Vol IV

No. 4

Fall 2014



The concept of infinity is explained vividly in Hilbert's Hotel. This graphic was created by Editor Cheryl Hughes using THINGLINK.com, a convenient tool to combine many media on one topic. Check out all the links as you mouse over the image.

President's Message
by Dan Funsch

I am always energized by the start of a new school year. The hubbub leading up to the first day, the unpacking, the new, clean materials, shiny waxed floors, the smell of new books, the students freshly back from a break – all these fill me with a deep hopefulness and optimism for the year ahead.

I hope that your summer has been one of refreshment and refocus; that you've been able to spend time letting any weariness from last year seep away as well as some time learning new things and preparing to better engage your students in some of the old things.

As you get down to work in your classrooms across the state, don't forget to reach out to your fellow mathematics teachers. Reach out with encouragement. Reach out with needs and questions. Be advocates for each other in the work room.

Have a great year!

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GCTM Honors and Awards
by Peggy Pool


Nominate a deserving educator today!

It is not too late! The deadline has been extended to allow more wonderful Georgia teachers of mathematics to be nominated for the awards listed below.

Please check out the qualifications for each award, as well as the steps to nominate yourself or your friends.

Dwight Love Award

John Neff Award

Teacher of Promise

Gladys M. Thomason Award

Teacher of Excellence

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NCTM Notes Fall 2014
by Dr. Dottie Whitlow

NCTM Notes is a featured column in each GCTM Reflections publication. GCTM is an Affiliate of National Council of Teachers of Mathematics (NCTM) & this column is intended to keep GCTM members informed of NCTM services & events. All material in this column is excerpted directly from NCTM e-blasts, updates and newsletters. It is compiled by NCTM Representative, Dr. Dottie Whitlow.

  1. For those of you who want to receive NCTM news directly, you can visit the web site for news & announcements and/or become a member of NCTM. Visit NCTM.org for information.

  2. You can also get ideas and math news from NCTM's Math Education SmartBrief. Sign up at mathed@smartbrief.com. Please be aware that this is supported by advertisers and that there are many ads in the issues.

  3. NCTM Regional Conferences & Expositions for the remainder of the 2014 calendar year are in Indianapolis (Oct.29-31), Richmond (Nov. 12-14) and Houston (Nov. 19-21). Registration is open now and early registration can save you money!

  4. 2015 NCTM Annual Meeting & Exposition will be held in Boston, April 15-18, 2015. Housing & Lodging for the conference became available on August 26, 2014. It is open now!

  5. NCTM has issued a position statement in support of the Common Core State Standards. To review the document & compare the points outlined (regarding professional learning, slowing down teacher evaluation systems that include CCSS accountability, ample funding, funding research for CCSS assessments, & adequate state funding) to what you think is occurring in your school or district, visit Supporting the Common Core State Standards for Mathematics.

  6. NCTM has a new book titled Principles to Actions: Ensuring Mathematical Success for All. It describes what it will take to turn the opportunity of the Common Core into reality in classrooms, schools & districts. Visit Principles to Actions for various tools, guides & webcasts that can be accessed without a purchase, as well as links if you wish to purchase the book.

  7. Procedural Fluency in Mathematics is another of NCTM's position statements. Procedural fluency is mentioned throughout the grades in the Common Core and this paper gives insight & meaning to what it means.

  8. See all the topics that NCTM has a provided a position statement on. Topics include but are not limited to Assessment, Algebra, Calculus, Early Childhood, Equity and Teacher Quality.

  9. It is already time to be thinking about GCTM's annual Georgia Mathematics Conference! It is October 15-17, 2014! Plan to visit the NCTM booth at the conference to view various NCTM publications for sale or to place an order. Bring your NCTM membership card number to ensure your discount!

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2014 GaDOE Summer Mathematics Academies
by Peggy Pool

During the months of June and July, the Georgia Department of Education held a 2-day Summer Mathematics Academy at seven sites across the state. The Georgia Council of Teachers of Mathematics was able to have representatives present during Day 1 at each site to raise awareness of our organization and solicit membership.

As teachers stopped at our table, all expressed their joy in learning many valuable things and were excited about using the new ideas in their classrooms this fall.

It was good to see many familiar faces and to get acquainted with teachers that were not acquainted with GCTM. We look forward to seeing many of our new friends at Rock Eagle in October.

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The Georgia Mathematics Conference 2014
by Kaycie Maddox, Georgia Mathematics Conference 2014, Program Chair

It's time to make preparations to join us for the Georgia Mathematic Conference 2014 at Rock Eagle 4-H Conference Center on October 15-17, 2014. The theme this year is Mathematics for ALL: Let's Build Bridges, focusing on the need for access and equity for all students as we enter the third year of implementation of the Common Core Georgia Performance Standards for mathematics. According to a recent publication from the Equity Assistance Centers, "The Common Core provides a clear, consistent definition of what students are expected to learn and what is needed to prepare all students for success in postsecondary college or career preparation and life in the 21st century. The EACs support effective implementation of the standards so that, as individual states and as a country, we may finally ensure success for all students, regardless of their race, national origin, linguistic background, physical abilities, or economic status."  In the past, mathematics has often been used as the gate-keeper for entry to and success in college and career opportunities following high school. Now, the challenge from the authors of the Common Core State Standards as well as the Georgia Department of Education is to open up mathematics instruction to enable and support all students to be college and career ready.

"Access typically refers to the ways in which educational institutions and policies ensure—or at least strive to ensure—that students have equal and equitable opportunities to take full advantage of their education. …Generally speaking, the widespread use of the term access in education, along with related terms such as equity or at-risk, reflects increased national attention to the needs of students who have historically been underserved by schools, who have failed to take full advantage of their education, whose learning needs have been overlooked, or who have otherwise 'fallen through the cracks.'"  Equity is the term used for the need to provide a high-quality education for all students, though not necessarily in the exact same manner, to address gaps that occur in areas such as opportunities, instruction, and assessment for particular sub-groups of students.

However, the goal to provide a high-quality education for all students is not easily attainable, especially in the area of mathematics. As a 26-year veteran of the middle and high school mathematics classroom, I often have felt the desire for resources and ideas to guide me in this endeavor to open up the doors for my students to become mathematical thinkers who can use their mathematical knowledge to make sound decisions. This desire is one shared by most mathematics educators, leading the program committee members for GMC to solicit national as well as local experts who can, in very specific ways, address this need for access and equity for all students. Our keynote speakers are Cindy Moss, an expert on STEM education for all students, Janna Peskett, a researcher for Mindset, and mathematics educator, and Kati Haycock, the president of The Education Trust, the organization whose "goal is to close the gaps in opportunity and achievement that consign too many low-income students and students of color to lives on the margins of the American mainstream" . All three presentations will offer new challenges and opportunities to our beliefs about teaching and learning for all students. The featured speakers will each, in turn, offer practical ways to provide accessible and equitable instruction in Georgia mathematics classrooms.

Please plan to attend this year's Georgia Mathematics Conference, and bring some friends with you. Share the program with your administrators so they may encourage others to be in attendance as well. We look forward to seeing you among the pines of the Oconee National Forest at Rock Eagle 4-H Center in October!

Other speakers include:

Learn more about the conference and/or register...

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Things I Learned Teaching AP Calculus
by Chuck Garner

I started teaching AP Calculus in 2002 at a magnet school for science and technology. I was not very good at it, which was a shame, since every student in my school had to take calculus to graduate, and I was the only calculus teacher. I realize now why I was not good at it. I taught calculus to my students the way I was taught calculus: heavy on symbolic manipulation and algebra skills; more concerned with finding antiderivatives than with understanding what the definite integral means; and only focused on those applications which could be approached in a rote fashion...

Read the rest of this article...

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Tips to Begin the Year from Master Teachers

From Bill Marsh

  1. First day of class have students complete a form with name, course they took preceding year, name, phone number and email of parent or guardian.

  2. Email or call each parent during that first week. This opens lines of communication in a positive setting. (If you have to call later for something negative, you will be glad it is not your first contact with them!)

  3. Give clear expectations concerning attendance and homework from Day One!

From Tom Ottinger

  1. When planning, don't focus on what YOU will do. Think about what the STUDENTS will do.

  2. When planning, don't focus on what YOU're going to do. Think about what the STUDENTS will do.

  3. Ask questions that probe for understanding rather than for right answers.

  4. You don't have to know everything. Don't be afraid to learn from your students, and let them know you're learning from them.

  5. ALWAYS plan more than you expect to need because you often need more than you expect.

From Linda Oliver

  1. Put up pictures of generic kids having fun in school at the beginning of the year, and then gradually replace them with work from YOUR students all during the year.

  2. Make your expectations crystal clear.

  3. Show your students that you are excited about mathematics.

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There is a New Graphing Calculator in Town!
by Rebecca Gammill

Have you ever been inspired by a professional development workshop only to return to your classroom and find you and your students cannot access the same technology? In particular, graphing technologies, such as software programs and graphing calculators, can vary drastically, be cumbersome to learn, and offer slightly different functionalities. Desmos.com offers one solution for your graphing needs that is cost effective, accessible, and easy to use – the Desmos graphing calculator.

This elegant, online graphing calculator is accessible to anyone who has internet access and is available as an app for iPad users. When users launch the calculator, they can type in functions in multiple formats to instantly produce colorful, polished graphs of dotted lines, various colors, or shaded regions. Users can also easily graph vertical lines, write domains piecewise functions, and determine critical points, such as axis intercepts, intersection points, and extrema, by simply clicking on the location within the graph itself.

One of the most dynamic functions of this calculator is sliders.  Sliders can be easily created for variables within equations that are not understood to be independent variables.  For example, if one typed in the equation \(y=a|x-h|+k\), Desmos would prompt the user to create manual or animated sliders for the variables a, h, and k so that educators or students can explore the relationship of the variable with the corresponding function's graph.  Users can also graph polar and parametric equations, construct tables data, and label axes in either degree or radians.  Common mathematical variables such as \(\theta\) and \(\pi\) can be typed as "theta" and "pi" without having to use a special keyboard.  However, a keyboard is available for superscripting and subscripting variables and typing special mathematical symbols, variables, functions, and commands (i.e., square roots, inequalities, trigonometric and logarithmic functions, derivatives, series, and much more).   While spectacular, the graphing still maintains some room for improvement.  For example, the Desmos graphing calculator cannot graph lines of best fit, determine correlation coefficients, or calculate more complex forms of regression.  Other statistical graphs – such as box plots, histograms, and pie charts – are also not possible.  In terms of compatibility, importing the graphs into a Word document requires the user to save the image first and then import the image into other programs where it is no longer an editable file. 

Desmos.com provides a one-page "Quick Start Guide" to help new users start graphing within minutes. The program's flexibility in mathematical syntax and layout promote an enjoyable and intuitive graphing experience for newcomers. The Desmos staff also features a plethora of complex graphs to explore, such as "Math Examples," "Creative Art," or "Recently Saved Graphs."

Best of all, users do not need to provide any personal information to get started. However, if graphers want to save and share sketches, a personal email address and password are required for a personal account. With this simple interface, students and teachers are able to create, save, and share colorful mathematical animations online, and find this calculator to be an invaluable resource for generating classroom discussions, producing mathematical projects, and making mathematical discoveries.

We welcome Rebecca as the Publications Intern for this school year. Rebecca teaches at Kennesaw Mountain High School

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What's My Rule?
by Sandra Davis Trowell

Providing opportunities for students to develop number sense, operation sense, and algebraic reasoning is valuable at all levels of mathematics. An opening activity that provides such opportunities at various levels of mathematics is "What's My Rule?" (Wheatley & Reynolds, 2003). This activity also promotes Common Core Standards for Mathematical Practice as students have opportunities to problem solve, reason, and model (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010).

To begin this game the teacher must explain the "rules" to the students, as one goal is to provide all students an opportunity to consider a rule that might be viable. An example for negotiating this activity is as follows:


I am going to think of a mathematical rule to use and I will ask one of you for a number to use in "my rule". I will then use my rule and give you the answer using that rule. I will give several of you an opportunity to give me a number and will tell you the answer using these different numbers. When you think you have figured my rule, raise your hand. Do not state a rule!!! We want everyone to have a chance to figure out the rule. If you raise your hand letting me know that you have figured out the rule, then I will give you a number to try. When you give me the answer using your rule, I will let you know if that is the answer that I would also get. Okay, let's begin...

Suzy: 5
Teacher: 5 gives me 16
Tom: 2
Teacher: 2 gives me 10
Mary: 13
Teacher: 13 gives me 32
Maria: 3
Teacher: 3 gives me 12
Anna: I think that I know the rule.
Teacher: Okay, what would I get if I used the number 7?
Anna: I think that you would get 20.
Teacher: Yes, you would get 20.
John: I have it figured out.
Teacher: What would 10 give us?
John: 26!
Teacher: Yes, that is correct

The teacher could continue this until it appears that most or all of the students have figured a rule that will work. The next part will be to hold a discussion so that students can share their "rules". Ask students to share their rule as well as encourage them to share their strategies. In the above example, there are various ways that students might think about this:

Teacher: Anna, would you share how you thought about this?
Anna: The way that I thought of this was to add 3 to the number and then multiply it by 2.
Teacher: Thanks, John did you think of it this way?
John: No, I thought of doubling the number and then adding 6.
Teacher: What do others think? Does anyone have another way to think of this? Do both of these work?

As others share their ideas and thoughts on this, different ways of expressing an equivalent "rule" will emerge. These various ways in which one may communicate equivalent expressions, in both words and symbols, are important as students come to make sense of numbers and operations. In using symbols to express the above relationship, students may use any of the following, depending upon their level of sophistication:

$$2\Box + 6$$ $$2(\Box + 3)$$

Notice that both of these "rules" are equivalent expressions and this is an example of the distributive property. Remember that it is important that everyone have an opportunity to think about a rule before beginning a class discussion. Too many times we focus on speed rather than sense making. Students need time to consider the relationships among the numbers.

Choosing a rule will depend upon the level of mathematical sophistication of your students. For young students, one could start with a rule such as the number plus one. Students may describe this rule in various ways, such as:

$$\textrm{number plus one}$$ $$\textrm{one more than the number}$$ $$\textrm{the next number}$$ $$\Box + 1$$ $$1 + \Box$$

When asking students to share, record exactly what they say or ask them to record how they thought of a rule. This demonstrates that there are equivalent expressions for the same rule and promotes flexibility in mathematical thinking. It is also important to encourage students to share how they thought about a rule. Asking, "Did anyone think about it differently?", note the many different ways that participants may state or write their rules.

Another benefit of "What's My Rule?" is that students come to recognize that choosing a number to use when figuring out a rule that will work can make things easier. Do not suggest numbers to the students as that takes away their opportunity for this recognition. Students may begin to realize that 0 is a "good" number to plug into the rule, as well as smaller consecutive numbers.

There are many possibilities for rules to use. When working through these, one can begin to formalize the rules depending upon your students' level of mathematical sophistication. Possible rules are:

$$x + 5 \textrm{ or } \Box + 5$$ $$2x \textrm{ or } 2 \times \Box$$ $$3x + 2 \textrm{ or } 3 \times \Box + 2$$ $$x \div 2 \textrm{ or } \Box \div 2$$ $$x^2 \textrm{ or } x \cdot x$$

After students have been introduced to "What's My Rule", it can be used as an opening activity or as a short problem solving activity. This is a rich activity that builds upon students' number sense, operation sense, algebraic thinking, and imagery.


  • National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
  • Wheatley, G. H., & A. M. Reynolds. (2003). Coming to Know Number. Tallahassee, FL: Mathematics Learning.

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Exemplary Teacher: Bethany N. Ray
by Valerie Lemon

Happy New Year!!! Bethany N. Ray

To begin the school year we are already keeping the end in mind. We know that we are transitioning into a new world of testing in which the integration of content will be important more than ever before. But remember, many educators across the state have been doing this already. Due to national and state curriculum expectations, mathematics educators have been using resources such as CCGPS framework units and Exemplars. Speaking of exemplars, I am highlighting an exemplary teacher who is a very valuable asset to Bibb County.

Bethany N. Ray taught for eight years at James H. Porter Elementary School in Macon, Georgia in the Bibb County School District. She taught third grade for four years, second grade for one year, and for the past three years she taught first grade. Just this year, she has transferred to Heritage Elementary School in the Bibb County School District to teach third grade. She is excited about this new change and adventure!

Ms. Ray has a wealth of knowledge as she graduated from Wesleyan College in 2006 with a B.A. in Early Childhood Education and in 2009 with a M.A. in Early Childhood Education. In 2012, she graduated from Georgia College and State University with an Ed.S in Curriculum and Instruction in Early Childhood Education. In the spring of 2014, she completed the requirements for the Teacher Support Specialist and Teacher-Leader endorsements. She has presented at literacy conferences on Reading and Writing hosted by Georgia College and State University, Georgia Reading Association, Georgia Department of Education, and Kennesaw State University.

Ray says that over the past eight years, her path to discovery of new methods has been an exciting one. Her biggest discovery occurred in the past two years while teaching first grade. She had to learn how to simplify the learning process of mathematics to increase student achievement in math competency with her first grade students. She believes that if students learn how to add and subtract fluently, they are well equipped to complete more advanced math tasks.

During this process, she collaborated with her team [of coworkers and students] to increase their use of math strategies, manipulatives, and exemplars to understand the world of mathematics through the eyes of first grade and third grade students. Ms. Ray expressed,

"In first grade, students get the foundation that prepares them for the rest of their mathematic journey. It is through this process that allows for these students to be armed for the rest of their mathematical path of discovery."

Mrs. Ray's passion in teaching has always has been reading and writing. However, over the past two years she has focused on helping her students develop a love for math through math workshop, guided math instruction, and interactive math lessons and notebooks for her students. Using interactive notebooks creates a way for students to personalize and make meaning from the information taught in class by the classroom teacher, it is powerful study for students to use from year to year, it is a working portfolio to monitor the progress of students for teachers, parents, and administrators to see first hand, it appeals to the multiple intelligences, and it encourages students to take pride in their work. To create these with your students all you will need for each child are spiral or composition notebooks, pencils, highlighters, and glue sticks.

During the 2013-2014 school year, her biggest goal was to increase the school's scores in numbers and operations. It was the lowest percentage across all grade levels from K-5th. Ray and her coworkers knew that they had to research and implement effective practices to create a feasible action plan for increasing math competency among their students. To increase their scores, they used a variety of feasible interventions. They used math competency stations, small group instruction, interactive notebook addition and subtraction activities, practice and graded fluency AIMSweb probes, and a motivational piece to keep their students involved in being successful mathematicians. It was through all of these interventions that raised the ability level among my students. Ray stated, "After creating their plan and putting it into place, third grade scores increased by 18% and first grade scores increased by 14% in numbers and operations. How exciting is it that they improved in their mathematics abilities!"

Ms. Ray's enthusiasm transcends into the upcoming school year. She concludes by saying,

"Over the past year to help increase my students' achievement more in mathematics for the upcoming school year, I developed an interactive math notebook to use with my students. I created this tool in order to help my students become more involved in their journey of learning through the world of mathematics. Using interactive notebooks in my classroom has helped my students to take ownership of their own learning. If you are interested in learning more, see http://www.teacherspayteachers.com/Store/Bethany-Ray or email me at bethanynray@gmail.com."

Ms. Ray and Bibb County would like to wish you a Happy New Year... School New Year that is! We look forward to all of the new ideas and strategies that our educators have to share with each other in Bibb County and in Georgia.

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GCTM Executive Board

President – Dan Funsch

President-Elect – Kaycie Maddox

Executive Director – Tom Ottinger

Membership Director – Susan Craig

NCTM Representative – Dottie Whitlow

Secretary – Debbie Kohler

Treasurer – Nickey Ice

IT Director – Paul Oser

REFLECTIONS Editor – Cheryl Hughes

VP for Advocacy – Shelly Allen

VP for Constitution and Policy – Patti Barrett

VP for Honors and Awards – Ned Colley

VP for Regional Services – Valerie Lemon

VP for Competitions – Chuck Garner

Conference Board Chair – Debbie Poss

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Table of Contents

President's Message - by Dan Funsch

GCTM Honors and Awards - by Peggy Pool

NCTM Notes Fall 2014 - by Dr. Dottie Whitlow

2014 GaDOE Summer Mathematics Academies - by Peggy Pool

The Georgia Mathematics Conference 2014 - by Kaycie Maddox

Things I Learned Teaching AP Calculus by Chuck Garner

Tips to Begin the Year from Master Teachers

There's a New Graphing Calculator in Town! - by Rebecca Gammill

What's My Rule - by Sandra Davis Trowell

Exemplary Teacher: Bethany N. Ray - by Valerie Lemon

GCTM Executive Board


Georgia Council of Teachers of Mathematics | PO Box 5865, Augusta, GA 30916 | 1-855-ASK-GCTM