Stages of development for the concept of inverse in abstract algebra
John Paul Cook
Oklahoma State University
Rosaura Uscanga
Oklahoma State University
In this study, we conducted a teaching experiment with two students to investigate the development
of a generalized concept of inverse in abstract algebra. In particular, we document the stages through
which the students’ reasoning progressed, initiating with an understanding of the additive inverse of
an element as the result of a procedure applied to that element, and concluding with a generalized
understanding of inverse that was broad enough to identify instances of inverses in various and
unfamiliar algebraic structures. Of critical importance was the development of and coordination with
a corresponding concept of identity.
Key words: student thinking, abstract algebra, Realistic Mathematics Education, teaching
experiment
Introduction
The concept of inverse is prevalent in abstract algebra and is used to define such foundational
algebraic structures as ring and field. Inverses are a key concept in secondary algebra as well and
have been stated as a direct connection that pre-service teachers should make between their advanced
coursework in abstract algebra and the algebra they will be teaching (CBMS, 2001). Accordingly, as
abstract algebra is intended to be “the place where students might extract common features from the
many mathematical systems that they have used in previous mathematics courses” (Findell, 2001, p.
12), the inverse concept – like the other properties that characterize rings and fields – is certainly
familiar to abstract algebra students from previous courses. What makes inverse unique, perhaps, is
the wide variation it exhibits across different contexts. Consider, for instance, the following
examples of inverse from the undergraduate curriculum that reappear in abstract algebra:
𝑎
𝑏
 Inverse of a complex number: if 𝑎 + 𝑏𝑖 is nonzero, then (𝑎 + 𝑏𝑖)−1 = 𝑎2+𝑏2 − 𝑎2+𝑏2 𝑖.
 Inverse of an integer modulo 𝑛: if gcd(𝑎, 𝑛) = 1, then 𝑎𝜙(𝑛)−1 ≡ 𝑎−1 (mod 𝑛).
1
𝑎 𝑏
𝑑 −𝑏
 Inverse of a matrix: if 𝐴 = [
] and 𝑎𝑑 − 𝑏𝑐 ≠ 0, then 𝐴−1 = 𝑎𝑑−𝑏𝑐 [
].
𝑐 𝑑
−𝑐 𝑎
These few examples are widespread and varied, each largely dependent on its mathematical context.
Without coordinating the binary operation with these examples of inverse and their respective
identities, there are few surface-level indications that these are all direct implementations of the same
overarching concept. The key notions of the generalized inverse concept – that the inverse of a given
element is the element that, when combined with the given element, yields the identity – might be
obscured within specific examples of inverses. Though students would likely recognize such
examples of inverse within each context, research suggests that it might be challenging to recognize
commonalities amongst these examples that reflect a generalized inverse concept. In particular,
familiarity with algebraic properties prior to abstract algebra does not guarantee proficiency with
these properties in abstract algebra (e.g. Larsen, 2010). Additionally, when working with examples,
students attempt to overgeneralize in order to reduce the level of abstraction (e.g. Hazzan, 1999) and
struggle to focus their attention on productive facets of the example structure (e.g. Simpson &
Stehlikova, 2006). Thus, it is unclear if and how introductory abstract algebra students might
identify these examples as instances of a general notion of inverse. This observation motivated our
central research question: how might students in abstract algebra come to understand a generalized
notion of inverse in a way that enables them to recognize instantiations of inverse within various
structures? We propose that it is the coordination of the inverse concept with a corresponding
concept of the identity that is critical to using and recognizing inverses in abstract algebra.
Analyzing results from a ring theory teaching experiment, we document three stages of development
for the inverse concept through which two undergraduate students progressed as we guided their
reinvention of the concept of ring.
Literature
Much of the research pertaining to inverses provides characterizations of students’ understanding
of particular kinds of inverses, such as the reciprocal of a rational number (e.g. Tirosh, 2000), inverse
functions (e.g. Even, 1992), and inverse matrices (e.g. Wawro, 2014). We focus on those papers
describing student activity with inverse (1) across multiple contexts and (2) in abstract algebra.
Use of the Superscript −𝟏 Symbol Across Multiple Contexts
Consistent with our observation above that the varied manifestations and contexts in which
inverses arise might prevent students from noticing that each is an instance of the same underlying
concept, Zazkis and Kontorovich (2016) pointed out a discrepancy related to potential interpretations
of the superscript −1 symbol in a study of students’ lesson scripts. For example, 5−1 can be
interpreted as the reciprocal of the rational number 5, whereas 𝑓 −1 refers to the inverse of the
function 𝑓, a concept that itself admits various interpretations. For example, students might interpret
𝑓 −1 algebraically (switching 𝑥 and 𝑦 and solving for 𝑦), graphically (reflecting the graph of 𝑓 about
the line 𝑦 = 𝑥), or as the result of reversing or undoing the function 𝑓 (Carlson & Oehrtman, 2005;
Even, 1992). In accordance with these contextually varied interpretations, many students (15 of 22)
described the superscript −1 as “the same symbol applied to different, unrelated ideas” (p. 103). All
other students viewed the superscript as having “different but related” (p. 107) meanings, each
depending on the context in which it is used. One of these students wrote that “we have an example
of using the same symbol or same word in different contexts. Both are inverse … a multiplication
inverse means 1 over, and an inverse function is when you switch x any y.” There are no substantial
commonalities amongst her descriptions; her use of the word “inverse” seems to be the only link
between “1 over” and “switch x and y.” Notably absent from any of the above interpretations of
inverse is any mention of an identity element.
While Zazkis and Kontorovich noted the extent to which the students attempted to account for
the apparent variation in meaning across different instances of inverse, their study left for future
research the question of how students’ might develop a generalized, coordinated understanding of the
concepts of inverse and identity that enables them recognize instantiations of these concepts within
various structures.
Students’ Use and Understanding of Inverse in Abstract Algebra
Even though inverses are a familiar concept from school algebra, there is reason to believe that
employing the inverse axiom in abstract algebra can be challenging for students. Research on
student thinking about inverses delineates two lines of thinking that might explain these challenges.
First, students might not carefully attend to closure and the binary operation(s) of an algebraic
structure (Nardi, 2001). This could lead to determining inverse pairs by overextending the familiar
operations of the real numbers (Hazzan, 1999) or identifying a potential inverse for an element but
not verifying that the inverse is in the given set (Brown et al, 1997, p. 207). Second, students might
not carefully attend to the algebraic properties on which their reasoning hinges. Indeed, there is
evidence that students fail to verify or directly acknowledge inverses (Brown et al., 1997), even if
they are implicitly invoking inverses by cancelling adjacent inverse pairs in a calculation (Larsen,
2013).
Theoretical Perspective
We adopted the perspective of Realistic Mathematics Education (RME) to guide our inquiry into
how students might develop a generalized notion of inverse because of its view that mathematics can
and should “link up with the informal situated knowledge of the students” (Gravemeijer, 1998, p.
279) in order to “enable them to develop more sophisticated, abstract, formal knowledge” (ibid). We
leveraged the RME principle of guided reinvention, the goal of which is to design tasks that
encourage students to “formalize their informal understandings and intuitions” (Gravemeijer, Cobb,
Bowers, & Whitenack, 2000) en route to developing formal mathematical concepts themselves. The
reinvention principle shaped both the overarching objective of the teaching experiment (to
investigate how students might be guided to reinvent the concepts of ring, integral domain, and field)
and also our specific objectives in this paper (to investigate how students might develop generalized,
coordinated concepts of identity and inverse from their own intuitive understandings).
We propose that a robust understanding of the inverse concept necessarily involves coordination
with the relevant binary operation and a generalized concept of identity. This is based upon
suggestions in the literature (e.g. Brown et al., 1997) and also the hypothesis that such a coordination
elucidates the common inverse structure across different binary operations and algebraic structures
(as opposed to comparing the formulas or procedures to compute inverses across contexts). By
generalized concept of the identity, we mean that a student should understand that an identity, if it
exists, can only be conceptualized with respect to a particular binary operation (this is particularly
important when studying rings, which have 2 binary operations and, therefore, two potential identity
elements). An identity element, then, if it exists, is an element of that structure such that, when
combined (in any order) with any element of that structure (under the relevant binary operation),
leaves that element unchanged. That is, given an algebraic structure 𝑅 with binary operation ∗, an
identity is an element 𝐼 ∈ 𝑅 for which 𝑎 ⋅ 𝐼 = 𝑎 = 𝐼 ⋅ 𝑎 for any 𝑎 ∈ 𝑅. This understanding should be
broad enough to accommodate instances of identity elements beyond the familiar 0 and 1 (such as an
identity matrix or the element 8 in the ring 4ℤ12 ). It should also be broad enough to employ in
symbolic arguments for a general (unspecified) algebraic structure. Accordingly, a generalized
concept of inverse depends not only upon a particular binary operation but also the aforementioned
understanding of identity. Moreover, an inverse should be understood as an inverse element of the
algebraic structure in question (as opposed to an inverse operation), and therefore the student needs
to understand that, if an element 𝑎 of an algebraic structure 𝑅 with binary operation ∗ has a
corresponding inverse element 𝑎−1 , then 𝑎−1 is also an element in 𝑅, and combining 𝑎 with 𝑎−1 (in
any order) yields the identity, i.e. 𝑎 ∗ 𝑎−1 = 𝐼 = 𝑎−1 ∗ 𝑎. This understanding should be broad
enough to accommodate unfamiliar instances for which there is not a canonical procedure or formula
that determines the existence of an inverse or produces the inverse element itself, and should also be
broad enough to employ in symbolic arguments for an unspecified algebraic structure. A student
must also understand that some elements in certain algebraic structures might not have a well-defined
inverse, and that some elements might have an inverse that exists only outside the algebraic structure
under consideration.
Freudenthal (1973), the founding father of RME, provided suggestions about how students might
be guided to reinvent such generalized notions of algebraic structure. Arguing that there is a
hierarchy of levels of mathematical activity, he noted that “the means of organization of the lower
level become a subject matter on the higher level” (1973, p. 123). He defended this claim by
describing the historical development of the concept of group. In the 19th century, he noted, groups
were only implicit in mathematicians’ intuitive reasoning, which led to explicit formulation of the
group properties and, eventually, to the axiomatic abstraction of the definition of group. This
characterization provides an operational, hypothesized model of how a student might abstract
algebraic properties by leveraging his/her own activity:
(1) The property is implicit in the student’s activity with an example structure;
(2) The property appears explicitly as a student’s general description of his/her activity with
the structure;
(3) The student uses the property as a lens to classify and investigate other structures.
Larsen (2013) verified that a learning trajectory proceeding in such a manner could indeed be
leveraged to support students’ reinvention of the group concept, and earlier studies provided similar
evidence for the efficacy of this approach for reinventing rings (Cook, 2012). In this study, we
hypothesized that students would be able to achieve the desired, coordinated understanding of
identity and inverse by engaging in a task sequence informed by Freudenthal’s characterization of the
emergence and gradual formalization of an algebraic property in a student’s activity.
Methods
We adopted the teaching experiment methodology (Steffe & Thompson, 2000) in order to
construct models of students’ thinking about the concepts of inverse and identity. We also sought to
discern how their understanding of these concepts might evolve to a coordinated, generalized
understanding of these concepts as they engage in mathematical activity in response to our teaching
actions. It should be noted that such explanatory models of student thinking “may not, and probably
cannot, account for students’ mathematics” (Steffe & Thompson, 2000, p. 268), and instead reflect a
researcher’s best attempts to provide a rational frame of reference for a student’s observable
behaviors in response to a mathematical scenario. Thus, we deemed the models of student thinking
that we constructed as valid insofar as they provided a viable explanation for students’ utterances and
written work.
Participants and Data Collection
Two undergraduate mathematics majors – Josh and Meagan (pseudonyms) – at a large
Midwestern research university participated in this study. Both Josh and Meagan were juniors and
had completed courses in linear algebra and number theory but had no prior exposure to concepts in
abstract algebra. We selected these students for participation not only because their respective stages
of mathematical preparation and abilities (as indicated by their own reports of their mathematical
course experience and by a survey administered prior to participation) were suitable for this study,
but also because of our perception of their enthusiasm for participating and willingness to openly
articulate their mathematical thoughts. The teaching experiment consisted of 7 sessions of up to 2
hours apiece. The first author served as the interviewer and the second author served as an observer.
We conducted two sessions per week to allow sufficient time for ongoing analysis (explained below).
Each session convened on the campus of the students’ institution. We recorded students’ verbal
utterances and written work using Microsoft Surface tablets, which recorded video-like capture of
their written work with synchronized audio.
Data Analysis
We employed two methods for data analysis: ongoing and retrospective. The ongoing analysis
occurred during and between sessions as we attempted to construct more stable models of student
thinking in situations involving the concepts of inverse and identity. Ongoing analysis involved the
first author’s intuitive responses to students’ activity in the form of asking questions or introducing
tasks that would provide additional insight into their thinking and test the viability of our proposed
explanations for their mathematical behavior. The models constructed during sessions were
necessarily fluid and in need of additional refinement. Therefore, between sessions, we both viewed
the videos from the previous session and, in particular, looked for evidence (in the form of the
students’ observable behaviors) that would either confirm or disconfirm the hypothesized models that
we had developed during that session. Any additional hypotheses that arose would inform the
instructional tasks administered in the next session. At the conclusion of data collection, we
engaged in a retrospective analysis (Steffe & Thompson, 2000) in order to look for conceptual
change across the entire teaching experiment.
Results
While it is beyond the scope of this proposal to comprehensively document the evolution of Josh
and Meagan’s understanding of the inverse concept throughout the entire teaching experiment, in this
section we simply describe the first few stages of development:
Stage 0: Entrance Survey
Prior to selection for this study, Josh and Meagan, working independently of each other,
answered the following questions as part of an entrance survey (the questions that appear here are
only those that involved inverses; they have also been renumbered for easier reference):
 Consider the set {0, 1, 2, … ,11} with addition and multiplication modulo 12. To familiarize
yourself with this set and its operations, construct an operation table for both addition and
multiplication modulo 12, and then answer the following questions.
o Q1: Does this set contain an identity with respect to addition? Explain.
o Q2: Does this set contain an identity with respect to multiplication? Explain.
o Q3: Does this set contain an additive inverse for 4? If so, identify it. If not, explain.
o Q4: Does this set contain an additive inverse for 5? If so, identify it. If not, explain.
o Q5: Does this set contain a multiplicative inverse for 4? If so, identify it. If not,
explain.
o Q6: Does this set contain a multiplicative inverse for 5? If so, identify it. If not,
explain.
 Consider the set {0,4,8} with addition and multiplication modulo 12. To familiarize yourself
with this set and its operations, construct an operation table for both addition and
multiplication modulo 12, and then answer the following questions.
o Q7: Does this set contain an identity with respect to addition? Explain.
o Q8: Does this set contain an identity with respect to multiplication? Explain.
o Q9: Does this set contain an additive inverse for 4? If so, identify it. If not, explain.
o Q10: Does this set contain a multiplicative inverse for 4? If so, identify it. If not,
explain.
The results from this survey indicate that Josh and Meagan’s understanding of inverse seemed more
tied to notions of inverse as an operation (i.e. the inverse of addition is subtraction) or a procedure
(i.e. take the reciprocal), as their additive inverses were initially negative numbers (Q3, Q4, and Q9),
the multiplicative inverses were reciprocals (Q5, Q6, Q10). It appears that they both strongly
associated the number 1 with multiplicative identity, and, moreover, there was no evidence that they
coordinated these inverses with the respective identities. Had they been coordinating with an
identity, we would have expected their explanation in response to Q10, for example, to focus on their
assertion in Q8 that a multiplicative identity did not exist in the set.
Q1
Q2
Yes; Yes;
0
1
Q3
Yes;
-4=8
Q4
Yes;
-5=7
Meagan Yes; Yes;
0
1
Yes;
-4=8
Yes;
-5=7
Josh
Q5
No; 1/4
not in
set
No; 1/4
does
not
make
sense
Q6
No; 1/5
not in
set
No; 1/5
does
not
make
sense
Q7
Yes; 0
Yes; 0
Q8
No; 1
not in
set
No; 1
not in
set
Q9
Yes;
-4=8
Yes;
-4=8
Q10
No; 1/4
not in
set
No; 1/4
does
not
make
sense
Stage 1: Additive Inverse of 𝒂 as −𝟏 ⋅ 𝒂
Notions of inverse first appeared in the teaching experiment in response to the first task, which
prompted them to use addition and multiplication modulo 3 to determine the total number of distinct
elements that could be generated, starting from 1 + 2𝑖. The task permitted them to add and multiply
1 + 2𝑖 by itself as many times as desired; they could also add and multiply results of any calculations
they performed. The idea behind this task was that the algebraic properties that define a field would
emerge implicitly in their activity as they acclimated to the elements and operations of what we knew
to be the field ℤ3 [𝑖]. Soon after they generated the element −1, Meagan concluded that the existence
of −1 enabled them to include the “opposite” of every element in their list because they could
multiply any element by −1:
Meagan:
Josh:
Meagan:
Josh:
So, like … if we … Isn’t negative one, is negative one one of our numbers?
Yeah, because since negative one is one of our numbers …
It’s always going to be there.
Every one will have an opposite.
This characterization persisted unchanged into session 3, at which time they were completing
operation tables to keep track of their calculations. In doing so, Josh and Meagan noticed that the
addition table was a Latin Square (every element appears exactly once in each row and column).
Following Larsen’s (2013) recommendation regarding the role of proof in furthering students’
reasoning, we prompted Josh and Meagan to prove this result, which essentially amounts to
justifying (1) existence (that each equation of the form 𝑎 + 𝑥 = 𝑏 has a solution) and (2) uniqueness
(that 𝑎𝑥 = 𝑎𝑏 ⇒ 𝑥 = 𝑏). Their existence proof involved the cancelling of additive inverses. At this
point, we prompted Josh and Meagan to articulate their “opposite” rule. Josh wrote:
Figure 1: Josh and Meagan’s initial characterization of additive inverse.
Notice that their demonstration of their use of this rule involves a cancellation, but does not
acknowledge the role of the identity (or any of the other algebraic properties involved). We asked
them to develop a “test for inverses” so that we could determine if cancelling featured prominently in
their characterization of inverses or if it was just a consequence they derived from a stable −1 ⋅ 𝑎
conceptualization:
Meagan:
Multiply them by the negative, and then according to our mod three, um,
determine if they’re, equal. And you can do that with any equation.
Thus, their operational characterization of additive inverse still centered on −1 ⋅ 𝑎.
Stage 2: Beginnings of a Coordination of −𝟏 ⋅ 𝒂 with the Additive Identity
It was not until Josh and Meagan proved the “uniqueness” component of the Latin Square
property that the additive identity began to emerge in their discussions of additive inverses.
Figure 2: Josh and Meagan’s proof of uniqueness for the Latin Square property.
Upon writing out the proof (Figure 2), I asked them to justify that 𝑎 + (−1 ⋅ 𝑎) = 0:
Researcher:
Josh:
Researcher:
Meagan:
Josh, what you’ve written there seems to be minus one times a?
Right.
Ok, how do you know that minus one times a gives you zero? Is that true?
I mean, it might be another rule. An inverse plus itself should always equal
Josh:
Right?
Yeah.
zero.
We asked Josh and Meagan to update their statement of the inverse rule:
Figure 3: Josh and Meagan’s statement of their additive inverse rule includes 0.
This concludes the results section, which is admittedly brief due to space constraints. The conference
presentation (pending the acceptance of this proposal) will provide a much more comprehensive and
detailed analysis of Josh and Meagan’s understanding of inverse and identity and how it evolved
throughout the teaching experiment.
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