What Group Theory Can Do for You: From Magmas to Abstract Thinking in School Mathematics

Abstract

In this paper, we present a novel concept for an abstract algebra course at the secondary school level. Against the backdrop of findings from mathematics education research, we discuss benefits that arise from teaching abstract algebra to secondary school students. We derive suitable content items as well as educational ideas on how to implement them into mathematics classrooms. We build on frameworks from mathematics education research to design a teaching-learning sequence that enables a beneficial learning trajectory across three units—the result is a coherent teaching concept that encompasses Magmas in a hands-on way. It can be used for either extra-curricular activities or advanced courses in mathematics classes, and can be of benefit for mathematics learners.

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Citation: Veith, J.M.; Bitzenbauer, P.
What Group Theory Can Do for You:
From Magmas to Abstract Thinking
in School Mathematics. Mathematics
2022,10, 703. https://doi.org/
10.3390/math10050703
Academic Editor: Jay Jahangiri
Received: 18 January 2022
Accepted: 22 February 2022
Published: 23 February 2022
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mathematics
Article
What Group Theory Can Do for You: From Magmas to Abstract
Thinking in School Mathematics
Joaquin Marc Veith 1,* and Philipp Bitzenbauer 2
1Institut für Mathematik und Angewandte Informatik, Stiftung Universität Hildesheim,
31141 Hildesheim, Germany
2Physikalisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany;
philipp.bitzenbauer@fau.de
*Correspondence: veith@imai.uni-hildesheim.de
Abstract:
In this paper, we present a novel concept for an abstract algebra course at the secondary
school level. Against the backdrop of findings from mathematics education research, we discuss
benefits that arise from teaching abstract algebra to secondary school students. We derive suitable
content items as well as educational ideas on how to implement them into mathematics classrooms.
We build on frameworks from mathematics education research to design a teaching-learning sequence
that enables a beneficial learning trajectory across three units—the result is a coherent teaching concept
that encompasses Magmas in a hands-on way. It can be used for either extra-curricular activities or
advanced courses in mathematics classes, and can be of benefit for mathematics learners.
Keywords: algebra; groups; magmas; secondary school; mathematics education
1. Introduction
Over the past two decades, numerous studies have shown how abstract algebra is con-
nected to school mathematics and how studying the concepts of abstract algebra influences
one’s view on mathematics in general. For example, with a course for mathematics teachers,
Wasserman [
1
] has shown that presenting K-12 teachers with the ideas of abstract algebra
not only increased their content knowledge but also influenced their beliefs and even their
classroom practices. In other words, it vastly transformed their way of thinking about
mathematics. The participants of the course argued that learning about group properties
made them realize underlying concepts of school mathematics (specifically in arithmetic
and algebra), putting them into perspective, which in turn helped them paint a bigger
picture. This specific reasoning was also touched upon by McCrory et al.’s [
2
] in-depth dis-
cussion of knowledge for algebra teaching. In another study by Even [
3
] multiple teachers
in the context of an in-service training voiced the opinion that dealing with these concepts
advanced their understanding about what mathematics is and what doing mathematics
actually means. On the other hand, approaching mathematics disregarding structures can
lead to misconceptions, as has been shown for different mathematics topics (cf. [
4
]). So not
only does abstract algebra provide learning opportunities for the nature of mathematics
itself, it also improves knowledge about specific subdisciplines (cf. [
5
]). In addition to
arithmetic and algebra, for exmaple, the study of group theory marks great value for the
context of geometric symmetries in school mathematics [6,7].
These connections were studied even further by Wasserman [
8
], looking at all possible
content areas across elementary, middle and secondary mathematics that may be influenced
by knowledge about concepts of abstract algebra. Exploring abstract algebra yet again
through the lens of school mathematics, this work constructed a progression of four content
areas (Solving Equations, Structure of Sets, Inverses and Arithmetic Properties) that are
potentially transformed by the perspective abstract algebra provides.
Mathematics 2022,10, 703. https://doi.org/10.3390/math10050703 https://www.mdpi.com/journal/mathematics

Mathematics 2022,10, 703 2 of 24
However, simply because teachers view their field of education differently does not
mean these changes also translate to the students or help students understand school
algebra better in any way. In fact, multiple empirical studies have shown that this is not
the case, the constructs teacher knowledge in abstract algebra and student achievement in school
algebra do not seem to correlate significantly (cf. [
3
,
9
–
16
]). This raises the question whether
the mentioned benefits that come with abstract algebra are used to its full potential when
leaving students out of the equation. In addition, since the ultimate goal of mathematics
education is always enhancing students’ perceptions and understanding of mathematical
concepts, these observations result in one pressing question: Can these abstract algebra
courses be tweaked so that students themselves can profit from them? And if so, do they
profit in a similar way?
This question is by no means a new one. It was the driving force behind the educational
reform ’new math’ which overhauled the entire mathematical curricular infrastructure
during the 1960s and 1970s. It was eventually dismissed, but during the time a lot of
educational work has been produced on the topic and it is been laying waste for the
better part of a century now. This unsatisfying coexistence of the numerous possibilities
of abstract algebra (better understanding of arithmetic properties, discovering underlying
concepts of equations, positive impact on structural knowledge, better understanding
of concatenations) and the lack of ambition to make use of it led us to design a new
teaching concept for abstract algebra at secondary school levels which in the following we
will introduce.
At first, we will present a brief historical overview of the ‘new math’ reform as well as
present the key ideas and notions that have been produced by numerous didacts at that
time before comparing them to modern research on this topic. One may interpose that these
ideas are partially over 50 years old and thus outdated, but the mathematics has remained
the same and it is the first and only attempt ever of bringing abstract algebra down to school
level at a large scale. Additionally, since empirical studies on these ideas are also dated
and new educational ideas have been produced over the years, they can be merged and
studied thoroughly with modern methods, potentially yielding new conclusions. Therefore,
the guidelines for our proposal will not only take modern approaches into account but
we will also pick up ideas of the past and try to synthesize both bodies of work to a
homogeneous concept.
2. The New Math Era: A Brief Historical Overview
The purpose of this section is to provide unfamiliar readers a quick perspective on the
new math movement to better contextualize the referenced literature in the next sections.
A full historical in-depth look can be found in [17].
On 4 October 1957, the Soviet Union succeeded in launching a satellite into Earth orbit
for the first time. This historic achievement and clear demonstration of technical progress
triggered numerous impulses that ultimately penetrated the education system of the US
and many European countries. In response to what is now called the “Sputnik shock”,
several attempts were made to raise the level of education, specifically in scientific related
subjects, in order to remain internationally competitive. In his article “The New Math
and Its Aftermath”, David Rappaport [
18
] writes: “Sputnik was also responsible for the
development of the new math programs in the elementary schools and in the high schools.
That the Soviet Union could be ahead of the United States in some area of technology came
as shock to most Americans. The result was a demand for drastic change in the mathematics
and science curricula in the school programs. The National Science Foundations, as well
as a number of private foundations, made available millions of dollars to schools and to
individuals do create new mathematics programs” (p. 563).
This gave rise to the SMG (School Mathematics Study Group) as a major project aimed
at revising the traditional curriculum. Numerous writers produced books and learning
materials for mathematics teaching; a new mathematics was emerging under the name
“New Math”. In 1959, two years after the Sputnik Shock, the OEEC (today OECD) convened

Mathematics 2022,10, 703 3 of 24
a two-week seminar in Royaumont, France. At the center of the discussions held there is one
topic alone: new thinking in mathematics education; the topics of this seminar are publicly
documented in [
19
], key figures are Hans-Georg Steiner from Germany and Jean Dieudonné
from France. The entire subject of mathematics is to be restructured and reoriented. New
ways of thinking are to be introduced, and the structures behind mathematical concepts
are to be brought to light. In concrete terms, this means, among other things: structural
mathematics (today called abstract algebra), which has been relegated to university, must
be integrated into the school curriculum.
Since our teaching concept will be studied in Germany we will specifically highlight
the movements development in Germany: In 1966, about 7 years after the Royaumont
Seminar, the project manifests and the reform points discussed in the seminar get published
in volume form under the title “Synopsis for Modern School Mathematics” [
20
]. The
educational reform is made binding nationwide, teacher training and curricular changes
are implemented and school literature is revised. However, with teachers and parents
being unable to cope with the sudden changes, the reform quickly receives a bad image in
public and its purpose is heavily discussed and critiqued (cf. [
21
]). This ultimately leads to
the reform being dismissed in the late 1970s, ending its short-lived existence.
Many mathematics educators at the time, however, argued that the abolition of the
reform was as hastily as its implementation and many studies claimed that students
who engaged in abstract algebra performed better in school algebra tests than their peers
(cf. [
22
]). A brief description of the movements’ development in the US including a
comprehensive overlook of all the covered topics of abstract within the new math reform
can be found in [22].
3. Historical and Contemporary Ideas on Teaching Abstract Algebra
In this section, we present and discuss didactical ideas and research on the topic of
teaching abstract algebra. We will highlight both the work of researchers from the ’new
math’ era as well as contemporary work. The goal is to identify learners’ difficulties with
topics of abstract algebra, how to avoid them and what contents of abstract algebra are
suited when dealing with this subject for the first time.
3.1. Educational Key Findings of the ‘New Math’ Era
Mathematicians and mathematics educators have been very productive during the
‘new math’ era. So productive in fact, that presenting all the developed ideas will be an
impossible task in the context of this contribution. We therefore decided to present two
recurring themes of the literature that seemed to be the least controversial and thus provide
the most common ground.
At first, however, one has to note that bringing abstract algebra all the way down to
school mathematics is without any doubt an ambitious goal. It requires careful consider-
ation of contents and compelling educational ideas in order to be fruitful to the student.
These requirements can be categorized in a broad manner, resulting in basic landmarks
when introducing abstract algebra to pupils. In this regard, Nöbauer [
23
] enunciated three
fundamental rules:
1.
Algebraic structures must not be introduced in a formal way or independent of
tangible models. The axiomatic nature can be discussed if an opportunity presents
itself, but it is of secondary importance.
2.
Mathematical terms may only be introduced if they can be demonstrated or illustrated
on the basis of already existing terms.
3.
The goal of abstraction is only possible if the underlying concepts have been stud-
ied thoroughly.
To understand the third rule, one has to keep in mind that originally algebraic struc-
tures were planned to be introduced in the lower level of secondary schools. For our
concept, however, only the upper level is targeted, so familiarity with arithmetics and
geometry can be assumed.

Mathematics 2022,10, 703 4 of 24
The first and second rule can be explained by looking at the mathematics of the time:
The axiomatic approach to mathematics which was developed by the Bourbaki group a few
years prior to the ‘new math’ era developed a fundamentally new view on mathematics in
general. Introducing mathematical structure was a new way of unifying concepts, but it
required a more general thinking. Demanding from school students who are just learning
about the basics of mathematics to form this perspective was highly controversial. If at all,
it was seen to be possible if the abstract concepts of structures appeared explicitly, meaning
that they are made visible and physically engaging via concrete models. This is, by far, the
most recurring condition among the entire body of research during the 1960s.
3.1.1. No Axiomatic Group Theory
As mentioned above and in the first fundamental rule by Nöbauer, axiomatic group
theory was neither a goal nor a desideratum. In an important paper Freudenthal [
7
] argued
that when exploring groups one should not start by defining it in an abstract way and
then look at examples (basically the way every mathematical content is introduced in
university). Instead it should happen vice versa: The educational process should move
from the particular to the general. However, Dubinsky et al. [
24
] noted that it still might be
difficult for students to abstract concepts they’ve seen in specific and concrete examples.
After all, the group concept is not a single abstract idea, but rather a collection of multiple
abstract ideas.
It is worth mentioning that not all mathematics educators voiced their opinions against
an axiomatic treatment. For example, Liermann [
25
] argued that the concept of groups
is ideal for a first look at the axiomatic nature of mathematics, since it is less convoluted
compared to other school mathematics areas such as geometry or set theory. To provide
another example, Kropp [
26
] also took a decisive stance in favor of treating axioms in class.
He reasoned that the demands on the ability to abstract are only high when compared to
the usual requirements in conventional contents of school mathematics or other subjects.
He further argued that the transition towards abstraction is just a matter of acclimatisation
and that the more one gets used to the notions the less abstract they appear.
Since the empirical research that was used to back up those statements in most cases
would not suffice modern research standards, we do not dare to derive a clear “yes” or “no”
for the axiomatic approach from the literature. For reasons mentioned in the next paragraph,
we decided to take the less abstract approach and leave the axioms of group theory out of
the concept for now. For more in-depth discussions of the time as to how the axiomatic
method can or should be treated in class, we refer the reader to the articles [27–32].
3.1.2. Magmas Instead of Groups
As mentioned in the second and third rule, the abstract concepts have to be made
very explicit by demonstrating their manifestations visually and making them engaging.
Steiner, a key figure in the ’new math’ era and participant of the Royaumont Seminar,
had the idea in this regard to not approach abstract algebra by looking at groups right
away, but looking at magmas first. He reasoned that set theory is deeply rooted in every
discipline of mathematics, even in school mathematics, i.e., sets of points in the euclidean
plane or space, number sets in arithmetic, domains and ranges of functions or solution sets
of equations. So even though their applications may vary, the mathematical object itself
is a recurring theme, suggesting that students have a better understanding of it than of
any other aspect of groups [
33
]. Binary operations, on the other hand, are neither known
nor made explicit at any point. For instance, multiplication and addition are operations
solely applied to real numbers and their structural similarity as maps
R×R→R
is never
discussed. Therefore, before even thinking about tackling groups, it is vital that students
have a sufficiently abstract idea of binary operations [
34
]. For our concept, we thus deduce
that while still working with groups it might be better to just view them as magmas first and
embezzle additional properties. To support this, Steiner [
34
] derived two main principles
for establishing algebraic structures:

Mathematics 2022,10, 703 5 of 24
1.
How many models in the direct mathematical periphery tap the algebraic structure?
In other words, which structures draw the most from visual demonstrations?
2.
Which of those algebraic structures in 1. allow for an adequate (in relation to school
mathematics), deductive approach as well as an exploratory approach?
Very similar main principles were also developed by Dieudonné (cf. [
35
]). From these
principles one can derive groups that satisfy the conditions and are, thus, suited for a first
introduction. For example, Griesel [27] derived the following list:
•
Dihedral groups but also the groups of symmetries of rectangles, rhombuses, rhom-
boids, kites, etc.
• Frieze groups
• Cyclic groups
• The additive groups of Z,Qand R
• The multiplicative groups of Q>0,{2n:n∈Z}, etc.
• The orthogonal groups
This list was further refined by Leppig [
36
] to
D3
,
D4
and
Zn
. In this article, cyclic
groups are defined as groups that are generated by a single element, and, hence, can be
written as
G=hgi={g,g2,g3, . . . }
for some
g∈G
. Furthermore, the Dihedral group
Dn
of order 2
n
is defined as the subgroup
of the Symetric group Sngenerated by
σ=1 2 . . . n−1n
2 3 . . . n1and τ=1 2 3 . . . n
1n n −1 . . . 2
where
σ
can be interpreted as a rotation and
τ
can be interpreted as a reflection of the
regular
n
-gon. For the refined list Leppig reasons as follows: On the one hand, congruent
mappings are already part of the geometry curriculum in secondary schools, so transforma-
tions of equilateral triangles and squares can be studied without having to establish new
mathematics while at the same time allowing for a visually engaging entry to group theory,
making the dihedral groups
D3
and
D4
perfect candidates. On the other hand, the direct
mathematical periphery are number sets, so groups like
(Z
,
+)
,
(Q
,
+)
or
(R\{
0
}
,
·)
are
obvious candidates. It can be argued, however, that not much from those groups can be
learned and instead one should try to make use of the ubiquity of arithmetic by looking at
finite cyclic groups
(Zn
,
+)
. Those groups have more compelling applications and are more
abstract, so understanding them allows for a better understanding of the concepts of group
theory. For additional justification, we will see in Section 3.2 that the learning processes of
precisely those groups have been well studied and documented in recent research, so we
can pick up modern ideas on how to approach those structures.
For dihedral groups the exploratory approach mentioned in the second main principle
will not pose too much trouble. For cyclic groups however, some work needs to be done in
order to enable students to explore them. We will present our ideas on how to explore both
kinds of groups in a compelling way in Section 5.
3.1.3. Interim Conclusions
Summarizing the educational ideas above, from literature at the time we categorized
three educational key findings (cf. Table 1).
Table 1. Key findings derived from historic research on abstract algebra teaching.
Key Finding References
No axiomatic group theory [7,24,29]
Magmas instead of groups [33–35]
Best groups for introducution are D3,D4and Zn[27,36,37]

Mathematics 2022,10, 703 6 of 24
Very importantly, this already gives hints which groups are suited the most for an
introductory course as well as the fact that the binary operation is of such a relevance
that focusing on Magmas first is a crucial part of entering abstract algebra. This is also
important with regards to practicability of the teaching concept; exploring groups would
require additional activities for inverses, the neutral element and associativity. Additional
activities however require more time and thus more teaching units. On the other hand,
one can study properties of groups without actually naming them and generalize ideas. In
order to get to abstraction, it is necessary to use the ideas - but not necessarily the language
- of inverses, the neutral element and associativity [19].
3.2. Educational Key Findings in Contemporary Research
Modern research on the topic, as mentioned in the introduction, involved solely
prospective teachers. In addition, university students differ vastly from school students,
both in terms of previous knowledge and in terms of interests and beliefs. The participants
in the subsequently mentioned studies obviously had a richer math education to build
upon, and also are arguably somewhat drawn towards mathematics, having made the
decision to teach this subject in school. On the other hand, this allows for research to dig
deeper and explore more content with the participants, so a lot of insight on learning all
the numerous concepts of abstract algebra is gained. Lots of study has been made on
learning Groups, Subgroups, Cosets, Cayley Tables, Isomorphisms, Polynomials, Rings and
Fields. For example, a collection of the scientific output can be found in the edited volume
“Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics
Teachers” [
38
]. Although the researchers on the field touch on many different topics,
studying the literature suggests that there are three distinctive key findings for learning
abstract algebra which we will present in the following.
3.2.1. Linear Equations and Familiar Number Sets
Students have shown to prefer working with familiar number systems when presented
with objects of abstract algebra [
39
,
40
]. This result is not surprising since the most obvious
connection between abstract algebra and school algebra lies within the fact that the number
sets
Z
,
Q
and
R
are among the first groups one can study and they are also very present in
school mathematics - with the difference that in school they are just viewed as sets without
any further structure. Computations that arise from binary operations in these sets (i.e.,
adding, subtracting, multiplying, etc.) are familiar to the students and don’t require much
abstraction.
Wasserman [
1
] and Shamash [
5
] have already impressively demonstrated how abstract
algebra can be introduced through solving equations. At last, all of the axioms of groups
and rings can be traced back to the basic needs of solving linear equations, for example:
2+x=3r◦x=s
(−2) + (2+x) = (−2) + 3 (concatenate with inverse) r2◦(r◦x) = r2◦s
((−2) + 2) + x=1 (associativity) (r2◦r)◦x=r2s
0+x=1 (property of inverse) id ◦x=r2s
x=1 (property of neutral element) x=r2s
Thus solving linear equations works exactly the same in every group and parallels
can be demonstrated elegantly, as shown above where the left-hand side can be viewed in
(Z
,
+)
(or
Q
or
R
) and the right-hand side can be viewed in
(D3
,
◦)
(or any of its isomorphic
copies). When presenting students with new knowledge it seems apparent that the proven
affinity for familiar number sets should be used so it will be part of our teaching concept.
3.2.2. Geometrical Aspects of Group Theory
Picking up the idea of showing students concepts they are familiar with, studies
have shown that such connections between school algebra and abstract algebra can easily
be overlooked unless they’re made very explicit [
41
,
42
]. One has to keep in mind that

Mathematics 2022,10, 703 7 of 24
the general mathematical approach of unifying concepts and categorizing mathematical
objects in terms of their structure is an idea that’s completely absent in school mathe-
matics. Alternatively to solving equations, these underlying connections can be revealed
via geometrical groups. In a recent study by Suominen [
43
] where mathematicians and
mathematics educators where questioned about the connections of abstract algebra and
secondary school algebra, over half of the participants brought up links between abstract
algebra and secondary school geometry. This not only underpins the statements about
abstract algebra connecting seemingly disjoint topics within mathematics, it also shows
how important it is to include geometry when teaching abstract algebra. A full list of all
the connections can be found in the mentioned paper.
Additionally, according to Burn [
6
], permutations and symmetries are the fundamental
concepts of group theory. Consequently, while being deeply connected to group theory,
geometrical approaches allow for another opportunity to make connections between sec-
ondary school mathematics and abstract algebra. We conclude that studying groups that
consist of geometrical transformations such as the dihedral groups
D3
or
D4
offer a lot of
potential for a fruitful teaching concept.
On a side note, Burn further reports on a course on abstract algebra successfully
introducing groups in a pre-axiomatic manner via geometry. This suggests yet again that
the abstract axioms of group theory are not necessarily needed in order to work with groups.
Given that students in secondary school level have never faced any axiomatic structure
before it seems obvious that an approach without axioms will be much more accessible as
the threshold will be much lower. This is recurring aspect we will come back to.
3.2.3. Properties of Binary Operations
In school mathematics, properties of binary operations are often taken for granted.
For example, it is never questioned why addition of real numbers is commutative, and
the existence of zero is obvious. However, the idea of the identity transformation of an
equilateral triangle being an actual element of the set can be confusing; after all, nothing
happened to the triangle [
1
]. So when trying to generalize such concepts, one has to evoke
a sense of caution in the student. A study by Melhuish [
44
] has shown that students
overgeneralize and conflate properties such as associativity and commutativity. In addition,
they found that conceptual understanding of groups is tied to the understanding of binary
operations. Especially since geometric groups will be part of the concept and most of them
are non-abelian, it will be crucial to look at the properties in detail and compare them with
the other groups.
Also the inverse element poses many hurdles and can’t be neglected. Students have
trouble seeing the conceptual familiarity of an inverse function
f−1
and a multiplicative
inverse
a−1
as Wasserman [
45
] found using task-based interviews. The concept maps
produced in the context of his study generally showed that the participants had trouble
grasping the concept of inverses. In particular, it seemed that there were no differences
spotted between right and left inverses. However, one can easily construct examples where
such a differentiation is necessary. There is a fundamental difference between the equations
cos−1(cos(π)) = π
and
cos(cos−1(π)) = π
. The latter requiring complex numbers to be
made sense of. Also, the generic square root function
f:R≥0→R≥0,x7→ √x
ist not a left inverse of the square function
g:R→R≥0,x7→ x2,
for example
(f◦g)(−
1
) =
1. The study suggests that the inverse can’t properly be
understood by just looking at functions since the properties of the identity element and
commutativity in general seem to be disregarded or overlooked too easily in that case. So

Mathematics 2022,10, 703 8 of 24
for dealing with inverses it is suggested to use both function sets and number sets, which
is deeply connected to the first two key findings we presented.
Additionally, a large scale assessment by Brown et al. [
46
] has shown that an exper-
imental approach to abstract algebra is fruitful for learning about the concepts of binary
operations and groups. For our teaching concept we conclude that a hands-on approach
is required for dealing with the problems that properties of binary operations seem to
entail. Experimenting and physically engaging with the mathematical objects is suggested
to invoke a deeper understanding of the contents.
3.2.4. Interim Conclusions
Considering all the problems that have been excavated in contemporary literature, we
categorized 3 key findings for our teaching concept which later will justify the contents.
Not only does the research deliver crucial hints as to which challenges come with teaching
abstract algebra, the key findings also present solutions and ways of teaching it. To reference
the key findings, we established an abbreviation for each one (cf. Table 2):
Table 2. Key findings derived from modern research on abstract algebra teaching.
Key Finding Abbreviation References
Solving Equations,
working with familiar number sets (N,Z,Q,R)K1 [1,5,39,40]
Symmetries,
experimenting, engaging physically K2 [1,6,24,40,42,46,47]
Properties of binary operations K3 [1,44,45]
From the key findings in Table 2one can also immediately derive content areas of
abstract algebra and assign them to each finding (cf. Table 3).
Table 3.
Key findings derived from modern research on abstract algebra teaching as well as the
contents assigned to each finding.
Key Finding Suggested Contents
K1 Modular Arithmetic, Cyclic Groups
K2 Dihedral Groups
K3 Comparing (R,−),(R,+) etc.
It is noteworthy that the contents assigned to K1–K3 are precisely the ones presented in
Table 1. Moreover, the contents are very much in line with a list obtained by Wasserman [
8
]
where he explored abstract algebra yet again, only this time through the lens of school math-
ematics, constructing a progression of four content areas that are potentially transformed
by the perspective abstract algebra provides: Solving equations, structure of sets, inverses
and arithmetic properties. We observe that there is a striking consensus throughout both
bodies of work when it comes to selecting groups for an introductory course.
4. Design Principles and Structure of a New Teaching Concept
In this section, we want to take the contents previously distilled from literature and
arrange them in a certain order so that the resulting learning trajectory is the most fruitful
according to research. A very famous statement in this regard comes from the ’new math’
critique Why Johnny Can’t Add by Kline [
21
] and we take it as a general guideline for
our concept:
“Abstraction is not the first stage, but the last stage, in a mathematical develop-
ment.” [p. 98]

Mathematics 2022,10, 703 9 of 24
4.1. Realistic Mathematical Education (RME)
For purposes of bridging the gap between secondary school mathematics and ad-
vanced mathematics it is necessary to reverse the order in which mathematics is usually
presented [
48
]: We will first look at examples to draw on previously existing knowledge,
refining the concepts further and only in a last step present formal and abstract notions.
For our teaching concept we therefore adopt the instructional design principles of RME
(Realistic Mathematical Education, cf. [
49
]), meaning that the axiomatic and abstract nature
is not the starting point but the last goal of the teaching concept. The basic idea of RME is
that the construction of mathematical contents should be realistic, representing the process
of constructing a definition rather than just presenting the final result. In other words,
one should enter a mathematical domain with contexts that students can make sense of
and only later then generalize the underlying concepts, giving students time to gradually
develop an understanding. This method is especially suited for complex topics such as
abstract algebra [
50
], hence Freudenthal [
7
] coined three hierarchical levels of activity when
engaging with abstract algebra:
•Level 1:
An algebraic property emerges implicitly in the student’s intuitive activity
with an example structure.
•Level 2:
The property appears explicitly in the student’s general description of his or
her activity with the structure.
•Level 3: The student repurposes the property as a lens to classify other structures.
We intend on realizing this hierarchical process exactly as it is described above within
a time frame of three teaching units. Considering students have no prior knowledge of
the formal foundations this surely marks an ambitious goal that can only be achieved by
trimming back the axiomatic nature of the subject at hand. The main goal throughout the
entire process remains “from the concrete to the abstract”. Thus, it is recommended, as
described in Section 3, to start with visually accessible contents like geometrical groups.
Exploring the dihedral groups
D3
and
D4
(K2 and K3) will therefore be the contents of
lecture 1 and lecture 2 as they offer a great potential for visual stimuli, while exploring the
more abstract modular arithmetic via cyclic groups
Zn
(K1) and unifying all the concepts
will be the goal of the final lecture 3.
4.2. The EDUS-System
On an additional note, for justifying the didactical approach of our teaching concept,
we will use the EDUS-System. The EDUS-System is a framework designed by Lee [
48
],
specifically tailored for guiding the transition between school algebra to abstract algebra.
The acronym EDUS encompasses 4 characteristics that are central to developing a structural
perspective in mathematics:
•Extending (CE)
the context in which a set of existing understandings are situated. For
our concept, this is incorporated by considering geometrical transformations of the
equilateral triangle and the square.
•Deepening (CD)
the level of existing understandings of a certain, single mathematical
object. For our concept, this is incorporated by deepening the understanding of the
object map and inverse map.
•Unifying (CU)
existing understandings that were previously unrelated by the student
under a specific overarching mathematical object. For our concept, this is incorporated
by unifying the understandings of number sets and the properties of binary operations
defined on them.
•Strengthening (CS)
the links between existing understandings of more than one
mathematical object. For our concept, this is incorporated by looking at similarities of
the different sets and deriving the magma structure from those observations.
Merging the educational ideas of RME and EDUS now allows for a highly customized
teaching concept of which the schedule is presented in Figure 1.

Mathematics 2022,10, 703 10 of 24
unifying
deepening
Level 1
Level 2
Level 3
Lecture 1: D3
K2, K3, CE, CD
Lecture 2: D4
K2, K3, CE, CD
Lecture 3: Zn
K1, K3, CU, CS
Figure 1.
Schedule for the teaching concept, including the RME level, the EDUS characteristics as
well as the key findings and contents for each lecture.
As illustrated in Figure 1, lectures 1 and 2 will be located within both level 1 and
level 2 of RME. Looking at the non-abelian nature of these groups it will be clarified how
concatenation leads to non-commutativity for the given geometrical transformations so
this property of groups will first be implicit and later explicit. The identity element and
inverses of the groups will be explicit only in lecture 3 when all the contents can be looked
at simultaneously to reach unification. Only then we enter level 3 by comparing the new
structures to the ones students have already seen in school mathematics.
Also we have decided to make K3 part of every lecture since literature suggests that
problems and difficulties of binary operations seem to be the most crucial aspect about
learning group theory [44].
4.3. Connecting Abstract Algebra to Algebraic Thinking
Since teaching abstract algebra is at its’ core a teaching of the fundamentals underlying
school algebra it seems apparent that another priority of this teaching concept should be
the development of algebraic thinking. Although the notion of algebraic thinking in the
literature is mainly tied to the learning of primary school students and thus a concept for
the early stages of school mathematics, we argue that when dealing with abstract algebra
the same ideas and principles apply: For example, composing functions instead of numbers
is a sufficiently new step of generalization to pose new hurdles for learners, as seen in [
45
].
Even though the term Algebraic thinking is not a defined construct itself, much work
has been done to describe the aspects of mathematics it entails. For example, according
to [51], algebraic thinking encompasses among other things
• generalizing mathematical structures (cf. CE, K1, K2, K3)
• exploring and symbolizing functional relationships (cf. CU, K2)
• building mathematical arguments that reflect more generalized forms (cf. CE, K3)
• reasoning about abstract quantities (cf. CS, K1)
and according to [52] it develops from
• generalized arithmetic (cf. CE, K1, K3)
• functional thinking (cf. CU, K2)
• application of generalizations as a modeling language (cf. CS, K1, K3).
Here, in brackets we have assigned the characteristics according to the EDUS-System
(cf. Section 4.2) and our key findings (cf. Table 2) to each item in the two lists shown
above. This highlights the striking similarities between the introduced frameworks for
abstract algebra and the aspects of algebraic thinking. For example, dealing with modular
arithmetic builds on generalizing mathematical structures of secondary school arithmetic
such as
(Z
,
+)
or
(Q\ {
0
}
,
·)
. Or, as another example, modeling the transformations
of different polygons can be seen as direct applications of mathematics as a modeling
language. Thus, the teaching concept directly draws on the fundamental aspects that
promote algebraic thinking.
Another aspect often linked to algebraic thinking is that mathematical generalization
starts by identifying mathematical patterns and relationships [
53
] since they contribute

Mathematics 2022,10, 703 11 of 24
to the development of process-relational skills such as mathematical representation and
abstract thinking [
54
,
55
]. When dealing with transformations of polygons, for example,
mathematical patterns can be visually identified and recurring themes can be observed (i.e.,
for odd
n
the reflection lines of
Dn
pass through each vertex exactly once). Additionally,
since the subgroup of rotations of Dnis always isomorphic to Zn, functoinal relationships
between the different structures can be explored in an engaging way. We will elaborate
these synergies in more detail in the next section, when introducing the learning trajectories
of each lecture.
5. The Hildesheim Teaching Concept of Abstract Algebra
The concept is intended as an introductory course to abstract algebra for upper level
students of secondary school (grade 11/12, 16–18 years), focusing on exploring the concept
of magmas. A minimal time frame of three 90 min. lessons is required to implement it in
mathematics classrooms. In the following, we will outline our ideas in terms of didactical
approaches to the contents we derived from literature in the previous sections. A full
teaching manual as well as all the work sheets can be obtained from the corresponding
authors in either English or German. The learning trajectory derived from Sections 3and 4
looks as follows (cf. Figure 2):
School Mathematics
Magmas
Unit 1 (D3):
Composition, Inverses,
neutral El., Commutativity
Unit 2 (D4):
Cayley Tables,
Substructures
Unit 3 (Zn):
modular arithmetic,
Isomorphism
Figure 2.
The learning trajectory of the Hildesheim Teaching Concept of Abstract Algebra with its’
contents spanning three teaching units.
The core of the concept will be exploring groups via symmetries according to Burn [
6
]
and starting with looking at equilateral triangles as starting activities, synthesizing the
ideas of Wasserman [
1
] and Larsen [
47
], but building upon them and making the concepts
experienceable hands-on, keeping Dubinsky et al.’s [
24
] concerns in mind, and thus reduc-
ing the level of abstraction. The ’hands-on’ part will be carried out by providing students
with triangles made of acrylic glass, enabling physical engagement with the mathematical
contents. For reasons we will later elaborate, it is important that the material is transparent.
The students thus have the opportunity to perform abstract group actions in a non-abstract
way by rotating and flipping their triangle. This method of students actively engaging with
the important concepts supports an experiential basis for understanding ideas of abstract
algebra [40].
5.1. Unit 1: Introducing the Dihedral Group D3
5.1.1. Exploring the Transformations
In the beginning of the first unit, the students will explore all possibilities of how an
equilateral triangle can be mapped onto itself. To make this process more engaging, we

Mathematics 2022,10, 703 12 of 24
prepared acrylic glass triangles so the students can perform the geometrical transformations
hands-on (cf. Figure 3).
Figure 3.
The acrylic glass triangles can be used to perform the elements of
D3
hands-on (own
photograph). The triangles are sized exactly like the image containing all the elements on the
worksheet (cf. Figure 4).
Once the students have gathered some of the possibilities the question raises whether
those are all possibilities and if not, how to find the remaining ones. We introduce the first
step of abstraction via numbering all the vertices of the triangle with the numbers 1–3. The
material for the triangles was chosen to be transparent so that the numbers can be seen
when performing reflections, flipping the triangle on its’ backside.
It can now be observed that each mapping corresponds to a certain manipulation of
the vertices, and vice versa, each manipulation of the vertices corresponds to a mapping.
This turns the quest of finding all the mappings into a trivial one—one simply has to find
all combinations of numbers. For the first vertex, there are 3 possible numbers, for the
second vertex only 2 and then the number of the last vertex is fixed. This yields 3
!=
6
possible mappings. Of course, this method will not be successful when exploring the group
D4
in the second unit, because
D4S4
unlike
D3∼
=S3
. But by then the students will have
worked with D3already and have a better understanding of the contents.
Inevitably, the question will arise as to why “doing nothing” is also a legal choice of a
mapping—since the motivation of having an identity element in the set is not yet feasible,
we will simply describe the rotation by 360◦as identity mapping in id notation.
5.1.2. Describing the Transformations
Once all the mappings have been found the students’ next activity will be describ-
ing them geometrically, assigning symbols to them and in a last step gathering all the
descriptions in a tabular (cf. Table 4). The students’ notes in Larsen’s [
47
] study on stu-
dents exploring the dihedral groups suggest that students will have non-intuitive ways
to describe the elements of
D3
, making operations and operation tables very difficult to
read. This is no surprise, since they don’t know yet what content comes next and what
the established abbreviations will be used for. For our concept it is therefore important to
denote each element of
D3
with a very visual and self-explanatory symbol. For example,
the permutation
(
123
)
can be described as rotating the triangle counter-clockwise by 120
◦
,
so the symbol
r120
suggests itself. Analogically one has
r240
and
r360
. One might interject
here that only two symbols are required for describing all the elements of
D3
, since it is
generated by just two elements. But working with the symbol
r2
presumes knowledge
of the agreed notation
r2=r◦r
, and compositions have not been introduced yet, so the
symbol of each element has to be independent of the other symbols. For this reason, we
will not use the notations D3={r,r2,s,rs,r2s, id}used in mathematics literature.

Mathematics 2022,10, 703 13 of 24
Table 4. Two exemplary rows of the tabular containing all the elements of D3.
Mapping Description
r120
120°
12
3
31
2
Rotating the triangle
counter-clockwise by 120◦.
We will denote this mapping
with the symbol r120.
s1
12
3
13
2
Reflecting the triangle on the
angle bisector through
vertex 1. We will denote this
mapping with the symbol
s1
.
.
.
..
.
.
For the identity element of
D3
, one can argue that since the triangle remains identical
under
r360
, another plausible symbol might be
id
. When describing the permutations
(
12
)
,
(
23
)
and
(
13
)
however, we face a new problem: Since a triangle is part of the plane,
the mappings are geometrically viewed as reflections. A reflection, however, can’t be
carried out with physical objects such as our triangles. In fact, the only way to perform
the corresponding manipulations of the vertices is to rotate the triangle around a bisecting
line, which requires three dimensions. Thus it has to be made very clear that the physical
triangle is just a representation, and in the euclidean plane the mapping has to be described
as a reflection. We finally present the students with an image containing all the mappings
(cf. Figure 4), the triangle in the image is of the same size as the glass triangles, so one can
put the triangle on the paper and perform the transformations as shown by the arrows in
the image.
s1
s2
s3
r
Figure 4. Image containing the elements of D3.
Before moving on to the next step of the teaching unit, we gather all the mappings and
put them in a set: D3={s1,s2,s3,r120,r240, id}.
5.1.3. Exploring the Composition
Exploring the composition of the transformations in D3poses two central problems:
1. The (for students) somewhat unintuitive definition of composition, and

Mathematics 2022,10, 703 14 of 24
2. compositions involving reflections.
Firstly, a composition of two maps
f:A→B
and
g:B→C
is defined by
(g◦f)(a) =
g(f(a))
for all
a∈A
. This means, even though
g
comes first when reading
g◦f
, it comes
last in the mathematical sense, because arguments of maps are usually denoted on the right-
hand side. One could write images as
(a)f
and then proceed with
(a)( f◦g) = ((a)f)g
, so
the order of execution is coherent with the order of reading, but over the years the notion
(a)f
has more or less completely vanished. This dispute has to be addressed before any
compositions in D3can be computed.
Secondly, compositions with reflections can be very misleading. Since the reflection
lines in the descriptions are tied to the corresponding vertex number, it seems obvious that
the reflection lines also get transformed. For example, when computing
s1◦r120
one may
assume that the rotation also rotates the bisecting line, effectively turning
s1
into
s2
if it is
performed afterwards (cf. Figure 5).
12
3
31
2
21
3
r120 s1
s3
Figure 5.
When also rotating the object used to describe
s1
, the reflection can change under certain
compositions, leading to fundamental errors. Obviously s1◦r120 6=s3.
It is therefore crucial to note that the reflection lines never change positions, i.e., if the
left vertex is assigned the number 1 yielding the name
s1
for the corresponding reflection,
the reflection line will always go through the left vertex (cf. Figure 6). To avoid this error
we instruct the students to put their triangles on the sheet with the image containing all
the elements (cf. Figure 3)–this way the reflection lines stay in position when rotating
the triangles.
12
3
31
2
32
1
r120 s1
s2
Figure 6.
Keeping in mind that the reflection lines don’t change yields the correct result
s1◦r120 =s2
.
Another crucial part when working with compositions for the first time is noting that
in general geometrical transformations do not commute. In our concept we immediately

Mathematics 2022,10, 703 15 of 24
address this fact by letting students compute both
r120 ◦s1
and
s1◦r120
and compare
the results.
The last crucial part when exploring compositions is noting that when composing
any two elements of
D3
it will always result in another element of
D3
, meaning that the
set is closed under composition. One can verify this explicitly by computing all possible
36 combinations and filling of a Cayley Table. In our concept however, this will happen in
the second teaching unit. For now, one can justify this fact with a compelling argument: If
every element of
D3
maps the triangle to itself, then composing any two of those elements
will also map the triangle to itself—since there are only 6 such possible mappings and we
identified all of them, the result must again be an element of D3.
5.1.4. Exploring the Group Structure of D3
In the last section of the first teaching unit, we will explore some aspects of the group
structure of
D3
for purposes of later generalizing our findings in the last unit. In this regard,
we can explore two more aspects with the students: Inverses and the neutral element.
When computing compositions of the form
id ◦r120
,
s1◦id
, etc. it will quickly become
obvious that
id
has no impact on the composition. Leaving the triangle identical always
means that the outcome of the composition is solely determined by the other component.
The identity element is unique in this regard, and we can state that for any
x∈D3
we have
x◦id =id ◦x=x
. With having established the neutral element, we can move on and
observe that each transformation of the triangle can be reversed. Exploring the reverse
transformations with the students and gathering them in a table will yield Table 5.
Table 5. The elements of D3and their inverses.
Transformation s1s2s3r120 r240 id
Reverse
Transformation s1s2s3r240 r120 id
This can be further formalized by saying that for each element
x∈D3
there is an
element
x−1∈D3
reversing it, meaning that
x◦x−1=id
and
x−1◦x=id
. By looking at
the table it is also clear that the reverse transformations are unique for each element. It is
expectable for students to argue that the inverse of
r120
is
r−120
. By letting them perform
the transformation
r−120
and looking at the vertices of the resulting triangle, it will be easy
to identify r−120 =r240.
To conclude the teaching unit one can deepen the concepts by computing more ad-
vanced compositions, such as
(s2◦r240)−1
or finding all
x∈D3
that solve the equation
s1◦x=r120. For the latter mentioned exercise, the students may either
• try out all six possible x∈D3by drawing the transformations as seen in Figure 6,
• try out all six possible x∈D3physically by taking the glass triangle or
• note that one can reverse the reflection s1and thus compute x=s−1
1◦r120.
If time allows it, in a last activity the students can explore that each element of
D3
can
be expressed by using only
r120
and
s1
, leading to the conclusion that all the transformations
of the triangle are built out just of two transformations, so in a sense
r120
and
s1
can be
viewed as the “atoms” of triangle transformations.
5.2. Unit 2: Introducing the Dihedral Group D4
In the second of the three teaching units, we will first recap the learnings of the first
unit and then guild upon them by exploring the group
D4
. The students are by now familiar
with the concept behind the geometrical transformations dihedral groups consist of. This
means that the contents of composition, inverses, the neutral element and commutativity
do not require an equally in-depth discussion anymore. Many of the educational ideas

Mathematics 2022,10, 703 16 of 24
elucidated in Section 5.1 can be implemented in the same way so we will not go into detail
anymore with regards to those.
5.2.1. Cayley Tables
The first activity in the second unit will consist of establishing the Cayley Table for
D3
,
this both acts as a recap so the relevant concepts are present within the students minds and
as a setup for looking at substructures at a later point of this unit. As computing all possible
combinations might be time consuming and end up being a dull repetitive exercise, we
shortcut this by presenting a Cayley Table that’s already halfway filled (cf. Table 6).
Table 6. Cayley Table for D3.
◦id r120 r240 s1s2s3
id id r240 s1s3
r120 r120 r240 id s3
r240 id s3
s1s1s2id
s2s2id r120
s3s3s2r120 id
When filling the table we of course have to explain how this is done. One has to decide
for either of two possibilities: rows first or columns first. For our concept we decided to
go row first and then column, meaning that, for example, in the row of
r120
and column
of
s1
we have
r120 ◦s1
. Once the table is filled, the students can discuss on how to read
directly from the table that the composition
◦
is not commutative and what the table would
look like if it was. Also, it can be discussed how such tables can be used to quickly identify
reverse transformations.
5.2.2. Exploring the Transformations
Even though the students are now familiar with the concepts, the group
D4
will pose
two new challenges. The first challenge lies within the fact that
D4S4
, as mentioned
before. This means that the task of finding all possible transformations can’t simply be
solved by looking at all the 4
!
positions of vertex numbers. To supply students with an
alternative, we refer to the fact that all the elements of
D3
could be obtained by taking one
rotation and one reflection and looking at all possible compositions and that the same is
true for D4.
We argue that a lot can be learned from clarifying the question why the method of
finding all positions of the vertices does not work in the case of the square. For example,
we can discuss as to why the map represented in Figure 7can never be a transformation of
the square.
12
3
4
12
4
3
Figure 7. The map above is not a transformation of the square, or, in other words, (34)∈S4\D4.
When working with their acrylic glass squares, the students will find out that it is
impossible to perform the transformation in the figure above–the positions of 3 and 4 can
only be achieved by rotating the square, however, this will also permute 1 and 2. Thus the
students will see that the alternative way presented by the teacher has to be chosen. If, for
example, we take r90 and s1, we get to Table 7.

Mathematics 2022,10, 703 17 of 24
Table 7. Generating the group D4with r90 and s1.
Transformation
r90 r180 r270 id s1s2s3s4
Construction
r90 r90 ◦r90 r3
90 r4
90 s1r90 ◦s1r2
90 ◦s1r3
90 ◦s1
with the corresponding image (cf. Figure 8).
s1
s2
s3
s4r
Figure 8. Image containing the elements of D4.
The second challenge comes with finding symbols for the reflections. Since the re-
flections are no longer tied to a specific vertex, a single number can’t be assigned to
them in a plausible manner. We therefore decided to start with the lower left corner
and counting the reflections counter-clockwise as seen in Figure 8, resulting in the set
D4={id, r90,r180,r270,s1,s2,s3,s4}.
5.2.3. Exploring a Substructure
After confronting students with similar activities of categorizing and describing all the
elements in a table and finding the neutral element as well as all the reverse transformations,
we proceed with taking a closer look at the Cayley Table of
D4
. If the first row and column
of the tables are arranged in a certain order, one can identify that the rotations build a
smaller Cayley Table within the larger one (cf. Table 8).
Table 8. Cayley Table of D4.
◦id r90 r180 r270 s1s2s3s4
id id r90 r180 r270
r90 r90 r180 r270 id
r180 r180 r270 id r90
r270 r270 id r90 r180
s1
s2
s3
s4
In other words, the rotations build a closed subset where we have the neutral element
and reverse transformations. The students can then explore the table to see if they find other
subsets with the same properties, obvious candidates are
{id}
,
{id
,
s1}
,
{id
,
s2}
,
{id
,
s3}
and
{id
,
s4}
. It can be discussed that, for example, any subset containing both
r90
and
s1
already
leads to the entire
D4
or that the subset of reflections
{s1
,
s2
,
s3
,
s4
,
id}
is not closed under
composition. Since in the third teaching unit the group
({id
,
r90
,
r180
,
r270}
,
◦)
will reappear
in the form of
Z4
, we will establish the “Rotation Set”
R4={id
,
r90
,
r180
,
r270} ⊂ D4
to later
compare it to Z4.

Mathematics 2022,10, 703 18 of 24
5.3. Unit 3: Introducing the Cyclic Groups Zn
An introduction of cyclic groups can happen in many different fashions, although
usually is done be constructing it via the quotient group
Z/(n)
. Since
Z
is cyclic and thus
abelian, every subgroup is normal and one doesn’t need to bother with checking whether
the construction is well-defined. For dealing with this subject for the first time however,
this requires far too many technical tools, so we will adopt the elementary construction by
Kirsch [56]. For Znhe simply defines the operation ⊕on the set {0, . . . , n−1}via
⊕:(a,b)7→ Remainder of a+bin the division by n.
This marks an introduction to cyclic groups solely based on arithmetic of primary
school. While still maybe abstract for students, this can be made sense of by referring to
mathematical tools they’re familiar with.
5.3.1. Exploring Residue Classes
Introducing cyclic groups with this approach basically translates to introducing residue
classes—Residue classes can be motivated by examples from everyday life:
1.
Let’s imagine waiting at a station for a train. It is already late, 10:00 p.m. to be precise,
and according to the schedule the train will not arrive before 01:00 a.m. A quick
mental calculation tells us we need to wait for three more hours, since 10 +3=1.
2.
Then we remember that we have to study for a test. Today is 26 November, and
the test is precisely in two weeks. Another mental calculation tells us that because
November has 30 days, the test will be on 10 December, since 26+14 =10.
Discussing these examples with the students results in two observations: The calcula-
tions in the examples are obviously wrong, however, they provide a plausible answer that
in the real world makes sense. Thus the question rises whether these weird additions can
also be made sense of mathematically. In order to further explore these ideas, we pick up
the first example and simplify it: We imagine that one of the moons of Jupiter rotates faster
than the earth so that a day there only lasts 5 h. If we lived on this moon, our clocks would
therefore only have the 5 digits 0, 1, 2, 3 and 4. If it was 4 o’clock, then in 3 h it would be
2 o’clock (cf. Figure 9).
0
4
32
1
+3
Figure 9. A clock on the fictitious Jupiter moon yielding to 4 +3=2.
A time table on this moon would thus look like presented in Table 9.
Table 9. Timetable on the fictitious Jupiter moon.
⊕01234
001234
112340
223401
334012
440123
From this we derive that this table looks very similar to the Cayley tables explored
in the previous units, evoking the presumption that with similar structures we can also

Mathematics 2022,10, 703 19 of 24
make mathematical sense of the given examples. In our concept we have decided to
first just look at the residue classes of the natural numbers
N
and once the foundations
has been laid, expand the concept to
Z
. Otherwise the minus-symbol gets involved,
interfering with the goal that we look at a structure with a single binary operation. Also,
negative representatives of the residue classes are not needed at any point in the way
described above.
In order to further approach the residue classes, in a next step we will start to explore
which numbers are treated as the same in the above mentioned calculations and observe
that writing the natural numbers down in a certain way yields a distinctive pattern (cf.
Figure 10).
[0] [1] [2] [3] [4]
0
5
10
15
20
.
.
.
1
6
11
16
21
.
.
.
2
7
12
17
22
.
.
.
3
8
13
18
23
.
.
.
4
9
14
19
24
.
.
.
Figure 10. The residue classes visually demonstrated.
Conflating all the numbers in each column to one set gives the sets
[
0
]
,
[
1
]
,
[
2
]
,
[
3
]
and
[
4
]
. With the students we then can explore that all the numbers in the sets have a gap that is
a multiple of 5 and thus we conclude that they all have the same remainder when divided
by 5:
2=0·5+2
7=1·5+2
12 =2·5+2
17 =3·5+2
.
.
.
This suggests that these sets are named residue sets with respect to 5 and we can note
R5={[0],[1],[2],[3],[4]}. For further specification one can also write R5={[0]5,[1]5,[2]5,
[3]5,[4]5}when intending to work with other residue classes as well.
5.3.2. Establishing the Addition of Residue Classes
With the residue classes established, we can finally start to tackle the examples given
at the beginning of the class. For this, we let the students explore different additions. For
example, from the equations
8+9=17
3+4=7
9+13 =22
we can derive that a number in
[
3
]
plus a number in
[
4
]
always resulted in a number in
[
2
]
and we can generalize this observation. Every number
a∈[
3
]
can be written as
a=
3
+k·
5
and every number b∈[4]can be written as b=4+`·5 for some k,`∈N, resulting in
a+b= (3+k·5) + (4+`·5) = 2+ (k+`+1)·5∈[2].

Mathematics 2022,10, 703 20 of 24
We conclude that we can write
[
3
]+[
4
]=[
2
]
and view this as an addition of two sets rather
than just two numbers. So one entry of the time table on our Jupiter moon is made sense of
and the others work exactly the same. It can be further observed that
[
0
]
acts as a neutral
element and the table of inverse elements looks like Table 10.
Table 10. The inverses in Z5.
Residue
Class [0] [1] [2] [3] [4]
Inverse Class [0] [4] [3] [2] [1]
5.3.3. Modular Arithmetic
With the visual ideas of the Cayley table and the Jupiter clock in mind, it is now
possible to generalize the findings and also look at multiplication and other residue classes.
However, it must be pointed out to students that multiplication does not result in the same
structure. For example, in
R4
the class
[
2
]
does not have an inverse class.With this in mind
we can now look at equations that look similar to equations on
Z
but have an entirely
different behaviour. For instance:
•
The equation
x2= [
0
]
has only one solution when viewed in
Z
but two solutions when
viewed in R4.
•
The equation
[
2
]x= [
1
]
has no solution when viewed in
Z
but one solution when
viewed in R5.
The Cayley tables can also be used to solve more advanced equations such as
[
3
]·(x⊕
[
2
]) = [
2
]
in
R4
. Similarly to solving equations in
D3
and
D4
, the students can either try
to make use of the algebraic structure equipped on the sets, or they can simply try out all
possible combinations.
5.3.4. Generalizing the Findings
With having seen so many different operations and different sets within the three
teaching units, we use this for a final take on generalizing the findings, trying to unify the
concepts at least to some degree. We do so by putting the Cayley Tables of the rotation
subset
R4⊂D4
and the residue set
R4
next to each other and slowly manipulate the
symbols we invented. By embezzling the brackets, the table of the residue set can be
written as Table 11.
Table 11. The Cayley Table of Z4with slightly changed notation of the elements.
⊕0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
As for the rotations of the square, we note that it is not necessary to use the measure
of the rotation in the symbols. When rotating the square by 90 degrees, each vertex moves
exactly one position forward, so one could also write the rotation as
r1
and change
r180 →r2
,
r270 →r3
, and
id →r0
respectively. In the next step, since we’re dealing just with rotations
anyway, it is no longer necessary to use the letter
r
, the index number sufficiently describes
how the square is rotated. So we further simplify the notation to
r1→
1,
r2→
2,
r3→
3
and r0→0. The Cayley table thus can be written as seen in Figure 11.

Mathematics 2022,10, 703 21 of 24
◦id r90 r180 r270
id id r90 r180 r270
r90 r90 r180 r270 id
r180 r180 r270 id r90
r270 r270 id r90 r180
◦r0r1r2r3
r0r0r1r2r3
r1r1r2r3r0
r2r2r3r0r1
r3r3r0r1r2
◦0123
0 0123
1 1230
2 2301
3 3012
Figure 11. Transformation of the Cayley Table of R4when changing notations.
We observe that, apart from the operation symbol, both tables look exactly the same—
since the tables encompass the entire information of the structure, this means that rotating
squares or calculating sums of residue classes follows the exact same mathematical rules.
In other words, the sets we in both cases denoted with
R4
, consist of different symbols
but in the context of describing real-life problems the structure of the sets was identical.
With the remaining time of the final unit we can coin the name magma for structures
equipped with an operation and together with the students search for magmas hidden
in the conventional school curriculum, such as
(R
,
+)
,
(Z
,
−)
and so on. Depending on
the previous knowledge of the students, we can even illustrate the structure similarity of
(R,+) and (R>0,·)by using the exponential function (cf. Figure 12).
+. . . x. . .
.
.
..
.
.
y. . . x+y. . .
.
.
..
.
.
·. . . ex. . .
.
.
..
.
.
ey. . . ex+y. . .
.
.
..
.
.
Figure 12.
Structural similarities of
(R
,
+)
and
(R>0
,
·)
made visible by the exponential function.
Both structures just differ in symbolism.
We can conclude the final unit with the observation that many different aspects from
the real world look different at the surface, but when describing them mathematically
they reveal to be structurally identical. We have explicitly seen this with the examples of
rotating polygons and the calculations of times and dates—while different symbols and
ways of speaking are used to describe what’s going on, on a mathematical level there is
absolutely no difference. We can enlighten the students that from this observation an entire
mathematical theory emerged and that this theory is finding numerous applications outside
of mathematics, namely in computer sciences, physics, chemistry and even music theory.
6. Conclusions and Outlook
Even though the contents of abstract algebra may pose challenging hurdles, the re-
wards can be very beneficial, as pointed out in the first section. Additionally, fostering
process-relational skills in mathematics classrooms becomes a more and more relevant issue
in education research, and engaging creatively with new mathematical concepts as presented
in this course provides an ideal opportunity to train those skills. After all, less reproduction
of algorithms is involved, less solving problems to which the solution is already known
and just has to be rehashed with slightly different parameters. Mimicking solutions and
imitating patterns are no longer strategies that lead to immediate success [
40
,
57
], creating the
necessity to work with new and creative ways. Abstract algebra emphasizes mathematical
understanding in two ways according to Bass [
58
]: curricular, by making explicit often under-
developed or overlooked connections among different topics; and cognitive, through inviting
new requirements and methods in problem solving activities: “The power of mathematical
abstraction is its generality, thus having the potential to conceptually unify many apparently
distinct mathematical contexts” ([
58
], p. 127). Another discussion of the modes of relevance
of the term structure in the context of school algebra can be found in [
48
], describing how
problems can be transformed and solved much easier with a structural thinking, recognizing
that form and structure should be emphasized in school algebra [59].

Mathematics 2022,10, 703 22 of 24
And on a side note, the most recent mathematical tool in school mathematics is that
of differentiation, which by now is over 350 years old. In other words, the thinking in
school mathematics is very much a thinking of the past and what mathematics actually
encompasses vastly remains hidden throughout the entire school career. Changing this
certainly poses many educational problems for learners (cf. [
39
,
44
,
60
,
61
]), but, as we will
see, they can be overcome. An acceptance survey [
62
] of the Hildesheim Teaching Concept
for Abstract Algebra is currently being conducted with individual learners in the laboratory
setting—the results of which we will report in a follow-up paper. In the future, however,
the concept will be tested empirically, and a transition beyond mere laboratory studies is
planned, in order to evaluate the teaching concept in real teaching scenarios at secondary
schools in field studies. In the sense of the Design-Based-Research paradigm [
63
], we will
use insights gained from such research to refine the ideas behind the concept as well as
revise it both in terms of content and didactical approaches.
Author Contributions:
Conceptualization, J.M.V. and P.B.; writing—original draft preparation, J.M.V.
and P.B.; writing—review and editing, J.M.V. and P.B. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: No new data were created or analysed in this study.
Conflicts of Interest: The authors declare no conflict of interest.
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