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Why should teachers engage in research studies? As a community, teachers and researchers are concerned with addressing critical issues in math education. NCTM's web resources and conferences, as well as the pages of this journal, give evidence of a growing community and an expanding body of work supporting NCTM's (2012) position of linking research and practice—a “border crossing” between the world of research and the world of teaching (Silver 2003). Despite these initiatives, an emerging issue remains: How do we work together to cultivate a two-way exchange of professional knowledge (Heid et al. 2006)?
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462 MATHEMATICS TEACHER | Vol. 110, No. 6 • February 2017
to teaching
CONNECTING RESEARCH
Connecting Research to Teaching appears
in alternate issues of Mathematics Teacher
and brings research insights and findings to
the journal’s readers. Manuscripts for the
department should be submitted via http://
mt.mtsubmit.net. For more information,
visit http://www.nctm.org/mtcalls.
Department editor
Dana Cox, dana.cox@miamioh.edu, Miami
University, Oxford, Ohio
Nicole L. Fonger, Lindsay Reiten,
Susanne Strachota,
and Zekiye Ozgur
Engaging in Research:
Why? How? Now!
Why should teachers engage in
research studies? As a commu-
nity, teachers and researchers
are concerned with addressing critical
issues in math education. NCTM’s web
resources and conferences, as well as
the pages of this journal, give evidence
of a growing community and an expand-
ing body of work supporting NCTM’s
(2012) position of linking research and
practice—a “border crossing” between
the world of research and the world of
teaching (Silver 2003). Despite these
initiatives, an emerging issue remains:
How do we work together to cultivate
a two-way exchange of professional
knowledge (Heid et al. 2006)?
Based on our experiences as math
educators, we address the following
questions:
Why should teachers engage in
research?
What might teachers’ roles be in a
research project?
How do teachers get involved?
From a perspective of community
building toward a shared goal and our
experiences in collaborating with sec-
ondary math teachers, we showcase
some of the diversity of teachers’ roles in
research. Reflecting on lessons learned,
we share potential challenges, benefits,
and practical suggestions for teachers
who wish to engage in research and pro-
fessional inquiry.
WHY SHOULD TEACHERS
ENGAGE IN RESEARCH?
The goal of providing all students access
to quality math education requires that
teachers and researchers collaborate
and engage together as a community.
Researchers working alone lack the
information needed to effectively
address problems of practice that matter
most—problems that are highly contex-
tual and based on teachers’ day-to-day
experience. For example, Ms. B won-
dered, “How can I strengthen my stu-
dents’ understanding of ratios and pro-
portions?” Ms. L wondered, “How can
I better support students’ learning with
multiple representations using technol-
ogy?” Mr. D wondered, “What cur-
riculum might I draw on to enhance my
instruction? I want students to engage in
inquiry-oriented practices.” Such ques-
tions, grounded in teachers’ experiences,
may be best addressed through collabor-
ative efforts of teachers, researchers, and
other stakeholders committed to improv-
ing math teaching and learning.
WHAT MIGHT TEACHERS’ ROLES
BE IN A RESEARCH STUDY?
A teacher may choose to participate in
a study because it piques her interest or
because her voice, context, and experi-
ence would make an important contribu-
tion. Or she might actively collaborate
with a researcher on addressing a shared
problem of practice. Or a teacher might
research his own problem of practice to
address an important goal. Across these
roles teachers influence the findings
of research, co-shape the direction of
research, or determine the very question
to be studied. To illustrate the diversity
of teachers’ roles in research, we offer
three vignettes of teachers engaged in
research.
Teacher as Participant
and Collaborator
Ms. B, an eighth-grade math teacher,
was part of a cohort of middle school
math teachers taking courses at a local
university through a district partner-
ship. She was invited to participate in
a research project aimed at teacher-led
implementation of an instructional
unit designed by researchers to support
students’ algebraic reasoning through
quantitatively rich situations. The goal
of the unit was to understand that a rela-
tionship is linear if the rate of change
of one quantity compared to another is
constant. To this end, a set of tasks was
developed using gears as a context; a
sample gears task is shown in figure 1.
Before implementing the instruc-
tional unit that focused on linear rela-
tionships grounded in gear ratios, Ms. B
met with the research team (including
authors Reiten and Ozgur) to discuss the
goals of the unit and supports for imple-
menting lessons in her classroom (e.g.,
how to use physical gears as manipula-
tives). Throughout the unit, Ms. B modi-
fied the lessons based on her students’
needs (e.g., adding additional directions)
and found that the lessons extended
Copyright © 2017 The National Council of Teachers of Mathematics, Inc. www.nctm.org.
All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Vol. 110, No. 6 • February 2017 | MATHEMATICS TEACHER 463
her students’ thinking. Reflecting on
the unit, she said it gave students an
opportunity to think deeply about linear
relationships in terms of proportional
thinking before going to the book, which
“just jumps in and goes and doesn’t dig
that deep.”
After about two weeks of teaching
the unit, she increasingly felt that her
students were not ready for the tasks,
and she ended her participation in the
research project. When asked about her
participation, Ms. B remarked that she
was glad she had done the unit and that
there was value in the unit. After seeing
how much her current on-grade-level
students’ understanding of proportions
and linear relationships had improved
compared to students in years past, she
decided to implement the unit with her
algebra students later in the year.
Teacher as Collaborator
Some teachers may choose to collaborate
with researchers. Such was the partner-
ship between author Fonger, a math
education researcher, and Ms. L, a ninth-
grade teacher. Motivated by NCTM’s
linking research and practice initiative
(Arbaugh et al. 2010), Fonger searched
for a collaborating teacher who expressed
interest in a problem of practice: What
are productive supports for algebra stu-
dents’ representational fluency in solving
problems involving linear equations with
computer algebra systems (CAS) and
with paper and pencil?
Initially, Ms. L and Fonger met to
develop learning goals for three math-
ematical ideas: equivalence of expres-
sions, understanding one-variable equa-
tions as relations between equivalent or
nonequivalent expressions, and solving
equations. They also planned a potential
instructional path for addressing these
goals through the use of CAS and paper-
and-pencil graphs, tables, symbols, and
words. They worked to find a balance
between the district-adopted curricu-
lum and alternative research-informed
approaches.
During the study, Ms. L and Fonger
collaborated on designing lessons:
Ms. L taught all lessons, and Fonger
supported instruction by identifying
students’ work to discuss and facilitat-
ing technology use. Ms. L and Fonger
debriefed after each lesson to discuss
what went well, what they could build
on for the next lesson, and how they
were meeting their goals. Ms. L’s use of
“warm-ups” and “exit tickets” was use-
ful for reflecting on students’ emerging
understandings.
Fonger shared results of her analy-
sis with Ms. L during and after the
study. They learned about the power
of encouraging students to “predict,
act, reflect, and reconcile” their activ-
ity with CAS and with paper and pencil
(Fonger 2014), an activity structure
that informed Ms. L’s teaching after the
research project.
Ms. L openly shared her professional
knowledge to support the collaboration.
Beyond this study, Ms. L used predic-
tion questions and reflection questions
to probe student thinking. Fonger also
gained practical knowledge of support-
ing secondary school students’ learning
with CAS, which has informed her work
with preservice teachers and her presen-
tation of this research.
Teacher as Collaborator
and Researcher
Mr. D, an eighth-grade science and math
teacher, assumed a researcher role in
a study conducted in partnership with
the same research team as Ms. B’s. Like
Ms. B, Mr. D had support to implement
the gears unit both before and during
instruction.
Intrigued by the premise of sup-
porting students’ algebraic reasoning
through quantitatively rich situations,
Mr. D expressed interest in being part of
the study because he wanted to “inform
his own understanding” of an inquiry
approach to teaching mathematics. He
wanted to investigate how he could use
a research-based unit in a response to
intervention (RtI) class while also work-
ing toward an advanced degree.
Mr. D was motivated to explore on
his own whether this research-based
unit could better support students’
learning and wondered whether a more
“constructivist approach to mathematics
and mathematical reasoning [may] be a
better fit for the RtI [class]” rather than
what he described as currently occurring
in RtI classrooms (i.e., “we’re just going
to kill them with worksheets”).
Mr. D took on an independent role as
a practitioner-researcher. While imple-
menting the unit, Mr. D analyzed his
students’ reasoning and reflected on his
instructional support. This investigation
informed his implementation of subse-
quent iterations of the unit with a dif-
ferent group of students. Mr. D pursued
questions that interested him, enabling
him to take ownership as a researcher.
LESSONS LEARNED
These stories tell of how experiences
and professional knowledge shape and
are shaped through engagement in
research. Across all forms of engaging
in research, teachers play important and
central roles and navigate a complex
web of potential benefits and concerns.
Even teachers who know the benefits
of engaging in research may be hesitant
to add more to already high demands
on their time. We suggest that teachers
advocate for their needs and discuss
potential direct and indirect benefits for
their local and broader context. Across
the vignettes we see examples of how
teachers managed their roles in research
and addressed issues that interested
Say you had two gears that had 5 teeth and 7 teeth,
instead of the 8 teeth and 12 teeth like we have been
working with. What would be the relationship between
the number of turns the small gear makes and the
number of turns the big gear makes in this situation?
Fig. 1 A sample gears task (from http://tinyurl.com/hy6drdm) highlights a gear ratio of 7/5.
464 MATHEMATICS TEACHER | Vol. 110, No. 6 • February 2017
them. For example, Ms. B decided to
shorten her participation to better meet
the needs of her class, Ms. L worked
with Fonger to shape the direction and
duration of the study, and Mr. D took
ownership of the research to address his
professional goals.
Another concern that teachers and
other school stakeholders may have is
that research is a one-way street that
helps researchers only. However, we
argue that both teachers and research-
ers (and the wider math education
community) may benefit from teach-
ers’ engagement in research. As the
vignettes demonstrated, taking part in
research gave the teachers access to
supports and provided researchers with
an opportunity to test and refine ideas.
Research can serve as a professional
learning opportunity for both teach-
ers and researchers. For example, both
Ms. L and Fonger benefited from daily
reflection in which they learned from
each other’s varied expertise. This pro-
fessional learning continued beyond the
research study. Although some benefits
to teachers may be indirect or take time
to occur, we encourage all teachers to be
open and explore where their participa-
tion in research might lead them.
A final approach to bolstering ben-
efits for teachers as well as researchers
is to frame and design research studies
that address shared problems of practice
(Arbaugh et al. 2010), a foundational
motivation for the collaboration between
Fonger and Ms. L. Finding common
interest among stakeholders is impor-
tant so that the goals of research and
the goals of teachers’ practice are well
aligned, as in the case of Mr. D’s collab-
orative and researcher roles.
HOW CAN TEACHERS GET
INVOLVED? A CALL TO ACTION
Teachers’ engagement in research aligns
with NCTM’s (2012) position linking
research and practice, in which the
importance of community and collabora-
tion is a priority. This position requires
that teachers and researchers work
together to figure out how best to meet
the needs of all stakeholders, including
teachers, students, administrators, and
parents. In this article we have tried
to show why teachers might engage in
research and what their engagement
might look like in practice.
We offer some possible next steps for
how to get involved in linking research
and practice:
Reflect—Take an inquiry-oriented
and reflective stance toward teaching
and learning (e.g., What stands out to
me as a problem of practice?).
Be open—Keep in mind that
research opportunities take on a vari-
ety of forms, styles, time scales, and
levels of commitment both within
and across schools and institutions
(e.g., How might new ideas impact
my practice?). Also see http://www.
researchandpractice.org.
Start local—Share your desire to
engage in research with colleagues,
school and district leaders, and math
education professors and researchers.
Share goals—Advocate for issues
that matter to you, your students, and
your local context.
Network—Attend conferences, con-
tact journal authors, or capitalize on
social media for support and inquiry
into issues (e.g., the Facebook® group
“Mathematics Education Researcher”
or MT’s Twitter chats on the fourth
Wednesday of each month).
Ultimately, linking research and prac-
tice involves collaboration among teach-
ers, researchers, and other stakeholders
in ways that are mutually beneficial.
There is also great value in teachers
doing independent research. We encour-
age you to consider the factors listed in
table 1 and ask yourself what your role
might be in the growing community of
teachers engaged in research.
REFERENCES
Arbaugh, Fran, Beth Herbel-Eisenmann,
Nora Ramirez, Eric Knuth, Henry
Kranendonk, and Judith Reed Quander.
2010. Linking Research and Practice:
The NCTM Research Agenda Conference
Report. Reston, VA: National Council of
Teachers of Mathematics. https://www.
nctm.org/uploadedFiles/Research_and_
Advocacy/Linking_Research_and_
Practice/Linking_Research_20100511.
pdf
Table 1 Guidance for Addressing Issues in Linking Research and Practice
Issue Guiding Question Role of Teacher(s)
Relevance to practice Is the research helping
teachers better under-
stand and/or address an
important problem of
practice related to cur-
riculum, instruction,
assessment, or student
learning?
Help uncover a problem
of practice that is shared
among stakeholders.
Level of participation How are stakeholders
involved in the design
and implementation of
the research?
Participate in and
negotiate your role and
level of participation.
Time commitment Is the research something
that is already a part of
teachers’ practice or an
add-on that requires
additional time?
Help decide the level of
commitment based on
the context and needs of
the district, school, and
classroom.
Contribution to research What is the role of
existing research in shap-
ing the study? How might
this work contribute to
math education?
Discuss how research
might inform practice,
and potential short-term
and long-term benefits
for both research and
practice.
Vol. 110, No. 6 • February 2017 | MATHEMATICS TEACHER 465
Fonger, Nicole. 2014. “Equivalent Expres-
sions Using CAS and Paper-and-Pencil
Techniques.” Mathematics Teacher 107
(9): 688–93.
Heid, M. Kathleen, Matthew Larson,
James T. Fey, Marilyn E. Strutchens,
James A. Middleton, Eric Gutstein,
Karen King, and Harry Tunis. 2006.
“The Challenge of Linking Research
and Practice.” Journal for Research in
Mathematics Education 37 (March):
76–86.
National Council of Teachers of Math-
ematics. 2012. “Linking Mathematics
Education Research and Practice.”
http://www.nctm.org/Standards-and-
Positions/Position-Statements/Linking-
Mathematics-Education-Research-and-
Practice/
Silver, Edward A. 2003. “Border Cross-
ing: Relating Research and Practice in
Mathematics Education.” Journal for
Research in Mathematics Education 34
(3): 182–84.
NICOLE L. FONGER,
nfonger@wisc.edu, is a post-
doctoral fellow of Mathemati-
cal Thinking, Learning, and
Instruction at the Wisconsin
Center for Education Re-
search at the University of
Wisconsin-Madison. She re-
searches students’ learning of
algebra and the nature of cur-
ricular and instructional sup-
ports for learning. Fonger
aims to cultivate links be-
tween research and practice
in her work as an educator.
LINDSAY REITEN, reiten@
wisc.edu, is a doctoral candidate at the
University of Wisconsin-Madison. A
former middle and high school mathe-
matics teacher and university lecturer,
she is interested in teacher education
and supporting teachers’ modification
and integration of technology activities.
SUSANNE STRACHOTA, sstrachota@
wisc.edu, is a doctoral candidate in Cur-
riculum and Instruction with a focus on
mathematics at the University of Wiscon-
sin-Madison. Before graduate school, she
taught high school math in Boston. She
researches algebraic reasoning, specifi-
cally how students generalize functional
relationships and justify those general-
izations. ZEKIYE OZGUR, zozgur@wisc.
edu, is a doctoral candidate at the Uni-
versity of Wisconsin-Madison. She is in-
terested in student cognition and the
ways in which teachers support students’
learning, particularly in the area of alge-
braic reasoning and proof.
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Article
Full-text available
A research-informed perspective on teaching equivalence of expressions within a CAS and paper-and-pencil learning environment in a ninth-grade algebra classroom is presented. A Task-Technique-Theory framework is a pedagogical tool used to organize two approaches: (a) develop representational fluency with multiple representations and (b) reconcile differences between symbolic representations.
Article
This article discusses the challenge of improving the interrelationships between research and practice in mathematics education, and it outlines actions being taking to respond to that challenge. The need for improvement is bidirectional. The practice of classroom mathematics teaching needs to be better informed by an understanding of the implications of existing bodies of research, and researchers need to learn more from the insights and knowledge of practitioners. Building on its series of initiatives designed to use research to guide mathematics teaching and learning, NCTM has made a new commitment to a flexible, nimble, and sustainable initiative that will strengthen the bidirectional link between research and practice. This initiative includes the development of Research Analyses, Briefs, and Clips (ABCs), research syntheses designed through collaboration of teacher leaders and researchers to inform instructional leaders and policymakers about research perspectives on critical issues of practice. (Contains 1 figure and 8 footnotes.)
Linking Research and Practice
  • Fran Arbaugh
  • Beth Herbel-Eisenmann
  • Nora Ramirez
  • Eric Knuth
  • Henry Kranendonk
  • Judith Reed Quander
Arbaugh, Fran, Beth Herbel-Eisenmann, Nora Ramirez, Eric Knuth, Henry Kranendonk, and Judith Reed Quander. 2010. Linking Research and Practice: The NCTM Research Agenda Conference Report. Reston, VA: National Council of Teachers of Mathematics. https://www. nctm.org/uploadedFiles/Research_and_ Advocacy/Linking_Research_and_ Practice/Linking_Research_20100511. pdf