The focus of this research project is to
identify best practices in eighth-grade mathematics teaching and
learning. Our methodology is to identify school systems who face
high poverty, yet whose students reach high levels of
achievement as indicated by the Georgia Milestone Assessment
System (GMAS). Within the 29 Georgia counties with the highest
poverty rates in the state, we identified four counties with the
highest percentage of students performing proficient and above
on the GMAS. Teacher survey results at the high-performing
middle schools in these counties reveal the importance of
continuous assessment, grouping strategies, high expectations,
technology use, strong curriculum, and a variety of
instructional methods. In addition, the surveys indicate that in
order to meet the specific needs of students in poverty, schools
need to: (1) address students' nutritional needs, (2) provide
supplies to students, (3) seek out families in need and
immediately provide help, (4) demonstrate to students that you
care about them, (5) increase the curriculum at school in order
to decrease homework expectations, and (6) obtain external
From 2003 to 2015, the state of Georgia has
continuously closed the gap with the nation in 8th-grade math
performance as measured by the National Assessment of
Educational Progress (NAEP). During the past 2 years, this
improvement has stalled. So, how can we start narrowing the
performance gap again?
One way of improving overall performance is
to identify low performing groups of students and determine how
to better reach them. Children living in poverty are more at
risk of underperforming when compared to their counterparts not
living in poverty. Research conducted by the NAEP showed
nationally that only 18% of 8th grade math students who were
eligible for the National School Lunch Program reached the level
of at or above proficient, while the overall percentage of 8th
grade math students in the nation who reached the level of at or
above proficient was 32% (NAEP Report Cards). In Georgia, only
15% of 8th-grade math students eligible for the National School
Lunch Program performed at the level of at or above proficient,
while the overall percentage of 8th-grade math students in
Georgia who reached the level of at or above proficient was 28%
(NAEP Report Cards). Clearly, poverty is a major impediment to
success in mathematics.
Is it possible that the performance gap
between Georgia and the nation could be narrowed again if more
students in poverty began to perform at a proficient level? By
identifying high performing schools in high-poverty areas, we
can determine the reasons for this high performance, in order to
better reach math students in Georgia who live in poverty.
In the U.S. Quick Facts: Persons in Poverty,
the U.S. Census Bureau divides the counties of Georgia into five
categories based on the percentage of people living in poverty
with the lowest percentage category of 5.7-14.9% and highest
percentage category of 26.8-39.2%. There are 30 Georgia counties
in the category with the highest level of poverty. According to
the Quick Facts, Clarke County is a high poverty county.
However, research indicates that poverty rates are over-reported
in towns near a large university, as in Clarke County's case,
University of Georgia. Therefore, we excluded Clarke County from
our study (Benson & Bishaw).
NAEP assessment data does not break down
scores by county. Therefore, to identify which of the 29
counties have high-performing schools, we studied the Georgia
Milestone Assessment System (GMAS) from the Georgia Department
of Education, which publishes the results of the GMAS by county.
In order to select the high-performing counties, we chose
counties with at least 40% of students scoring at or above
proficient on the GMAS. We chose this minimum value as it is
significantly higher than the overall state's percentage of
34.5% (Georgia Department of Education). According to the
2016-2017 Georgia Milestones Statewide Scores Spring 2017 School
Summaries, there are four counties of the 29 high-poverty
counties which significantly exceeded the state of Georgia's
percentage of students at or above proficient (Georgia
Department of Education). See the table below.
Proficient and Above
Table 1- Georgia Milestone High-Performing Counties
The Georgia Department of Education also
provides assessment data for all the schools within these
high-performing counties. The Spring 2017 Georgia Milestones
End-of-Grade Assessment for these schools is shown in Table 2.
Proficient and Above
Bethune Middle School
Johnson County Middle School
Glennville Middle School
Collins Middle School
Reidsville Middle School
Telfair County Middle School
Table 2- Georgia Milestone High-Performing Schools
We emailed the survey below to the eleven 8th
grade math teachers at these schools.
If we did not receive a response, we repeated
the request two more times. After sending the survey request
three times, five of the eleven teachers responded.
Number of Responses
Bethune Middle School
Johnson County Middle School
Telfair County Middle School
Results of the Surveys
When asked about
instructional methods or other aspects that the teachers deemed
important to their students' success, we found that most of the
teachers stressed the importance of assessments, grouping
strategies, high expectations, and technology. Some also
referenced their curriculum and actual instructional methods.
The first main focus was
assessments, mainly formative assessments. These formative
assessments varied in timing at different locations in the
lesson. For example, Mauri Jarrard, an 8th-grade math teacher at
Telfair County Middle School, uses an informal or formal
assessment prior to covering new content. This enables her to
determine the student's background knowledge and fill in any
missing skills. Sabrina Rentz, another 8th-grade math teacher at
Telfair County Middle School, utilizes formative assessments
during the lesson. Rentz stated, "Students have numerous
opportunities to show what they know during the lesson by
utilizing an array of assessments, such as stick-its,
whiteboards, thumbs-up/down, agree/disagree, Socrative app, and
Quizizz. These results are used to gauge my students'
understanding of the content and adjust instruction."
Eighth-grade math teacher, Siterro Wheeler, mentioned another
opportunity to use formative assessment. After exposing her
students to the new content one day, the next day Wheeler uses a
formative assessment to determine daily grouping for students.
This enables the students to practice the new ideas learned on
the first day of the lesson. Another location of formative
assessment use is at the end of the class. Not only does Jarrard
begin the new content with an assessment, she also ends each day
with one in order to assess the students' learning of the
content introduced that day. Some of her practical examples are:
ticket out the door, think-pair-share, writing assignment to
explain their learning for the day, and demonstrating
understanding of the content via a hands-on activity.
Not only are teachers at the
high-performing schools conducting formative assessments, they
are also continuously assessing informally during the lesson.
Wheeler is constantly rotating throughout the classroom. This
allows her to be able to assist any student as needed while they
complete their independent practice. She also repeats and asks
questions about the topic throughout the lesson. Additionally,
Wheeler has students summarize the main vocabulary terms and
main concepts as a class. Likewise, Rentz informally assesses
while students work in groups together. Informal assessment
allows both of these teachers to determine what their students
know or where they are lacking understanding so they can adjust
Secondly, the teachers
heavily utilize grouping strategies. The surveyed teachers use
formatives assessment to group students. Rentz said, "All groups
are created with a purpose. Sometimes I utilize heterogeneous
groups to allow for peer tutors and at other times I use
homogenous groups with leveled work that matches their
abilities." This method allows the grouping to be fluid and
benefit all students. Wheeler also describes her use of
formative assessment for grouping as, "A formative assessment is
given to the students to determine daily grouping. Based on the
formative assessment, students either work independently on a
rigorous assignment or they work in a small group led by me to
strengthen their understanding of the concept."
Thirdly, the teachers
attributed their students' success to high expectations. Rentz
stated, "It is my belief that another major impact on my
students is the high expectations that I hold for them
regardless of their ability levels. My classes are grouped based
on ability and my lower achieving class is exposed to everything
my other classes are. They are expected to be able to succeed
with the same content as everyone else. I use the assessments to
see what parts of it I need to differentiate, but I never
decrease work amount as differentiation. The type of
differentiation that I utilize has students learning the same
material and practicing on the same material, but maybe in
different ways. I require them to dig deeper into the content
and expand on that. High expectations play a tremendous part in
student success. They are going to perform to the expectations
you have for them." A teacher, who asked to remain anonymous,
echoed Rentz by saying, "Students must be held to a high
standard. Expect every student to excel." Not only should
individual teachers convey high expectations, the whole school
should also hold each student to a high standard. Wheeler
praised her school for conducting a major school-wide initiative
that "raised the bar for all students." This included
implementing PBIS school-wide and having teachers create lessons
that are "rigorous and challenging while meeting the needs of
every student." From reading the teacher responses,
communicating clear and high expectations to the students is a
major key to their success.
The fourth major aspect of
student success mentioned by many of the teachers is the use of
technology. Technology is a way to keep students engaged.
Additionally, technology is very mobile and versatile. Walker
uses it for students to practice tests online through the
academic software Study Island. Rentz and Jarrard stress the
importance of giving every student access to some form of
technology. "We expose our students to various forms of
technology. Our students have access to Chromebooks, iPads,
SmartBoards, and calculators. The SmartBoard is used almost
daily to provide instruction and occasionally used for student
activities. Each math teacher has a class set of iPads that can
be used for assessment and activities", stated Jarrard.
Likewise, Rentz expresses, "We use technology on a daily basis
in the classroom to increase student success. The iPads are used
for different performance tasks, review activities, and
assessments. By incorporating technology, we are able to reach
more of our students and keep them actively engaged."
Fifth, the teachers stressed
the importance of a strong curriculum. When discussing
curriculum, every teacher varied in his or her response. Some
use only the Georgia Performance Standards and supplement with
various resources. One teacher mentions, "When looking for
resources, the focus is to solidify the understanding of the
content and to provide appropriate rigor." Similarly, another
teacher revealed that she uses the Georgia Performance Standards
and the Engage New York (ENY) curriculum. "I really like the ENY
approach because it is more rigorous, and the methods are
mathematically sound. The curriculum is based on the deep
understanding of mathematical concepts rather than just
procedures," she said. On the other hand, one uses Glencoe Math
by McGraw-Hill as her main math curriculum. She states, "It
helps me as a teacher to be better prepared to differentiate my
instruction, which in turn will help the students achieve
success." After reading all of these responses, there was
another aspect that was intriguing. Bethune Middle School has
created two math classes for the 8th graders. Walker states,
"Our principal made a crucial decision in 2014-2015, he added a
support math class to 8th grade during the first year of GMAS,
so 8th graders had two math classes. In one class, students
learn the standards through units. The support math class is
used to reinforce concepts and do extra practice. In my opinion,
the addition of the second math class has been the reason for
our higher than the state's percentages in the proficient and
exceeding levels of the mathematics GMAS. I really believe the
reinforcement during the 2nd math class in addition to the extra
hour of practice has made the biggest difference in the scores."
Lastly, the teachers reported
using a variety of instructional methods to meet the needs of
their students, such as:
(1) Direct instruction.
(3) Discovery of patterns
to aid in the development of algorithms.
(4) Modeling procedures.
(5) Thinking maps.
(7) Constant review
Addressing the Needs of Students Living In
In order to address the needs
of students living in poverty, the high-performing schools,
oftentimes in collaboration with their school systems, seek and
utilize external partners and consider the special needs of
students living in poverty. In particular, the respondents
credit the following for impacting student learning and success
at their schools:
(1) Providing for the
nutritional needs of their students.
All respondents stressed the importance of meeting the
nutritional needs of their students. Students are provided a
free breakfast and lunch. According to Rentz, "Students
living in poverty may only receive a warm meal at school. By
providing students with the nutrition that their bodies
need, we are helping fuel their brains and increase focus
and memory retention in the classroom." Some schools provide
food beyond the regular work week to ensure their students
are not in want. Wheeler states, "Several students are given
sack lunches to take home on the weekends because of lack of
food within their homes."
(2) Providing supplies to
In partnership with members of the community, employees of
the Telfair County schools host a beginning of the year
kick-off event at the county recreational department. At
this event, parents receive the supplies their child will
need for the school year. In addition, Jarrard shares, "Our
school has a supply room with supplies provided by various
local churches, businesses, and people throughout the
community. Our students have access to these supplies
whenever they are needed."
(3) Seeking out families
in need and immediately providing help.
At Johnson County Middle School, Wheeler states, "All
faculty and staff go above and beyond to ensure that
students are learning and succeeding. If there is an issue,
we quickly act upon it by contacting the necessary people to
help find a solution to the issue. We have a representative
that seeks out families in need and she makes sure those
students needs are taken care of immediately."
(4) Demonstrating to your
students that you care about them.
One teacher stressed that "you must have empathy for what
your students go through every day. Students must know that
you care about them and understand that you expect them to
excel" in order to break the poverty cycle.
(5) Increasing the
curriculum at school in order to decrease the homework
expectations based on the situation at home.
One teacher states, "I rarely give homework because most of
the time there is no help at home." At Bethune Middle
School, Walker states, "the factor that has had the major
impact on test scores besides sound math instruction is the
extra math class. The higher rigor of common core math
requires students to spend more structured time using and
applying what they learned. Before, students had to find
time at home to review, but most parents are unable to help
their children in this new math."
(6) Obtaining external
Telfair County Middle School was awarded a technology grant
that provided each math classroom with a SmartBoard and a
classroom set of iPads. According to Rentz, "Most students
who live in poverty are not fortunate enough to have access
to this type of technology.
By studying schools in
high-poverty areas that have achieved high levels of student
success, we have found the following key components of success:
continuous assessment, grouping strategies, high expectations,
technology use, strong curriculum, and a variety of teaching
methods. We believe that all students at all schools in Georgia
will benefit from these components of success. Further, all
schools have at least some students who live in poverty. In
order to address their specific needs, our study found that
schools and school systems need to: (1) address students'
nutritional needs, (2) provide supplies to students, (3) seek
out families in need and immediately provide help, (4)
demonstrate to students that you care about them, (5) increase
the curriculum at school in order to decrease homework
expectations, and (6) obtain external technology grant.
Acknowledgments: We wish to
thank the 8th-grade math teachers Mauri Jarrard and Sabrina
Rentz at Telfair County Middle School, Suraya Walker at Bethune
Middle School, Siterro Wheeler, and an anonymous teacher for
both their dedication to their students as well as their
willingness to take the time to provide well-thought-out
responses to our survey. We also thank the principals Danny
McCoy at Bethune Middle School, Christopher Ellis at Telfair
County Middle School, and Elaine Merritt for connecting us with
these excellent teachers.
Benson, C., & Bishaw, A.
(2017, December 07). Examining the Effect of Off-Campus College
Students on Poverty Rates. Retrieved April 09, 2018, from
Georgia Department of
Education. (2017, July 20). Georgia Milestones 2016-2017
Statewide Scores. Retrieved February 19, 2018, from
The Nation's Report Card. (n.d.).
NAEP Mathematics. Retrieved February 10, 2018, from
United States Census Bureau.
(n.d.). U.S. Census Bureau QuickFacts: United States. Retrieved
March 01, 2018, from
Gregory Harrell is a Professor in the Department
of Mathematics at Valdosta State University. He is
interested in facilitating success for all
mathematics students from kindergarten to college
graduation. He earned his M.A. in Mathematics at the
University of Georgia and Ph.D. in Instruction &
Curriculum with a specialty in Mathematics Education
from the University of Florida.
Anna Joy Holton is
currently attending Valdosta State University. She is pursuing a
bachelor's degree in Middle Grades Education with concentrations
in Mathematics and Social Studies. She is currently completing
her student teaching in an 8th grade Pre-Algebra class and will
graduate with Honors in December of 2018.
Back to Top
Four Cs and Mathematics
by Dr. Julie
Carter and Dr. Rhonda Amerson; both assistant professors
within the Department of Teacher Education and Social Work
at Middle Georgia State University
A new school year always brings with it the
excitement of a new group of students and renewed energy to plan
and teach engaging lessons that not only meet the standards but
also prepare students for their life after high school. Teaching
the traditional "Three R's", Reading, Writing, and Arithmetic,
leaves students lacking the skills they need for the 21st
century. In addition to subject matter knowledge, today's
students must be skilled in critical thinking, communication,
collaboration, and creativity (the "Four Cs") (National
Education Association, 2010). Educators must complement their
subject matter content with the "Four Cs" to prepare young
people for citizenship and the global workforce (National
Education Association, 2010). The "Four Cs" can be incorporated
into any classroom, but it requires educators to be intentional
and purposeful in their planning. If students are going to be
equipped with 21st-century skills when they graduate high
school, they must have been provided opportunities to develop
these skills throughout their schooling.
Unfortunately, there is not a single resource
that educators can implement in their classroom that will
address all of the 21st century needs of their students. The
flood of technology resources provided for educators is vast.
Sorting through the plethora of resources in order to find the
perfect combination that will address all of the "Four Cs" in
one lesson will most likely leave the educator feeling defeated
and overwhelmed. However, the skills students need for the 21st
century, critical thinking, communication, collaboration, and
creativity, can be addressed by taking advantage of technology,
utilizing group activities, assigning project- and problem-based
learning, and most importantly, sharing resources.
to think critically and problem solve are essential in the
mathematics classroom. These learning experiences push students
from "doing" mathematics to "understanding" mathematics.
Problem-based learning and project-based learning are great ways
for math educators to increase critical thinking in their
classroom. These learning opportunities should incorporate
inductive and deductive reasoning, require students to interpret
information and draw conclusions, and solve unfamiliar problems
in both conventional and innovative ways. Robert Kaplinsky's
is an excellent source for a list of free real-world
problem-based lessons that encourage critical thinking.
represents a variety of interactions in the classroom.
"Expressing thoughts clearly, crisply articulating opinions,
communicating coherent instructions, motivating others through
powerful speech--these skills have always been valued in the
workplace and in public life. But in the 21st century, these
skills have been transformed and are even more important today"
(National Education Association, 2010, p. 13). While important,
face-to-face communication is no longer the primary way students
interact with each other.
must continue to be taught how to communicate with someone
face-to-face, but must also be equipped with the skills to
communicate virtually and with people from different
backgrounds. In a mathematics classroom, students should be
communicating with the educator and other classmates through
multiple platforms. These could include email, Google classroom,
class website, making individual and group presentations inside
and outside of the classroom, creating a blog, and discussing
assignments with students in another state or country, to name a
few. The G Suite for Education by Google provides numerous ways
for students to communicate in and outside of the classroom.
the "Four Cs" individually can be difficult. Through many of the
communication examples listed above, students are engaging in
collaboration. In a mathematics classroom, students must be
required to work in diverse teams to develop a solution to a
real-world problem. Fifty years ago, many jobs could be
completed individually. However today, much work is accomplished
in teams, and in many cases, global teams (National Education
Association, 2010). Through the use of resources similar to
Google docs and Padlet, students can collaborate with classmates
or students across the world. These learning experiences provide
students with the opportunity to strengthen their collaboration
Time constraints often impede the ability to
incorporate opportunities to strengthen students' creativity.
However, in the math classroom, creativity can easily be
integrated into the class by creating an environment where
students are encouraged to develop innovative ways of solving a
problem and are praised for sharing their original solution.
Math educators should incorporate problems into their classroom
that can be solved using multiple methods or have multiple
solutions. This encourages students to use a wide range of
skills and think outside of the box. Students should have access
to materials inside the classroom so that they can creatively
problem solve. Additionally, students should be encouraged to
use a variety of technology resources, such as building a
website through Google sites, creating a movie using an iMovie,
or comic strip through
Integrating the "Four Cs" into the
mathematics classroom does take some research and planning, but
these efforts will benefit students as they prepare to enter the
current workforce. Educators should not feel overwhelmed by
attempting to address all of the "Four Cs" in one lesson.
Instead, educators should focus on one or two of the "Four Cs"
and slowly integrate new technology resources into their
classroom. Incorporating technology resources, group activities,
project-based, and problem-based learning in the mathematics
classroom will help to develop the skills students need to be
successful in the 21st century.
National Education Association (2010).
Preparing 21st Century Students for a Global Society: An
Educator's Guide to 'The Four Cs'. [online] Washington DC:
National Education Association. Available at:
[Accessed 24 Jul. 2018].
||Dr. Julie Carter is an assistant professor
within the Department of Teacher Education and
Social Work at Middle Georgia State University. She
has experience teaching high school math. Her
research interests include instructional technology
Dr. Rhonda Amerson is an assistant professor
within the Department of Teacher Education and
Social Work at Middle Georgia State University. She
has 23 years of classroom experience in grades 2-8,
and her research interests include instructional
practices, integrated curriculum, and classroom
Back to Top
Implementing open tasks that reveal students'
misconceptions can be unnerving for any teacher, and supporting
students' discourse about those misconceptions can be equally
challenging. Such tasks and discourse can be especially difficult
with younger students who are still developing communication skills,
but engaging in their use can be rewarding for students and
teachers. In this paper, we describe the design of, enactment of,
and reflection on a lesson in which technology was used to revisit
ideas from a previous lesson, to support first and second-grade
students in arguing and confronting valid and invalid strategies for
measuring length in order to develop deeper understandings of
aspects of measurement.
We describe the lesson in more detail below, but
first, give an overview of its affordances and limitations. This
lesson, collaboratively developed by a mentor teacher, an intern
teacher, and a teacher educator (first author), was productive:
students tried out strategies, discussed their ideas, were surprised
when they saw strategies that resulted in different answers, and
argued about why the answers were different and why they were not
correct. At the same time, the lesson was messy: we knew not all
students left the lesson with a perfect understanding of length
measurement, and it was difficult to measure new growth or new
understandings for each student. The teachers and the first author
reflected on the lesson and considered potential improvements. We
all felt confident that the benefits from the discussions,
arguments, and confrontations of misconceptions had at least added
to the students' foundations for later understandings.
We describe the development, enactment, and
reflection on this lesson in this paper, with one focus on
acknowledging and accepting that the mathematics and learning
outcomes that emerge from this approach can be messy and difficult
to measure, but are also valuable. A second focus is on the
complexity of interaction between pedagogy, mathematics, and
technology. That is, the pedagogy involved implementing an open,
multi-level task and facilitating productive discussion at points in
the task, the mathematics involved attention to sophisticated
aspects of length measurement such as unit-attribute relations,
partitioning, and tiling (explained below), and the technology was
used to build on a previous, concrete task and to focus student
attention on particular strategies.
Background and Method
Ms. Holst, an intern teacher in a full year
teaching field experience, worked with a mentor teacher who had
taught for ten years. The elementary school was a Title 1 school
with students at a wide range of mathematics proficiency levels
based on standardized mathematics testing. The classroom in this
paper combined 28 first- and second-grade students (14 at each
level). Following recommendations from Mathematics Teaching
Practices from Principles to Actions: Ensuring Mathematical Success
for All (National Council of Teachers of Mathematics, 2014), Ms.
Holst's mentor teacher incorporated challenging tasks and student
discourse into mathematics lessons. She valued developing classroom
norms encouraging students to make and test conjectures, make
claims, and support claims with evidence to support their
development of conceptual understanding and procedural fluency. She
supported students' metacognition by asking them to reflect on their
thinking and to notice when it changed. She also asked her students
to engage in challenging tasks. After a task, she frequently asked
students to reflect on their strategies by viewing examples of work
done by unknown (imaginary) students that illustrated different ways
of thinking and misconceptions. The mentor teacher asked students to
explain what they thought the imaginary student was thinking and
then to revise their initial solutions based on their new thinking.
In collaboration with her mentor teacher, Ms.
Holst designed and taught a lesson focusing on length measurement to
her combined first- and second-grade students. The lesson
incorporated van de Walle et al.'s (2013) Crooked Path activity (see
Figure 1). Ms. Holst created crooked paths similar to those in
Figure 1 using masking tape on the floor and on some tables.
Students worked individually. They used physical objects to measure
paths, recorded their measurements, and moved to the next path. Ms.
Holst felt her lesson engaged her students in experimenting with
length measurement. However, because of the classroom management and
logistics involved in facilitating students' movement from path to
path, and because students were measuring and recording
individually, Ms. Holst felt she missed important aspects of student
thinking and insight in the end-of-lesson discussion.
Figure 1. Examples of "Crooked Paths" (Van de Walle, 2013).
Planning the Lesson
Ms. Holst and the first author planned a
follow-up length measurement lesson for the initial physical object
implementation of the Crooked Paths task. Ms. Holst's goals for the
follow-up lesson were for students to build on the Crooked Paths
experience by allowing them to (a) engage in a task designed to
reveal misconceptions, (b) use a dynamic applet to view and then
reflect on two common strategies, each revealing a different
misconception, and (c) use the dynamic applet as a tool to support
mathematical discourse about the meaning of units in measurement. As
part of the planning, the author adapted the task based on
discussions with Ms. Holst, creating a set of three applets to be
incorporated into the lesson. We designed the lesson to support and
elicit student thinking about unit-attribute relations,
partitioning, and tiling based on Barrett & Clements' (2003) and
Lehrer's (2003) discussion of these components for learning length
measurement. Unit-attribute relations refers to students'
ability to recognize the relationship between the unit that
is used to measure and the attribute being measured. In the
digital Crooked Paths task, unit-attribute relations emerged as
students used a two-dimensional shape (e.g., a tile) to measure the
one-dimensional attribute (e.g., length) of a path. Common
misconceptions about the relationship between the tiles and the
length of a path are allowed to emerge when corners are part of the
path (see Figure 2). Students may place tiles around the outside of
the corner, counting one extra unit because a tile fits outside the
corner (see a). Students might place tiles around the inside of the
corner, counting a tile in the corner once despite the two unit
lengths (see b). Attending to the unit-attribute relationship, a
student would skip the corner (green) or double-count the corner
(yellow) to count 4 unit lengths (see c).
Figure 2. Counting tiles (2D object) compared to
counting lengths (1D part of a 2D object).
Partitioning refers to the idea that the
length must be divided into some number of equivalent spaces. That
is, the lengths along a path identified by placing tiles must be the
same size. When using concrete objects to measure length,
partitioning means attending to precision to create unit lengths
that are the same size; that is, avoiding gaps and overlaps (see
Figure 3). The other side of partitioning, tiling, refers to
the concept of filling the space completely (e.g., no gaps). This
applet deliberately allows gaps and overlaps to allow misconceptions
to be revealed and to allow students the opportunity to actively
work on precision (when they notice the gaps and overlaps as
problematic). The task supports students in developing their
conceptual understanding of partitioning and tiling by
allowing them to leave spaces or overlaps and then to confront the
mathematical consequences when they compare solutions.
Figure 3. Mathematical consequences of gaps and overlaps
include the possibility of infinite, distinct measures of the same
The lesson structure used a
Launch-Explore-Connect-Explore-Connect format. Ms. Holst
introduced the task by showing students how to open the applet and
move the pieces. In teams of three or four, students found lengths
for paths in Activities 1 and 3 (see Figure 4).
Figure 4. Activities 1 and 3 are shown on the left and right,
Ms. Holst led the class to compare solutions.
They found groups had different solutions. She showed the students
the video of Yellow over-counting and Green under-counting (see
Figure 5). Ms. Holst asked students to discuss Yellow's thinking and
Green's thinking in their small groups. She asked them to revise
their strategies as needed. Finally, Ms. Holst led a whole class
discussion where students shared their thinking and guided her
through the tiling process to discover the misconceptions about
Yellow's and Green's thinking.
Figure 5. Jagged Paths animation showing two common (but
incorrect) strategies: underestimating and overestimating.
The lesson was the last of the day and took one
hour. As students worked on their exploration of Activities 1 and 3,
Ms. Holst asked groups to take screenshots of anything they thought
was interesting. Of the nine groups, six groups took screenshots
within a few minutes of each other. All screenshots are shown in
Figure 6, in no particular order. The screenshots show different
strategies used by students: placing tiles along the inside of the
path, placing tiles along the outside of the path (with corner
blocks and without corner blocks), and placing tiles that were
centered on the path.
Figure 6. Screen shot images taken by six of the nine groups.
(Some images are smaller or larger depending on student actions, but
otherwise differences in size have no significance.)
Groups A, C, D, and F placed blocks along the
inside of the path, while group B placed blocks along the outside
and omitted the corners, and group E placed blocks centered on the
line. Students' attention to placement without gaps or overlaps can
be seen. Student groups shared the answers they had found, with most
answering 15 tiles, one group answering 17, and one group answering
Ms. Holst showed the green squares placement
(Green's strategy) animation in Activity 2 (shown in Figure 7a),
which results in the placement of 19 squares. She asked them, "What
was Green's answer? What did the Green student do? Write about it!"
After a few minutes of writing, she showed the yellow squares
placement (Yellow's strategy) animation in Activity 2 (shown in
Figure 7b), which results in 15 squares. Students acted surprised
and several hands shot in the air, impatient to share. After a few
minutes of writing, Ms. Holst asked students to discuss their
reactions in their groups.
Figure 7. Screenshots from Activity 2. (a) Image shows
Green's (wrong) placement and (b) Image shows Yellow's (wrong)
placement of squares in contrast to Green's.
Students described their thinking in different
ways showing a range of various understandings and various levels of
ability in communicating their own thinking. A student in one group
pointed to Green's corner squares and said, "Corners -- they put one
in the corners." Another student answered, "The line on the inside
is shorter. The one on the outside is longer." I asked him if the
line was different lengths on different sides and he agreed,
pointing to Yellow's path and Green's path. A student (Student B in
Table 2) in a different group pointed to Green's path and said,
"That's what we did. That's why we got 19. But the yellow is right.
I have proof -- because more [groups] got 15."
After some time spent in group discussion, Ms.
Holst called for students' attention. She asked them to share
thoughts about Green's and Yellow's strategies. One student
referenced the previous lesson in which students measured with
objects. In that lesson, because the students kept counting with the
first counted object as "zero," Ms. Holst explained to them that
zero means "a nothing unit." Ms. Holst showed me that she had
indicated the nothing unit by squishing her fingers tightly
together. In explaining why the squares placed outside the corners
of the path (and thus touching only a tiny bit), the student
referenced that discussion by also pushing her fingers together
while talking about the space at the corner being counted as zero.
Ms. Holst did not feel satisfied with the students' explanations,
because they had seemed to accept the inside (yellow) strategy. At
the moment, she made a decision to continue the class discussion by
asking students to be "Line Detectives." She opened Activity 1,
using a computer behind the class. She wanted to support students in
noticing that measuring the path was not counting the squares;
rather, measuring the path was counting the number of side lengths.
First, she focused students' attention on the
side lengths by asking them, "What is the unit we are using to
measure?" Students responded with multiple responses, including
"Inch" and "Kind of an inch like we used to measure things." Ms.
Holst asked again, "Read the instructions again: 'Find the length of
the jagged path by dragging the unit squares.' Hmmm. '...dragging
the unit squares...' Maybe the unit isn't an inch. We just have a
line that we're measuring -- is the unit square the unit?" One
student answered, "On the side." Another student agreed, "She said,
and I agree with her, to measure on the side." Ms. Holst responded,
"Is the unit the whole square or the side? [moves a square] When am
I measuring part of the line? You will be line detectives as I move
Next, Ms. Holst placed the green and yellow
squares, following the same paths as the animation. She asked
students to carefully watch and tell her when a square touched the
jagged path once, twice, or not at all. Students watched closely as
Ms. Holst moved each square, placing it on the outside of the path.
Frequently, a chant of "Gap!-Gap!-Gap!-Gap!-Gap!" went up. Finally,
Ms. Holst stopped and said, "You guys are so good at noticing the
gaps. But I'm doing the best I can with this mouse so just look for
whether I'm measuring part of the line or not." When Ms. Holst
reached a corner and placed a square at the corner, students yelled,
"No!" When she finished placing the squares, the students counted
with her to reach 17 squares touching the line. She repeated the
process with the inside of the square, asking them "For the inside,
is each square only measuring one unit of the line?" As she placed a
square in the corner, she asked, "Is this square only measuring one
unit of the line?" Some students said, "Yes" and others said, "No."
Ms. Holst explained, "Actually, the square is measuring two units
here -- so we need to count it twice." She and the students counted
how many times each square touched the line. They counted twice at
the corners to reach an answer of 17. A girl said, "What!"
Ms. Holst instructed them, "Write on the paper.
What is your answer and how can you prove it's right?" Some students
had already written an answer. One student asked, "Do you want our
thinking now or from before?" Ms. Holst responded, "What is your
answer right now and how can you prove it's right?" As I watched
students writing, I asked some students to explain their answers to
me. One student told me the answer was 17 and that he could prove it
because it was how many times each square touched the line. His
written answer only indicated that the answer was 17 because we had
counted it twice for Green and for Yellow.
The enactment of the lesson had mixed results.
Ms. Holst pointed out that one benefit of the lesson was providing
students with the opportunity to show off the new thinking acquired
from other measurement lessons, such as their thinking about the
value of precision, avoiding gaps and overlaps, and applying the
"zero unit." This attention can be seen in the groups' screenshots
shown in Figure 4, where squares are relatively aligned and fitted
closely together. One student even showed this value by remarking,
"He's so good! He's being very precise!" Students also showed their
attention to avoidance of gaps and overlaps during the last class
discussion when they chanted "Gap!-Gap!-Gap!" as soon as they saw
any space between squares. Another benefit of the lesson was
exposing misconceptions of some students, especially with respect to
gaps and overlaps and not connecting the act of measurement with the
characteristic of the line that was being measured. These
misconceptions are discussed in the next section.
Complex mathematical ideas cannot be learned in
one lesson, but over time through messy work that allows student
misconceptions to be revealed, and through discourse that allows
students to confront their own misconceptions, show their newly
acquired knowledge, and change their thinking. Building a
measurement lesson that supports students' meaningful engagement in
making sense of measurement ideas, arguing, thinking critically, and
changing their own thinking is important to developing this
knowledge but may problematize the measurement of students' gains.
As Ball, Lubienski, and Mewborn (2001) wrote, "With focused, bounded
tasks, students get the right answers, and everyone can think that
they are successful. The fact that these bounded tasks sometimes
results in sixth graders who think that you measure water with
rulers may, unfortunately, go unnoticed" (p. 436). A lesson that
ends even though not all students have reached the learning goal is
not a failed lesson. Understanding measurement takes time, repeated
opportunities, and opportunities to engage with mathematics in
Ball, D. L., Lubienski, S., & Mewborn, D.(2001).
Research on teaching mathematics: The unsolved problem of teachers'
mathematical knowledge. In V. Richardson (Ed.), Handbook of
research on teaching (pp. 433-456). New York, NY: Macmillan.
Barrett, J. E., & Clements, D. H. (2003).
Quantifying Path Length: Fourth-Grade Children's Developing
Abstractions for Linear Measurement. Cognition and Instruction,
21(4), 475-520. doi:10.1207/s1532690xci2104_4
Lehrer, R. (2003). Developing understanding of
measurement. A Research Companion to Principles and Standards for
School Mathematics, 179-192.
National Council of Teachers of Mathematics (NCTM).
2014. Principles to Actions: Ensuring Mathematical Success for
All. Reston, VA: NCTM.
Van de Walle, J. A. (2013). Elementary and
middle school mathematics: teaching developmentally (8th ed.).
Van de Walle, J. A., Karp, K. S., & Bay-Williams,
J. M. (2013). Elementary and middle school mathematics: Teaching
developmentally (8th ed.). Upper Saddle River, NJ: Pearson.
||Dr. Eryn M. Stehr is an Assistant Professor
of Mathematics Education at Georgia Southern
University, and a 2018 fellow in the Association of
Mathematics Teacher Educators' (AMTE's) Service,
Teaching, and Research (STaR) program for
mathematics education faculty. Her research
interests focus on developing teacher autonomy and
decision-making in mathematics teaching and
learning, with a special focus on integrating use of
technology with rich tasks and mathematical
discussion. She earned her M.A. in Mathematics from
Minnesota State University and her Ph.D. in
Mathematics Education from Michigan State
Dr. Ha Nguyen is an Assistant Professor of
Mathematics Education at Georgia Southern University
and a Blue'10 Fellow in the Mathematical Association
of America's (MAA's) Project NExT (New Experiences
in Teaching) for mathematics faculty. She is
interested in students' understanding and thinking
of mathematics and how to deepen their mathematical
knowledge. She earned her Ph.D. in Mathematics from
Back to Top
The 59th annual Georgia Mathematics
October 17-19, 2018
2018 Georgia Math Conference at Rock Eagle, will be held
Wednesday October 17 - Friday, October 19, 2018. We hope you
will join us! This year promises to be a great conference! The
theme this year is "Embracing Productive Struggle". Early
registration opened August 1st. Don't miss out --
register for this conference today! As an added bonus,
conference attendees will receive a voucher to get 20% off an
NCTM book order during the subsequent week of the conference.
GMC 2018 Keynote Speakers
||Dr. Thomasenia Lott Adams
Oct 17 (Wednesday Evening)
Thomasenia Lott Adams,
Ph.D. is a mathematics teacher
educator/researcher in the School of Teaching &
Learning and the Associate Dean of Research (ADR) in
the College of Education at the University of
Florida (UF), Gainesville, FL. Dr. Adams'
scholarship includes funded grants at the national
and state levels, including her role as Co-Principal
Investigator on the Florida STEM-Teacher Induction
and Professional Support Center funded by the
Florida Department of Education and the UF Noyce
Scholars Program: Raising STEM EduGators funded by
the National Science Foundation. She has a
commendable list of publications and conference
presentations is a senior author of the Go Math!
elementary mathematics textbook series published by
Houghton Mifflin Harcourt. She is also an author on
several publications of the National Council of
Teachers of Mathematics (NCTM) and Solutions Tree.
Service posts to her credit
include editor of the Mathematical Roots Department
in Mathematics Teaching in the Middle School and
co-editor of Investigations Department of Teaching
Children Mathematics both published by NCTM. Dr.
Adams has also served as board member of the
Association of Mathematics Teacher Educators board
member of the School Science and Mathematics
Association and past president of the Florida
Association of Mathematics Teacher Educators. Dr.
Adams is a previous recipient of the Mary L. Collins
Teacher Educator of the Year Award for the Florida
Association of Teacher Educators. She has
participated in major efforts to improve teaching
and learning of mathematics, including serving as a
mathematics coach across grades K-12, is a co-author
of K-12 mathematics professional development
programs and leader for the mathematics and science
job-embedded professional development and graduate
degree program for middle and high school teachers.
She is also an accomplished mentor of mathematics
education doctoral students.
|Dr. Matt Larson
Oct 18 (Thursday Evening)
Matt Larson, Ph.D. is
president of the National Council of Teachers of
Mathematics (NCTM), a 70,000-member international
mathematics education organization. Previously,
Larson was the K-12 curriculum specialist for
mathematics in Lincoln (Nebraska) Public Schools for
more than 20 years.
Dr. Larson began his career in
education as a high school mathematics teacher and
served as a member of the leadership team for the
National Science Foundation Math and Science
Partnership project Math in the Middle at the
University of Nebraska-Lincoln. Dr. Larson's long
history of service within NCTM includes chairing the
Research Committee, serving for three years on the
Board of Directors, serving for two years on the
Executive Committee, and chairing the Budget and
Finance Committee. He has contributed extensively to
NCTM journals and books. His two-year term as NCTM
president began in April at the conclusion of the
2016 NCTM Annual Meeting & Exposition in San
Dr. Larson is a frequent
speaker before mathematics education audiences, and
he has authored or co-authored several books,
including a series on professional learning
communities and Common Core Mathematics. He is a
co-author of Balancing the Equation: A Guide to
School Mathematics for Educators and Parents, and he
was on the writing team of Principles to Actions:
Ensuring Mathematical Success for All (NCTM, 2014).
Dr. Larson has taught mathematics at the elementary
through college level and has held an appointment as
an honorary visiting associate professor at Teachers
College, Columbia University. In 1994 he was
recognized for his teaching accomplishments with a
U.S. West Outstanding Teacher Award.
||Dr. James Tanton
Oct 19 (Friday Afternoon)
Believing that mathematics
really is accessible to all, James Tanton, Ph.D.
is committed to sharing the delight and beauty
of the subject. In 2004 James founded the St. Mark's
Institute of Mathematics, an outreach program
promoting joyful and effective mathematics
education. He worked as a full-time high-school
teacher at St. Mark's School in Southborough, MA
(2004-2012), and he conducted, and continues to
conduct, mathematics courses and workshops for
mathematics teachers across the nation and overseas.
Dr. Tanton is the author of
Solve This: Math Activities for Students and Clubs (MAA,
2001), The Encyclopedia of Mathematics (Facts on
File, 2005), Mathematics Galore! (MAA, 2012) and
twelve self-published texts. He is the 2005
recipient of the Beckenbach Book Prize, the 2006
recipient of the Kidder Faculty Prize at St. Mark's
School, and a 2010 recipient of a Raytheon Math Hero
Award for excellence in school teaching. He also
publishes research and expository articles, and
through his extracurricular research classes for
students has helped high school students pursue
research project and publish their results. Dr.
Tanton is currently an ambassador for the
Mathematical Association of America.
9:30-10:45 Kristopher Childs:
"Every Child Has Genius Level Potential -- Are You Giving
Them Time to Showcase It?"
(K-5) Participants will engage in the 6 Stages of Effective
Mathematics Instruction. Participants will make sense of a
lesson design process, develop an understanding of rich
problem-solving task selection and implementation, and
explore an effective instructional model. There will be a
keen focus on the impact of productive struggle during the
Thursday, 10/18/18, 1:45-3:00
Math Rx for a Productive Struggle -- "Do the Math"
(Grades 6-12) Do you often hear students saying, "I don't
get this!" or "How do we do this?" Do you wonder about ways to
help your students through the struggle they have in
mathematics? Well, the best medication for the symptoms
associated with Productive Struggle will be addressed in this
engaging and hands-on math session. Through active
participation, you will leave with a better understanding of how
to develop lessons that support a Productive Struggle in a
positive way using the "Do the Math" Protocol.
Thursday, 10/18/18, 1:45-3:00
"Using Data to Make Sense of Students Struggles"
(K-12) Participants will learn how to develop an effective
data-informed environment that is student-centered. Participants
will explore effectively assessing student learning, providing
meaningful feedback, and using data to inform instructional
decisions. In real-time participants will engage with an
assessment system and develop a plan of action to implement the
Friday, 10/19/18, 9:45-11:15
"The Power of Visualization in Mathematics"
(K-12) This session will demonstrate the astounding power --
and fun! - of visual thinking in mathematics. Let's see just how
fundamental and crucial pictures are to problem-solving, doing
joyful mathematics and engaging in deep learning.
Friday, 10/19/18, 9:45-11:15
a.m.: Michelle Mikes hosts a panel discussion,
Mathematics Leadership in Education and Instruction:
Promoting Productive Struggle
Dr. Brian Lack, Math Specialist
Dr. Brian Lawler, KSU Math Professor
Dr. Joanna McGaughy, Elementary Teacher
We are all stakeholders in the education of our students.
From the classroom teacher to legislators, how are you promoting
mathematical productive struggle in and beyond your class? Join
our panel of experts to learn how to pave the way for productive
struggle opportunities with our stakeholders at large. Walk away
with strategies in promoting productive struggle for success of
Friday, 10/19/18, 1:00-2:00
I See, I Hear, I Wonder -- Using Visual Literacy to Ignite a
(Grades 6-12) How do we get our students to persevere and
make sense of the mathematics when they cannot relate at all to
the mathematics being taught? Let's look at how Visual Literacy
can be used to foster mathematical understanding and reasoning
through thoughtful Math Talk and authentic work products.
Through various simulations, participants will leave with
tangible activities that can be incorporated in the mathematics
classroom to ignite a Productive Struggle.
SPECIAL CONFERENCE EVENT
Escape Rooms will be
available for educators of grades K-2, 3-5, 6-8, and 9-12 on
Wednesday, 10/17/18, as a special pre-conference activity
from 3:00-5:00 p.m.
State Superintendent Forum
On Thursday, October 18th, GCTM will
host state superintendent candidates Richard Woods and Otha
Thornton for a live Q&A session. This forum will take place
from 11:00-12:00 at Talmadge Auditorium at the Rock Eagle
Click here if you plan to attend and/or if you would like to
submit one or more questions for the candidates to address
during the forum.
Interested in becoming a GMC vendor?
Please contact Jennifer Peek, Director
of Exhibits at
Back to Top