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Citation: Veith, J.M.; Bitzenbauer, P.;
Girnat, B. Teaching and Learning
Group Theory: Empirical Insights
Facilitated by the CI2GT. Educ. Sci.
2022,12, 516. https://doi.org/
10.3390/educsci12080516
Academic Editors: James Albright
and Fien Depaepe
Received: 27 May 2022
Accepted: 26 July 2022
Published: 28 July 2022
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education
sciences
Article
Exploring Learning Difficulties in Abstract Algebra: The Case
of Group Theory
Joaquin M. Veith 1,* , Philipp Bitzenbauer 2and Boris Girnat 1
1Institut für Mathematik und Angewandte Informatik, Stiftungsuniversität, 31141 Hildesheim, Germany;
girnat@imai.uni-hildesheim.de
2Physikalisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany;
philipp.bitzenbauer@fau.de
*Correspondence: veith@imai.uni-hildesheim.de
Abstract:
In an earlier contribution to Education Sciences we presented a new concept inventory to
assess students’ conceptual understanding of introductory group theory—the CI
2
GT. This concept
inventory is now leveraged in a pretest-post-test design with
N=
143 pre-service teachers to enrich
this body of work with quantitative results. On the one hand, our findings indicate three recurring
learning difficulties which will be discussed in detail. On the other hand, we provide a summative
evaluation of the Hildesheim Teaching Concept and discuss students’ learning gain in different
sub-domains of group theory. Together, the results allow for an empirical perspective on educational
aspects of group theory and thus bridge the gap between qualitative and quantitative research in this
field which constitutes a desideratum to date.
Keywords: algebra; group theory; secondary school; mathematics education
1. Introduction
1.1. Literature Review
Over the past 10 years, research into educational aspects of abstract algebra has gained
increasingly more traction. With the works of Wasserman et al. [
1
–
4
] and Melhuish [
5
–
7
],
various studies have shown the benefits of learning abstract algebra and group theory in
particular, as well as different pitfalls for students—an overview of the research results in
this regard is presented in [8].
For example, from a subsample of
N=
286 undergraduate students it was derived
by Melhuish and Fagan [
6
] that learners tend to conflate and overgeneralize basic proper-
ties such as associativity and commutativity. This finding was substantiated both by (a)
Larsen [
9
] who found with the method of teaching experiments on
N=
5 undergraduate
mathematics students that associativity and commutativity have a potential to lead to
many errors in algebra education as both properties are related to order in ways that are
often not carefully distinguished and by (b) Zaslavsky and Peled [
10
] who in the context of
an in-service professional development course for
N=
67 mathematics teachers showed
that the participants felt the properties of associativity and commutativity were logically
dependent. Another learning difficulty was identified by Veith et al. [
11
], who used an
acceptance survey with
N=
9 secondary school students to identify linguistic preconcep-
tions which posed learning difficulties regarding binary operations and isometries of the
equilateral triangle.
On a more positive note, the concepts of abstract algebra also provide beneficial
opportunities for affective learner characteristics and mathematical stances in general.
Specifically, with a mathematics for teachers course presented in [
1
], the introduction to
algebraic structures such as groups exhibited a positive impact on
N=
12 K-12 teachers’
beliefs and intended practices. Moreover, in a further study by Even,
N=
15 mathematics
teachers participating in an advanced mathematics course voiced the opinion that dealing
Educ. Sci. 2022,12, 516. https://doi.org/10.3390/educsci12080516 https://www.mdpi.com/journal/education
Educ. Sci. 2022,12, 516 2 of 21
with the concepts of algebraic structures helped them develop more knowledge of the
nature of the discipline itself [12].
It can be concluded that the studies in abstract algebra education research so far have
two striking similarities, namely that (a) the body of research consists mainly of qualitative
investigations and (b) the samples are mostly comprised of secondary school teachers
and mathematics majors. The first statement can be ascribed to a lack of respective test
instruments. Therefore, to overcome this lack of quantitative insights into student learning
of abstract algebra, two concept inventories were developed in recent times—the GTCA
(Group Theory Concept Assessment) by Melhuish [
5
] and the CI
2
GT (Concept Inventory
for Introductory Group Theory) by Veith et al. [
13
]—their main difference being the target
group. While the GTCA is developed for mathematics majors the CI
2
GT is developed for
students who “only enter this area on a superficial level” [
13
] (p. 2). Nonetheless, with
the CI
2
GT operationalizing conceptual understanding of introductory group theory it is
now possible to study group theory education from an empirical perspective. Hence, in
this paper we make use of the CI
2
GT to enrich this body of research with quantitative
insights, tackling, among other aspects, the often described problems tied to associativity
and commutativity with new methods.
As for statement (b) the questions arise whether the benefits tied to dealing with con-
cepts of abstract algebra as well as the reported learning difficulties can also be observed
in samples consisting of primary school teachers. In this regard, Chick and Harris [
14
]
found in their 2007 study that the
N=
14 examined primary school teachers displayed
overall poor sense of how their mathematical contents build the foundation of later al-
gebra. They concluded that “for some teachers, this limited perspective may be due to
their own educational history” [
14
] (p. 133) and further demanded that “more needs to be
done to help teachers understand what the key aspects are and how they contribute to the
understanding that needs to be developed in the secondary school.” [
14
] (p. 133). Thus,
in order to better identify which content domains precisely profit from abstract algebra
concepts and how they may be transformed, Wasserman [
2
] explored the potential abstract
algebra offers for school mathematics instruction across the entire spectrum, ranging from
elementary school to high school content areas. In the case of elementary schools, it was
elaborated as to how inverse operations and arithmetic properties such as commutativity
and associativity manifest themselves in primary mathematics education. To this end,
Wasserman concluded that “transforming teachers’ knowledge regarding these content
areas through understanding more abstract ideas about algebraic structures likely is ac-
complished through fostering reflection on their connections to and their importance for
more elementary content in school mathematics.” [
2
] (p. 42), building on the the CCSS-M
(Common Core State Standards in Mathematics, cf. [
15
]). In the context of elementary
school mathematics, this reflection may be facilitated by group theory, which after all is
precisely the generalization of arithmetic properties of binary operations and inverses.
Thus, as teachers “need to know the mathematics they are teaching, as well as how to
teach it” [
16
] (p. 5), we argue that great potential may come with introducing pre-service
primary teachers to basic notions of group theory as they can be used to build upon in
geometry (e.g., Dihedral groups) and arithmetic (e.g., Cyclic groups) as well as in linear
algebra courses, where further algebraic structures are studied such as fields and vector
spaces.
In addition, in our prior research contributions (cf. [
8
,
11
,
13
]) we derived desiderata
regarding abstract algebra education from the literature (cf. [
12
,
17
–
22
]). This article con-
tributes empirical evidence regarding these desiderata, for example by asking (for our
research questions see Section 2):
• How should instructional elements be designed when teaching group theory?
•
Do learning difficulties found with qualitative methods also present themselves in a
quantitative setting? If so, which difficulties can be observed and how pronounced
are they?
Educ. Sci. 2022,12, 516 3 of 21
The investigation of these questions required analysis methods addressed in Section 3.3
which we adopted from physics education research – namely the normalized gain expressed
by Hake’s g[23] and Hasan et al.’s Certainty of Response Index CRI [24].
1.2. The Hildesheim Teaching Concept
The Hildesheim Teaching Concept is a teaching concept focusing on introductory
group theory elements aimed at secondary and undergraduate mathematics education.
Details regarding the Hildesheim Teaching Concept are presented in our earlier contribu-
tion [8]. Hence, we only outline the main aspects here.
The curriculum is the result of an in-depth literature review where viewpoints from
the new math era of the 1960s (cf. [
25
]) were merged with viewpoints from contemporary
works on abstract algebra education. In particular, the development process was guided
by Larsen’s TAAFU project (Teaching Abstract Algebra for Understanding), presented
in [
26
,
27
]. The main differences lie in (a) “exploring groups via symmetries” [
8
] (p. 12)
in a hands-on way using haptic learning material and thus translating the introduction
presented in [
26
] into a more physically engaging process, and (b) adjusting the content
depth to be more in line with the curricula pre-service primary school teachers are presented
with. For example, cosets, quotient groups, normal subgroups and kernels are cut. From
this, the multifaceted perspectives have been synthesized into a coherent teaching trajectory
spanning three units (of 90 minutes each) across multiple aspects of introductory group
theory (cf. Section 3.2). From mathematics education literature, it was derived that this
introduction should be guided by three groups specifically – the dihedral group
D3
of the
regular triangle, the dihedral group
D4
of the square and cyclic groups
Zn
(cf. [
28
]). We
refer readers unfamiliar with these mathematical concepts to the reference handbook [
29
]
where these notions are explained in rigorous detail.
The Hildesheim Teaching Concept has been subject to a formative evaluation using the
method of probing acceptance among students in a laboratory setting (cf. [
11
]). The results
of this pilot study suggested that the instructional elements of the concept are well accepted
by learners. In addition, the instructional elements were found to be potentially conducive
to fostering algebraic thinking (cf. [
11
]). These findings are now to be complemented and
substantiated in the course of a summative evaluation that...
•· · ·
examines the impact of the curriculum on students’ development of conceptual
understanding of group theory, and;
•· · ·
that explores possible learning difficulties that appear regarding introductory
group theory.
2. Research Questions
As elaborated in Section 1, we aim to clarify the following research questions with this
contribution:
RQ1:
Do learners achieve an adequate conceptual understanding of introductory group
theory when instructed with the Hildesheim Teaching Concept and which concepts
post the most hurdles for learners?
RQ2: Which learning difficulties regarding introductory group theory can be identified?
We elaborate on the operationalization of adequate conceptual understanding in Section 3.3.
3. Methods
3.1. Study Design and Samples
To clarify the research questions, two studies were conducted:
1.
An expert survey with
N=
9 experts from mathematics and mathematics education.
2.
A quantitative evaluation of the Hildesheim Teaching Concept with
N=
143 pre-
service teachers.
The quantitative evaluation was conducted as part of a two-week group theory pro-
gramme. The instructions of this programme were based on the Hildesheim Teaching
Educ. Sci. 2022,12, 516 4 of 21
Concept and the CI
2
GT was administered in a pretest-post-test design. The two weeks
of the programme were identical in the sense that they consisted of one lecture (90 min),
followed by an exercise session (90 min) and a problem sheet for the participants to solve
at home. The first week focused on the introduction of groups via the dihedral groups
D3
and
D4
, as suggested by the Hildesheim Teaching Concept, and the second week focused
on cyclic groups Znand applications of group theory.
The pre-service teachers participating in this study were pre-service primary school
mathematics teachers in their first semester. Thus, for the vast majority, it can be expected
that the participants had no prior knowledge in abstract algebra—this assumption was
tested with the pre-test.
3.2. Instruments
To assess the learners’ conceptual understanding of introductory group theory, we
used the Concept Inventory of Introductory Group Theory - the CI
2
GT [
13
]. The CI
2
GT is a
concept inventory consisting of 20 two-tier single-choice items, where exactly one out of
three answer options in tier one is correct. In the second tier, the respondents additionally
rate their answer confidence on a 5-point rating scale (1
=
guessed,
. . .
, 5
=
very confident).
A point was only assigned if the correct answer option was chosen and the respondent was
confident (4) or very confident (5). Consequently, if the respondent indicated uncertainty
(CRI of 1, 2 or 3) no points were assigned, regardless of which answer option was chosen.
The internal consistency is expressed by Cronbach’s
α=
0.76. All items of the CI
2
GT can
be found in the Appendix A. It is noteworthy, that for this article the answer options for
all items have been sorted such that option 1 is always the correct answer. For the test
administration during our study, however, answer options appeared in a randomized
order.
To analyze the strengths and weaknesses of the Hildesheim Teaching Concept, we
analyzed the students’ growth regarding conceptual understanding in different content
domains. These content domains were extracted from an expert survey (
N=
9). The
experts (mathematicians and mathematics educators) were asked to assign each item of the
CI2GT to one or more sub-domains of group theory, namely:
• D1: Naive Set Theory, Binary operations, associativity, commutativity;
• D2: Neutral element and inverses;
• D3: Cayley Tables;
• D4: Cyclic groups and dihedral groups;
• D5: Subgroups;
• D6: Isomorphism.
A free response option was included in case no domain seemed suitable by the expert.
In the first round, the experts’ assignments were summed up for each item (cf. Figure 1).
In the next step, the sub-domains D1 to D6 were merged such that each item could be
assigned precisely to one overarching domain. This resulted in a total of three domains:
Domain 1
including D1,
Domain 2
including D2 and D3,
Domain 3
including D4, D5 and
D6 (cf. Figure 2).
In summary, the domains that resulted from the expert survey can be described as
follows:
•Domain 1—Definitional Fundamentals:
Naive set theory, binary operations, associa-
tivity, commutativity;
•Domain 2—Beginner Concepts: Neutral element, inverses, Cayley Tables;
•Domain 3—Intermediate Concepts: Dihedral groups, cyclic groups, isomorphisms.
The CI
2
GT items corresponding to these sub-domains are shown in Table 1. As typical
for concept inventories comprised of large domains no high values of
α
are to be expected.
Thus, according to Lienert and Raatz [
30
], for concept inventories values of
α≥
0.55 are
considered sufficient. In combination with the adjustment
α?=n−1
n·α
for small scales of
length nby Bauer [31] this yields different thresholds α?.
Educ. Sci. 2022,12, 516 5 of 21
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
D1
D2
D3
D4
D5
D6
Item 5
Item 10
Item 15
Item 20
Figure 1.
Bar Chart for the first round of the expert survey showing what percentage of the experts’
votes account for each domain.
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Domain 1 Domain 2 Domain 3
Item 5
Item 10
Item 15
Item 20
Figure 2.
Bar Chart for the second round of the expert survey showing how much percent of the
experts’ votes account for each domain.
Table 1.
Sub-domains of group theory and the respective items of the CI
2
GT, as well as their internal
consistencies, expressed by Cronbach’s αand the adjusted Cronbach’s α?as explained above.
Domain Items Length Description α?α
1 1,2,3,7 4 Definitional Fundamentals 0.41 0.28
2 4,5,6,10,11,12,17,20 8 Beginner Concepts 0.48 0.54
3 8,9,13,14,15,16,18,19 8 Intermediate Concepts 0.48 0.50
3.3. Data Analysis
3.3.1. Analysis Carried Out to Answer RQ1
The students’ learning gain through the intervention with the Hildesheim Teaching
Concept was investigated using the CI
2
GT. The students’ pre- and post-test scores were
compared using Hake’s
g
[
23
] as is common practice for this study design [
32
]. The pretest
score itself was solely used to check prior knowledge. The idea behind Hake’s
g
lies in
taking into account the students’ possible learning gain: For example, if student A scored
Educ. Sci. 2022,12, 516 6 of 21
80 out of 100 possible points in the pretest and student B only scored 10 it will be impossible
for A to gain more than 20 points, but not for B. Additionally, an increase from 80 to 100 is
certainly more challenging than one from 10 to 30 as no room for errors is allowed. In other
words, the difference of pre- and post-test scores does not measure reliable at the ends of
the scale [33]. Hake’s gthus expresses the normalized gain
g=postscore% −prescore%
100% −prescore%
and takes values below 1, indicating how much of the overall possible learning gain was
achieved. Values of 0.30
≤g<
0.70 are considered medium learning gain according to
Hake [
23
] (p. 65). In this regard, it is noteworthy that high-
g
values are exceptionally
rare—in fact, of the 62 courses including
N=
6542 students analysed by Hake no course
lied in the high-
g
region, the average normalized gain was
hgi=
0.48
±
0.14 [
23
] (p. 66). In
summary, Hake’s
g
allows to compare learning gains of different teaching concepts and
will be used to investigate which domains show the greatest learning gain.
Lastly, in this article, we understand
adequate conceptual understanding
to be achieved
by students who scored at least 50% of the total post-test score as has been achieved in prior
research (cf. [
34
]). In this regard, it is important to note that for the clarification of the first
part of RQ1 the pretest score is not relevant. As mentioned in Section 3.1, the participants
had no prior instructions and thus a significant difference in the pretest and post-test scores
is to be expected by design of this research project (cf. Table 3). By defining adequate
conceptual understanding solely based on the post-test scores, however, the investigation
of this part of the research question is divorced from a difference in the test scores.
Lastly, to explore the difference in pre- and post-test scores we used a Wilcoxon signed-
rank test (cf. [
35
]) and to explore differences between the three different learning gains we
used a Kruskal–Wallis-Test (cf. [
36
]). The differences were further specified using dwass-
steel-critchlow-fligner pairwise comparisons [
37
]. These non-parametric tests have up to
95% test power of their parametric analogues [38].
3.3.2. Analysis Carried Out to Answer RQ2
In order to identify learning difficulties we utilized the Certainty of Response Index
(CRI) established by Hasan et al. [
24
]. As mentioned in Section 3.2, each question of
the CI
2
GT was accompanied by an additional request for the respondent to assess their
confidence with the given answer from 1 (guessed) to 5 (very confident). The Certainty of
Response Index is the option selected in this regard. This enables to classify the answers in
a matrix scheme (cf. Table 2).
Table 2.
Decision matrix based on combinations of correct or wrong answer and low or high CRI
adapted from [24].
Low CRI (≤3) High CRI (>3)
Correct
Answer
Correct answer and low CRI
Uncertainty of Knowledge
Correct answer and high CRI
Knowledge of scientific concept
Wrong
Answer
Wrong answer and low CRI
Lack of knowledge
Wrong answer and high CRI
Misconceptions
As seen in Table 2, wrong answer options that were confidently selected (CRI
>
3)
indicate the presence of misconceptions or learning difficulties (cf. [
39
]). When investi-
gating learning difficulties for larger sample sizes this method may be utilized in two
different ways:
1.
Calculate the average CRI for each wrong answer option and investigate options with
hCRIi>3.
Educ. Sci. 2022,12, 516 7 of 21
2.
Calculate for each wrong answer option the number of responses that were given
confident (CRI = 4) or very confident (CRI = 5).
While in [
24
] the first method was used, we argue that using just the average CRI has
a potential to embezzle learning difficulties. For example, if every respondent selected the
wrong option 2 for some item and half of those selections were due to guessing (CRI
=
1)
while the other half was given confidently (CRI
=
4) the average CRI for option 2 would be
hCRIi=
N
2·1+N
2·4
N=2.5.
Thus, in this case, the answer pattern will be seen as unproblematic even though
N
2
participants confidently gave a wrong answer. The drawback of the second analysis
method, however, is that no thresholds are established in the literature so far. To leverage
the second method nonetheless, we set a lower threshold of 10% of total responses for a
wrong answer option that were given confidently or very confidently. Hence, we combined
both methods to analyze answer patterns. The results are provided in Table 5 and show
that each learning difficulty obtained from method 2 is also identified by method 1.
4. Results and Discussion
In the following, we will present the results and their discussion bundled for each
research question.
4.1. Results Regarding RQ1
The descriptives of the pre- and post-test scores are provided in Table 3alongside the
statistics of the Wilcoxon signed-rank test to ensure that the difference is significant.
Table 3.
Mean value
µ
and standard derivation
σ
for the pre- and post-test scores as well as the test
statistics of a Wilcoxon Signed-Rank Test.
µ σ Wilcoxon Signed-Rank Test
pretest 1.28 2.74 Z(136) = 147, p<0.001; r=0.965
post-test 8.99 3.54
The normalized gain as well as the different normalized gains for each of the three
domains are presented in Table 4and Figures 3and 4. A Kruskal–Wallis-Test comparing
the different learning gains for the three domains was highly significant (
H(
2
) =
10.6,
p<
0.01). A Dwass-Steel-Critchlow-Fligner pairwise comparison further indicates that
the difference between
g1
and
g2
is significant
(p<
0.05
)
and the difference between
g1
and
g3
is highly significant
(p<
0.01
)
. In contrast, the difference between
g2
and
g3
is not
statistically significant (p=0.37).
Table 4.
Mean value
µ
and standard derivation
σ
for the normalized gain
g
for each of the three
domains as well as the total gain gtot.
Domain Normalized Gain µ σ
Domain 1: Definitional Fundamentals g10.37 0.26
(i.e., Naive set theory, binary operations, associativity, commutativity)
Domain 2: Beginner Concepts g20.44 0.26
(i.e., Neutral element, inverses, Cayley Tables)
Domain 3: Intermediate Concepts g30.47 0.26
(i.e., Dihedral groups, cyclical groups, isomorphisms)
Overall CI2GT gtot 0.40 0.21
(i.e., all domains)
Educ. Sci. 2022,12, 516 8 of 21
g
0
0.5
1
Domain 1:
Definitional
Fundamentals
Domain 2:
Beginner
Concepts
Domain 3:
Intermediate
Concepts
Total
Figure 3.
Boxplots of the normalized gains for each of the three domains as well as the total learning gain.
g
0.2
0.4
0.6
Domain 1 Domain 2 Domain 3 Total
?n.s.
??
Figure 4.
Mean values of the normalized gains for each of the three domains as well as the total
learning gain. The error bars indicate the 95% confidence intervals. The asterisks indicate statistical
significance (?) and high statistical significance (??).
4.2. Discussion of RQ1
The total normalized gain at 0.40
±
0.21 is satisfactory and comparable to similar
research projects (cf. 0.35
±
0.21 in [
34
] (p. 156) or 0.37
±
0.18 in [
33
] (p. 68)). The smallest
learning gain was approximately 6% as indicated by the boxplot of
gtotal
in Figure 3,
thus a non-negative impact can be recorded for all participants. Consequently, with the
Hildesheim Teaching Concept, all students increased their conceptual understanding of
group theory and on average reached a reasonable learning gain. In addition, 85 out of the
143 participants (59%) reached an adequate understanding of group theory as defined in
Section 3.3.
For the different domains, it can be observed that, while close, the first domain (on
Definitional Fundamentals) records the smallest increase and the third domain (on Intermediate
Concepts) records the highest increase (cf. Table 4and Figure 4). As presented in Section 4,
the differences of
g1
and
g2
, as well as
g1
and
g3
, are statistically significant. Thus, it can
be stated that learners recorded the most significant gain in advanced concepts while the
lowest progress is attributed to the fundamentals of group theory, i.e., naive set theory,
binary operations, associativity and commutativity (cf. Section 3).
Lastly, to ensure that higher learning gain is not due to higher prior knowledge
(measured via pretest score) we divided the sample into two groups. The median was not
suitable to split the sample as the overwhelming majority of participants scored 0 in the
pretest. Thus, we established one group (
N1=
49) consisting of students with a pretest
Educ. Sci. 2022,12, 516 9 of 21
score greater than 0 and the rest (
N2=
94). The average normalized gain for the group with
prior knowledge was
g>0=
0.37 (
σ=
0.19) and for the group without prior knowledge
g0=
0.44 (
σ=
0.17). Thus, higher learning gain is not directly related to a higher prior
knowledge.
Summarizing all results with reference to research question 1, we conclude that the
Hildesheim Teaching Concept seems to be conducive to learning about abstract algebra.
Overall, the participants achieved an adequate conceptual understanding of introductory
group theory. The strength of our teaching-learning-sequence lies in fostering intermediate
concepts while it can be improved regarding definitional fundamentals. This insight will
be used to further refine the teaching concept by reworking the instructional elements
concerning basic notions.
4.3. Results Regarding RQ2
Table 5displays the number of respondents that selected either one of the items’
distractors but also stated to be confidently or very confidently that their given answer was
correct. As described in Section 3.3, answers of this type serve as fruitful indicators for
identifying learning difficulties and systematic errors. We see that 16 out of 20 items have
an option with
hCRIi>
3 and the lower threshold of 10% concerning method 2 is relevant
for 12 out of 20 items. Thus, the data gathered with the CI
2
GT provide opportunities to
uncover learning difficulties. In Section 4.4, we will demonstrate particularly conspicuous
examples and how they tie in with similar findings from abstract algebra education research,
addressing our second research question.
Table 5.
Number of responses regarding the wrong answer options 2 and 3 chosen confidently or
very confidently by our study participants for each item. The first column provides the total number
of responses (tot. #), the second column provides the relative number of responses (rel. #) and the
third column provides the average CRI.
Option 2 Option 3
Item tot. # rel. # hCRIitot. #rel. #hCRIi
1 71 50% 3.86 4 3% 3.50
2 10 7% 3.48 0 0% 2.83
3 19 13% 3.77 15 10% 3.48
4 7 5% 3.50 54 38% 3.44
5 15 10% 3.15 7 5% 3.10
6 13 9% 2.00 1 0% 2.54
7 25 17% 3.37 15 10% 3.02
8 7 5% 2.58 2 1% 3.00
9 25 17% 4.13 39 27% 4.15
10 13 10% 3.13 2 1% 2.75
11 17 12% 3.74 15 10% 3.56
12 5 3% 3.00 36 25% 3.48
13 25 17% 2.90 13 9% 3.25
14 26 18% 3.11 0 0% 1.86
15 8 6% 2.66 5 3% 3.27
16 10 7% 3.50 5 3% 3.27
17 3 2% 2.92 5 3% 2.69
18 13 9% 3.00 13 9% 2.82
19 15 10% 3.58 15 10% 3.67
20 9 6% 3.46 20 14% 3.56
Educ. Sci. 2022,12, 516 10 of 21
4.4. Discussion of RQ2
To identify learning difficulties we analyzed the response behaviour as presented in
Table 5. If one of the wrong answer options was selected confidently or very confidently by
at least 10% of the participants the respective option was investigated more thoroughly to
identify learning obstacles regarding introductory group theory among our participants.
14 options qualified regarding these standards. Within these learning difficulties, three
categories emerged that summarize similar obstacles and which we will present in the
following. Each of these categories may be associated with precisely one of the content
domains (obtained from the expert survey) that are represented in our concept inventory
(cf. Section 3). Thus, the separation of group theory by contents also reflects different
themes of learning difficulties that come along with the established domains. The themes
will be presented in ascending order to match the hierarchical structure of the domains and
an overview is presented in Table 6.
Since we cannot present every item of the CI2GT in detail in this discussion, we refer
the reader to Appendix A. For readability, answer option x of item y will be abbreviated by
option y-x (etc.).
Table 6.
An overview of the three recurring themes in learning difficulties. The percentages describe
how many of the participants selected the respective answer option confidently (
CRI =
4) or very
confidently (CRI =5).
Domain Domain Description Learning Difficulty Item Option 2 Option 3
1
Definitional Fundamentals
(Naive set theory, associativity,
commutativity, etc.)
Problems with associativity
1 50% –
3 13% 10%
4 – 38%
7 17% 10%
2Beginner Concepts
(Neutral element, inverses, etc.)
Problems with inverses and the
neutral element
5 10% –
10 10% –
12 – 25%
20 – 14%
3
Intermediate Concepts
(Dihedral and cyclic groups,
isomorphism, etc.)
Problems with visualizing
abstract notions
9 17% 27%
13 17% –
14 18% –
4.4.1. Problems with Associativity and Commutativity
The most glaring learning obstacle is directly observed with option 1-2. Roughly
50% of all participants stated confidently that the purpose of associativity is to be able
to neglect order of concatenation. In other words, commutativity and associativity get
mixed up which is substantiated by results from Larsen [
9
] who found that students
struggle to differentiate between those properties. Additionally, associativity is not checked
when verifying the properties of a group structure (cf. option 3-3) as 10% of the students
confidently marked
(Z
,
−)
as a group. Tirosh et al. [
40
] have shown that in some cases
students may even see associativity as a direct consequence of commutativity which is
clearly not the case as the example
◦:N×N→N,(a,b)7→ 2a+2b
illustrates. On the other hand, commutativity is often left unchecked (cf. option 7-2 and
option 7-3) or assumed even when explicitly excluded (cf. option 4-3), leading to multiple
different hurdles. Thus, it can be summarized that associativity and commutativity are
Educ. Sci. 2022,12, 516 11 of 21
properties whose purpose is somewhat unclear for many learners – they are confused with
one another and often falsely generalized. This particular finding was also reported on
by Melhuish and Fagan [
6
] who researched introductory group theory with the concept
inventory GTCA (cf. [5]).
4.4.2. Problems with Inverses and the Neutral Element
The next recurring theme is located in the second domain and concerns the role of
inverses and the neutral element. Similar to associativity these concepts have a potential to
be overlooked when studying groups. Option 3-2 shows that 13% of participants confidently
stated that
(Q
,
·)
is a group, missing that 0 does not have an inverse in this structure.
Furthermore, option 5-2 and option 10-2 suggest that starting examples in introductory
courses such as
(Z
,
+)
,
(Q
,
+)
or
(R\ {
0
}
,
·)
get overgeneralized in the sense that 0 and 1
are a priori special elements and always self-evident candidates for neutral elements, even
when dealing with completely different binary operations. In addition, the concepts of
inverse elements and the neutral element are mixed up in more abstract scenarios as option
12-3 shows where an inverse element had to be extracted from a Cayley Table (cf. Table 7).
Table 7.
The Cayley Table from item 12 of the CI
2
GT: The task lies in finding the inverse of
z
with
respect to ◦.
◦a t w z
a z w a t
t w z t a
w a t w z
z t a z w
Here, it is clear that
w
must be the neutral element for
◦
, and thus, we have to look
for
w
in the row (or column) of
z
to find its inverse. A total of 25% of students, however,
detected
z◦z=w
and confidently jumped to the conclusion that
w
must be the inverse of
z
, reversing the roles of inverses and the neutral element. This learning difficulty might be
tied to Cayley Tables as a similar problem can be observed with option 20-3 (cf. Table 8).
Table 8.
The tables from item 20 of the CI
2
GT: The task lies in identifying which of the three presented
tables is a Cayley Table.
◦a b c
a a c b
b c a b
c b b a
◦a b c
a c a b
b b c a
c a b c
◦a b c
a c a b
b a b c
c b c a
Here, the first table can be ruled out immediately as the column of ccontains btwice.
However, the second table also does not make any sense as the row of
c
suggests that it is
the neutral element, contradicting the column of
c
. The fact that neutral elements in groups
are always left- and rightneutral was disregarded. A total of 14% confidently stated that
the second table is a Cayley Table.
These observations result in two central aspects for instructors: Firstly, the usual
mathematical simplification of using the symbol 1 generally for multiplicative identities
and the symbol 0 for additive identities is to be treated with care, especially in introductory
courses. Secondly, leaving vivid and concrete examples such as
(Z
,
+)
for more abstract
ones poses hurdles for beginners that are expressed by mistakes even in the most basic
fundamentals such as inverses and the neutral elements.
Educ. Sci. 2022,12, 516 12 of 21
4.4.3. Problems with Visualizations of Abstract Notions
The last theme is more subtle and concerns abstract concepts such as isomorphisms
and their relation to symmetry. In this regard (cf. option 9-2 and option 9-3), learners
confidently think that two groups being isomorphic means that they are identical (27%) or
that that their Cayley Tables are identical (17%). On the surface level, this does not look too
harmful, however, the whole concept of isomorphisms is to enrich this sense of uniqueness
with mathematical precision. This level of abstraction is vital not only for group theory but
for mathematics in general, as equivalence relations often present the highest degree of
distinction possible, thus phrases such as “up to isomorphism”, “up to homeomorphism”,
“up to congruency”, etc. are ubiquitous and further differentiation is neither necessary nor
possible from a mathematical point of view.
This observation carries over to option 13-2 and option 14-2 where respondents have
to classify the symmetry group of a given figure up to isomorphism. For both figures (cf.
Figure 5), the wrong options
D2
and
D3
, respectively, seemed attractive. A total of 17%
were confident that the left figure has symmetry group
D2
and 18% were confident that the
right figure has symmetry group D3.
Figure 5.
The figures of items 13 (
left
) and 14 (
right
) of the CI
2
GT. The task lies in classifying their
symmetry group.
From this, we assume that the students focused on the circles in the left figure and on
the resemblance with a triangle of the right figure and concluded that they must have the
same symmetry as the regular 2-gon and 3-gon, respectively. The underlying structural
components of these groups were, however, neglected. The left figure does not have a
rotational symmetry other than the identity and the right figure does not have an axial
symmetry other than the identity. Thus, with the structure of the dihedral group in mind
one can quickly rule out these options.
In both cases, a component of gestalt simplified the underlying functionality of the
mathematical object when modeling a mathematical concept–uniqueness in the first exam-
ple and symmetry in the second. An explanation for this observation could be found in
a separation of two cognitive dimensions. In this regard, Ubben and Heusler [
41
] empir-
ically extracted two cognitive dimensions underlying students’ mental models from an
exploratory study in the context of physics education research, namely the Fidelity of Gestalt
and the Functional Fidelity. The first dimension is described as an understanding of ones’
mental models as “exact visual representations of phenomena or exact depictions of how
things look” [
41
] (p. 1356) while the second dimension constitutes how “much the mental
models’ underlying abstract functionality [...] is perceived as accurate” [
41
] (p. 1360). It is
noteworthy that this two-factorial structure of students’ mental models has been confirmed
in different thematic contexts and was supported by literature from educational psychol-
ogy and neurology (cf. [
42
]). Although this two-dimensional model stems from physics
education research, we argue that the psychological components accurately describe the
observations discussed in this article.
Educ. Sci. 2022,12, 516 13 of 21
With this in mind, the results suggest that the discussed learning difficulties arise
from a predominant gestalt thinking, indicating a lack of abstraction. Furthermore, the
learning difficulty observed with option 20-3 may also be described with a lack of functional
thinking (cf. Table 8): the second table follows the filling rules of Sudoku and thus looks
like a promising solution on the surface. However, as pointed out above, a deeper look
reveals how and why the group axioms are violated nonetheless.
Ultimately, we refrain from delving too deep into this characterization of mental
models at this point as the CI
2
GT was not designed with this specific cognitive structure in
mind. With findings indicating a possible learning opportunity for educational research
into group theory, however, we plan on incorporating this theory into instruments for future
investigations. After all, extracting a two-factorial structure empirically in mathematics
education might open up completely new possibilities for this research body.
5. Limitations of This Study
Before we present our conclusion it is necessary to address the limitations of the study
presented hereby. The most striking aspect is the lack of a control group that was treated
with a different teaching concept. This is due to a lack of other published material in
this field. The only other teaching concept of abstract algebra contents is provided by
Larsen’s TAAFU project (cf. [
26
,
27
]). However, as mentioned in Section 1, different topics
are covered and thus the CI
2
GT cannot be applied to a potential control group taught with
this concept. Conclusively, no direct comparisons are drawn in the results presented in this
article. Nonetheless, Hake’s
g
as an expression of normalized gain is designed to bypass
this problem to some extent. As mentioned in Section 3.3, this parameter was used to
compare learning gains of different treatments and with a value of
g=
0.40 the normalized
gain of the Hildesheim Teaching Concept can be classified as medium, aligning it with the
vast majority of interventions presented in the aforementioned study (cf. [23]).
With regards to RQ2, another crucial aspect arises from linguistic subtleties in the items
of the CI
2
GT, namely items 1 and 9. This will now be addressed to better contextualize our
discussion of this research question. Answer option 2 of item 1 reads
“The associativity property is required because we do not want the order of composition
to matter.”
This distractor is ambiguous. Associativity is a characteristic of binary operations that
is required for composing three or more elements. Since a priori the expression
a◦b◦c
(for
some
a
,
b
,
c∈M
and some set
M
) bears no mathematical meaning, one needs to interpret
it as either
a◦(b◦c)
or
(a◦b)◦c
. If
◦
is associative, both expressions are equal and the
abuse of notation
a◦b◦c
is justified. However, one could also describe associativity by
stating that the order implied by the brackets does not matter. Thus, this answer option
might be misinterpreted. It is to be revised (still hinting at commutativity) in future studies
to examine whether the identified misconception in Section 4.4 still holds.
Answer option 2 of item 9 reads
“The notion isomorphic means that the Cayley tables are identical.”
Here, even though one could argue that the word identical has a somewhat sacred
meaning within mathematics, it might be mistaken for similar, meaning that both Cayley
tables are only required to have a matching pattern, i.e., the Cayley tables presented in
items 10 and 12. This is crucial because in this case, the statement becomes seemingly true
(even though the notion pattern does not exist formally). Therefore, this answer option
should be revised to eliminate possible confusion of notions. The revision of this item’s
distractor may be influenced by recently gathered expert opinions on the topic of sameness
in mathematics, presented in [43].
Educ. Sci. 2022,12, 516 14 of 21
6. Conclusions and Outlook
In this contribution, we demonstrated how data assessed with the CI
2
GT may be
utilized to research various educational aspects of group theory. By measuring conceptual
understanding of introductory group theory we were able to (a) investigate different
subdomains assigned by experts in the field and the respective learning gains as well as (b)
leverage the CRI to uncover and discuss recurring learning difficulties.
On the one hand, this substantiates the qualities of the Hildesheim Teaching Concept
from an empirical point of view, complementing the formative assessment presented in [
11
]
and providing useful information regarding group theory instructions. However, the
discussed results further mark a fruitful starting point for future research and contribute
to theory building in abstract algebra education research in the sense of Design-Based
Research. Methods from neurology, physics education and mathematics education were
synthesized to describe learning barriers in detail. In this regard, we provided the first
empirical indications of Functional Fidelity as well as Fidelity of Gestalt being potentially
relevant cognitive dimensions to describe mental models and thus of great use when
characterizing learning difficulties within mathematics education. However, some of the
items’ distractors also possessed ambiguous qualities that might have interfered with the
observed learning difficulties. With this in mind, the following research questions should
be investigated in the future:
1.
Does a revision of items 1 and 9 lead to a disappearance of the observed learning
difficulties?
2.
How and to what extent do the expounded learning difficulties impede learning gain?
3.
Can these systematic learning obstacles also be observed in qualitative settings with
individual learners? If so, can they be characterized in greater detail?
4.
Can the cognitive structure of gestalt and functionality be extracted empirically to
enrich the understanding of learning processes in introductory group theory?
Author Contributions:
Conceptualization, J.M.V., P.B. and B.G.; writing—original draft preparation,
J.M.V.; writing—review and editing, J.M.V., P.B. and B.G.; supervision, B.G. All authors have read
and agreed to the published version of the manuscript.
Funding:
This research was funded by the Department of Mathematics and applied Informatics,
University of Hildesheim.
Institutional Review Board Statement:
Ethical review and approval were waived for this study due
to the fact that the study was in accordance with the Local Legislation and Institutional Requirements:
Research Funding Principles (https://www.dfg.de/en/research_funding/principles_dfg_funding/
research_data/index.html) and General Data Protection Regulation (https://www.datenschutz-
grundverordnung.eu/wp-content/uploads/2016/04/CONSIL_ST_5419_2016_INIT_EN_TXT.pdf)
(accessed on 15 April 2022).
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the
study to publish this paper.
Data Availability Statement:
The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
Educ. Sci. 2022,12, 516 15 of 21
Appendix A
Item 1: The associativity property is required because . . .
. . . otherwise it is not clear how to compose 3 or more elements.
. . . we do not want the order of composition to matter.
. . . along with distributivity and commutativity it is a fundamental rule of
mathematics.
Very sure
Sure
Undecided
Unsure
Guessed
Item 2: A binary operation on a set Mis . . .
. . . a map f:M×M→M.
. . . a map f:M→M×M.
. . . a map f:M×M→M×M.
Very sure
Sure
Undecided
Unsure
Guessed
Item 3: An example for a group is . . .
. . . (R,+)
. . . (Q,·)
. . . (Z,−)
Very sure
Sure
Undecided
Unsure
Guessed
Item 4: Let G= (M,◦)be non-abelian and a,b∈M. The inverse of a◦bis
. . .
. . . b−1◦a−1
. . . a−1◦b
. . . a−1◦b−1
Very sure
Sure
Undecided
Unsure
Guessed
Item 5: One can show that a?b:=a+b−5defines an operation on Zsuch that
(Z,?)is a group. The neutral element of this operation is . . .
. . . 5
. . . 0
. . . −5
Very sure
Sure
Undecided
Unsure
Guessed
Educ. Sci. 2022,12, 516 16 of 21
Item 6: One can show that a•b:=ab
7defines an operation on Q\{0}such that
(Q\{0},•)is a group. The inverse of x∈Q\{0}is given by . . .
. . . 49
x
. . . 49
x2
. . . 7
x
Very sure
Sure
Undecided
Unsure
Guessed
Item 7: Let f(x) = 2
x−1and g(x) = ex+1, then . . .
. . . (g◦f)(x) = ex+1
x−1
. . . (g◦f)(x) = 2
ex−1−1
. . . (f◦g)(x) = e2
x
Very sure
Sure
Undecided
Unsure
Guessed
Item 8: In the group D4the equation s1◦(x◦s1) = s1is solved by . . .
. . . x=s1
. . . x=r90
. . . x=id
Very sure
Sure
Undecided
Unsure
Guessed
Item 9: The notion isomorphic means that . . .
. . . the groups are indifferentiable from a mathematical point of view.
. . . the Cayley tables are identical.
. . . the groups are identical.
Very sure
Sure
Undecided
Unsure
Guessed
Item 10: The operation ⊕within the group (Z4,⊕)has been altered to ◦such that
[
0
]is no longer necessarily the neutral element. Find the neutral element with the
help of the Cayley table.
◦[0] [1] [2] [3]
[0] [1] [3] [0] [2]
[1] [3] [2] [1] [0]
[2] [0] [1] [2] [3]
[3] [2] [0] [3] [1]
[2]
[1]
[3]
Very sure
Sure
Undecided
Unsure
Guessed
Educ. Sci. 2022,12, 516 17 of 21
Item 11: A group structure is to be established on the set {0, π, 55}where the
following Cayley table is given. Which element must be at ??
◦0π55
0?0
π π
55 0 π55
?=π
?=0
?=55
Very sure
Sure
Undecided
Unsure
Guessed
Item 12: The set {a,t,w,z}has been equipped with a group structure by the
following Cayley table. What is the inverse of z?
◦a t w z
a z w a t
t w z t a
w a t w z
z t a z w
z−1=z
z−1=t
z−1=w
Very sure
Sure
Undecided
Unsure
Guessed
Item 13: What is the symmetry group of the figure?
Z2
D2
D4
Very sure
Sure
Undecided
Unsure
Guessed
Educ. Sci. 2022,12, 516 18 of 21
Item 14: What is the symmetry group of the figure?
Z3
D3
Z6
Very sure
Sure
Undecided
Unsure
Guessed
Item 15: What is the symmetry group of the figure?
D5
Z5
Z10
Very sure
Sure
Undecided
Unsure
Guessed
Item 16: Which two of the following figures have an isomorphic symmetry
group?
The first and the third.
The first and the second.
The second and the third.
Very sure
Sure
Undecided
Unsure
Guessed
Item 17: If a Group is commutative the Cayley tyble is . . .
. . . axially symmetric to the diagonal.
. . . point symmetric to the entry in the middle.
. . . axially symmetric to the anti diagonal (top left to bottom right).
Very sure
Sure
Undecided
Unsure
Guessed
Educ. Sci. 2022,12, 516 19 of 21
Item 18: Which of the following sets is a subgroup of (Z10 ,⊕)if equipped
with ⊕?
{[0],[2],[4],[6],[8]}
{[0],[1],[2],[5]}
{[0],[1],[3],[5],[7],[9]}
Very sure
Sure
Undecided
Unsure
Guessed
Item 19: Which of the following permutations does not describe an
isometry of the square?
π=1234
1324
σ=1234
2341
τ=1234
2143
Very sure
Sure
Undecided
Unsure
Guessed
Item 20: Which of the following tables is a Cayley table?
◦a b c
a a c b
b c a b
c b b a
◦a b c
a c a b
b b c a
c a b c
◦a b c
a c a b
b a b c
c b c a
The third.
The first.
The second.
Very sure
Sure
Undecided
Unsure
Guessed
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