A Reflection on the Process and Value of Studying Our Own Professional Growth After class, my students were still talking. Even though I decided to skip the closure activity, the discussion that took place after class ended up providing appropriate closure to the lesson. I believe their post-test explanations of the relationship between a function and its inverse would have been even more insightful if I could have let them talk first. They had questions about the pre- and post-tests, questions about inverses, and many opinions to share. I was barraged with questions from four students in particular: But why are there inverses? What was the answer to those problems on the pre-test? How could we possibly use this? I was able to point back to a student's comment from before: Inverses are looking at a relationship from a different perspective, and we frequently want to do this in the real-world. We talked for a few minutes on many real-world examples, including cricket chirps. One student even brought up the relationship of sales to profits, noting that by using the concept of inverses it would be easy to solve for a different variable (items sold or profits) to gain a different perspective. Most of my students have math anxiety and a general distaste for the subject, so it is always nice to show the simplicity of a concept that seems foreign. I showed my students a simple code that my seven-year-old son is using (as most kids do), that assigns numbers to the alphabet. The inverse is decoding that function. Though my students complained that question five, "Why switch the variables?" on the preand post-test was open-ended, they now seemed more comfortable with inverses and seemed to understand the relationship between a function and its inverse and the impact on variables. Step 7: Reflect My seventh step was to reflect on my lesson and my experiences. One of the goals of action research is to improve teaching practices and learning; the reflection process is vital to reaching this goal (Raymond & Leinenbach, 2000). During my reflection process, I found room for improvement. First, I recognized two revisions I could make to the pre- and post-test. I realized my wording on question three of the test, "Explain the relationship between inverses," was confusing (see Figure 2). Next time, to get more thoughtful and revealing answers from the students, I would ask, "From your knowledge of inverses and the graph above, provide three specific things you notice about the relationship between a function and its inverse." This question would also better connect to our in-class activity-patty-paper folding over y = x. Additionally, I might include questions measuring student attitudes toward learning mathematics (Norton, 2019). Second, next time I would like to incorporate a discussion about whether or not the inverse of Dolbear's Law is useful. Third, while I used the example of decoding as a simple real-world use of inverse functions, next time I would like to include more examples that anyone, even young children, could understand. For example, as an introductory activity, I might have each student write the directions from their home to school and then from school to home, or the steps for tying then untying their shoes. www.txmathteachers.org "Even my students were excited to be guinea pigs as we explored how to help them learn and retain nformation better." Fourth, instead of ending with a chaotic (though thoughtprovoking) whole class discussion, I will be sure to make time for the think-pair-share closure activity. This would allow students to cement their understanding by summarizing and explaining newly-learned ideas to their peers and to ensure that the final whole-class discussion would be more organized and efficient. So that our whole class discussion would revolve around their interests and understandings, I would revise my think-pair-share questions to ask, "Can you think of more ways that we use inverse functions in the real world? What do you still have questions about?" I was surprised how this lesson highlighted my students' struggles with retention. As I looked into the topic of a retention of understanding in students' learning, I found that many of the approaches I used in this lesson are beneficial for retention, such as making connections between mathematical representations, using realworld problems, and metacognitive strategies that make students aware of their own thinking processes (Foong & Ee, 2002). One area I can improve upon is reinforcing new concepts throughout the semester. Even my students were excited to be guinea pigs as we explored how to help them learn and retain information better. The next day, after they brainstormed ways to combat their difficulties with retention, they enthusiastically asked me if we could plan to review prior concepts before each class. Step 8: Share the Results Invigorated, encouraged, and challenged by the results, I look forward to repeating this process again in the upcoming semester as part of my own growth and professional development. I profited by informally sharing the activities and results of my lesson with my co-teacher and academic chair, and have already incorporated their feedback as I begin to take my lesson through a second iteration of the eight-step process. As I complete the second cycle, I would like organize a small group of college algebra teachers with whom to share my experiences and resources, and in turn, glean from their expertise. Ultimately, after completing a few more iterations, I hope to disseminate my successes to the mathematical community, including the larger body of mathematics instructors at my college and beyond. In conclusion, my students' learning and my teaching benefited from this practice of action research. This lesson was fun to research and plan, and it was well-received by my students. I enjoyed the process, and my students even enjoyed being such a big part of my research. It is important for my students to see that I am still learning and growing as an educator. Fall/Winter 2020 | 11http://www.txmathteachers.org

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