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). www.txmathteachers.org 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 | 7http://www.txmathteachers.org

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