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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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