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Citation: Veith, J.M.; Bitzenbauer, P.;
Girnat, B. Towards Describing
Student Learning of Abstract
Algebra: Insights into Learners’
Cognitive Processes from an
Acceptance Survey. Mathematics 2022,
10, 1138. https://doi.org/10.3390/
math10071138
Academic Editor: Michael Gr.
Voskoglou
Received: 14 March 2022
Accepted: 30 March 2022
Published: 1 April 2022
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mathematics
Article
Towards Describing Student Learning of Abstract Algebra:
Insights into Learners’ Cognitive Processes from an
Acceptance Survey
Joaquin Marc Veith 1,* , Philipp Bitzenbauer 2and Boris Girnat 1
1Institut für Mathematik und Angewandte Informatik, Stiftungsuniversität Hildesheim,
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 Mathematics, we presented a new teaching concept for abstract
algebra in secondary school mathematics, and we discussed findings from mathematics education
research indicating that our concept could be used as a promising resource to foster students’ algebraic
thinking. In accordance with the Design-Based Research framework, the developed teaching concept
is now being revised in several iteration steps and optimised towards student learning. This article
reports on the results of the formative assessment of our new teaching concept in the laboratory
setting with
N=
9 individual learners leveraging a research method from science education: The
acceptance survey. The results of our study indicate that the instructional elements within our new
teaching concept were well accepted by the students, but potential learning difficulties were also
revealed. On the one hand, we discuss how the insights gained in learners’ cognitive processes
when learning about abstract algebra with our new teaching concept can help to refine our teaching–
learning sequence in the sense of Design-Based Research. On the other hand, our results may serve
as a fruitful starting point for more in-depth theoretical characterization of secondary school students’
learning progression in abstract algebra.
Keywords: algebra; groups; secondary school; mathematics education
MSC: 97H40
1. Introduction
Abstract algebra, as a research field of mathematics education, has become increasingly
popular in recent years, as more and more benefits of its notions are being discovered. For
example, Wasserman [
1
] showed that dealing with abstract algebra vastly transformed the
beliefs and classroom practices of K-12 teachers. The participants stated that learning about
group theory made them view school mathematics in a different way, especially regarding
arithmetic and algebra. A similar study by Even [
2
] reflected this observation. Using an
abstract algebra course for in-service teachers, an investigation of how mathematics learned
could be connected to aspects of the mathematics school curricula. The teachers stated that
the notions of abstract algebra advanced their understanding about mathematics in general
and, on a more epistemological level, stated that abstract algebra progressed their views on
what mathematics is and what doing mathematics means.
However, simply because teachers draw great benefits from their field of work does not
imply that these benefits also translate from teachers to students in the classroom. Indeed,
it has been shown that there is no significant correlation between a teacher’s knowledge in
abstract algebra and students’ achievement in school algebra [
2
–
9
]. This conflict raises the
question of whether students themselves can benefit from learning abstract algebra if they
Mathematics 2022,10, 1138. https://doi.org/10.3390/math10071138 https://www.mdpi.com/journal/mathematics
Mathematics 2022,10, 1138 2 of 22
are confronted with it instead of only their teachers. To investigate this question, the new
Hildesheim Teaching Concept for Abstract Algebra was invented, and we reported on this in an
earlier contribution to this journal Mathematics [
10
]. By following the works of Griesel [
11
],
Leppig [
12
], and Freudenthal [
13
,
14
] as well as those of Wasserman et al. [
15
] and Lee [
16
],
a new teaching–learning sequence spanning three teaching units with a time frame of 90
min each, targeted at grade 11/12 students (16–18 years), was constructed. The details of
this teaching concept can be found in [
10
], so we will briefly reiterate the key notions here:
1. Key Notion 1:
In the first unit, the students explore the dihedral group
D3
of the equi-
lateral triangle and get in touch with the basic notions of group theory, such as inverses,
the identity element, and commutativity. They do so by working with a plexiglass tri-
angle, so the isometries can be experienced in a haptic way (Figure 1). The isometries
are described from a geometrical point of view (i.e., rotation by 120
◦
or reflecting on
the bisecting line through vertex 1), resulting in D3={s1,s2,s3,r120,r240, id}.
2. Key Notion 2:
In the second unit, the dihedral group
D4
of the square as well as
its substructures and the Cayley Tables are explored. The fact
D4S4
provides
new problems to solve, i.e., discussions of why certain permutations do not describe
isometries of the square.
3. Key Notion 3:
The concluding unit deals with modular arithmetic and isomorphisms
by looking at the cyclic groups
Zn
. Examples from everyday life, such as calculating
times and dates, serve as the motivation to introduce “new” additions. A crucial point
in this regard is realizing that Z4only differs in terms of notation from the subgroup
D4generated by r90.
Figure 1.
The plexiglass triangles can be used to perform the isometries of the triangle in a hands-on
way (own photograph). The teaching materials can be obtained from the corresponding author in
either English or German.
The next goal is to investigate how the instructional elements and didactical ap-
proaches developed for this concept are perceived by students. Additionally, we want to
gain the first insights into the thought processes underlying the conceptual development of
abstract algebra along the way (cf. Sections 4and 5). To achieve this goal, we first locate
the status of this research project within a Design-Based Research framework (cf. Section 2.1)
and, from its principles, derive a survey method that is suitable for addressing these goals
(cf. Section 2.2).
Mathematics 2022,10, 1138 3 of 22
2. Theoretical and Methodological Framework
2.1. Design-Based Research
Education occurs in multifaceted settings and involves many variables [
17
]. Therefore,
educational research should not be limited to evaluation research aimed at finding out
whether a certain intervention or method has a positive impact on learning, but educational
research should also contribute to the development of a (local) theory about student learning
in a given field [
18
]. A research framework that is aimed at “producing new theories,
artifacts, and practices that account for and potentially impact learning and teaching in
naturalistic settings” [
19
] (p. 2) is the Design-Based Research paradigm, sometimes also
referred to as a Design experiment [
20
,
21
], Design research [
22
–
24
], Development research [
25
],
or Developmental research [
26
,
27
]. Design-Based Research is widely ascribed to Ann Brown.
In her 1992 paper [
28
], she states that a “critical tension in our goals is that between
contributing to a theory of learning [
. . .
] and contributing to practice” (p. 143) and demands
that ”an effective intervention should be able to migrate from our experimental classroom
to average classrooms operated by and for average students and teachers, supported by
realistic technological and personal support” (p. 143). Thus, Design-Based Research “seeks
to increase the impact, transfer, and translation of education research into improved practice”
and “stresses the need for theory building and the development of design principles that
guide, inform, and improve both practice and research in educational contexts” according
to [29] (p. 16). An overview of the Design-Based Research paradigm is given in Figure 2.
Design
Implementation
Evaluation
Re-Design
Implications for TheoryTheory
Requirements
Formative Evaluation
(laboratory setting) Summative Evaluation
(field setting)
Figure 2.
Visualizing the process of Design-Based Research orientied to [
30
] specified for our research
project. This figure serves as a structuring element both for (a) our research project in general, and (b)
this paper in particular: While the design of a new teaching concept for abstract algebra including a
literature review (cf. theory, requirements) was presented in our earlier contribution [
10
], we now
focus on the second DBR cycle (implementation and evaluation) through a formative assessment.
The illustraced circles of development can be repeated, and of course with different methods.
In the literature, Design-Based Research (DBR) is described by five widely-accepted
characteristics (cf. [19,31,32]):
1.
Sequential cycles of design, evaluation and re-design: DBR is characterized by an iterative
process. DBR projects consist of sequential cycles of design, implementation, and
evaluation followed by re-design cycles which are based on the evaluation results of
the previous cycle [31], as shown in Figure 2.
2.
Real educational contexts: DBR ensures that research findings are transferable to real
educational contexts [
19
,
29
,
33
], which is why DBR projects do not only comprise
(experimental) laboratory studies but also field studies, e.g., research questions “about
the nature of learning in authentic learning environments” [
32
] (p. 3). In particular,
Mathematics 2022,10, 1138 4 of 22
researchers often conduct qualitative surveys in the early stages of their DBR projects
to (a) identify possible learning difficulties, and (b) refine their innovation based on
the results of such formative assessments. This may serve as a starting point for field
studies to evaluate the innovation’s learning effectiveness in a later DBR cycle.
3.
Use of mixed-methods: In addition to the abovementioned point, in DBR projects,
researchers take advantage of a pluralism of quantitative and qualitative methods to
complement insights into learning processes and complex educational situations from
different perspectives, e.g., via triangulation [34].
4. Theory building and design of educational innovation: In DBR projects, researchers build
on theories from the literature for the design of an educational innovation (e.g., a
new teaching concept, method or media) on the one hand (cf. Figure 2). On the other
hand, the evaluation of such innovations in authentic classroom settings may lead to
refinement of the developed instructional elements. However, these evaluation results
may also result in the derivation of new research questions, and hence, in stimuli for
new research questions in the field under investigation [35].
5.
Close interactions between researchers and practitioners: Lastly, DBR is characterized by
a close collaboration between researchers and practitioners [
19
]. ”This contributes
toward creating ownership and commitment from teachers and learners”
[32] (p. 2)
.
Furthermore, such close cooperation may help to uncover how new teaching innova-
tions are used in educational practice [
36
]. These insights are particularly important,
since empirical research has revealed that teachers often use innovative materials in a
different way than intended by the material developers [37].
The development and evaluation of the Hildesheim Teaching Concept for Abstract Algebra
is realized in the sense of the Design-Based Research paradigm. In our earlier contribu-
tion [
10
] published in Mathematics, we reported on the first design cycle and discussed
results from mathematics education research that indicate the potential for mathemat-
ics learning at the secondary school level that may emerge from introducing students to
concepts of abstract algebra. We, consequently, took into account the use of theoretical
frameworks from mathematics education for the development of a new teaching concept
for abstract algebra in a first step of our research project. A brief summary of the theoretical
background and the key ideas involved in our teaching–learning sequence was given in
Section 1of this paper. In this article, we focus on the second step of our Design-Based
Research program, namely the first formative assessment of our new teaching innovation
in the laboratory setting (cf. the next Section 2.2).
2.2. Acceptance Survey as a Method for Formative Evaluation
In the context of curriculum evaluation, Scriven [
38
] distinguishes between different
roles that evaluation can play. On the one hand, it can be used during the development of
curricula, and on the other hand, it can be used to evaluate a final product. In this context,
he introduced the concepts of formative and summative evaluation:
“Evaluation may be done to provide feedback to people who are trying to improve some-
thing (formative evaluation); or to provide information for decision-makers who are
wondering whether to fund, terminate, or to purchase something (summative evalua-
tion).” [38] (pp. 6–7)
Consequently, the implementation of a formative evaluation primarily serves the aim
of the developers to improve the development of the concept. Hence, before a teaching
concept is subjected to a large-scale empirical survey in the field, examination of the concept
in the laboratory setting by conducting an acceptance survey, sometimes also referred to as
aTeaching experiment [
39
] has proven fruitful. The technique of probing acceptance can be
traced back to Jung [
40
] and “gives insight into the plausibility of an information input in
terms of whether it makes sense to students. Probing acceptance thus means identifying
elements of the instruction that students accept as useful and meaningful information and
that they can successfully adapt during the one-on-one interview” [
41
] (p. 856). Hence,
Mathematics 2022,10, 1138 5 of 22
acceptance surveys are realized via one-on-one interviews, most often comprising four
interview phases (cf. [42]), as shown in Figure 3:
1.
Providing information: The interviewer provides information input in a similar way
as in a classroom lesson. Media can be used and any questions from the interviewee
are addressed.
2.
Survey of acceptance: The interviewer asks the interviewee to assess the information
presented in terms of whether the explanations were comprehensible and understand-
able: “What do you think about this topic?” or “Was there anything you could not
understand?” [
41
] (p. 857). The interviewee can, hence, also express criticism at
this point.
3.
Paraphrasing: In this phase, the interviewer asks the interviewee to paraphrase the
presented information in their own words or to independently repeat previously
heard explanations.
4.
Application: In this final phase, the respondent is given a short task that allows the
researcher to observe the student in a problem-solving situation. The student uses
the information provided in previous phases in their problem-solving process, which
enables the researcher to identify learning difficulties that might occur with specific
instructional elements.
Once all four phases have been conducted for one of the key ideas involved in the
concept under investigation, where the key ideas may be regarded “as elementary steps for
the topic” [
41
] (p. 855), the procedure is being repeated for the next key idea (cf. Figure 3).
Key
Idea 1
Key
Idea 2
Information
Acceptance
Paraphrasing
Application
Information
Acceptance
Paraphrasing
Figure 3. Visualizing the cyclic process of an acceptance survey (own illustration).
The cyclic process of acceptance surveys allows the researcher to evaluate students’
learning progression, on both conceptual and abstract levels. In particular, the method
allows the analysis of interaction effects between instructional elements based on learners’
conceptions [
43
]. This method has already been used in numerous research projects and
in the early stage of designing teaching–learning sequences in science education research
(cf. [44–48]), in particular, within Design-Based Research programmes (cf. Section 2.1).
Consequently, we conducted an acceptance survey to be used in the early stage of the
development of the Hildesheim Teaching Concept for Abstract Algebra for individual learners
working in a laboratory setting (cf. Section 3). In this article, we report on the results of
this formative assessment of our teaching concept, and we demonstrate how the results of
this study (cf. Section 4) can help to (a) refine the teaching concept’s instructional elements
on the one hand (cf. Section 5.4) and (b) derive implications for a theory about learning
abstract algebra (cf. Section 5) on the other hand. After the re-design of our innovation
based on the results presented in this article, we will, in a later cycle of our Design-Based
Research program (cf. Figure 2), evaluate its learning effectiveness in authentic learning
environments, i.e., by conducting a field study (cf. Section 6).
Mathematics 2022,10, 1138 6 of 22
2.3. Research Questions and Key Ideas
In order to gain insight into the usefulness of the instructional elements involved in
our teaching concept from a learners’ perspective, we sought to clarify how students accept
those elements as well as how and what criticism is being voiced. Since the two main
magmas introduced are the dihedral groups
D3
and
D4
and the amount of work that has
been put into visualizing the concepts, we address this acceptance by looking at how the
learning of isometries and permutations of the equilateral triangle are supported by the
material. The Cayley Tables are also part of this investigation, as they serve to link finite
magmas visually. Furthermore, we wanted to find out whether students prefer to work
with abstract symbols or permutations when dealing with isometries, and we also looked
into which approach is best for exploring symmetries of regular n-gons. Lastly, we want
to use this information to enhance the concept by looking at possible learning difficulties
students might encounter.
In summary, through this research, we aimed to clarify the following research ques-
tions with regard to the instructional elements within the Hildesheim Teaching Concept of
Abstract Algebra:
RQ1:
How do students accept the instructional elements within the Hildesheim Teaching
Concept of Abstract Algebra...
(a)
. . .with regard to introducing dihedral groups?
(b)
. . .with regard to introducing permutations as a tool to describe isometries?
(c)
. . .with regard to introducing the Cayley Tables?
RQ2:
Do students prefer to work with the abstract symbols of the dihedral group
D3
, or do
they prefer to work with permutations presented by matrices?
RQ3:
What learning difficulties can be expected in the implementation of the Hildesheim
Teaching Concept of Abstract Algebra?
To address the research questions with this acceptance survey more purposefully, we
derived six key ideas that match the research questions (RQ1–RQ3). The key ideas were
directly derived from the key notions of the teaching concept presented in Section 1. Table 1
shows the key ideas as well as the research questions they aim at clarifying:
Table 1. Key ideas of the acceptance survey.
Key Idea Description Research Question
K1 Isometries of the triangle RQ1 (a), RQ2, RQ3
K2 Composition of isometries RQ1 (a), RQ2, RQ3
K3 Permutations RQ1 (b), RQ2, RQ3
K4 Composition of permutations RQ1 (b), RQ2, RQ3
K5 Cayley Tables RQ1 (c) RQ3
K6 Magmas RQ3
Key ideas 1/2 and 3/4, respectively, describe the same mathematical content but from
different perspectives. This is necessary to address RQ2, i.e., to find out whether student
learning is better supported by the use of. . .
• . . . abstract symbols, such as r90 or s1, etc., or
• . . .concrete mapping rules presented in matrices, such as
r90 =1234
2341,s1=1234
1432, etc.
in the course of our teaching concept. For each key idea presented above, a four-item cycle
was developed in accordance with the structure of the acceptance survey presented in
Section 2.2. The questions asked by the interviewer in steps 2–4 varied in quantity and
Mathematics 2022,10, 1138 7 of 22
depended on the respective key idea involved. For example, only one task was prepared for
K1 (isometries of the triangle), whereas three exercises were prepared for K2 (composition
of isometries). This was necessary to confront the students with different types of problems,
enabling a deeper look into the learning process. The full interview guide consisted of the
units presented in Tables A1–A6 in the Appendix A.
3. Methods
3.1. Study Design and Sample
To clarify the research questions, we conducted an acceptance survey with
N=
9
high school students in grade 12 (ca. 17–19 years old), following [
41
,
44
,
49
]. The acceptance
survey was conducted in the setting of one-on-one interviews, each of which lasted about
60 to 90 min. Subjects were purposively selected so that 3 participants had a mathematical
profile (science branch), 3 had a partial mathematical profile (economics branch), and 3
had a non-mathematical profile (agricultural branch). Here, the branches refer to different
backgrounds that students in Germany can choose from grade 7 onwards. The branch,
among other aspects, determines how the subject of mathematics is weighted throughout a
student’s remaining school career. By cooperating with the headmaster, we ensured to that
each type was adequately represented in our study. None of the participants had received
any prior instruction in abstract algebra. As an external criterion, the last two report card
scores in mathematics were collected (
m=
10.11,
SD =
2.58). It is noteworthy that scores in
German high schools range from 0 to 15, where 0 is the worst and 15 is the best score. The
anonymized participants are shown in Table 2. All students participated on a voluntary
basis, according to ethical standards. Informed consent was obtained from all participants.
Table 2.
Anonymized participants involved in the acceptance survey. The profiles are abbreviated
with M (mathematical), PM (partial mathematical), and NM (non-mathematical).
Participant Score 1 Score 2 Profile
A 6 7 PM
B 7 6 PM
C 11 10 PM
D 10 11 NM
E 9 11 M
F 12 14 M
G 8 10 NM
H 10 11 NM
I 15 14 M
3.2. Data Analysis
The interviews were tape-recorded, transcribed, and evaluated using the coding
manual (cf. Tables A8–A14 presented in the Appendix A). For the transcription, the
rules established in the transcription system of Dresing and Pehl [
50
] were used. For the
evaluation of students’ responses regarding acceptance, paraphrasing, and the transfer
task, we used a scaling content analysis in accordance with [
51
]: a three-level ordinal scale
system was implemented, as is widely done in acceptance surveys to code the students’
responses. The evaluation of the transcripts was carried out by two independent raters,
and the intercoder reliability was evaluated via Cohen’s κ(cf. Table 3).
Mathematics 2022,10, 1138 8 of 22
Table 3.
Intercoder reliability for the different phases of our acceptance survey. Based on our coding
manual, the accepted standards according to [52] were met.
Phase of the
Acceptance Survey κ%-Agreement Judgement
According to [52]
Acceptance 0.77 94.4
substantial agreement
Paraphrasing 0.59 85.7 fair agreement
Application 0.78 95.8
substantial agreement
The research questions were addressed by looking at the different phases of the
acceptance survey. RQ1 was investigated through an evaluation of the acceptance phase,
RQ2 was investigated through an evaluation of the application phase, and RQ3 was
investigated by combining all phases. The analysis methods are described in more detail in
the following Sections 3.2.1–3.2.3.
3.2.1. Rating Acceptance
In order to evaluate the level of acceptance, the participants answered questions about
the comprehensibility of the mathematical content as well as about plausibility of notations
and symbols. The answers were classified according to a three-level ordinal scaled coding
system that distinguished between full acceptance,restricted acceptance, and no acceptance
(cf. [48]):
1.
Perfect acceptance (coded with numeric value 1): The explanations were accepted by the
participant without reservation and classified as plausible or understandable.
2.
Partial acceptance (coded with numeric value 0.5): The explanations were accepted by the
participant, but criticism was voiced.
3. No acceptance (coded with numeric value 0): The explanations were not accepted by the
participant. This means that the contents were not explained in a comprehensible way
or seemed implausible.
If the mean value of a acceptance level for one key idea was above the cut-off value of
0.5, it was seen as acceptable (cf. [
53
] (p. 72)). To illustrate the coding guidelines above, two
examples are provided in the context of K1 in Tables 4and 5, where (I) is the interviewer
and (S) is the student.
Table 4. Perfect acceptance of Participant Dand A1-2 (cf. Table A3 in the Appendix A).
Anchor Example for Perfect Acceptance
(I):
Do you find the symbols we’ve chosen to describe the isometries plausible?
(S): Yes, of course. For each vertex, we have
s1
,
s2
, and
s3
depending on which vertex we reflected,
and the other thing with the rotation was also plausible.
Table 5. Partial acceptance of Participant Gand A1-2 (cf. Table A3 in the Appendix A).
Anchor Example for Partial Acceptance
(I):
Do you find the symbols we’ve chosen to describe the isometries plausible?
(S):
Yes, but only with the descriptions next to them in the table, because in mathematics,
everybody uses their own symbols, which can be confusing. But with the description in the
table, it’s fine.
3.2.2. Rating Paraphrasing
In order to evaluate the levels of paraphrasing, the participants were asked to rephrase
the description of the mathematical contents presented earlier in their own words. Analo-
gous to the evaluation of the level of acceptance, the answers were classified according to a
Mathematics 2022,10, 1138 9 of 22
three-level ordinal scaled coding system that distinguished between fully adequate,partially
adequate, and not adequate:
1.
Fully adequate (coded with numeric value 1): The paraphrasing included all core aspects
of the content according to the coding guide.
2.
Partially adequate (coded with numeric value 0.5): The paraphrasing mentioned some
core aspects of the contend but misses others. The mathematical object was only
partially successfully described.
3.
Not adequate (coded with numeric value 0): The paraphrasing mentioned no core aspect
of the content or was wrong.
If the mean value of a paraphrasing level for one key idea was above the cut-off value
of 0.5, it was seen as acceptable (cf. [
53
] (pp. 75–76)). To illustrate the coding guidelines
above for each level of paraphrasing, examples are provided in Tables 6–8where, again, (I)
is the interviewer and (S) is the student. The complete coding guide can be found in the
Appendix Aof this article (cf. Tables A8–A14), following [44,49].
Table 6. Fully adequate paraphrasing of Participant Iand P3 (cf. Table A3 in the Appendix A).
Anchor Example for Fully Adequate Paraphrasing
(I):
Can you describe in your own words what a permutation is?
(S):
For me a permutation is, let’s say, some kind of table where you can look up for each vertex
how it got manipulated. For example, we can see where vertex 1 went as well as vertices 2
and 3.
Table 7. Partially adequate paraphrasing of Participant Hand P6 (cf. Table A6 in the Appendix A).
Anchor Example for Partially Adequate Phrasing
(I):
Can you describe in your own words what a Magma is?
(S):
Well, if we have a set, and composing two things gives another thing of the set, then we have
a set which we composed.
Table 8. Not adequate paraphrasing of Participant Eand P1-1 (cf. Table A1 in the Appendix A).
Anchor Example for Not Adequate Phrasing
(I):
Can you describe in your own words what an isometry of the triangle is?
(S):
An isometry of the triangle is a form of the triangle that has the same orientation and the
same area as before, but we can move it and change it.
3.2.3. Rating Application
In order to evaluate the levels of application, the participants were given tasks to
carry out on their own. Again, the performance of each student was classified according to
a three-level ordinal scaled coding system that distinguished between successfully solved,
solved with help and not solved:
1.
Successfully solved (coded with numeric value 1): The problem was solved independently
and without any help. The application was also evaluated as 1 if the student made an
error but immediately realized it and corrected it.
2.
Solved with help (coded with numeric value 0.5): Solving the task independently was not
possible. A correct solution could be found, however, after one clue was given by the
interviewer.
3.
Not solved (coded with numeric value 0): Solving the task was either impossible or
required more than one clue from the interviewer.
To ensure standardized interventions by the interviewer, a list of clues was developed
for each key idea beforehand. The list of clues provided for each task can be found in the
Appendix A(cf. Table A7).
Mathematics 2022,10, 1138 10 of 22
4. Results
In this section, we present the results of the acceptance survey. Table 9contains all
numeric values obtained for the acceptance phase, Table 10 contains all numeric values
obtained for the paraphrasing phase, and Table 11 contains all numeric values obtained for
the application phase. An in-depth discussion of these results with regard to the research
questions follows in Section 5.
Table 9.
Acceptance levels for each key idea. Ax stands for the acceptance level of the student with
regard to key idea x, cf. Tables A1–A6. The responses are color coded such that 1 is green, 0.5 is
yellow and 0 is red.
A1-1 A1-2 A2-1 A2-2 A3 A4 A5-1 A5-2 A6
A 1 1 1 1 1 1 1 0.5 1
B 1 1 1 1 1 1 1 0.5 1
C111111111
D111111111
E111111101
F111111101
G 1 0.5 1 1 1 1 1 1 1
H111111111
I111111101
Mean 1.00 0.94 1.00 1.00 1.00 1.00 1.00 0.56 1.00
Table 10.
Paraphrasing levels for each key idea. Px stands for the paraphrasing level of the student
with regard to key idea x, cf. Tables A1–A6. The responses are color coded such that 1 is green, 0.5 is
yellow and 0 is red.
P1-1 P1-2 P2 P3 P4 P5 P6
A 1 1 0 1 1 0.5 0.5
B 1 1 0.5 1 0.5 1 1
C 0.5 1 0.5 0.5 1 1 1
D 1 1 0.5 1 0.5 1 1
E 0 0.5 1 1 1 1 0
F 0.5 1 1 1 1 1 1
G 1 0.5 1 1 0.5 1 1
H 1 1 1 1 1 1 0.5
I 0.5 1 1 1 1 1 0.5
Mean 0.72 0.94 0.72 0.94 0.83 0.94 0.72
Table 11.
Application levels for each key idea. Ex stands for the application level of the student
with regard to key idea x, cf. Tables A1–A6. The responses are color coded such that 1 is green, 0.5 is
yellow and 0 is red.
E1 E2-1 E2-2 E2-3 E3-1 E3-2 E4 E5-1 E5-2 E5-3 E6-1 E6-2
A 1 1 0.5 1 1 1 0.5 1 1 0 1 0
B 1 1 1 1 1 0.5 1 1 0 0.5 1 0
C 0 1 1 1 1 1 1 1 1 0 1 0.5
D 1 1 1 1 1 0.5 1 1 1 1 1 1
E 1 1 1 1 1 0.5 1 1 0.5 1 0.5 0.5
F 1 1 1 1 1 1 1 1 1 0.5 1 0.5
G 1 1 1 1 1 1 1 1 1 1 1 0.5
H111110111111
I111111111111
Mean
0.89 1.00 0.94 1.00 1.00 0.72 0.94 1.00 0.83 0.67 0.94 0.56
Mathematics 2022,10, 1138 11 of 22
4.1. Results of the Acceptance Phase
Table 9gives the impression that, overall, the instructional elements were well per-
ceived, with the exception of A5-2 (reading Cayley Tables), which is part of the discussion
for RQ1 (c). The mean values are all well above the cut-off value of 0.5. We propose that the
explanations and learning material were perceived as meaningful. For example, regarding
the isometries, A argued that
“the arrows and reflection lines that were drawn inside the triangles helped me understand.
[. . .] With the arrows and the angles where you reflect, one could well imagine it. That’s
how it was for me at least.”
Simmilarily, H mentioned that
“I liked that we always had these vivid illustrations. That’s good for starters. I also liked
that we summarized all isometries in the first tabular, so one could take a peek every now
and then.”
H further highlighted the instructional elements regarding the composition of permu-
tations, saying that
“the pictures used when composing permutations were very helpful for understanding it.”
The plexiglass triangles appeared to be helpful. C stated
“I liked the visualization with the real triangles. [. .. ] It was very comprehensible. I think
it’s good that everything is first presented pictorially—first the non-mathematical level
and then later it’s easier to understand the mathematical level”.
Interestingly, C regarded the first two key ideas (isometries) as being at a non-
mathematical level and key ideas 3 and 4 (permutations) as being at a mathematical level.
This means that describing the isometries geometrically and only with abstract symbols was
perceived as somewhat “non-mathematical” compared to working with the permutations.
Thus, even though it is completely sufficient to compose isometries in their abstract form
to determine the result of the composition, this was not seen as mathematically rigorous.
C further mentioned that they prefer to work with “the abstract symbols, because they are
easier to understand”. In conclusion, working with just symbols was perceived as more
abstract but less mathematical by C. We assume that this view was caused by the absence
of numbers and concrete computations.
Regarding the entire learning process from key idea 1 all the way to key idea 6, E
stated that
“if it is designed in a consecutive way, it is very comprehensible”,
hinting that skills and knowledge required to understand the key ideas has always been
established previously.
4.2. Results of Paraphrasing
Overall, the paraphrasing level was acceptable (cf. Table 10). The mean values were all
well above the cut-off value of 0.5. However, in contrast to the acceptance levels, the aver-
age paraphrasing levels were much lower. This is, to some extent, to be expected, since an
entirely new technical language cannot be adopted within a session of just
60–90 min.
How-
ever, especially regarding the paraphrasing of isometries, systematic student difficulties
were observed. These are addressed in the discussion of RQ1 (a) (cf. Section 5).
Magmas were seen more as a result of a process than a distinct mathematical object.
For example, C explained that
“you get a magma if you take two elements of a set and you compose them or multiply or
add them, and by doing so, you again get an element of the set.”
Similarly, participant I explained magmas as
“a set of elements, in our example isometries, that is presented by an action, and if we
compose those actions, for example by reflecting along a bisecting line and then rotating,
we get a magma.”
Mathematics 2022,10, 1138 12 of 22
By looking at those explanations, it is clear that some students were well aware of the
fact that a magma has two components, namely a set and an operation; however, it is still
unclear how two such different mathematical objects can be put together and not be seen
as separate. In this regard, student H elaborated that a magma is
“when you have a given set and you compose two elements you get another element of the
set. Then, we have a composed set.”
The phrase Then we have a composed [=linked, combined] set, while mathematically
nonsensical, shows that the motivation behind studying magmas was well understood: it is
about equipping a set with an additional structure so that elements within the set become
mathematically related or linked. In conclusion, a magma was seen by some students as an
object that comes to existence when doing something with the elements of its underlying
set, revealing a crucial aspect of the learning path.
4.3. Results of Application
A closer look at Table 11 immediately draws attention towards problems E3-2, E5-3,
and E6-2. The two latter problems, E5-3 (What would a Cayley Table look like if
◦
was
commutative?) and E6-2 (Can you give an example for
◦
such that
R2
becomes a magma?)
were more challenging problems in terms of determining whether transfer of learning
was achievable after the short information phases for key ideas 5 (Cayley Tables) and 6
(Magmas). E3-2 was the inverse problem of E3-1—instead of finding a permutation to
a given isometry, the students had to find the isometry for a given permutation. The
conclusions drawn from the students’ underperformance with those items are presented
in Section 5. For the remaining items, it was observed that the students did not have
much trouble with solving the problems independently. In addition to the high acceptance
levels (cf. Table 9), perceived understanding can, therefore, be contrasted with applicable
understanding. In summary, the students not only accepted the explanations but were also
able to make sense of them and use them to solve problems. The students were asked to
rate the perceived difficulty and plausibility of this notation in the last step of the interview.
4.4. Perceived Difficulty and Plausibility of Notation
In the final phase of the interview, the participants were asked to rate the the key ideas
on a three-level ordinal scale (0, 0.5, and 1) in two ways:
1. What is the perceived difficulty of each content area? (0 = highest difficulty)
2.
How plausible were the notations and symbols of each content area? (0 = unplausible)
It should be noted that perceived difficulty is a highly subjective method of assessment
where the purpose is complementation of the acceptance, paraphrasing, and application
ratings, and as such, should be treated with care. The participants’ performance levels in
the exercises delivered a clearer picture of the actual difficulty level. However, perceived
difficulty, to some extent, shows how comfortable the students felt when dealing with the
content, which is certainly a relevant factor. The results for the perceived difficulty and
the plausibility of notation are shown in Table 12 (not including K4 for the plausibility of
notation, since no new symbols were introduced).
Mathematics 2022,10, 1138 13 of 22
Table 12.
Perceived difficulty and plausibility of the notation for each key idea. Kx stands for key
idea x, cf. Table 1. The responses are color coded such that 1 is green, 0.5 is yellow and 0 is red.
Perceived Difficulty Plausibility of the Notation
K1 K2 K3 K4 K5 K6 K1 K2 K3 K5 K6
A 0.5 1 0.5 0 1 0.5 1 1 0.5 1 0.5
B 1 1 1 0.5 1 1 1 0.5 1 1 1
C 1 1 1 0.5 1 1 1 1 1 1 1
D 1 1 0.5 1 1 0.5 1 1 0.5 1 0.5
E 1 1 1 0.5 1 0.5 1 1 1 1 0.5
F 1 1 0.5 0.5 1 0.5 1 0.5 1 1 0.5
G 0.5 1 0.5 0.5 1 1 1 1 1 1 1
H 1 1 0.5 1 1 1 1 0.5 1 1 1
I 1 0.5 0.5 1 1 0.5 1 1 1 1 0.5
Mean
0.89 0.94 0.67 0.61 1.00 0.72 1.00 0.83 0.89 1.00 0.72
5. Discussion
In this section, we discuss the results with regard to the research questions and refine
the teaching concept. The research questions formulated in Section 2.3 were as follows:
RQ1:
How do students accept the instructional elements within the Hildesheim Teaching
Concept of Abstract Algebra...
(a)
. . .with regard to introducing dihedral groups?
(b)
. . .with regard to introducing permutations as a tool to describe isometries?
(c)
. . .with regard to introducing Cayley Tables?
RQ2:
Do students prefer to work with the abstract symbols of the dihedral group
D3
, or do
they prefer to work with permutations presented by matrices?
RQ3:
What learning difficulties can be expected in the implementation of the Hildesheim
Teaching Concept of Abstract Algebra?
5.1. Discussion of RQ1
5.1.1. Discussion of RQ1 (a)
The students showed almost perfect acceptance for all acceptance questions regarding
isometries of the triangle and their compositions (cf. Table 9). Only participant G voiced a
slight critique regarding the notations of the elements (cf. Table 5). Additionally, Table 12
shows that, overall, the students felt very comfortable when dealing with K1 and K2 and
found the instructional elements to be comprehensible. However, it has to be noted that
paraphrasing the introduced contents with their own words revealed some deficiencies
(cf. Table 10). The paraphrasing result for P1-1 (Can you describe in your own words what
an isometry of the triangle is?) was, while still being above the cut-off value of 0.5, among
the lowest in the entire table. The answer of participant E can be seen in Table 8, and
participants C and I, for example, described isometries as follows:
C: “An isometry tells us which area the triangle covers, and if we manipulate the triangle,
the area still remains the same.”
I: “An isometry for me is the projection of the triangle onto the paper, and it doesn’t
matter how I rotate the triangle, it will always have the same projection. It doesn’t matter
which vertex is where, because the triangle will always cover the same area.”
Here, it becomes clear that C and I did not fully grasp the concept of isometries. While
the conception that the triangle will always cover the exact same area is true, the isometry
itself is confused with the area, and the students do not view the isometries as actions or
maps on the triangle. In other words, the manipulations of the triangle were seen as a
different part not belonging to the isometries.
While hints on how to revise the concept were given in Section 5.4, these conceptual
misunderstandings should not be overevaluated. After all, we can see in Table 11 that
Mathematics 2022,10, 1138 14 of 22
almost all students were able to find and describe the missing isometry correctly as well as
compose the isometries without making any mistakes.
5.1.2. Discussion of RQ1 (b)
Complementary to the key ideas K1 and K2 were the key ideas K3 and K4, describing
the same mathematical ideas in terms of permutations. Again, the students showed perfect
acceptance for all acceptance questions regarding permutations and their compositions
(cf. Table 9). In terms of paraphrasing, it should be noted that, overall, the students tried to
stick very close to the instructions given previously. While some students mixed up small
aspects of the explanation, no systematic errors were observed. We argue that this was the
case because the process here is less abstract and more formalized. However, the level of
perceived difficulty of K3 and K4 (cf. Table 12) was by far the highest among all key ideas
and was close to the cut-off value of 0.5, which shows that students perceived permutations
to be far more complex. This also manifested in the application phase (cf. Table 11), where
E3-2 posted the biggest hurdle of any exercise for the students. The inverse problem of
describing an isometry given as a permutation led to many mistakes.
5.1.3. Discussion of RQ1 (c)
The acceptance of the instructional elements regarding Cayley Tables was split. On
the one hand, the concept of Cayley Tables was, in general, perfectly accepted among the
students and perceived as the easiest content overall, with a perfect score of 1.00 obtained
in both Tables 9and 12. On the other hand, the instructions on how to read such tables
were met with much disregard (see A5-2 in Table 9), making this the least accepted item.
The reason for this low level of acceptance can be retraced to a strong association with
coordinate systems when looking at tables:
A: “To be honest, I first thought about coordinate systems [. . .] If you search for a point,
you first look for the x coordinate and then y.
E: “Because it is maths, I think I would have read it just like a coordinate system, so first
x and then y.”
F: “I learned in primary school that if you have a coordinate system you first look in
which house you are and then in which floor you are. [. .. ] I would have first looked at
the rows and then the columns.”
I: “I’m unsure because in a coordinate system, you first go horizontally.”
This is a rather surprising result, indicating that the students had little to no experience
with reading tables. However, it is also a problem that can easily be fixed. We touch upon
this in Section 5.4.
5.2. Discussion of RQ2
Since the levels of acceptance, paraphrasing, and application only differed slightly
when comparing K1 and K2 with K3 and K4 (cf. Tables 9–11) we answer this question
by looking at the students’ perceived level of difficulty (cf. Table 12). It is noticeable that
K3 and K4 were perceived as the most challenging among all key ideas, meaning that
describing isometries via permutations and composing those permutations was seen as
more difficult than working with the abstract symbols
r90
,
s1
and so on. This observation is
underpinned by the fact that E3-2 had the lowest score overall (cf. Table 11), hinting that
describing isometries of a triangle in terms of matrices did not help to reduce abstraction
but, instead, posed a new hurdle. In this regard, participant H commented that, when
looking at just the isometries themselves, it helped to visualize them with arrows and
reflecting lines, but the permutations
“got me confused with all the numbers”.
When presented with the question about which approach to the isometries they
preferred, 6 out of 9 students argued in favor of the abstract symbols. We conclude that,
Mathematics 2022,10, 1138 15 of 22
even though permutations are less abstract in the sense that they directly describe all the
vertices’ paths with numbers, the lack of visual stimuli when working with them made
them appear more difficult.
5.3. Discussion of RQ3
The most striking learning difficulty observed was that students, to some extent, did
not associate maps or operations with isometries. Instead, they often used words like
shadow or area to describe isometries (cf. Table 8), even though they had the plexiglass
triangles in their hands while giving the description. They did not incorporate the tech-
nical terms given by the interviewer in the information phase, which led to insufficient
paraphrasing overall. However, the subsequent application phases were successful. Thus,
it remains to be investigated whether the students just had trouble describing isometries
from a mathematical point of view and actually understood the concepts, or whether
they were simply copying strategies from the information phase without having a deeper
understanding.
Another observable difficulty is tied to the German language. The German word for
operation is Verknüpfung, which more directly translates to link or combination and thus has
deeper roots in common parlance. This meant that students had preconceptions with the
term, which seemed to impede an abstract understanding. We picked up on this issue by
asking for an operation on the set
R2
in the final phase of the interview (see E6-2 in Table 11).
Indeed, 6 out of 9 participants first suggested that a line should be drawn between two
points, ’linking’ or ’connecting’ them. It was only after the interviewer reiterated that the
result of the operation has to be an element of
R2
that they realized that their suggestion
did not make any sense. This linguistic barrier certainly has to be kept in mind in further
design.
Another insight gained from this is that the immediate generalization of magmas is
not immediately possible by just looking at one dihedral group. This, however, does not
impact the teaching concept since more magmas and operations are studied throughout
the three teaching units.
5.4. Implications for Revising the Concept in the Sense of Design-Based Research
Looking at the results of this study, we drew the following conclusions regarding the
refinements needed for our teaching concept:
1.
For the introduction of dihedral groups, the approach via permutations has turned out
to be inferior. Composing isometries was possible without ever introducing permuta-
tions, and students perceived them as more complex compared to just working with
the abstract symbols. We conclude that this content can be dropped if time restrictions
enforce a selection.
2.
Since students had strong associations with coordinate systems when dealing with
tables, they reversed the order in which such tables are usually read (rows first,
columns second). We did not anticipate this confusion. However, since the reading
order is more or less arbitrary from a mathematical point of view, we can simply
reverse it for our teaching concept, avoiding this problem by simply adapting the
reading order to the experience of the students.
3.
The biggest learning difficulties were caused by linguistic disparities. Mathematical
terms like image, map, and (the German version of) operation that are used with a
different meaning in common parlance caused students to transfer those different
meanings into mathematics, resulting in incorrect descriptions/paraphrasing and
misconceptions. We conclude that, when implementing the teaching concept, the
instructional elements need to address those disparities from the beginning and clearly
outline the differences.
Mathematics 2022,10, 1138 16 of 22
5.5. Comparison with Related Research
We conclude the results section by placing our findings within the body of work on
abstract algebra learning. First, however, it should be noted that the field is still relatively
new and unexplored. For an in-depth literature review on contemporary research, we
refer the reader to [
10
,
54
]. Additionally, we want to point out that the research presented
in the following text is solely focused on prospective and in-service teachers as well as
mathematics students (cf. [
55
]). Thus, since the sample in the study presented here consisted
of secondary school students, the comparison is to be treated with care.
One of the great benefits of abstract algebra lies in its use for generalizing concrete ideas
and notions to more abstract concepts [
56
]. However, this approach does not necessarily
translate to learners, as students have a tendency to make analogies that are not based on
mathematical structures [
57
,
58
]. In other words, learners of abstract algebra may encounter
problems when trying to connect new to previous knowledge. This can be traced back
to a lack of understanding of fundamental objects [
59
]. To guide this transition, different
approaches have been developed, i.e., the EDUS-framework (cf. [16]) as well as the ISETL
(Interactive Set Language) program, which is based on APOS (Action-Process-Object-
Schema) theory within problem-based learning (cf. [
54
]). In our study, we saw a similar
tendency where composing elements of
R2
was done in the wrong way when the visual
stimuli dominated the mathematical structure.
Furthermore, a study by Melhuish [
60
] showed that learners overgeneralize and
conflate properties such as associativity and commutativity and that the conceptual un-
derstanding of those properties is tied to the understanding of binary operations. This
was further substantiated by Wasserman [
61
] who, through task-based interviews, found
that students did not connect composition of functions to the composition of elements
in general, which led to such properties being neglected under the false assumption that
commutativity will hold.
Lastly, on a positive note, in the context of an abstract algebra course for prospective
mathematics teachers, Baldinger [
62
] found that dealing with abstract algebra caused
learners to more deeply implement the method of ”special cases“ into their problem-
solving inventory. Since the abstract nature of algebraic notions, like groups, allow for less
interactive or visual approaches, generally speaking, a need for the reduction of abstraction
arose, and the participants of his study did so by looking for concrete examples. Among
other things, this led to common school mathematics problems related to proving and
reasoning to be solved much more easily, since special case examples were constructed
beforehand. Thus, in this context abstract algebra helped to develop process-relational
skills.
6. Conclusions and Outlook
In this paper, we reported on a formative assessment of the Hildesheim Teaching Concept
for Abstract Algebra. For this, we adopted the method of acceptance surveys from non-
mathematical science education into mathematics education. By investigating how students
accept instructional elements and letting students explain mathematical ideas in their own
words, we obtained a deeper understanding of how the mathematical objects of abstract
algebra were being constructed in the learners’ minds, contributing to the development of
a (local) theory about student learning (cf. [
18
]). Unexpected learning difficulties were iden-
tified, such as linguistic associations, which are hurdles of a completely non-mathematical
nature. With those insights in mind, the teaching concept can be revised and refined to be
more specifically tailored towards students’ needs and thought processes.
This completes the second design cycle according to the Design-Based Research
paradigm laid out in Section 2.1 (cf. Figure 2). In the next step of the iterative process, the
concept will be tested empirically to enable a transition towards field studies, where the
learning of abstract algebra can be explored more thoroughly, in accordance with [
19
,
29
,
33
].
In addition, further qualitative analysis regarding the technical language involved needs to
be conducted. The following questions need to be investigated:
Mathematics 2022,10, 1138 17 of 22
•
To what extent does the technical language impede learning processes? In other words,
what is the magnitude of the problem entailed by linguistic preconceptions?
•
How can associations from common parlance be avoided so that misuse of the mathe-
matical language can be prevented?
Author Contributions:
Conceptualization, J.M.V., P.B. and B.G.; writing—original draft preparation,
J.M.V., P.B. and B.G.; writing—review and editing, J.M.V., P.B. and 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, Uni-
versity of Hildesheim and also by the Open Access Publication Fund of the University of Hildesheim.
Institutional Review Board Statement:
Informed consent was obtained from the participants in-
volved in the study.
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the
study prior to publication of 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.
Appendix A
Table A1.
Overview and descriptions for all units included in the interview regarding key idea 1
(Isometries of the triangle, cf. Table 1). A stands for acceptance, P for paraphrasing, and E for exercise.
Abbreviation Description
A1-1 Do you understand what an isometry of the triangle is?
A1-2 Are the symbols to describe the isometries plausible to you?
P1-1 Can you describe in your own words what an isometry of the triangle is?
P1-2 Can you describe in your own words what the isometry s1does?
E1 Can you find the missing isometry in the table?
Table A2.
Overview and descriptions for all units included in the interview regarding key idea 2
(Composition of isometries, cf. Table 1). A stands for acceptance, P for paraphrasing, and E for exercise.
Abbreviation Description
A2-1 Do you understand how to compose isometries of the triangle?
A2-2 Is it plausible to you that composing isometries yields isometries?
P2 Can you describe in your own words what a composition is?
E2-1 What expression describes “I first use s1on my triangle and then r90 ”?
E2-2 Please compute s1◦r90 .
E2-3 Previously, we computed r90 ◦s1. Do you notice anything?
Table A3.
Overview and descriptions for all units included in the interview regarding key idea 3
(Permutations, cf. Table 1). A stands for acceptance, P for paraphrasing, and E for exercise.
Abbreviation Description
A3
Do you understand what a permutation is and how it is used to describe isometries?
P3 Can you describe, in your own words, what a permutation is?
E3-1 Can you describe s2with a permutation?
E3-2 Which isometry is described by 123
312?
Mathematics 2022,10, 1138 18 of 22
Table A4.
Overview and descriptions for all units included in the interview regarding key idea 4
(Compositions of permutations, cf. Table 1). A stands for acceptance, P for paraphrasing, and E for
exercise.
Abbreviation Description
A4 Do you understand how to compose permutations?
P4 Can you describe, in your own words, how two permutations are composed?
E4 Can you compute 123
231◦123
321?
Table A5.
Overview and descriptions for all units included in the interview regarding key idea 5
(Cayley Tables, cf. Table 1). A stands for acceptance, P for paraphrasing, and E for exercise.
Abbreviation Description
A5-1 Do you understand what a Cayley Table is?
A5-2 Do you find it plausible to choose columns first?
P5 Can you describe in your own words what a Cayley Table is?
E5-1 Can you complete this Cayley Table?
E5-2 Can you tell me where in the Cayley Table the composition id ◦s1is?
E5-3 What would a Cayley Table look like if ◦was commutative?
Table A6.
Overview and descriptions for all units in the interview regarding key idea 6 (Magmas, cf.
Table 1). A stands for acceptance, P for paraphrasing, and E for exercise.
Abbreviation Description
A6 Do you understand what a Magma is?
P6 Can you describe, in your own words, what a Magma is?
E6-1 Can you give an example of a Magma that you already know from school
mathematics?
E6-2 Can you give an example for ◦such that R2becomes a magma?
Table A7. Overview of all clues prepared for each task in the key ideas application section.
Abbreviation Clues
E1 Did you find a position for the vertices that we haven’t seen yet?
We have already seen a rotation by 120 degrees. Does that give you an idea?
E2-1 There are only two possibilities. You have to figure out which one it is
E2-2 First, apply s1to your triangle, and then apply r90 .
E3-1 Compare the positions of the vertices before and after the isometry.
E3-2 We can see, for example, that vertex 1 switched positions with vertex 3. This
already excludes some isometries.
E4 First, only focus on vertex 1. Which final position does it go to?
E5-1 Remember that we read rows first and columns second
E5-2 There are only two possible cells. The correct one is determined by the
reading order.
E5-3
If the composition order did not matter, then
a◦b
and
b◦a
would be equal and,
thus, the reading order would not matter. How would the table look like in that
case?
E6-1 We have already seen some examples. Can you maybe switch out sets or
compositions and still get a magma?
E6-2 Which geometrical construction would yield a third point by two given points?
Mathematics 2022,10, 1138 19 of 22
Table A8. Coding manual for the paraphrasing of key idea 1.
P1-1: Can You Describe in Your Own Words What an Isometry of the Triangle Is?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Mentions that it is an operation/
manipulation of the triangle
• Use of the keywords rotation
or reflection
• Acknowledgment that the
triangle is mapped to itself
• Mentions that it is an
operation/manipulation of
the triangle
• Uses only one of the keywords
rotation or reflection
• No or wrong mention that it is
an operation/manipulation of
the triangle
• No or wrong use of the
keywords rotation and reflection
Table A9. Coding manual for the paraphrasing of key idea 1.
P1-2: Can You Describe in Your Own Words What the Isometry s1does?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Mentions that it is an
operation/manipulation of
the triangle
• Use of the keyword reflection
• Acknowledgment that vertex 1
is fixed and vertices 2 and 3
switch places
• No mention that it is an
operation/manipulation of
the triangle
• Use of the keyword reflection
• Acknowledgment that vertex 1
is fixed and vertices 2 and 3
switch places
• No or wrong use of the
keyword reflection
•
No acknowledgment that vertex
1 is fixed and vertices 2 and 3
switch places
Table A10. Coding manual for the paraphrasing of key idea 2.
P2: Can You Describe in Your Own Words What a Composition Is?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Acknowledgment that two
maps get concatenated
• Mentions that two isometries
are composed to yield a new
one
• Acknowledgment that two
maps get concatenated
• No or wrong mention that two
isometries are composed to
yield a new one
• No acknowledgment that two
maps get concatenated
Table A11. Coding manual for the paraphrasing of key idea 3.
P3: Can You Describe in Your Own Words What a Permutation Is?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Acknowledgment that the
permutation contains
information about how the
vertices get switched
•
Mentions that a permutation is a
mathematical description of
a map
•
Uses the keywords description or
representation
• Acknowledgment that the
permutation contains
information about how the
vertices get switched
• No or wrong mention that a
permutation is a mathematical
description of a map
• No or wrong use of the
keywords description or
representation
• No acknowledgment that the
permutation contains
information about how the
vertices get switched
Mathematics 2022,10, 1138 20 of 22
Table A12. Coding manual for the paraphrasing of key idea 4.
P4: Can You Describe in Your Own Words How Two Permutations Are Composed?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Mention that the permutations
switch the vertices sequentially
• Use of the keywords traveled
path or sequentially
• Acknowledgment that each
vertex gets permuted twice but
only the final result is needed
• Mentions that the permutations
switch the vertices sequentially
• Use of the keywords traveled
path or sequentially
• No acknowledgment that each
vertex gets permuted twice but
only the final result is needed
• No or wrong mention that the
permutations switch the
vertices sequentially
• No or wrong use of the
keywords traveled path
and sequentially
Table A13. Coding manual for the paraphrasing of key idea 5.
P5: Can You Describe, in Your Own Words, What a Cayley Table Is?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Mentions that it is a clear
presentation of all
possible compositions
• Use of the keyword tabular
or summary
• Acknowledgment that, for each
possible composition, there is a
cell in the Cayley Table
representing the result
• Mentions that it is a clear
presentation of all possible
compositions
• Use of the keywords tabular
or summary
• No acknowledgment that, for
each possible composition, there
is a cell in the Cayley Table
representing the result
•
No or wrong mention that it is a
clear presentation of all possible
compositions
• No or wrong use of the
keywords tabular and summary
Table A14. Coding manual for the paraphrasing of key idea 6.
P6: Can You Describe in Your Own Words What a Magma Is?
Fully Adequate Partially Adequate Not Adequate
Criteria
• Mentions that it consists of
elements which can
be composed
• Uses the keywords set
and composition
•
Acknowledgment that the result
has two components
• Mentions that it consists of
elements that can be composed
• Uses only one of the keywords
set or composition
• No acknowledgment that it has
two components
• No or wrong mention that it
consists of elements that can
be composed
• No or wrong use of the
keywords set and composition
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