Professional Development Linking the Concept of Inverse in Abstract Algebra to Function
Inverses in the High School Curriculum
Melissa Mills
Oklahoma State University
Cara Brun
Oklahoma State University
Pre-service and in-service high school teachers often do not leverage their experience with
abstract algebra when interpreting the notation of inverse functions. For this study, we have
designed a professional development activity in which teachers can explore inverses in different
sets with different binary operations to elicit pseudo-empirical abstraction of the relationship
“element * inverse = identity.” We used a scripting task found in previous literature to measure
the impact of the activity on both the teacher’s understanding of inverses and how the teacher
would explain the inverse function notation to students. We claim that emphasizing the role of
the identity element when discussing inverses can help pre-service teachers overcome
misconceptions about inverse functions.
Key words: teacher education, abstract algebra, horizon content knowledge
Introduction and Literature Review
Students in high school or college algebra classes commonly misinterpret the notation
1
f 1 ( x) as
. From an advanced mathematical perspective, we understand that the root of
f ( x)
this issue is that the student is not attending to, or is not aware of, the fact that the notation
f 1 ( x) generally refers to the inverse with respect to function composition and not function
multiplication. Really, the fact that function composition is the binary operation at hand is
merely a convention and the student’s interpretation, while it does not conform to standard
mathematical interpretations, is not unreasonable.
As instructors who have had experiences with students at this level, we realize that the
student is probably not thinking about binary operations at all and is merely generalizing from
their experiences with negative exponents of real numbers. But how do high school teachers,
who have presumably had training in abstract algebra, think about such a situation? Zazkis &
Mamolo (2012) report that they informally asked ten in-service teachers how they might respond
to a student who has this confusion. Eight of the ten responded that the meaning of the □ 1 is
context-dependent. For example, a teacher may explain that the meaning of the symbol □ 1
changes depending on what it is “next to.” Only two of the ten teachers referenced an inverse
with respect to an operation.
Continuing this line of research, Zazkis and Kontorovich (2016) crafted a scripting task for
pre-service teachers to investigate how they would respond to a student’s question about the
symbol □ 1 in a classroom setting. The pre-service teachers were given the beginning of a script
between a teacher and a student (see Figure 1) with directions to finish writing the dialogue
between the teacher and student (or several students). Their analysis divided the 22 scripts into
two groups: those that explained the symbol □ 1 as always meaning inverse with respect to an
operation and those that explained that the symbol □ 1 changes meanings (inverse or reciprocal)
depending on the context. Fourteen of the 22 pre-service teachers fell into the latter category.
T: So today we will continue our exploration of how to find an inverse function for a given
function. Consider for example f ( x)  2 x  5 . Yes, Dina?
S: So, you said yesterday that f 1 stands for an inverse function.
T: This is correct.
S: But we learned that the power (-1) means 1 over, that is, 5 1  15 , right?
T: Right.
S: So, is this the same symbol, or what?
T:
Figure 1: Scripting Task (Zazkis & Kontorovich, 2016)
We have developed a professional development activity designed to help pre-service and inservice teachers make connections between inverses as they appear in the school curriculum and
inverses with respect to binary operations in a group theoretic context. To measure the impact of
our professional development activity, we utilized the same scripting task (Figure 1) as a preand post-test. From our results from the pre-test we will identify two misconceptions held by preservice teachers: an “opposite” scheme for inverse, and a “get to one” scheme for inverse. We
present evidence that group theoretic activities designed to help teachers pseudo-empirically
abstract the generalized property “element * inverse = identity” can help them move from a
“same-symbol different meaning” understanding to a “same symbol same meaning”
understanding. In particular, an emphasis on identifying the identity element with respect to an
operation before discussing inverses can help teachers overcome a “get to one” scheme.
Theoretical Perspective
The data from this study will be analyzed through the lens of Piagetian genetic epistemology
and von Glasersfeld’s radical constructivism. According to these theories, “…what [people] are
able to observe about the world is more dependent on what they already know – that is, on their
own special system of thinking - than it is on what actually exists” (Gallagher & Ried, 1981, p.
1). These structures of knowing are referred to as schemes or “units of generalized behavior (or
actions) that provide the basis for mental operations” (Driscoll, 2005, p.192). When a learner
encounters something that does not fit into an existing scheme, the learner must accommodate
this new object by expanding his existing scheme or by creating a new one (vonGlasersfeld,
1995). Piaget used theories of abstraction to describe how assimilation and accommodation can
occur. Pseudo-empirical abstraction can be defined as, “abstraction based on the observation of
perceptible results, with coordination drawn from activities exerted on objects, reflection on the
products of activity” (Ellis, 2016). The process of acting on objects, reflecting on these activities,
and coordination of those actions can cause perturbation, which can lead to the learner
accommodating their current scheme to encompass this new element.
Horizon content knowledge is one part of mathematical knowledge for teaching (Ball,
Thames, & Phelps, 2008). While some view horizon content knowledge as a connection between
the mathematics that students are doing and more advanced mathematics that the students will
encounter (Ball, et. al, 2008; Fernandez & Figueiras, 2014), others perceive it as connected to the
teacher’s knowledge of advanced mathematics (Wasserman, 2013; Zazkis & Mamolo, 2011).
Our working definition of HCK is the teacher’s advanced mathematical knowledge and the
threads that connect it to the students’ mathematical understanding that guide a teacher’s
planning and in-action instruction.
Task Design
The professional development tasks were designed to give teachers experience with several
different sets and operations. We chose three sets: the Real Numbers, the integers modulo 12 (the
numbers on a clock), and the set of functions. We chose the first two sets because the teachers
have experience doing calculations in both of those settings, and the latter set because it was
relevant to the discussion about inverse functions. We used colloquial language whenever
possible so that the activities are accessible to those who were not familiar with group theory.
For the real numbers, the participants were asked to begin with the binary operation of
addition. We gave the following definition: “The additive identity element is the element that
‘does nothing’ when you add it to another element,” and asked them to identify the additive
identity. Then we gave them the definition: “An additive inverse for an element in the set is the
element that you must add to get back to the additive identity,” and asked them to find additive
inverses of several different elements and then to write a statement describing additive inverses
in the real numbers in general. We then asked corresponding questions regarding multiplication.
For the set of integers modulo 12, we chose to use only the numbers appearing on the face of
a clock, because we wanted them to have experience with a set in which the additive identity is
represented with something other than zero. Since they may not be as familiar with computing in
this context, we first gave them some true statements (i.e. 8+9=5) and asked them to explain why
each one is true and to generate some of their own statements. Then they were asked to identify
the additive identity element, find additive inverses of several elements in the set, and write a
general statement describing additive inverses in this set. The tasks for clock multiplication were
similar, but before we talked about multiplicative inverses we gave them a completed
multiplication table and asked them to circle all of the times that the multiplicative identity
appears in the table. Then they were asked to identify some of the elements that had
multiplicative inverses.
At this point, we had the participants make a table in which they described the different sets,
operations, identity elements, and types of inverses that they had seen in the exercises. The goal
of this activity was for them to begin to abstract the notion: element * inverse = identity.
The final set was the set of functions. We began with function addition. We gave them a few
functions to work with and had them practice adding functions. We then asked them to identify
the additive identity function, and asked them to graph it. We had a discussion about why this
function was the additive identity, and asked them to find the additive inverses of the given
functions. Then we moved on to function composition, asking them to compose some given
functions and think about an identity function with respect to composition (most groups were
able to figure it out). We then proposed the function i( x)  x , and asked them to compose the
given functions with the function i. They then found inverses for the given (injective) functions
with respect to composition and composed the given functions with their inverses to see that that
composition results in the identity function i( x)  x .
Methods
Participants were recruited via email and word of mouth from a mathematics department at a
large Midwestern public university and a local high school. Participant groups included two
graduate students in mathematics who piloted the activities, three senior level pre-service
secondary teachers who were doing their student teaching, three sophomore level pre-service
secondary teachers, and three in-service teachers from a local school district. We implemented
the professional development in groups of two or three. The activity took approximately two
hours. First, we gave the participants the scripting task (Figure 1) and had them write a script
individually. Then we collected the scripts and went directly into the professional development
activities, which they worked on in groups. At the end, we gave their scripts back and had them
individually write a reflection on their script, identifying things they might change in their script
and areas in which their own understanding had changed as a result of the activities. All groups
were video recorded and their written work was collected.
Results
The results reported here will focus on an analysis of the scripts that the participants wrote
before they did the group theory activity, and the reflections on the scripting task that they wrote
after the activity. We found, similar to Zazkis’s previous work, that only a few of our
participants (3 out of 11) explained the notation □ 1 as referring to an inverse with respect to an
operation: both of the graduate students and one in-service teacher. The scripts also revealed that
several of the participants held misconceptions about inverses. We identified two schemes that
displayed underdeveloped understandings: the opposite scheme and the get-to-one scheme.
The opposite scheme is characterized by the vague idea that “inverse means opposite.” They
pay no attention to the identity element at all in this scheme. When asked to find an inverse of an
element, they will do something to make it an “opposite,” whether that is changing the sign or
finding the reciprocal, and often cannot move smoothly between the “two types of opposites.”
One participant fixated on the symmetry that additive inverses have on the number line. He then
had difficulty thinking about multiplicative inverses because he couldn’t construct a visual
representation. Others referred to the inverse of an operation: “An inverse means opposite, so the
opposite (if you’re thinking about multiplication) is division. So 51 means you would divide by
five. An inverse function is different, but it still essentially means opposite.”
The get-to-one scheme is characterized by the idea that “the inverse is what gets you to one.”
Participants who displayed this scheme began to attend to the role played by the identity element,
but incorrectly generalized that the identity is always 1. One participant, Ben, showed in his
script that composition of a function and its inverse results in x. Ben seemed to be unsatisfied
with the x , so he wrote that x  1x and drew a box around the coefficient 1. We interpret this
action as his way to assimilate this result into his get-to-one scheme (see Figure 2).
T: So, the inverse of an object, be it a fraction or a function, is the thing that turns the thing into
1.
S: Okay?
T: Remember when we found the inverse of a fraction? We got the reciprocal, which was the
flipped fraction and when we multiplied it with the original fraction we got one.
S: Yeah…
T: So, that is what the inverse fraction does. And I will show you exactly what I mean by this.
So, for f(x) the inverse is 12 x  52 . Now… plug it in 2( 12 x  52 )  5  x  5  5  x  1x
Figure 2: Ben’s Scripting Task Exhibiting a Get-to-One Inverse Scheme
Five of the six undergraduates and two of the three in-service teachers reported that they
think differently about inverses after the group theoretic activity. All but two mentioned that
their understanding about the identity changed. Dan said, “I had forgotten how to find the inverse
of a function because I hadn’t done them in so long and I was only taught the process, not the
reasoning.” Markus also reported, “Yes, this session was a good refresher for me on what inverse
was.” Sloan explained, “Before this session, I never really realized how many different types of
inverses were possible depending upon the operations and sets. Seeing the relationships between
the various inverses and correlating identities is enlightening…”
It is interesting that although they had found inverse functions in the past and had likely
encountered the computation f ( f 1 ( x))  f 1 ( f ( x))  x , the pre-service teachers didn’t know
how to interpret this result. Jane said, “I now think of inverses as connected to the identity” and
Kim notes “the composition of functions having the identity x is very interesting and I hadn’t
thought of the idea of the operation actually having an identity.”
Dan, Sloan, and Ben all showed evidence of a get-to-one scheme in their first scripting task.
After the exercises, both Sloan and Ben said that they had never thought about different types of
identity elements and had expanded their understanding of inverses.
Six of the eleven participants said that they would change how they talked about inverse
functions to students. Sloan said, “I would make an effort to clarify which operation [and]
identity I would use as I explain.” Kim said, “The word ‘identity’ would be used much more.
However, the same concept regarding ‘neutralizing’ the function I believe I would continue to
use. When the composition [gives] x, the composition is neutral.”
Discussion
Previous research has shown that many pre-service teachers do not leverage their
understanding of abstract algebra when interpreting the notation □ 1 . Because of teacher
shortages, it is quite possible that many teachers have never had a course in abstract algebra at
all. Thus, professional development activities designed to broaden teachers’ mathematical
experiences can be beneficial. Zazkis and Mamolo (2011) suggest that a teacher could make use
of her horizon content knowledge by explaining the situation in terms of an inverse with respect
to an operation. Our work contributes by incorporating professional development tasks aimed to
help teachers make these connections. Our tasks emphasize the role of the identity element when
thinking about inverses. This emphasis seems to help teachers overcome the misconception that
an inverse is the element that “gets you back to 1.” Having the teachers experience finding
identity elements and inverses in different sets with different operations helped many of them to
generalize the relationship “element * inverse = identity” which can lead them to interpret the
notation □ 1 as meaning an inverse in general. This study is limited by the small number of
participants, however, the goal of this work was to design and conduct a professional
development activity focusing on a particular connection between an advanced mathematical
concept and secondary school mathematics in a way that teachers can leverage in their
classrooms. Future research is needed to continue the development of such professional
development activities.
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