Texas Mathematics Teacher Fall/Winter 2020 - 7

A Reflection on the Process and Value of Studying
Our Own Professional Growth
Michelle Branstetter & Dr. Beth Cory

Action research refers to the research that teachers
undertake in their own classrooms. It is an "ongoing
process of systematic study in which teachers examine
their own teaching and student learning through
descriptive reporting, purposeful conversation, collegial
sharing, and reflection for the purpose of improving
classroom practice" (Eisenhower National Clearinghouse,
2000, p. 18). The defining factors of action research are
the following (Loucks-Horsley, 2003): The teachers
themselves formulate the research questions. They
collect data to answer these questions. They use an
action-research cycle of planning, acting, observing, and
reflecting. They work collaboratively when possible. They
have access to outside sources of knowledge. Finally, they
document and share their research.
Action research is a powerful tool that K - 16 educators
can employ on a regular basis to inform their teaching.
Of course, it is unreasonable to suggest that teachers must
conduct in-depth research as part of each and every daily
lesson; however, I (the first author) can see the benefits of
action research in my classroom and in my development
as a teacher. While I implemented my action research
study with college algebra students, this model of action
research can be empowering for K - 12 teachers as well.
With the guidance and encouragement of a mentor, the
second author, I implemented an eight-step process,
which incorporated various aspects of action research
in order to design and study the impacts of a lesson on
students' understanding of inverse functions. The eight
steps were as follows: 1) Conduct a literature review
on the teaching of inverse functions; 2) Identify the
objectives for my lesson based on the literature review and
formulate my research question(s); 3) Devise an informed
lesson plan based on my literature review; 4) Administer
a pre-test and analyze the results; 5) Teach the lesson; 6)
Analyze the effectiveness of the lesson through post-test
data collection and analysis; 7) Reflect on the process;
and 8) Share the results. By carrying out this eight-step
process, I gained insight into effective methods of teaching
inverse functions and common misconceptions. As I
detail my journey of using this model of action research,
my hope is that the reader is informed and encouraged
to implement this practice in their own classroom. We
as teachers informally use action research daily as we
observe our students and adjust our teaching to their
needs, but providing a formal structure to this process can
yield richer findings that may allow us to learn far more
about student learning. Taking charge of my professional
development through action research put me at the
center of my own learning and enabled me to grow as an
educator (Norton, 2019).

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Background
I teach developmental mathematics at a local community
college. Students who are not prepared for College
Algebra are placed in a co-requisite course which I taught
for the past year with a colleague who taught the college
algebra component. This semester, our co-requisite class
was small-only 10 students. Our class was made up of
students who struggle with math anxiety and learning
disabilities, and are consistently low-performing in
mathematics. Since my students attended their college
algebra class the hour before my co-requisite class, they
came to my class with some understanding of inverse
functions already.
Step 1: Conducting a Literature Review
My first step in designing my lesson plan was to conduct
a literature review on the teaching of inverse functions.
Using the internet and my library's resources, I gleaned
research and practitioner articles related to the teaching of
inverse functions. I was aware that textbooks commonly
tell students to switch the variables x and y and then solve
for y to find the inverse function; this process is in the
textbook I use as well. However, throughout my literature
review, I saw that this focus on the symbolic procedure
of switching variables to derive inverse functions can
result in students misunderstanding the meaning of the
variables in inverse functions. For example, they may
think that the variable x represents the same quantity in
y = f (x) as it does in y = f -1 (x) (Wilson, Adamson, Cox, &
O'Bryan, 2011). I learned that students can overcome these
misunderstandings by instead focusing on the undoing
process of finding inverses (Benson & Buerman, 2007;
Lim, 2016; Maida, 1997; Teuscher, Palsky & Palfreyman,
2018; Wilson et al., 2011;). By seeing operations as
actions to "reverse" or "undo," students can better
understand how functions and their inverses describe the
relationships between variables. Teachers can easily model
this with intentional verbiage like opposite, reciprocal,
and arctangent (Benson & Buerman, 2007).
My literature review also revealed that finding inverses of
functions in context is helpful for students. A real-world
context is useful for highlighting the undoing process
because students can see more clearly how to solve for the
other variable when the variables have meaning (Wilson
et al., 2011). For example, students can convert Euros
to U.S. Dollars or explore Dolbear's Law, a function of
temperature based on cricket chirps (Connally, HughesHallett, & Gleason, 2015; Teuscher et al., 2018). Because
context gives meaning to the variables x and y, it can help
address students' misconception that x and y represent
the same quantities in y = f (x) and y = f -1 (x) (Wilson
et al., 2011). Additionally, context can help students
understand the graphical relationship between a function
and its inverse (Lim, 2016; Teuscher et al., 2018; Wilson
et al., 2011). Furthermore, with meaningful context,
students can make inferences about domain and range
and better understand how inverses relate over many
representations-functions, tables, equations, and graphs
(Teuscher et al., 2018).
Fall/Winter 2020

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Texas Mathematics Teacher Fall/Winter 2020

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http://www.brightcopy.net/allen/txmt/66-02
http://www.brightcopy.net/allen/txmt/66-01
http://www.brightcopy.net/allen/txmt/65-02
http://www.brightcopy.net/allen/txmt/65-01
http://www.brightcopy.net/allen/txmt/64-02
https://www.nxtbook.com/allen/txmt/64-1
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