Summer Academies 2019 by Kristi Caissie, Summer Academies Coordinator

 

I need your help with making the 2019 Summer Math Academies a success. The Summer Academies have the potential to reach over 800 classroom teachers. Please share this information with anyone that you feel might be interested in this event or anyone that might help us advertise.

This two day workshop offers face-to-face professional development focused on a grade-band of your choice!

K-1st, 2nd-3rd, 4th-5th, 6th-8th, Algebra, Geometry, and Algebra II

• Improving teacher knowledge

• Supporting productive struggle

• Experience engaging, rich tasks aligned to grade-band GSE standards

• Classroom strategies to meet the needs of ALL students

Email questions to academies@gctm.org

Cost: $120 for non-member and $90 for member Registration Opens February 1st at: new.gctm-resources.org/gctm/dv7/?q=academies Locations and Dates: Chestatee High School in Hall County - June 18th-19th Belair K-8 School in Richmond County - June 25th-26th Flat Rock Middle School in Fayette County - July 9th-10th Brunswick County High School in Glynn County - July 16th-17th

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Volunteer Corner by Jeff McCammon, Ph.D., Member Liaison

Have you been looking for ways to expand your professional resume? Do you love the art of teaching mathematics and want to get more involved in helping others? Are you in need of a TKES goal to help with achieving a level IV in Professionalism?

Then we would love to have you as a volunteer in our organization. There is so much behind the scenes work before, during, and after our main events and we are ready to connect you with the appropriate GCTM representative/leader.

We currently have 5 recognized avenues of service/volunteer work:

Summer Academies
Each year, we provide three or four two-day training events throughout the state of Georgia. We would love for volunteers who can arrive a day early to unpack and set up, assist during the event by working in the registration booth or help after the event with take down and clean up.

GMC meeting in October at Rock Eagle
This event is our big gathering each year at Rock Eagle. There is much work to be done in preparation for this meeting. We also have several volunteers that help with check-in, assist with information booths, and support the various extra functions and social gatherings that occur during our time together.

Math Day at the Capitol
Each year we have an event in February at the State Capitol Building called Math Day at the Capitol. We usually provide informational gatherings for our representatives.

Reflections
An on-going area of volunteering is by submitting an article for our Reflections newsletter. We also need volunteers who are charged with taking pictures at events. If your article is selected for publication, you may earn the opportunity for a free membership to GCTM.

Local Representative
Our Region Representatives oversee the various districts in Georgia. However, one of the key ways to get the word out about issues concerning math education and the benefits of GCTM is through our Local Representatives. A Local Representative is available to receive emails and newsletters and gives math teachers in their school (district) the information about upcoming events and gatherings. The Local Representative does not have to attend any formal planning meetings but rather communicates information to their local peers.

Would you like more information on any of these events, positions, or ways to help at these events? Please email me @ jeff.mccammon@gctm.org and I will be glad to connect you to the appropriate chairperson.

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From Stress to Success: Supporting Teachers As They Teach Multiple Strategies by Dr. Jennifer Bay-Williams, Dr. Sue Peters, Dr. Lateefah Id-Deen

There is a conundrum in our mathematics Standards related to the use of strategies, and it is causing stress for teachers, students, and parents. The Common Core State Standards for Mathematics (CCSS-M) (NGA & CCSSO, 2010) (and similar state standards) have decreased the number of topics, yet the emphasis on using multiple solution strategies in problem-solving has increased what needs to be taught in mathematics classrooms. A study commissioned by the Thomas B. Fordham Institute surveyed over 1000 teachers and confirmed that more teachers are teaching multiple strategies now than before CCSS-M (Figure 1) (Author, 2016).

 

 

Figure 1. Survey results related to teaching multiple strategies.

 

As teachers increase their emphasis on multiple strategies, they are finding the "devil is in the details." Or more appropriately, the stress is in the strategies. In the same survey referenced above, 53 percent of teachers believe, "students are frustrated because they're being asked to learn many different ways to solve the same problem" and 85 percent believe "reinforcement of math learning at home is declining because parents don't understand the way math is being taught" (p. 43).

 

Standards, because they are not the curriculum, provide little guidance on answers to the following questions:

• Why teach multiple strategies?

• For which topics are there multiple strategies, and what are they?

• How do I sequence and develop the strategies?

• How do I address parent/guardian concerns?

The impact on teachers of not having answers to these questions is...stress! We suggest that we, as teacher leaders, need to employ multiple support strategies to help teachers effectively teach multiple solution strategies. In the sections that follow, we use the questions here and offer suggestions on how we might respond. We hope these ideas will support your efforts to relieve stress and increase teachers' and students' success in using multiple strategies to engage meaningfully in doing mathematics.

 

1. Why Teach Multiple Strategies?

 

Put simply, multiple strategies are necessary but not because CCSS-M includes a stronger focus on strategies. Knowing multiple strategies is prerequisite to developing procedural fluency and conceptual understanding.

 

Response 1a: Pursuit of Procedural Fluency

 

Procedural fluency requires more than knowing standard algorithms and using them quickly. It includes accuracy, efficiency, strategy selection, and flexibility (National Research Council, 2001; NGA & CCSSO, 2010).  Stakeholders (teachers, principals, parents, and students) may recognize the first two components, but not the latter two. Yet, these last two components are necessary for fluency and are critical to understanding Why teach multiple strategies? Figure 2 illustrates the interrelated aspects of procedural fluency.

 

 

Figure 2. Procedural fluency components and inter-relationships (Author, 2017).

 

Using grade-band appropriate tasks that can be solved in multiple ways is one way to highlight flexibility and strategy selection in instruction.

 

Solve these two problems using any strategy you like:

• (8)(3.5) =

• =

Discuss options for solving each problem. In the first problem, the standard algorithm is to multiply 35 x 8, then count back to place the decimal point. But, a student might mentally create an equivalent expression using a halving and doubling strategy: (4)(7). Or, a student may decide to apply the distributive property, mentally multiplying 8 x 3 and 8 x 0.5 then adding 24 + 4. Challenge teachers to apply these two mental strategies in the second problem.

 

Finally, connect the tasks to the rationale--students must develop procedural fluency! For either problem, students who automatically apply the standard algorithm without considering options are not fluent, even if they are fast and accurate at implementing the standard algorithm. Students who only use standard algorithms overlook quicker mental strategies, meaning they are not efficiently solving such problems. They also lack flexibility and strategy selection skills. Students' development of procedural fluency is supported when teachers (1) encourage multiple strategies and (2) emphasize strategy selection.

 

Response 1b: Strategies Build Conceptual Understanding

 

As described in the Teaching Practice, Build Procedural Fluency from Conceptual Understanding (NCTM, 2014), students' fluency is supported when they employ intuitive or concrete strategies that connect to conceptual understanding. In working with teachers, select a 'critical area' or key concept and 'dig in' to a problem that can be solved using the range of strategies for that topic. Though solving the same problem more than one way is often pointless, in this case, teachers solve the problem in all the ways they think students might use so that they can see the conceptual connections each approach requires and supports. Here we illustrate such an approach with a focus on ratios and proportions.

TASK: The ABC Baseball Card Store sells baseball cards in packs of 10, and each pack sells for $2.00. If a child wanted to have 270 cards, how much would the child need to spend to get the cards?

 

SOLUTION STRATEGIES:

1. Unit rates strategy: Notice that 10 cards cost $2.00, so one card costs $0.20, and 270 cards cost 270 * $0.20.

 

2. Between-ratios strategy: Notice the factor from 10 cards to 270 cards is 27, and therefore the cost must be 27 times as much as the cost for a single pack:

 

 

3. Within-ratio strategy: Notice that there are five times as many cards as dollars and that dividing the 250 cards by five results in a cost of $54.00 for the cards:

 

 

4. Ratio Tables: Notice that the relationship of 10 cards for $2 can be extended to other possibilities and a table can help to organize that list (See Figure 3(a)). Further, notice that the ratio holds true by repeatedly adding the initial ratio (Figure 3a), a multiplicative strategy by using multiplication (and/or division) (3b), a doubling strategy (3c), or other strategies that reflect an understanding of covariation.

 

 

Figure 3. Ratio tables to solve baseball card problem.

 

5. Double Number Line Strategy: Notice that there is a missing value proportional situation and set up two number lines to model the situation:

 

 

With the values, recorded, apply one of the earlier reasoning strategies to determine the missing value, or use the values to set up a proportion.

 

6. Set up a proportion:

 

With the collection of strategies, describe the value of reasoning strategies (the first three) Reasoning strategies are important because they (1) are often more efficient, (2) are less prone to error than the cross multiplication approach, and (3) build from the underlying understanding of equivalent ratios.  Ratio Tables and Double Number Lines strategies are more abstract than the reasoning strategies, but effectively illustrate the covariation of two objects in ratios, meaning they help strengthen students' understanding of covariation.

 

Having illustrated that mental strategies and models can strengthen conceptual understanding, these methods can be connected back to the need for procedural fluency. Pose a similar task changed in such a way that influences strategy selection. For the Baseball Card Task, change number of packs to 9 (instead of 10). Teachers will see that the unit rate strategy is no longer a good fit (one card costs $0.22). The within-ratio strategy ($2 to 9 cards) is also difficult because the divisor is a decimal. The between-ratio strategy (9 cards to 270 cards) is still a whole number multiplier, and therefore still works well. Had this relationship not existed, a double number line or proportion might be used to solve the problem.

 

Response 1c: Supporting Each and Every Student and Their Different Ways of Thinking

 

While this section is short, it is perhaps the most critical response to Why teach multiple strategies? Students of all ages naturally approach problems differently--this should be nurtured, not squelched! Attempts to have all students solve a particular problem type in a particular way works against students' procedural fluency and can damage their emerging mathematics identities. Teaching multiple strategies provides students with options. Some students like to solve problems mentally whereas others prefer using illustrations. And, some students prefer numeric solutions. We must take time to ensure that different ways are understood, provide opportunities for students to select strategies, and offer forums where students can justify why they picked the strategy they did. As students develop and share their own ways of thinking, they are developing positive mathematical identities and engaging in mathematics the way mathematicians do.

 

2. Which Topics and What Strategies?

 

Prioritizing which strategies to use, and for which topics, are some of the most important decisions teachers make and should be a constant topic of professional conversations. The responses to this section are short, but the time spent with teachers addressing this question will not be short! Teachers need time to discuss and explore critical areas, useful strategies, and strategies that support understanding.

 

Response 2a: Focus on Critical Areas

 

Because our goal for students is to develop procedural fluency, we can focus on critical areas and identify the most important procedures learned in each grade. Each grade level or course has major topics, and this is a good place to focus on a collection of strategies. Grade 7 certainly includes the proportional reasoning strategies described earlier. In the early grades, strategies for basic facts are critical. Fraction operations are priorities across numerous grades, and student understanding of these operations with fractions will be strengthened with a focus on reasoning strategies (Olson & Olson, 2012/2013). For example, using jumps up/back on an open number line, strategies learned for solving whole number problems, can be used for solving fraction operation tasks.

 

Response 2b: Identify Useful Strategies

 

We need to help teachers identify strategies and models that are likely to be used for many problems. For example, halving and doubling and applying the distributive property strategies, as illustrated in the fraction and decimal problems, can be used for a wide variety of numbers and problems. The distributive property may be the most important central idea in arithmetic. It supports and is supported by strong conceptual understanding and applies regardless of number type (whole, integer, decimal, fraction). Exploring how to apply the distributive property to solve problems in different ways is almost always time well spent.

 

However, there are also strategies for which the effort to teach them is not worth the time invested to help students understand them, such as the lattice method of multiplication. This strategy is "based on place value," but it can take several days to teach. And, many students find it harder and less intuitive than partial products or the traditional algorithm.

 

Within particular topics, some structured strategies are very useful and need to be taught. Proportions provide a key example of such a topic. Ratio tables and double number line strategies need to be taught because they build conceptual understanding and provide a bridge for accurately setting up proportions. Tables and number lines are useful for many, many topics from early number through high school algebra. Showing how they can be used for covariation helps students better understand covariation and become adept with mathematical models to support their thinking.

 

Questions to ask teachers as they make decisions about whether to teach or exclude a strategy include:

• Does it strengthen students' understanding of [key mathematics]?

• Is it more efficient than the standard algorithm or other strategies?

• Is it less prone to error than the standard algorithm or other strategies?

• Can be used for many types/sizes of numbers?

• Can it be applied to other topics in the curriculum?

If the answers to these questions are no, the strategy is likely not a good choice. If all or most answers are yes, then it is worth the time to teach it. Having fourth-grade teachers respond to these questions, for example, as they think about 'break apart' (distributive property) and lattice methods, can help them identify and prioritize useful strategies.

 

Response 2c. Use Alternatives that Support Understanding

 

Some strategies are important to teach, even if they are less efficient than standard algorithms because they make sense to students. These strategies may serve as scaffolds to standard algorithms or may replace the need for standard algorithms. For example, students might learn the common denominator strategy to divide fractions:

 

 

Fuson and Beckman (2012/2013) suggest that even with the standard algorithms, we have options for how student notate their process, some of them more conceptually-focused. They suggest these criteria for consideration:

• Include methods [strategies] that will generalize to and become standard algorithms.

• Show variations in ways to record the standard algorithm that support and use place value correctly.

• Emphasize variations that make single-digit computations easier.

[Note: Within their article, they provide illustrations of these ideas for multidigit addition, subtraction, multiplication, and division.]

 

3. Sequencing and Supporting Strategies

 

When teaching a topic that has multiple strategies, unit and lesson design features are important to minimize stress and maximize success for students.

 

Response 3a: Start with Reasoning and ADD on (not MOVE on).

 

We know to teach informal strategies first. For addition, this might include jump up strategies, adding larger place values first, compensation, and so on. Then, we teach the standard algorithm. But, we have not moved on to the standard algorithm. Rather, the standard algorithm is added to students' repertoire of strategies. Because students work hard to learn standard algorithms, they may think teachers prefer this method. We must be explicit about the importance of selecting an efficient strategy from a set of strategies (which is not always the standard algorithm). In fact, standardized and high stakes tests often use problems for which there are very efficient mental strategies, but that otherwise are cumbersome time-consuming problems. Sets of computation problems (e.g., 20 addition and subtraction of fraction problems) provide an excellent opportunity to ask students to use mental strategies if possible. Discuss which problems lend to counting up or counting back solution strategies, which ones lend to a break-apart strategy, and which ones will require finding a common denominator to add. Encourage teachers to ask students to think about when they will use the standard algorithm, and when they will use their other strategies. Knowing when to use which strategy is essential to developing procedural fluency.

 

Response 3b: Incorporate a Strategy Focus into Lesson Design

 

There are days when a lesson's focus is on learning one new strategy. But, there must be many days for which the goal is to select from known strategies or student-generated strategies to solve problems. Figure 4 offers three teaching moves have been successful in supporting students' strategy development and fluency (Author, 2017; McGinn, Lange, & Booth, 2015; Star, 2005).

 

First-step Only!

Ask students to read a problem and think only of what they could do as their first step. Share and compare first step strategies. As students compare the options, ask: What is the reasoning or intent of doing that step first? What conditions are needed for that first step to 'work'? What are the limitations of that first step? Then have students individually select a preferred solution strategy and solve the problem.

 

Worked Examples

A worked example is just as it sounds--a finished problem. Students analyze how a 'student' thought through the worked problem. Often used to highlight misconceptions, worked examples that are correct provide an excellent way to focus on strategies. Here are two possibilities:

• Show a partially worked example where the first step is not one that students typically use. For example, for 15 x 22, could have a first step where the student has written 15 x 20 and 15 x 2 and "got stuck." You ask, "What can she do next to solve the problem this way?"

• Show two correct problems side-by-side. Ask students if they both are correct. Discuss which strategy is more efficient and when they might use each strategy.

Share and Compare

After students have solved a problem that lends to multiple strategies, strategically invite students to share their way. You can order the strategies from least sophisticated to most sophisticated, or begin with the most common strategy to least used (Smith & Stein, 2011). But, don't stop there! Seeing different ways is not enough. Ask students:

 

Which strategies are efficient?

Why does the strategy work for this situation?

When is a particular strategy is a good choice?

Figure 4. Strategy-focused Instructional Moves

 

4.    How do I Address Parent and Guardian Questions?

 

As the data shared in the introduction indicate, the greatest stress teachers face with using strategies is in the feedback and pressures from families and from students. Even in a world of frequent and dramatic change, change in mathematics teaching seems to always generate concerns. Equipping teachers with ways to provide parents and guardians with strong rationales and homework strategies can really reduce their stress and help them to help their students.

 

4a: Provide a Rationale that Resonates with Parents and Guardians!

 

Larson (2016) explains that when parents and guardians are shown ways to engage with student thinking about mathematics and have better understandings of various strategies, their support increases. It is our responsibility to make sure parents and guardians understand why a 'new' strategy is taught, and our explanations must resonate with what they value. Saying, "This is what the standards require" is not helpful and can be harmful. What is helpful is providing examples (such as the two problems earlier in this article) to illustrate how helpful it can be to know more than one way and helping parents expand on their understanding of fluency to include strategy selection and flexibility. Oftentimes parents and guardians use these strategies, which opens up the opportunity to notice how important reasoning or mental strategies are for adults and for careers (see Question 1 for more ideas!).

 

4b: Be 'Strategic' about Homework

 

This scenario has become an increasing reality in U.S. homes: A child is solving a math problem and is stuck. The child's parent or guardian recognizes the problem and shows the way they learned to solve it. The child says, "That is not how my teacher showed me how to do it!" Parents and guardians naturally wonder why their method can't be used anymore. We must make homework as comprehensible as possible for parents and guardians (Larson, 2016). Here are three ways to help parents and guardians navigate new strategies without alarming them.

 

Choice: Provide a choice of strategies in homework to increase parents' and guardians' access to mathematics and reinforce fluency. Parents and guardians can share a strategy they understand and students can explain a strategy back to them. Avoid requiring a 'new' strategy.

 

Examples: Provide explanations and examples. Technology affords many ways to communicate with families, such as providing website links to videos explaining strategies and screenshots of worked problems. And, if recording a video for a strategy likely to be new or unfamiliar to parents and guardians, begin with a rationale (e.g., "We are learning this strategy because...").

 

Tips: Provide general suggestions for how parents and guardians can help with homework. For example, parents and guardians can model persistence and productive struggle by making comments to children such as, "we can figure this out if we keep trying" (Zimba, 2016). Even if parents and guardians know a way to solve a problem, they can first ask children questions such as what the problem is about, where to begin, and whether to make a drawing or table to think about the problem (Van de Walle, Bay-Williams, Lovin, & Karp, 2018).

 

From Stress to Success

 

As teachers face the stress of determining which strategies to teach, and how much time to spend on them, we must help them to keep the big picture in mind: We teach multiple strategies because multiple strategies are necessary for ensuring each student develops procedural fluency.   Having a repertoire of strategies allows students to build conceptual understanding, to pick strategies they understand, and to seek strategies that are a good fit for the numbers in problems. Teaching multiple strategies develops students' flexibility for solving wide varieties of problems, helping them to be more efficient, competent, and confident problem solvers. In other words, the use of multiple strategies all students provides access to mathematics.

 

Invite teachers to watch their students as they solve problems to see if their efforts to focus on strategy selection are working. A good measure is how students first respond when they see problems, whether computational or contextual. If they launch into algorithms without pausing to consider their options, the instruction is not there yet. If instead, the student pauses to consider which strategies to use and selects a reasonable one, then a teacher can see that they are using multiple strategies to support students' procedural fluency. Ultimately, for any given procedure, one student may know three strategies, whereas another one knows five. But, in every case, our goal should be for students to look at a problem and first ask themselves, "Which strategy do I want to use to efficiently solve this problem?" Which strategies and which topics is perhaps less important than developing this disposition in our students. And, perhaps this message can help teachers move from stress to success!

 

References

 

Fuson, K. C., & Beckman, S. (2012/2013). Standard Algorithms in the Common Core State Standards. NCSM Journal of Mathematics Education Leadership, 14(2), 14-30.

 

Larson, M. (2016, November 15). The need to make homework comprehensible. Retrieved from https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/The-Need-to-Make-Homework-Comprehensible/

 

McGinn, K. M., Lange, K. E., & Booth, J. L. (2015). A worked example for creating worked examples. Mathematics Teaching in the Middle School, 21(1), 26-33.

 

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

 

National Governors Association Center for Best Practices & Council of Chief State School Officers (NGA & CCSSO). (2010). Common Core State Standards for Mathematics. Washington, DC: Authors. Retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

 

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academies Press.

 

Olson, T. A., & Olson, M. (2012/2013). The importance of context in presenting fraction problems to help students formulate models and representations as solution strategies. NCSM Journal of Mathematics Education Leadership, 14(2), 38--47.

 

Smith, M., & Stein, M. K. (2011) 5 Practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.

 

Star, J. R. (2005). Reconceptualizing procedural knowledge." Journal for Research in Mathematics Education, 36(5), 404-411.

 

Van de Walle, J. A., Bay-Williams, J. M., Lovin, L., & Karp, K. (2018). Teaching student-centered mathematics: Developmentally appropriate instruction for grades 6-8. (3rd Ed.). Boston, MA:  Pearson.

 

Zimba, J. (2016, January 15.). Can parents help with homework? Yes. Retrieved from http://edexcellence.net/articles/can-parents-help-with-math-homework-yes

 

Dr. Jennifer Bay-Williams is a professor at the University of Louisville. She advocates locally and nationally for effective mathematics teaching in order to ensure every student has access to and is supported in learning, relevant mathematics.

	Dr. Sue Peters currently teaches middle and secondary mathematics education as well as graduate research topics in mathematics education at the University of Louisville. Her research interests include statistics education, teacher knowledge, and mathematics teacher education.
	Dr. Lateefah Id-Deen is an assistant professor at Kennesaw State University. Her research interests examine vulnerable students' experiences that affect their sense of belonging in mathematics classrooms in and out-of-school settings. Her research and teaching reflect her passion for creating equitable learning environments for vulnerable students in mathematics classrooms.

 

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Turning an Idea into a Reality: An Urban Garden Project by Jennifer Donalson, Washington Middle School, Southwest Region Representative, 2017-2021

Innovative thinkers often find joy in seeing their ideas turn into reality. To be able to turn an idea into a reality, it often takes an army of followers with a similar vision. When I first began my educational journey five years ago, I would have never imagined how my project involvement during the first few years of my career would impact my teaching. To understand the journey, I am going to take you back in time several years to where it all began.

In 2014 I began my teaching career at Washington Middle School in Cairo, GA teaching 6th grade Honors Math and Earth Science. It was important to me to teach at my hometown school, and this involvement drove my passion for my school and my students. My hometown only has cotton and peanut fields and, of course, the local hangout, Mr. Chick. I wanted to embed the same passion that I have for my community in the hearts of my students. Keeping that vision in mind, I started to delve into the curriculum of Earth Science. Being a mathematically-minded educator, Earth Science was a tough curriculum to teach. I had to find a way for my students, and myself, to become involved with the Science curriculum and include Math every chance I could.

I decided that I wanted to buy a greenhouse so that my students could conduct real-world experiments and collect data to analyze during Math class. Of course, my administrators loved the idea but did not offer any funds to help me start the process. I started to look for grant opportunities to help with the funding of my project. I spent most of my second year in education writing grants with no success. By my third year, I had almost given up hope on the greenhouse idea until I found a different way to turn my idea into reality: build my own greenhouse.

With the support of my administrators and colleagues, I proposed the idea to my students. Their task was to design a greenhouse made of recycled 2-Liter bottles and bamboo. They had to construct a prototype (to scale), draw engineering floor plans, and create a budget to show any more resources that would need to be purchased during construction. Each team of students then had to present their ideas in a SharkTank Experience held during class. A panel of judges listened to their designs and even asked questions concerning any parts of the design. The judges chose the design that we later used in the construction of the greenhouse.

But I was not satisfied with a greenhouse anymore. I wanted something bigger that would encompass the agricultural community that my students and I were a part of. With the help of my colleagues, we decided to create Tiger Terrace, a school-wide urban garden project. The project included a compost bin, raised planting beds, an outdoor classroom, an experiment station, an aquaponics feature, and, the centerpiece, a greenhouse. If the idea of constructing a greenhouse made of recycled 2-Liter bottles and bamboo wasn't crazy enough, we now added five more project features that would amplify the madness!

As with any great plan, there are always unexpected bumps along the way. After my third year in 6th grade, I began to teaching 7th and 8th grade Honors Math. Having started the Tiger Terrace, I handed off the idea to the afterschool STEM program that we implemented at WMS. Their goal was to pick up where I had left off and use the plans that I originally made to begin the construction of the Tiger Terrace. We had received a local grant from the community to be able to begin the construction. In May of 2018, we had made very little progress: a frame was built for the greenhouse and the raised beds were marked but not constructed. Four years into this process and I felt like we had not made any gains toward my original vision.

During the summer of 2018, I made it my mission to finish the Tiger Terrace by the Spring of 2019. The students that designed the greenhouse are now 8th graders, and I want them to be able to see all their hard work pay off before they leave WMS to go to Cairo High School. This year we also received a grant from the GA Department of Education for Rural Innovation Funds. This grant specifically was to purchase the equipment that would be used in the experimental station in the Tiger Terrace. After receiving word that we had received the funds from this grant, we were also surprised with the fact that the State Superintendent of Schools, Mr. Richard Woods, would be coming down to visit our school and present us with the check. At this point, panic set in. We wanted him to be able to see our idea as a reality.

We immediately started further construction on the Tiger Terrace. This included cutting bamboo, cleaning, cutting, and stringing bottles, and assembling the walls of the greenhouse. It also included building the frames for the raised beds and filling the beds with topsoil. So in the horrendous heat of the South Georgia Fall, my students and several colleagues set out to complete these tasks before the visit from the boss. It was never an easy task but having a group of students and teachers that believe in my vision makes the work so much easier.

We are now up to the present day. The Tiger Terrace has come a long way in five years. The greenhouse is almost completed. We have one and a half walls left to assemble. 2-Liter bottles are not as common as they once were. We have recycling agencies all across the south pitching in to help with our collection efforts but are still several hundred bottles short to finish the project. We completed three of the six raised beds including the two largest raised beds.

Teachers may now use the outdoor classroom and in the next few months, benches will replace the hay bales that are currently used for seating. We are now using a newly constructed the compost bin. We have ordered and will install the experimental station equipment. Even though we have a long way to go to "complete" the project, we have definitely come a long way from where it all began.

Innovation begins with an idea. Turning an idea into reality can be a long, winding road, but joy happens when the students see their hard work pay off and enjoy learning using the area that they designed and created. Educators do not teach for the pay or even the time off during the summer. Educators teach to instill values in the youth of America and help to build future leaders of our communities. The Tiger Terrace has taught my students more than just math and science standards. This project has taught my students the value of hard work, how to connect with community members and business leaders, and appreciation for the agriculturally based town that we are all a part of.

#gradyunbroken

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Breathing Life into Math Instruction: Fractions, Geometry, Tangrams, and Origami by Seyoung Holte, Director of Elementary Mathematics, Northeast GA RESA, Northeast Regional Representative, GCTM

A segment from GCTM Summer Math Academy and a preview of a GMC session "From Brownies to Serving Sizes: Understanding Fraction Progression and Demystifying Fractional (and Decimal) Computation

When I ask 4th and 5th-grade teachers "What is the hardest thing about teaching math?" The resounding responses sum up to these three topics: 1) Teaching students that are not interested in learning, 2) Students' lack of problem-solving skills, and 3) The dreaded FRACTIONS!!! I hope to address the problems with problem-solving in a future Reflection (or you can visit the session at GMC at Rock Eagle!). For now, let's explore factors that contribute to the fraction anxiety and ways to reclaim the glory of teaching and learning mathematics.

Years ago, my niece from Korea visited me for summer vacation. Sumin, a top student in her 5th-grade class and in her school, brought five books so she had to work on during summer. She would study an hour or two every day in between the activities with her cousins. One evening, she got stuck with one problem. When I asked, "How did you learn to solve this problem?" She said, "My teacher said just replace the letter with 1." And she found the answer by doing that. "Then what's the problem?" And here came the response I was hunting for. "I don't get why this works." Here's Sumin's problem. Try for yourself!

Did you find the solution? What was your thinking process? How did you feel? When I present this problem to the participants in academies, school/district level PLs, and conference, or to college students, less than half a group can come up with the solution immediately. More than one-third of the group report feeling anxious when faced with the problem. Of course, some faces lit up and proudly presented their algorithms. And some responded, "I saw that..." Well...?

What do you see? "Two circles and 3 half-circles." How about now?

"I see four half-circles and three half-circles." "Two wholes broken into halves and three more halves" "seven half-circles."

How about now?

And there's a big and long "Ah~~~~!" Whatever the "a"'might be - an apple, a brownie, a pizza, 1, 100, etc.,- we are trying to see how many half units of the "a"there are, which can also be translated into a linear model like this:

Once the participants (of course, Sumin too!) saw this, they were able to make sense of other fractional equations and inequalities. So, what's the implication? NCTM's Effective Mathematics Teaching Practices (MTP) articulates 3. Use and Connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem-solving. (Principles to Actions, 2014) One of the biggest and dangerous misconceptions students develop as they advance in grade levels is "Manipulative and drawings are for the week, the low, and the I can't." On the contrary, good mathematicians connect all forms of mathematical representations constantly - to make sense, to represent, to explain, to analyze, and to justify. See the diagram below:

Principles to Actions, 2014

Notice the arrows - it's not linear moving from physical to symbolic! We must constantly connect and reconnect different representations to build understanding. When we do that purposefully with solid content, we create meaningfulness, relevance, and equity. Students need access to the math we are teaching through multiple entry points, multiple representations, and multiple (types of) experiences. Here are some ways to engage students in fractional reasoning with these aspects in mind.

The Warlord's Puzzle

• Connect Origami and Tangrams, and Geometry to discuss fractions as relational proportions.

• Bonus Outcomes: Spatial, visual, & temporal reasoning, Exposing the students to the world of origami, tangram, and engineering

• Grade Levels: 2nd - 6th (possible for 1st and 7th)

• Misconception buster: Primary students may think that same fraction of a whole should look the same. By discovering that three different pieces with different shapes in tangrams are indeed the same fractions of a whole, they gain the understanding of fractions as relational, proportional sizes of units rather than congruent shapes in a bigger whole.

At a glance: Read the book The Warlord's Puzzle by Virginia Walton Pilegard. It the end, announce "We will break the tile just the way the tile in the story broke!" See the lesson ideas here.

Part 1: The folding and cutting - Follow the directions to create pieces, using origami techniques. (Note the pieces are labeled with numerals, but it's better to label with letters as in the above image.)

After exploring the newly created tangrams pieces and discussing the relationships between the pieces, facilitate the following task depending on your learning goals.

Part 2: Determine the value of each piece when the whole is worth 1. Facilitate students' discourse to explain how they found out the fraction of each piece. Discuss relationship findings.

Part 3: During Lunch or as a Fun Friday enrichment, watch Between the Folds, a PBS documentary. This mind-blowing video will engage the students and take them to a whole new world of origami and geometry! I have used this documentary as an introduction of Origami cluster/club with this title:

Students then explore different aspects of origami: become a folder, research on the history of origami or origamists, study geometry with origami, learn about engineering and STEM aspects of origami, etc. The benefits of origami do not end here. By experiencing this folding art, students learn the value of soft skills - perseverance, productive struggle, self-regulation and discipline, problem-solving, and respect towards intense focus and pursuit of something intriguing. Paul E. Torrance, the creator of the measurement for creativity in gifted teaching known as Torrance Test of Creative Thinking, said in his Children's Manifesto: Don't be afraid of falling in love with something and pursue it with intensity. Part of our job as educators is to open the window for our students to be inspired, intrigued, and challenged to experience, learn and do something. Once we are there, we no longer need tickets or rewards to coerce them into learning.

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