6 Teaching for High Potential | November 2021
“I
just know…because it’s right.”
Zan’s response to our attempt
to get him to further explain
his thinking in writing (see Figure
1) caught us off guard since he had
scored five grade levels above his peers
on a standardized test. Although he
could readily provide answers, we knew
that he was missing a critical aspect of
mathematics: communicating his reasoning
(Common Core State Standards
Initiative [CCSSI], 2010; National Council
of Teachers of Mathematics [NCTM],
2000). In fact, NCTM’s Providing Opportunities
for Students with Exceptional
Mathematical Promise Position
Statement (2016) noted, “Students with
exceptional mathematical promise include
those who…are particularly good
at explaining complex concepts to others
or demonstrate in other ways that
they understand mathematical material
deeply” (p. 1).
This experience and similar ones with
the other excelling students in our second-
and third-grade classes made us
consider how we could help them advance
their abilities to communicate their
mathematical reasoning. It so happens
that recent work being done in elementary
mathematical writing emphasizes
writing for the purpose of reasoning and
communicating mathematically (Casa et
al., 2016). Writing therefore was a way
we felt we could provide services for students,
like Zan, who were underachieving
in this area.
Planning to Teach
Argumentative Writing
We decided specifically to target argumentation,
one of the four types of
mathematical writing identified by the
Elementary Mathematical Writing Task
Force (Casa et al., 2016) because of its
direct connection to the Common Core’s
mathematical practices. Specifically,
Mathematical Practice 3 states, “Mathematically
proficient students are also
able to compare the effectiveness of two
plausible arguments, distinguish correct
logic or reasoning from that which
is flawed, and—if there is a flaw in an
argument—explain what it is” (CCSSI,
2016, p. 7). The core elements of an argument
include a claim and evidence to
support that position. An example of a
written mathematical argument is when
a student writes the claim, “I disagree
that there are enough goodie bags,” and
backs it by stating, “There are three blue
ones and two reds. I know two and two
is four, plus one is five. There are six kids,
so there’s not enough.”
Following NCTM’s position that “each
and every student has mathematical
promise” (2016, p. 1), we were committed
to providing all our students with the
opportunity to develop their abilities to
communicate their reasoning through
mathematical arguments. We also hoped
their writing would more accurately show
us whether there were gaps in their understanding
to push their thinking further
Making Things
‘Write:’ Supporting
Mathematically
Promising Students
Tutita M. Casa
University of Connecticut
Samantha R. Schaaf
W. B. Sweeney Elementary School
Erica L. Ambrogio
Redding Elementary School
Dr. Tutita M. Casa is an Associate Professor
of elementary mathematics education in the
Neag School of Education. She was the CoPI
and co-author on Project M2 and staff for
Project M3 that produced researched-based
advanced math units
Samantha and Erica are elementary school
teachers in Connecticut. They were master’s
students in the Integrated Bachelor’s/Master’s
Teacher Preparation Program when they collaborated
on this work and continue to implement
these ideas with their current students.
Figure 1. Zan found his numerical
answer in two minutes, but then spent
30 minutes putting nothing else on the
page, even when prompted
November 2021 | Teaching for High Potential 7www.nagc.org
and, at the same time, provide them with
the opportunity to showcase a unique
way of thinking about any given topic.
We therefore sought to differentiate our
supports as suggested by NAGC’s PreK-Grade
12 Gifted Programming Standards
(2019) Learning and Development
and Curriculum & Instruction strands. To
do this, we identified excelling, average,
and struggling students based on their
scores on mathematics district assessments
and unit tests. We further tweaked
these three groups based on our observations
of their performance and participation
in discussions during class.
This work with our second and third
graders took place across two months.
Prior to responding to each of six writing
tasks, students discussed the prompt.
They became accustomed to solving the
problem first, then deciding whom they
agreed with and why. We specifically
prompted our students to write mathematical
arguments by asking them to
choose one of two solutions we provided
from fictional students and defend their
argument by explaining their choice, as
recommended by Firmender, Casa, and
Colonnese (in press). It took students
about 30 minutes to complete each task,
which included time to talk and write.
Lessons Learned from Our Students
Despite our designation of the three
student groups, we attended to the
needs of our excelling students with an
aim to have the strategies that helped
them serve other groups. Nonetheless,
students are more successful in learning
when they are taught in ways that meet
the needs of their individual readiness
levels. We decided to differentiate our
instruction and identify strategies that
could help us best serve our students.
We therefore had all our students work
on the same task with different levels
of support. Additionally, we designed
a student-friendly rubric (Figure 2) that
outlined expectations. We used the rubric
to assess our students’ work and
provide them with feedback. They also
used it to evaluate their own writing and
that of their peers.
Over time, we found that different
groups of students benefited from word
banks, sentence starters, manipulatives,
and hints. We initially thought that our
excelling students would be able to write
their ideas clearly enough to build an argument
that would demonstrate their
reasoning, and therefore did not provide
them with any scaffolds. It turned out
that they needed more support in attending
to precision (CCSSI, 2010), and
a word bank allowed them to further their
abilities to write strong arguments. This
scaffold turned out to be beneficial for
our average group. Although they often
figured out the correct answers, they initially
were lost about what to write when
it came time to construct their argument.
We therefore also provided them with
sentence starters. Lastly, our students
who were struggling, not surprisingly,
needed help with the mathematics and
writing. Thus, in addition to using the
word bank and sentence starters, we
initially made manipulatives available to
them, and then hint cards.
Strategies That Supported Our
Students’ Argumentative Writing
We recognize that the strategies we
implemented across groups might be
applicable to other teachers whose students
are excelling at different levels
with regards to their command of mathematics
and ability to communicate
their arguments in writing. Therefore, we
present the strategies we used in more
detail. Our goal was to ensure all our
second- and third graders could reach
their highest potential in writing mathematical
arguments, and we hope these
insights support other teachers’ work in
this area.
1. Using Word Banks to Improve
Accuracy
Our goal for utilizing the word bank
was to help our students hone in on the
ideas they could address to help readers
understand the argument. While many
students had command of the mathematics
content, utilizing precise mathematical
language pushed them to describe
more accurately what they knew
clearly and efficiently. We provided students
terms written on large index cards
that had student-friendly definitions on
the back. Students engaged with the
cards, yet students who knew the definitions
already did not need to flip them
over. We changed the word banks to
correspond to the content, such as including
“numerator” and “whole” for a
prompt about fractions. Eli’s response
in Figure 3 shows how he incorporated
relevant fraction terms and their definitions
to convince others why his claim
was correct.
2. Using Sentence Starters to Bridge
Between Student Solutions and Their
Writing
We needed to help some students
connect their solutions with their written
arguments. We therefore provided them
Figure 2. The student-friendly language in this rubric made it possible for students
to self- and peer-assess each other’s writing.
Figure 3. Eli used the fraction related
vocabulary words and their definitions
to explain to the reader why Evan was
right.
8 Teaching for High Potential | November 2021
with sentence starters (Figure 4) to help
them initially realize that instead of listing
the steps they took to solve the problem,
the structure of a written argument
includes a claim and evidence supporting
that claim.
Figure 5 shows how although Stephanie
wrote a relatively long piece, she only
wrote out her computational steps and
did not attend to the misconception introduced
in the prompt. In contrast, Lily
(Figure 6) adapted the “I knew ____ was
wrong because…” sentence starter to
structure her argument in a way that
provided important details about how
she chose which claim to agree with and
why the other one provided was invalid.
3. Using Manipulatives to Solve the
Problem
We recognized that having command
of the mathematics was necessary for
students to write a compelling argument.
Therefore, one way we tried to
support our students was to provide
them with the option of using manipulatives
like base-ten blocks (for regrouping)
and square tiles (for area) to help
them determine a solution prior to writing
their arguments. Although students
who chose to use the manipulatives
eventually settled on a valid answer,
they ran out of time to construct their
argument. While utilizing manipulatives
was a strategy that was beneficial for
some of our students, we wished to
have our whole group focused on one
task at the same time. We therefore
considered another alternative that still
would give them the foundation they
needed to figure out the mathematics
and time to write their arguments.
4. Using Hint Cards as a Reference of
Content
Eventually, we designed hint cards to
give our students a jumpstart on thinking
about the argument presented in the
prompts. These cards served to remind
students about key ideas we had discussed
in class to allow them to apply
it to the specific problem they were given,
such as the one in Figure 7 that we
used with a prompt focused on place
value. We passed out a hint card with
the prompts, which allowed students to
choose whether they would like to use
it. Students initially resist using it and instead
asked us for help. We suggested
they check the hint card. Over time,
students learned to be more independent
about using the cards when they
needed it while thinking on their own
whenever possible.
Final Thoughts
Although our students had differing
needs, our work taught us that all students,
including our excelling ones, may
need some level of support to start the
process of mathematical writing. Focusing
on writing mathematical arguments
required our students to go beyond the
answer and communicate their reasoning.
We envisioned all our students
as having mathematical promise and
providing them with scaffolds to allow
them to delve more deeply into the content
through writing seemingly empowered
them. Even for excelling students,
like Zan, just one support may make a
drastic difference in their writing. While
Zan was stumped about what to record
when we began asking our students to
write mathematical arguments, in the
end, he realized he had the voice to convey
a solution path that was advanced
among his classmates (see Figure 8).
Opening the possibility for our students
to communicate their unique reasoning
through mathematical writing is something
we plan to continue to provide all
our students. THP
References
Casa, T. M., Firmender, J. M., Cahill, J., CarFigure
5. Stephanie simply listed her
steps rather than defending an
argument.
Figure 6. Lily gives important details
about how she chose which claim
to agree with and why Veronica was
wrong.
Figure 7. These are hint cards that
students were given for a prompt about
place value.
Figure 8. While every other student
decomposed the shape into two rectangles
to find the area, Zan’s creative
approach showcased a strategy he
used that was more sophisticated.
Figure 4. Students received a copy
of these sentences starters with their
prompts.
	 detti, F., Choppin, J. M., Cohen, J., Zawod-
	 niak, R. (2016). Types of and purposes for
	 elementary mathematical writing: Task
	 force recommendations. Retrieved from
	 http://mathwriting.education.uconn.edu
Common Core State Standards Initiative.
	(2010). Common core state standards for
	mathematics. Authors: Washington, DC:
	 The National Governors Association Cen-
	 ter for Best Practices and Council of Chief
	 State School Officers. Retrieved from
	 www.corestandards.org/assets/CCSSI_
	Math%20Standards.pdf
Firmender, J. M., Casa, T. M., & Colonnese,
	 M. W. (in press). Write on: Reasoning
	 through mathematical writing. To be pub-
	 lished in Teaching Children Mathematics.
National Association for Gifted Children.
	(2010). Pre-K-Grade 12 gifted program-
	 ming standards. Retrieved from www.
	 nagc.org/resources-publications/resourc
	 es/national-standards-gifted-and-talent
	 ed-education/pre-k-grade-12
National Council of Teachers of Mathemat
	 ics. (2000). Executive summary: Principles
	 and standards for school mathematics.
	 Retrieved from www.nctm.org/Standards-
	 and-Positions/Principles-and-Standards/
National Council of Teachers of Mathematics.
	(2016). Providing opportunities for students
	 with exceptional mathematical promise:
	 A position of the national council of teach-
	 ers of mathematics. Retrieved from www.
	 nctm.org/Standards-and-Positions/Posi
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	 for-Students-with-Exceptional-Promise/
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