith and Stein, et al. Five Practices for Orchestrating Productive Mathematics Discussions brings the work of all these publications together in a cohesive profes-sional learning guide that is extremely useful for individual or large numbers of teachers.

Corinne MageeInstructional CoachAlexandria City Public Schools


Beyond Pizzas and Pies: 10 Essential Strategies for Supporting Fraction Sense, Grade 3-5Julie McNamara and Meghan M. ShaughnessyIndividual Book Price $33.95 paper. ISBN 978-1-935099-13-0. Math Solutions; www.mathsolutions.com.

Beyond Pizzas and Pies is broken into eight chapters that each focus on a different instruction strategy to sup-port the development of a “deep and flexible understanding of fraction concepts”. Each instructional strategy is linked to Curriculum Focal Points and NCTM Principle and Standards.

The chapters are structured to provide the reader with the opportunity to review research that supports the tar-geted instructional strategy, classroom scenarios to determine the strategy rationale and ready to use activities to facilitate the strategy. The activity section includes teacher overviews, black line masters, materials list, student misconception discussions and potential questions. Each chapter concludes with a Study Questions section that encourages a reflective approach to the reading and implementation of the focused instructional strategy.

I strongly recommend this book for classroom teachers, teacher leaders, or math coaches to use in grade level planning, book talk or lesson study format. Those responsible for mathematics professional development may also find this book to be a great tool when developing workshops or training sessions. While this book was writ-ten to support grades 3 -5, I have found it helpful when working with the comparison and operations of fractions at the middle school level.

Alfreda R. JerniganMathematics Teacher SpecialistNorfolk Public Schools

Review of My Math Lab by Kari Everett, Western Kentucky University, Department of Mathematics and Computer Science, Bowling Green, KY

http://www.mymathlab.com or http://www.coursecompass.com by Pearson Education.

Cost: For instructors, it is free when you sign up through a Pearson representative. For students, it can be bundled with a textbook for a course or purchased separately from the web site about $80 includes an e-book.

MyMathLab is a series of online courses that accompany Pearson’s textbooks in mathematics and statistics. MyMathLab provides an online environment for homework, quizzes, and test using a question bank that corresponds to the textbook being used. Instructors are able to set up all the components with the problems which they want the students to work. Homework can be setup by chapter or sections depending on what will be assigned with different due dates. Quizzes and test can be setup to be timed or untimed, but once started must be completed and graded before being allowed to continue with any other assignments. Instructors have control over what tools the students are allowed to use from viewing videos to being able to see the steps for the problem being worked.

MyMathLab is less papers to grade for an instructor. It provides step by step explanations on how to work a problem, and students can email problems to instructors if they have a question about something. There are training videos available as well as Getting Started guides for instructors and students.

My current students and in past semester like MML because it gives them more control over the pace of their homework and they like the tools available, i.e. videos, step by step explanations, and animations.


GENERAL INTEREST

Problem CornerRay Spaulding


GENERAL INTEREST

Ten Strategies for Maximizing Opportunity with Instructional Coaching

Theresa Wills and Molly Rothermel We, as coaches, have the unique opportunity of influenc-ing teachers to grow and become more reflective in order to increase student achievement. The job can be challenging, demanding, exciting and rewarding! Coaching teachers is a skill that can be learned and with these ten strategies; we hope that you find it more manageable and enjoyable. The opportunities and challenges that face coaches vary from day to day, from teacher to teacher, amid being stretched in multiple directions. In the experience of the au-thors, we have compiled ten strategies that have influenced our work as we focus on opportunity and hope and in turn, we experience more success and joy. Lucy West (2011) notes an instructional coach wears two hats; a pedagogical hat and a psychological hat. As we reflect on our experi-ence of wearing a variety of hats, we have had the most success when we build trusting relationships with teachers with a focus on opportunity and hope. Throughout this article, we identify ten strategies that we consider to be key components of coaching to build capaci-ty and grow teachers and students through opportunity. We embrace opportunities of collaboration, growth, and fun, while recognizing and believing in the potential in teachers, students, administrators, and coaches. The top ten list we have compiled is based on the foundation of learning with teachers through trust and opportunity.

1. Set standards and review them frequently. In our experience when we meet with teachers individu-ally or in learning teams, we have the most success when we set the standards for the meeting by frontloading. We start our meetings with “When we are working together, we are going to work on ideas to develop teaching and learn-ing. We will not blame kids. We’re focused on learning and growing.” This routine was challenging initially, but has paid off immediately, making a more enjoyable and successful meeting for everyone. We also use six standards during our learning teams and other meetings. These are:

• Be on time.• Be prepared and ready to learn.• Work at learning.• Ask for support.• When in doubt, ask a question.• Respect your rights and the rights of others to learn.

When standards (sometimes called group norms) are dis-cussed and set, there is opportunity for reflection and people to monitor their own contribution to the group. These stan-dards can also be used in a classroom for students because we want to model that learning takes time and focused ef-fort. By consistently using and referring to the standards, we have more focused and productive meetings.

2. The words we use matter! To influence teachers and students to think and take responsibility for their learning, it is important to allow op-portunities for others to have the success. For example, the language used in our standards is focused on what to do (as opposed to what not to do). As coaches, it is our job to influence instructional change and reflection through the words we chose. Through opportunities to grow and learn, coaches influence teachers to plan instruction that engages students where students are doing more of the talking and more of the thinking. To influence growth and reflection, here are some phrases we use.

Instead of.... use... Should Can, could I know that I wonder if thank you (for doing Appreciate (celebrate them & your job) give a positive specific) What is missing is... What if we tried... That student can’t do I wonder what we could do to anything support them better I want to change I want to influence (and in spire them with actions)

I have a concern We have an opportunity Struggling teacher A teacher with more opportunity for growth But And This last pair of words aligns with what Lucy West says, “when someone says a sentence with but, I just listen to whatever is after “but” because that is really what they are most concerned about” (2011). To influence reflection in both teachers and students, we use “and” instead of “but”. For example, instead of “you did a thorough job on that ex-planation, but you didn’t develop the introduction” a teacher could say “you did a thorough job on that explanation, and for even more success, you can focus on developing the introduction.” A word that we use in place of “concern” is opportunity. An interesting activity is to open a file or document on which you wrote concerns that you have on a teacher or student. Replace the word “concern” with “opportunity”, and read it again. We have found that it gives the same message, with a chance for growth, which is the cornerstone of our position. With the change of a couple words, rather than pointing out what is missing or wrong, there is opportunity for reflection and growth.


3. Your facial expressions and tone reveal your inten-tions. As we focus on our word choice to allow for opportu-nities, we also focus on delivering a message where our face reflects a positive and approachable presence and our tone of voice is non-judgemental. We do this by: rais-ing the eyebrows, smiling, and making sure that we take a brief pause before the last syllable in your sentence, mak-ing sure the tone goes up. This is a strategy adapted from Dennis McLoughlin (2011). Practice doing this in front of a mirror. Try both, keeping a flat tone and face, then eye-brows and tone raised, and see if you notice a difference. Repeat the process and notice the difference in perception of the message you are giving. We found that even a tough conversation is more productive and less combative if we remember to maintain non-judgmental tone of voice and positive facial expressions.

4. Give advanced notice before visiting a classroom. We feel strongly that to build trusting relationships, teachers should be given advanced notice for classroom visits. Giving notice provides teachers with the opportunity to plan a lesson deeply, include student centered learning, and any components of a lesson that they believe are good for teaching and learning. This allows you insight on their philosophy of teaching and learning. As coaches, we believe it is more efficient to see teach-ers do their best teaching rather than surprise teachers with unannounced visits. If we see the best they can do, then we have a better idea of our entry point. We can see what a teacher values, and we can learn what they consider a well taught class to look like. We have found this to be far more efficient than a surprise visit and hearing excuses for a less successful lesson. Another benefit for giving advanced notice is we avoid complaints and unproductive stories about how they do not normally teach like that or today was an off day for stu-dents. Instead, there is an opportunity to highlight work that teachers are doing. Classroom visits can be a spring-board for working collaboratively with the teacher. In our experience, once there is a trusting relationship, teachers invite you to stop by anytime and advanced notice is not necessary. This strategy provides great opportunities for building trust!

5. Ask questions--Don’t assume anything. To influence students and teachers to think and explain their behaviors and actions, it is important to give them the opportunity to share their perspective and ideas. When framing an open-ended question, ask yourself, “Do I have a ‘right’ answer in mind?” If so, revise your question or don’t ask it” (author unknown). Jumping to conclusions about why someone is late or why a person is unprepared will often lead to frustration, anger, or that person shutting down. We have had great success with students, teachers, administrators using this strategy of asking questions and influencing thinking (rath-er than telling or assuming). Here are some examples of how questions could be asked to influence thinking.

Assumption Question(s) to ask She is being lazy. When can we work on it together?

He skipped our Can you let me know if something meeting. comes up when we are scheduled to meet? When can we meet again? She did not do We talked about doing ____. How what we talked can we work together to complete about in our it?planning session.

He’ll never use How can we incorporate different manipulatives. learning styles for students who are kinesthetic?

The second part of this strategy is to ask questions to in-fluence teachers and students to think cognitively through higher order questioning. This builds their stamina for rea-soning and can result in a variety of answers which lead to more dynamic discussions. This gets us out of the habit of asking low-level, “guess what’s in my head” type of ques-tion and into discovery and open-ended questions. Asking questions leads to thinking and one of our goals as coach-es is to influence others to think and take responsibility for their learning.

6. Ask “If things were perfect …” In our experience, we have found that people generally complain about the problem, rather than go to solution. Since it is our job to find solutions and inspire change, we use a strategy that influences others to think positively as they brainstorm ideas that move toward positive desired re-sults. Suppose a teacher said that she doesn’t know what to do with a particular student. Our next move is to ask the teacher the following, “If things were perfect, what would he be doing?” In our experience, the teacher focuses on what she does want to have happen as she lists positive characteristics or behaviors. These end-in-mind visions be-come our goals. We write down all these characteristics, and then ask: “How can we support him better so that we see these behaviors.” The conversation will likely turn into a brainstorming session of solutions. On the rare occasion that the teacher doesn’t know what to do to support the student, we suggest, “this is a great list so far, let’s continue it the next time we meet.” This allows the coach and the teacher time to brainstorm ideas without the coach appear-ing to be the “fixer”. We have found that this extended wait time supports the teacher to think and provides a message of teamwork where all members contribute.

7. Don’t take things personally.When our colleagues act in a negative way, we have learned that it is often coming from a place of fear or uncer-tainty. We remember (and remind each other) that when a teacher or colleague lashes out, it is always about him or her--not us! In our experience, not taking things personally is much easier said than done, and so here are some rem-edies to keep a positive mindset working towards growth.


• Keep (and read) a journal of short reflections about suc-cesses that you have had with teachers and students

• Visit one of your favorite classrooms• Spend five minutes with your principals sharing about a success

• Meet with another coach and share a story of success• Come up with a list of “if things were perfect...” and then determine which is in your control - and go do it!

These ideas are based on the premise that when you are achieving, and re-living those achievements, you build confidence and belief in your ability to coach. This belief will keep you motivated and give you stamina to tackle more opportunities that will help you grow and develop as a coach. 8. Have the teacher observe first We have found that offering to teach a lesson in a class-room is the gateway to building trust with teachers. Teach-ers are more likely to see the coach as a peer rather than one who simply critiques others. Another benefit of mod-eling a lesson is that the coach and teacher can start to develop a common language. After the lesson, the coach and teacher can debrief the lesson. In our experience, pedagogical terms such as “turn-and-talk”, “discourse”, and “collaborative groupings” are buzzwords that, when used in a school, may not have the same definition among educators. This setting gives us the opportunity to set common language and definitions for the terms that we will be focusing on together. We have found that the teacher often observes a practice that s/he would like to work on together, and since we modeled it, the teacher is only choosing from practices that we show-cased. 9. How to have fun with your colleagues and enjoy yourself! Have you ever noticed a “burn-out” pattern in the months of December and February? Paula Rutherford (2005) notes these months are the most challenging for teachers because of the holiday stress and the long haul through the winter. As a coach, it is important to know about these challenging months so that you have the skills to support and celebrate your teachers. This is a great time to have some celebrations of successes. Some ideas include orga-nizing a small party or pot luck during a department meet-ing. Teachers rotate around different stations to grab new ideas and to socialize with colleagues while learning new content-related activities. Having black-line masters and materials for teachers will make great party favors. What-ever you choose to do during these months, make it re-juvenating, and exciting so that teachers feel fueled with positive energy and ideas. 10. Solution-Focused Moves Our final recommendation ties many of our other ideas to-gether. In our experience, building a culture of learning--for both teachers and students--is a challenge. Through a focused purpose, using specific language, and supporting

teachers to grow, there is opportunity for students to grow and learn. Below are some “get out of jail free cards”. In other words, when we are faced with an action that halts growth, we have used these ideas to get out of the situation and into a positive-oriented stance. Comments and Actions Solutionthat Halt Growth (question to ask) I don’t get it. Can you work with me on this problem? My kids can’t do this. Can we brainstormMy Sped kids can’t do this. some ideas to supportThis is too hard.s those students doing ____________ better?

There is too much going Can we look at the quarteron. I just don’t have the at a glance, and determinetime for that. which standards could be taught together, which would give us more time?

This isn’t going to last - What do you value? (Makenext week there will be a list and pick somethinga new initiative. off the list that you both value to work on.) Regardless of what our job description says, we know that our job is to influence teachers to grow as learners and teachers, so they can influence their students to think and be responsible, which in turn increases student achieve-ment. We cannot force teachers to change or do some-thing: however, we can inspire and influence them to think and work hard because the work we do is so important. Building trusting relationships with these strategies has made coaching successful and enjoyable. We have the best job in the world. Enjoy it because it is not about wait-ing for the storm to pass, it is about learning how to dance in the rain.

ReferencesMcLoughlin, D. M. (2011, March 7). High Trust: Acceler-

ating Student Achievement. Lecture presented at High Trust Workshop in George Washington Middle School, Alexandria.

Rutherford, P. (2005, April). The 21st Century Mentor’s Handbook. Just Ask Publications. Alexandria, VA.

West, L. (2010, September). Content-Focused Coaching. Mini-lecture presented at George

Washington Middle School, Alexandria.

THeReSA WILLS is a Mathematics Instructional Coach at Fran-cis C. Hammond Middle Schools in Alexandria, VA. MOLLY RO-THeRMeL is a Math Resource Specialist in Fairfax County.


GENERAL INTEREST

Meet BillyCandace Standley

Meet Billy...a bright and energetic little boy who has been counting and comparing numbers since he started talking. Billy’s parents always thought he was probably gifted in math. He counted his Cheerios when he was two. As he ate each one he counted again. By four, Billy was easily combining numbers. He knew his age was 4, and he knew that next he would be 5. He loved playing math games with his mom. They would grab a handful of M&Ms or Skittles and count to see who had more. Billy always knew who had more or less. He could count to 30, and he was al-ways telling someone that 1 and 1 is 2 or 3 and 2 is 5. Billy couldn’t wait to go to school. During the first few days of kindergarten, the class was introduced to a calendar routine. The teacher asked if any-one knew the date if it was a 2 and a 1. Billy knew, but when he said 3, the teacher told him to pay attention. He was paying attention. Why was that wrong? Later the date was 1 and 6. Billy thought that would be 7. He also be-lieved that 1 and 6 would be more than 2 and 1, but that’s not what the teacher said. She said when we see a 2 and 1 it’s more than 1 and 6. None of this made sense to Billy, so he twirled on his foot or sang to himself during calendar. The teacher told his parents he couldn’t pay attention. The parents decided he must be bored. Another curious thing about kindergarten was the way they counted. He and his mom always said 1, 2, 3, 4, 5, etc. His teacher taught them some different ways like 2, 4, 6, 8, or 10, 20, 30, 40 or 5, 10, 15, 20. Billy didn’t know why they counted that way, but it was fun when the class shouted the numbers together. Later in the year, the class began adding numbers. When the teacher asked if anyone knew what 2 and 1 made, Billy said 21. He remembered the calendar routine, and he was ready. The teacher scolded Billy and called his parents again. Billy’s parents couldn’t figure out what was wrong. One day the children played a game with dice. They were told to put the numbers on the dice together. The person with the highest number got a point. Billy rolled a 2 and a 3 which he now knew was 23. His partner got a 1 and a 6 which is 16, right, or maybe it was 61? Billy had known since he was 4 that 23 is greater than 16, but when he tried to claim his point a fight ensued. The partner said that 2 and 1 is 3 and 1 and 6 is 7, and 7 is more than 3. Billy threw the dice, and the teacher called his parents again. No one could understand why Billy was struggling so much with math. In first grade, Billy’s class started counting money. When the teacher laid out a pile of coins, Billy counted the coins 1, 2, 3, 4. The teacher tried to explain that each one repre-sented a different amount. She reminded the class about the skip counting they had done in kindergarten. Billy re-membered that, so he counted the pile of coins a different way--5, 10, 15, 20. The teacher raised her voice and told Billy that pennies had to be counted by ones, nickels by 5s and dimes by 10s. He didn’t understand that. He hated

math. None of it made sense. Next, the teacher said, “Let’s talk about fractions.” She said one over two was one half. Billy thought it was 12 or 3, but maybe it was 1/2. He didn’t know, and honestly, he didn’t care. He drew pictures or gazed out the window. Then the teacher said they would learn about place val-ue. She said 16 was 1 ten and 6 ones. Billy didn’t know the difference between a ten and a one. When he saw numbers like 76, he couldn’t remember if it was 6 tens and 7 ones or 7 tens and 6 ones. He still kind of wondered why a 1 and a 6 weren’t 7. Just about the time Billy understood that a 7 and a 6 were seventy-six or 7 tens and 6 ones, the teacher said 7 and 6 is 13. She said use a double like 6 and 6 to help figure it out. She said put 6 and 6 to-gether, which Billy had learned was 66 and add 1 and then you would know that the answer is 13. Math was awful. Why did it have to be so hard? During homework time Billy cried. When his mom tried to explain or play a game to help him learn like they used to, he told her that’s not what the teacher said. Billy’s parents feared that the son they once thought might be gifted in math was really disabled. Or maybe he had ADHD. The teachers had always said he didn’t pay attention. At the end of second grade, the teacher introduced num-bers like 321. She said that number was bigger than 199, but Billy didn’t understand that at all. How could little num-bers like 3 and 2 and 1 be bigger than big numbers like 9 and 9. She also showed Billy how to subtract. She said cross off the number in the ones place and borrow a ten. Make the number a 13 or 14 or whatever. Everything was running together in Billy’s mind. He crossed off the num-bers like he was told, but he couldn’t remember what to write above it. Sometimes he wrote 10 because she said borrow a ten. Sometimes he wrote 31 or 41 because he re-ally didn’t know what adding a ten meant. In kindergarten, he had said 10, 20. He guessed a ten was just one more. Things went from bad to worse in third grade. The teach-er wrote up strings of numbers like 786,221 and 114,456. She tried to explain that each number meant something different. Billy didn’t get it. And…this teacher said 7 and 6 were 42, not 13 or 76. He misbehaved and talked during math. The teacher had a lot of material to cover. She had 21 other students to teach and an SOL test to prepare for. Billy’s understanding of our number system ends here. We missed our window of opportunity for him. He will most likely struggle with math for the rest of his life, and he will always see himself as one who didn’t get math. Billy could be any child. His story represents what hap-pens to lots of children, and it brings to light some impor-tant points for teachers. We need to understand where our children are when they come to us. We need to lis-ten to what they are thinking and provide opportunities for them to resolve their misconceptions and to formulate new ideas. Children need to have a solid understanding of the numbers from 1 to 20 before they are even introduced to


bigger numbers. The concept of ten and the patterns in our number system should be at the center of every discus-sion. Skip counting groups like 2s, 5s, or 10s should be done with manipulatives until children fully understand the concept of counting groups. A child’s disposition toward math, the foundation for un-derstanding math concepts, and the child’s view of him/herself as a mathematical thinker is all developed is devel-oped in the early grades. We should be taking that respon-sibility very seriously.

Schifter, Debroah. Building a System of Tens: Casebook. Parsippany, NJ: Dale Seymour Publications, 1999. Print.

“Process Standards.” National Council of Teachers of Mathematics. Web. 27 June 2011.

CANDACe STANDLeY is a math specialist at Richardson El-ementary School in Culpeper, VA.


GENERAL INTEREST

Math & What I Know for Sure: Reflections of a Soon-to-BeMath Teacher

Antares Winslow

Math and I have had a contentious relationship for most of my life. That isn’t to say that I didn’t want to do well. Deep mathematical understanding was just too elusive for me. Math quickly became an obstacle in my education when I bombed the math section of the SAT. I wanted to continue on after I got an Associate’s degree, but I knew there was no getting around my enemy, math. I finally faced my fear, fumbled through the math and got my Bachelor’s degree in psychology. At 37 years old, I decided I wanted a Master’s degree in education from Mary Baldwin College. While I was looking forward to practicum, I was dreading taking the Praxis and required inquiry into math class. Little did I know how this one class would completely change my feelings about mathematics. As I was looking at the pages of notes from the recent practicum, I was reminded of the lump in my throat that I felt as I watched some of the third-graders struggle through the math portion of the class. I felt helpless because I didn’t know enough to help them. Or so I thought. I had just started taking an inquiry-based math class and I had seen some discrepancies between my practicum and the inqui-ry-based class. The inquiry class was comprised of mostly hands-on activities that emphasized coherency, math rea-soning, and deep understanding. I realized that I wanted to articulate the differences so I wouldn’t teach the way I was taught. I wanted to be a voice for students by resolving to articulate why the way I was taught math doesn’t work and committing myself to use the research-based inquiry methods that were changing my own negative feelings. My new motto: I can’t do everything, but I can do something. I won’t let what I can’t do stand in the way of what I can do. The greatest discrepancy, I realized, was the actual ap-proach to mathematics. The subtle (and sometimes not so subtle) messages of: 1. Math is hard 2. Math is only for smart students 3. Math must be attacked 4. Math isn’t intended to be enjoyable or interesting I know, for sure, that these are messages can be poten-tially harmful. I was told as a child that math was hard as though that was the end of the story. By saying that, my teacher, unintentionally, communicated a low expectation in my ability to be successful. I know now that it isn’t really important whether math is or isn’t inherently hard. What is important is that students receive support, resources and the confidence of their teacher that they can be successful. I was that math student who couldn’t memorize abstract algorithms. In fact, most algorithms made absolutely no sense to me. Since math is cumulative, I simply fell further and further behind, eventually failing all together. Only re-cently have I realized that I wasn’t the only one who failed. My math teachers had failed to believe in my ability to learn math.

I know for sure that students need adequate support to study math. Through inquiry methods, I have learned that math can be challenging, rather than hard. Hard translates to intimidating and defeating. Hard is an uphill battle with little hope of victory. Challenging implies that given the right support, guidance and practice, I can meet the chal-lenge. I can be successful. I remember trying to figure out how many squares were on a checker board in my inquiry-based class. It was more challenging than I first thought. However, I had the support of my teachers and classmates. As a class we worked and reworked the problem. All the while, I broadened my thinking beyond the checker pattern. The reward for meeting that challenge was sweet satisfac-tion. I know for sure that math doesn’t discriminate. As I sat in my inquiry class on the floor working on a mathemati-cal Venn diagram, math wasn’t rooting against me. Quite the opposite was true, in fact. Using the inquiry methods, I had the benefit of my fellow students discussing the prob-lems with me, helping me clarify my thinking. Again, I had the full support of my teachers. We were using hands-on materials and I had my own brain engaged and eliminat-ing wrong guesses. No one was being ridiculed, chastised or made to feel stupid. Together we worked to get the an-swers. Math isn’t just for the super smart. Math was for anyone who felt safe and interested enough to try. This was drastically different from how I was taught as a student. I remember standing at the chalkboard alone without a clue how to solve the problem. The other kids snickered and the teacher looked annoyed at my incompetence. I believed then that I just wasn’t smart enough to do math. I was con-vinced that my math teachers felt the same way. I believed this for the next thirty years. I know for sure that math isn’t a war. I have been so engrained by negative experiences to think of it as such that I still put on my math “armor.” My armor is the fear: that I won’t know the answer, that I will never learn, and that I will be embarrassed. It takes only a few minutes in my inquiry class for my armor to melt. We listen to a math story; we discuss its strengths, weaknesses and possible applications. The book is seguing to our lesson. Our lesson is hands-on, coherent and I am never left to fend for myself. We discuss, consider, analyze and work hard. I am relaxed and comfortable by design of my teachers. I no longer need armor. I know for sure that math can be enjoyable! The phrase that I will steal from my inquiry-based math professor is, “This is interesting!” Whenever I heard that phrase I was never disappointed. The same person who detested math found shapes I could hold in my hand, using manipulatives, cutting arrays and making an attribute train very interest-ing! What piqued my interest most was that it also made sense. And it didn’t even have to make sense immediately.


The suspense of impending knowledge was actually intoxi-cating! The activity was intriguing enough that I wanted to solve it. I was the proud new owner of mathematical in-trinsic motivation! Who knew that meta-cognition could be exhilarating? Finally, I know for sure that there are students who are still being sent the same counter-productive messages that I was thirty years ago. They are discouraged and fearful. They are frustrated, embarrassed and confused. The end-less worksheets and drills are reinforcing these messages. Too many students put on their own armor, attacking those boring worksheets in silence, and facing mathematics as their enemy. They are missing out! They deserve to see mathematics as something interesting to be discovered, rather than a chore to be tackled. They deserve to be in-

trigued, curious, and engaged. They need to know that there are ways to make math make sense and that there is often more than one way to solve a problem.I will be the first to admit that I still have a lot to learn. But I have a strong foundation on which to build. I have learned that I have a responsibility to help all my students enjoy math by making it coherent and relevant. And that all stu-dents deserve to feel fully supported and safe. That inqui-ry-based class changed how I feel about math as both a student and a future teacher. I am determined to break the cycle of teaching math how I was taught. I still don’t know everything, but I do know something. And I won’t let what I don’t know stop me from doing what I do know.

ANTAReS WINSLOW is an MAT student at Mary Baldwin College.


GENERAL INTEREST

Overcoming Resistance to Change: Why Isn’t It Working?Ted H. Hull, Ruth Harbin Miles, and Don S. Balka

Mathematics educators – whether teachers, coaches, supervisors, principals, or others – have sufficient and sig-nificant research on learning at their disposal to guide and support purposeful change. The lack of improvement in mathematics achievement, then, is not due to inadequate information. Most likely, the lack of change is due to the inability to transfer research knowledge into practical use in classrooms and schools. This is certainly not a revolu-tionary idea, yet still a significant one. Regardless of an intense, identified need for mathematics success for all students, achievement remains elusive. Is overcoming re-sistance to change unattainable?

Change Formula In general, people resist change, or at least avoid it as much as possible. For significant areas of our lives, we de-velop a comfort zone, and attempt to live within the bound-aries of this zone. Pressure from outside sources occasion-ally forces us to move beyond these zones. At that point, our goal is to regain a comfort zone in the new circum-stances as soon as possible. Educators responsible for change need to understand this dynamic of human nature. Change can occur as pres-sure is exerted over time. Pressure is the result of effort, and the formula can be written as:

Time + Effort = Change

Another way to state this formula, perhaps more mean-ingful for leaders who are considering change initiatives, is the negation. This formula is stated as:

No Time + No Effort = No Change

In other words, if the desired or requested change is not worthy of leaders’ attention, then it is not important to teach-ers either. Just as students take cues from their teachers about what mathematics content is important, teachers take cues from leaders about which change initiatives are important.

Overcoming Resistance Even when changes are deemed important, requested changes tend to be routinely ignored or rejected, then re-sisted to some degree. Change is finally adopted if, and only if, effort over time is provided. Many programs or ini-tiatives have no serious effort or time attached, and conse-quently never move beyond the ignored or rejected stage. Change does not occur, and this result is often translated as resistance. This is invaluable information in the struggle of overcom-ing resistance to change. In planning for and implement-ing change, leaders need to ensure that the program or

initiative is actually feasible and beneficial. Oddly enough, ignoring or rejecting many change initiatives has probably been a blessing in the past. Since numerous change re-quests seem to be occurring simultaneously, effective plan-ning and implementation are lacking. Examples of some of the changes are calculator usage, block scheduling, ma-nipulatives, and interactive whiteboards. These changes, by themselves, can prove to support student learning if implemented as designed.

Screening Factors Leaders’ awareness of “effort over time” is a tremendous burden and responsibility since change initiatives must actually be worthwhile. This was not the case in many in-stances. If change initiatives require time and effort, then the results achieved by implementing the change must be worth the time and effort exerted. This demands that change initiatives be carefully considered and controlled. During pre-planning, leaders must ensure that the change program or initiative slated for implementation is carefully screened. If leaders cannot positively address these screening factors, then the program or initiative will most likely fail – wasting time, sapping energy, squander-ing resources, and eroding confidence. The screening fac-tors are: 1) Counting your chickens (Program selection) 2) Building a bridge (Program support) 3) Creating a village (Program culture)

By not carefully considering these factors, instructional leaders have practically guaranteed rejection or resistance.

Counting Your Chickens Selecting the program or initiative that is truly aligned to the identified need is quite obvious, but also routinely over-looked. Leaders frequently consider three factors: (1) On the surface the program or initiative sounds reasonable; (2) It probably has had some level of reported success, and; (3) The program is assumed to work. Meeting these fac-tors does not provide adequate screening. The program or initiative must have a direct impact on student achievement before any other factor is considered.

Why Isn’t It Working? Three rules of thumb arise when it comes to unsuccess-ful programs or initiatives. One rule is that the program or initiative is actually ineffective – it doesn’t work. Ineffective programs may not work because they are not directly tied to student achievement. They may also be ineffective due to certain requirements, such as the program or initiative requires more time than is available, the program or initia-tive requires supporting strategies that are not used, or re-source materials are not accessible. With these omissions, the program is not effective.


The second rule is that the program or initiative is effec-tive, but it does not address the identified need. In other words, it fixes the wrong thing. For example, data analysis scores on the Virginia SOLs are low, so the school divi-sion purchased supplementary units for this standard. The materials targeted bar graphs; however, the SOLs tested line graphs. Finally, the program or initiative does not work because it is not implemented. Implementation is a serious issue. During program or initiative screening, instructional leaders need to be able to clearly articulate the actions needed to ensure adequate implementation. If these ac-tions cannot be carried out, the program or initiative, re-gardless of how effective it may be, should not be adopted.

What Is Recommended for Leaders?• Have all relevant parties study the research and the pro-gram or initiative.

• Ensure that there is an alignment between data specific to the identified need and division program goals.

• Ensure the data generated by the program or initiative is compatible to division and school level data.

• Pilot the program or initiative with “reluctant” volunteers. Willing volunteers tend to make the program or initiative be successful in spite of its failings, while resistant partic-ipants will undermine the program in spite of its benefits. Leaders need accurate, honest, and unbiased results.

Building a Bridge Program support compared to teacher evaluation cre-ates tremendous confusion. During implementation of a program or initiative, teachers need training and support. Regardless of the number of years of experience teachers may have, change requires a learning curve. If teachers are not learning new strategies or approaches, what is the purpose of the change?

Why Isn’t It Working? Teachers need support in order to make changes in their classroom routines and instructional techniques. With the information about learning curves, individual teacher evalu-ations undermine successful implementation. Why would teachers want to try something new just to be graded down on their attempt? Teachers do need feedback on their per-formance on using the strategy. Also, the program needs to be evaluated for degree of implementation and results. However, individual teachers should not be “scored” in their early efforts at enacting change. In many cases, evaluation is the default method for working with teachers. Mathematics coaches may have only experienced evaluation, and therefore when they find themselves responsible for coaching, they resort to an evaluative approach. As an example, coaches feel their job is to enter a classroom, move to be back of the room, observe and record teacher behaviors, and then provide a critique to the teacher. Principals usually receive training only in evaluation methods. They find it difficult to move between support and evaluation.

What Is Recommended for Leaders?• Analyze the program or initiative in order to identify the critical components present. These critical components make the program work. For instance, if the program re-quires student collaboration while working on challeng-ing mathematics problems, then various grouping strate-gies are critical components (pair-share, groups of three or four, and whole group).

• Provide continual training, and in-classroom support for use of the critical components.

• Continually gather data concerning both the quantity and quality of the use of the critical components.

• Regularly evaluate the degree of implementation and measureable program results.

Creating a Village Schools and school divisions have distinct cultures. These cultures are evident in every classroom. In order to work successfully, the program or initiative must fit within the parameters of established school and classroom cul-tures including policies, practices, and procedures. Lead-ers need to understand that rarely, if indeed ever, does a program or initiative designed to promote change not re-quire actual change to policies, practices, or procedures – changing the culture. There is great wisdom in recognizing where policies, practices, and procedures conflict with the intended program or initiative.

Why Isn’t It Working? Many times, embedded practices and procedures pro-vide significant impediments to the successful adoption and implementation of a program. Leaders may feel that only the mathematics teachers, or mathematics classrooms are impacted by the change. This oversight can cause serious problems. For instance, isolation is a huge impediment to success-ful adoption of a desired program or initiative. Teaching in isolation is a school culture. During early use of the strate-gies or components required by the program, teachers are unsure of both what it is supposed to look like, and what it is supposed to do. As a result, teachers need to collaborate and share. Organizing times for mathematics teachers to meet, or for mathematics schedules to coincide can have a ripple effect throughout the school. Furthermore, since iso-lation is the norm, teachers within the school are unfamiliar with strategies for effective collaborative planning. Another issue is that training and support frequently do not match the needs of the user. As teachers gain experi-ence with the components and strategies contained within the program or initiative, they move from novice users to more mastery. To actually gain proficiency and expertise, teachers need different knowledge and skills than they needed when first using the strategies.

What Is Recommended for Leaders?• Understand and be able to recognize appropriate uses of the critical components of the program or initiative.


• Recognize novice, mastery, and expert levels of usage of the critical components and strategies.

• Be open and unbiased in reviewing school and class-room culture shifts and impediments to change.

• Sustain positive pressure until the program or initiative is fully adopted by all appropriate staff, and the identified need has been addressed.

Closing Change takes time. Change also takes energy and effort. If instructional leaders and coaches do not exhibit these el-ements, then teachers will not exhibit them either. Leaders and coaches do untold damage to rapport and trust when they send the tacit message to teachers that, “while I am too busy to be bothered with this change, you are not.” The formula for change may perhaps best be stated as:

Adequate Time + Focused Effort = Desired Change

By appropriate analysis of the recommended program or initiative, leaders and teachers are better prepared to screen the programs or initiatives for potential success, an-ticipate required shifts in practices, and plan for correct im-plementation. All the while, teachers and leaders maintain a clear focus on student achievement. Through a careful screening process of pre-planning and analysis, resistance to change is greatly reduced.

About the authors:Ted H. Hull completed 32 years of service in public ed-

ucation before retiring and opening Hull Educational Consulting. He served as a mathematics teacher, K-12 mathematics coordinator, middle school principal, direc-tor of curriculum and instruction, and a project director for the Charles A. Dana Center at the University of Texas in Austin. While at the University of Texas, 2001 to 2005, he directed the research project “Transforming Schools: Moving from Low-Achieving to High Performing Learn-ing Communities.” As part of the project, Hull worked directly with district leaders, school administrators, and teachers in Arkansas, Oklahoma, Louisiana, and Texas to develop instructional leadership skills and implement effective mathematics instruction. Hull is a regular pre-senter at local, state, and national meetings.

Ruth Harbin Miles coaches rural, suburban, and inner-city school mathematics teachers. Her professional experi-ence includes coordinating the K-12 Mathematics Teach-

ing and Learning Program for the Olathe, Kansas Public Schools for over 25 years; teaching mathematics meth-ods courses at Virginia’s Mary Baldwin College and Ot-tawa, Mid America Nazarene, St. Mary’s, and Fort Hays State universities in Kansas; and serving as president of the Kansas Association of Teachers of Mathematics. She represented eight mid-western states on the Board of Directors for the National Council of Supervisors of Mathematics (NCSM) and has been a co-presenter for NCSM’s Leadership Professional Development National Conferences. Miles is the coauthor of Walkway to the Future: How to Implement the NCTM Standards (Jan-sen Publications, 1996), and is one of the main writers for NCSM’s PRIME Leadership Framework (Solution Tree Publishers, 2008).

Don S. Balka, a former middle school and high school mathematics teacher, is professor emeritus in the Math-ematics Department at Saint Mary’s College, Notre Dame, Indiana. During his career, Balka has presented over 2,000 workshops on the use of manipulatives with elementary and secondary students at national and re-gional conferences of the National Council of Teachers of Mathematics, state mathematics conferences, and at in-service training for school districts throughout the United States. In addition, he teaches classes in schools throughout the world, including Ireland, Scotland, Eng-land, Saudi Arabia, Italy, Greece, Japan, and the Mari-ana Islands in the South Pacific. Balka has written over 20 books on the use of manipulatives for teaching K-12 mathematics and is a coauthor of the Macmillan K-5 ele-mentary mathematics series, Math Connects. Balka has served as director for the National Council of Teachers of Mathematics, the National Council of Supervisors of Mathematics, and School Science and Mathematics As-sociation.

Together, Ted H. Hull, Ruth Harbin Miles, and Don S. Balka have written 3 books published by Corwin Press Publish-ers: Visible Thinking in Mathematics Class-rooms K-8: The Key to Student Engagement and Achievement; A Guide to Mathematics Coaching: Processes for Increasing Student Achievement; and A Guide to Mathematics Leadership: Sequencing Instructional Change. Their 4th book Over-coming Resistance to Change: A Guide for School Leaders & Coaches is published by LCM: Leadership, Coaching, and Mathematics. The authors may be contacted at [email protected] or 512-913-4738.


A Professional Learning Community (PLC), consisting of educators from Pomona who had been in other PLCs and / or were outspoken at their sites about their beliefs around student learning, came together in March 2010 with like-minded people to write a belief statement regarding the teaching of mathematics. The task was to

develop a statement of your beliefs about teaching and learning mathematics that will guide us as we teach students in Pomona. Begin each statement with the following: “We believe all students can learn rigorous mathematics when...” and then, “We be-lieve all students can be taught rigorous mathematics when...”

GENERAL INTEREST

Mathematics Leaders’ Belief StatementsDiane Kinch

The group perused a variety of documents that included the Mission Statement of the California Mathematics Coun-cil and position papers on related topics from the National Council of Supervisors of Mathematics and the National Council of Teachers of Mathematics. We also addressed issues around belief systems related to:

✓ digital immigrants and digital natives,✓ cultural proficiency needed for teaching a culturally di-

verse population,✓ teaching to learn versus teaching to a test,✓ every mathematics teacher as a literacy teacher,✓ improving teacher practice,✓ appropriate student placement, and✓ matching teaching methods to student needs.


Reprinted with permission from CMC ComMuniCator Vol. 35, No. 4, June 2011, an official publication of the California Math-ematics Council.


VCTM 2012 William C. Lowry Mathematics Educator of the Year Award

To: VCTM Members From: Brenda P. Barrow

Virginia Principals VCTM William C. Lowry Mathematics Educator of the Year Committee

Math Department Heads 1311 E. Bayview Blvd.

University Department Heads/Deans Norfolk, VA 23503

Email: [email protected]

Phone: 757 – 617 - 0984

Each year the Virginia Council of Teachers of Mathematics may recognize a classroom teacher on the elementary, middle,

secondary, university and math specialist/coach level for his/her outstanding work in the field of mathematics. One teacher

selected from each of the five categories may be awarded the VCTM William C. Lowry Mathematics Educator of the Year

Award. All awards will be announced in the spring of 2012.

Past winners and current elected VCTM Board members are not eligible for nomination.

The qualifications for this award are as follows:

* The nominee must be a current member of VCTM.

* The nominee must have a minimum of five years teaching experience and be a current classroom teacher, work with students as

a math resource teacher or be a math specialist.

* The nominee must have made notable accomplishments in teaching mathematics.

* The nominee may be nominated by a sponsor or may make a self-nomination. (Anyone who is a member of VCTM, a school

division superintendent, a school principal or headmaster, a supervisor, director of instruction, a college dean or department head

or the president of any NCTM affiliated group may sponsor a candidate.)

• Details about the nomination and information needed from the nominee will be mailed to the nominee.

You are encouraged to nominate an outstanding mathematics educator that you feel is deserving of this award.

Complete the form below and return it to the address on the form. Electronic nominations are acceptable.

The awards committee will contact the nominee upon receiving the nomination to request additional information.

Nominations must be postmarked no later than Oct. 1, 2011 or electronically submitted no later than Oct. 5, 2011.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Nomination Form

VCTM 2010 William C. Lowry Mathematics Educator of the Year Award

Nominee information – Please PRINT or TYPE.

Date: ______________________

Name of Nominee: ______________________________________________________________

Home Address: ___________________________________________ Email: ____________________________

____________________________, VA ________ Home phone: ( ____ ) _________________

Nominee’s Position and School: ______________________________________________________________

Nomination Category: Elementary ____, Middle ____, High ____, University ____, Math Specialist ____

Nominee’s School Address: _________________________________________________________________

___________________, VA __________ School phone: ( ____ ) ________________

Sponsor Information - Please PRINT or TYPE.

Name: ____________________________________ Position or Title: __________________________________

School Division, College or University: ____________________________________________________________

Business Address: ______________________________________________________________________________

Phone: ( ____ ) ________________________________ Email: _______________________________________

A letter of recommendation DOES NOT have to accompany the nomination. The nominee will ask that you submit a letter to

him/her that can be included in the response packet with the other two letters of recommendation that he/she must submit.

Nominations must be postmarked on or before October 1, 2011 or electronically submitted on or before October 5, 2011.

Please mail to: Brenda P. Barrow 1311 E. Bayview Blvd. Norfolk, VA 23503

Electronic nominations are welcome. Send to: Brenda Barrow at this email address. [email protected]

THANK YOU FOR MAKING THE NOMINATION!


Virginia Council of Teachers of Mathematics

33rd

Annual Conference Friday, March 9 – Saturday, March 10, 2012

Hotel Roanoke

Conference Registration Form

Name _________________________________________________________________________________ Address _______________________________________________________________________________ Street City State Zip

email (Please include)_____________________________________________________________________ Home Phone (_____)__________________________School Phone (_____)________________________ School and School Division ______________________________________________________________ Level (please circle all that apply): K-2 3-5 6-8 9-12 College

Math Specialist Administrator Other _____________

Pre-registration Deadline is February 9th

. It must be received by the 9th

.

Registration fee includes lunch on Friday.

On-site registration is an additional $10.

Registration fees are non-refundable. Registrations may be transferred.

________ $120.00 Conference Registration fee for current members. ________ $140.00 Conference Registration fee for non-member or renewal of membership. This price

provides a 1 year membership in VCTM. ________ $70.00 Full time college student registration fee – includes a one-year membership in VCTM ________ $30.00 Friday Awards Dinner. Buffet dinner with several choices including vegetarian.

Register online by February 9 at www.vctm.org or make check payable to VCTM and return to

Diane Leighty, Treasurer

VCTM PO Box 73593

Richmond, VA 23235

[email protected]

Thank you! We look forward to seeing you.

Virginia Council of Teachers of MathematicsP.O. Box 714Annandale, VA 22003-0714 Pat Gabriel, Exec. Secretary

Non-ProfitU.S. Postage

PAIDBlacksburg, VA

#159

DATE AND NOTE POSTANNuAL VCTM CONFErENCE

roanokeMarch 9-10, 2012

rEGIONAL CONFErENCES

Atlantic City, NJOctober 19-21

St. Louis, MOOctober 26-28

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virginia mathematics teacher · 2011. 9. 28. · volume 38, no. 1 fall, 2011 virginia mathematics...

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