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
Email questions to
|Cost: $120 for
non-member and $90 for member
Registration Opens February
Locations and Dates:
- Chestatee High
School in Hall County - June
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in Richmond County - June
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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
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.
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.
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 @
email@example.com and I will be glad to connect you to
the appropriate chairperson.
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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
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
How do I address parent/guardian
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
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
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
Response 1b: Strategies Build Conceptual
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
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?
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
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
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
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
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
Can it be applied to other topics in the
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
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
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
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
Response 3b: Incorporate a Strategy Focus into
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,
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.
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
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
When is a particular strategy is a good
Figure 4. Strategy-focused Instructional Moves
4. How do I Address Parent and Guardian
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
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
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!
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
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
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),
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),
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
||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,
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.
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
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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
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.
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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
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
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
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,
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
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)
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
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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