Towards Describing Student Learning of Abstract Algebra: Insights into Learners’ Cognitive Processes from an Acceptance Survey

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.

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 [2,3,4,5,6,7,8,9]. This conflict raises the question of whether students themselves can benefit from learning abstract algebra if they 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:

• Key Notion 1: In the first unit, the students explore the dihedral group $D_3$ of the equilateral 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 triangle, 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 ${120}^{\circ }$ or reflecting on the bisecting line through vertex 1), resulting in $D_3=\{s_1,s_2,s_3,r_{120},r_{240},id\}$.
• Key Notion 2: In the second unit, the dihedral group $D_4$ of the square as well as its substructures and the Cayley Tables are explored. The fact $D_4\ncong S_4$ provides new problems to solve, i.e., discussions of why certain permutations do not describe isometries of the square.
• Key Notion 3: The concluding unit deals with modular arithmetic and isomorphisms by looking at the cyclic groups ${\mathbb{Z}}_n$. 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 ${\mathbb{Z}}_4$ only differs in terms of notation from the subgroup $D_4$ generated by $r_{90}$.

The next goal is to investigate how the instructional elements and didactical approaches 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. Section 4 and Section 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).

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 [22,23,24], 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.

In the literature, Design-Based Research (DBR) is described by five widely-accepted characteristics (cf. [19,31,32]):

• 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.
• 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, 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.
• 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].
• 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].
• 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 innovations 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 contribution [10] published in Mathematics, we reported on the first design cycle and discussed results from mathematics education research that indicate the potential for mathematics 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 1 of 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 something (formative evaluation); or to provide information for decision-makers who are wondering whether to fund, terminate, or to purchase something (summative evaluation).”

[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 a Teaching 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, acceptance surveys are realized via one-on-one interviews, most often comprising four interview phases (cf. [42]), as shown in Figure 3:

• 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.
• Survey of acceptance: The interviewer asks the interviewee to assess the information presented in terms of whether the explanations were comprehensible and understandable: “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.
• 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.
• 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).

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. [44,45,46,47,48]), 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).

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 $D_3$ and $D_4$ 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 questions 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 $D_3$, 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. 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

shows the key ideas as well as the research questions they aim at clarifying:

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 $r_{90}$ or $s_1$, etc., or
• …concrete mapping rules presented in matrices, such as

  $$r_{90}=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right),s_1=\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 4& 3& 2\end{array}\right),\mathrm{etc}.$$

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 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

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 $s_1$ does?
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 $s_1$ on my triangle and then $r_{90}$”?
E2-2	Please compute $s_1\circ r_{90}$.
E2-3	Previously, we computed $r_{90}\circ s_1$. 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 $s_2$ with a permutation?
E3-2	Which isometry is described by $\left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}\right)$?

,

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 $\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right)\circ \left(\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\end{array}\right)$?

,

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 $id\circ s_1$ is?
E5-3	What would a Cayley Table look like if ∘ was commutative?

and

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 ${\mathbb{R}}^2$ becomes a magma?

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. 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

. All students participated on a voluntary basis, according to ethical standards. Informed consent was obtained from all participants.

3.2. Data Analysis

The interviews were tape-recorded, transcribed, and evaluated using the coding manual (cf.

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${\mathit{s}}_1$ does?
	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

,

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

and

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

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 $\kappa$ (cf.

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	$\mathit{\kappa }$	%-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 Section 3.2.1, Section 3.2.2 and Section 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]):

• Perfect acceptance (coded with numeric value 1): The explanations were accepted by the participant without reservation and classified as plausible or understandable.
• Partial acceptance (coded with numeric value $0.5$): The explanations were accepted by the participant, but criticism was voiced.
• 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

Table 4. Perfect acceptance of Participant D and 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 $s_1$, $s_2$, and $s_3$ depending on which vertex we reflected, and the other thing with the rotation was also plausible.

and

Table 5. Partial acceptance of Participant G and 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.

, where (I) is the interviewer and (S) is the student.

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. Analogous to the evaluation of the level of acceptance, the answers were classified according to a three-level ordinal scaled coding system that distinguished between fully adequate, partially adequate, and not adequate:

• Fully adequate (coded with numeric value 1): The paraphrasing included all core aspects of the content according to the coding guide.
• 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.
• 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

Table 6. Fully adequate paraphrasing of Participant I and 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 H and 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.

and

Table 8. Not adequate paraphrasing of Participant E and 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.

where, again, (I) is the interviewer and (S) is the student. The complete coding guide can be found in the Appendix A of this article (cf. Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14), following [44,49].

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:

• 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.
• 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.
• 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. 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 $s_1$ to your triangle, and then apply $r_{90}$.
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 $a\circ b$ and $b\circ a$ 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?

).

4. Results

In this section, we present the results of the acceptance survey.

Table 9. Acceptance levels for each key idea. Ax stands for the acceptance level of the student with regard to key idea x, cf. Table A1, Table A2, Table A3, Table A4, Table A5 and Table 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
C	1	1	1	1	1	1	1	1	1
D	1	1	1	1	1	1	1	1	1
E	1	1	1	1	1	1	1	0	1
F	1	1	1	1	1	1	1	0	1
G	1	0.5	1	1	1	1	1	1	1
H	1	1	1	1	1	1	1	1	1
I	1	1	1	1	1	1	1	0	1
Mean	1.00	0.94	1.00	1.00	1.00	1.00	1.00	0.56	1.00

contains all numeric values obtained for the acceptance phase,

Table 10. Paraphrasing levels for each key idea. Px stands for the paraphrasing level of the student with regard to key idea x, cf. Table A1, Table A2, Table A3, Table A4, Table A5 and Table 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

contains all numeric values obtained for the paraphrasing phase, and

Table 11. Application levels for each key idea. Ex stands for the application level of the student with regard to key idea x, cf. Table A1, Table A2, Table A3, Table A4, Table A5 and Table 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
H	1	1	1	1	1	0	1	1	1	1	1	1
I	1	1	1	1	1	1	1	1	1	1	1	1
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

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.

4.1. Results of the Acceptance Phase

Table 9 gives the impression that, overall, the instructional elements were well perceived, 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 permutations, 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 average 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. However, 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.”

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 ${\mathbb{R}}^2$ 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:

• What is the perceived difficulty of each content area? (0 = highest difficulty)
• 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. 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

(not including K4 for the plausibility of notation, since no new symbols were introduced).

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 $D_3$, 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 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 Table 9 and Table 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. Table 9, Table 10 and Table 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 $r_{90}$, $s_1$ 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, 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 technical 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 ${\mathbb{R}}^2$ 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 ${\mathbb{R}}^2$ 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:

• For the introduction of dihedral groups, the approach via permutations has turned out to be inferior. Composing isometries was possible without ever introducing permutations, 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.
• 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.
• 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.

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 ${\mathbb{R}}^2$ 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 understanding 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 identified, 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:

• 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 mathematical language can be prevented?