Texas Mathematics Teacher Fall/Winter 2020 - 11

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

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Texas Mathematics Teacher Fall/Winter 2020

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