# Exploring Learning Difficulties in Abstract Algebra: The Case of Group Theory

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

- How should instructional elements be designed when teaching group theory?
- Do learning difficulties found with qualitative methods also present themselves in a quantitative setting? If so, which difficulties can be observed and how pronounced are they?

#### 1.2. The Hildesheim Teaching Concept

- ⋯examines the impact of the curriculum on students’ development of conceptual understanding of group theory, and;
- ⋯that explores possible learning difficulties that appear regarding introductory group theory.

## 2. Research Questions

**RQ1**:- Do learners achieve an adequate conceptual understanding of introductory group theory when instructed with the Hildesheim Teaching Concept and which concepts post the most hurdles for learners?
**RQ2**:- Which learning difficulties regarding introductory group theory can be identified?

## 3. Methods

#### 3.1. Study Design and Samples

- An expert survey with $N=9$ experts from mathematics and mathematics education.
- A quantitative evaluation of the Hildesheim Teaching Concept with $N=143$ pre-service teachers.

#### 3.2. Instruments

- D1: Naive Set Theory, Binary operations, associativity, commutativity;
- D2: Neutral element and inverses;
- D3: Cayley Tables;
- D4: Cyclic groups and dihedral groups;
- D5: Subgroups;
- D6: Isomorphism.

**Domain 1**including D1,

**Domain 2**including D2 and D3,

**Domain 3**including D4, D5 and D6 (cf. Figure 2).

**Domain 1—Definitional Fundamentals:**Naive set theory, binary operations, associativity, commutativity;**Domain 2—Beginner Concepts:**Neutral element, inverses, Cayley Tables;**Domain 3—Intermediate Concepts:**Dihedral groups, cyclic groups, isomorphisms.

#### 3.3. Data Analysis

#### 3.3.1. Analysis Carried Out to Answer RQ1

#### 3.3.2. Analysis Carried Out to Answer RQ2

- Calculate the average CRI for each wrong answer option and investigate options with $\langle CRI\rangle >3$.
- Calculate for each wrong answer option the number of responses that were given confident (CRI = 4) or very confident (CRI = 5).

## 4. Results and Discussion

#### 4.1. Results Regarding RQ1

#### 4.2. Discussion of RQ1

#### 4.3. Results Regarding RQ2

#### 4.4. Discussion of RQ2

#### 4.4.1. Problems with Associativity and Commutativity

#### 4.4.2. Problems with Inverses and the Neutral Element

#### 4.4.3. Problems with Visualizations of Abstract Notions

## 5. Limitations of This Study

“The associativity property is required because we do not want the order of composition to matter.”

“The notion isomorphic means that the Cayley tables are identical.”

## 6. Conclusions and Outlook

- Does a revision of items 1 and 9 lead to a disappearance of the observed learning difficulties?
- How and to what extent do the expounded learning difficulties impede learning gain?
- Can these systematic learning obstacles also be observed in qualitative settings with individual learners? If so, can they be characterized in greater detail?
- Can the cognitive structure of gestalt and functionality be extracted empirically to enrich the understanding of learning processes in introductory group theory?

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Item 1: The associativity property is required because ⋯ | |||||

□ | ⋯ otherwise it is not clear how to compose 3 or more elements. | ||||

□ | ⋯ we do not want the order of composition to matter. | ||||

□ | ⋯ along with distributivity and commutativity it is a fundamental rule of mathematics. | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 2: A binary operation on a set M is ⋯ | |||||

□ | ⋯ a map $f:M\times M\to M$. | ||||

□ | ⋯ a map $f:M\to M\times M$. | ||||

□ | ⋯ a map $f:M\times M\to M\times M$. | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 3: An example for a group is ⋯ | |||||

□ | ⋯$(\mathbb{R},+)$ | ||||

□ | ⋯$(\mathbb{Q},\xb7)$ | ||||

□ | ⋯$(\mathbb{Z},-)$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 4: Let $G=(M,\circ )$ be non-abelian and $a,b\in M$. The inverse of $a\circ b$ is ⋯ | |||||

□ | ⋯${b}^{-1}\circ {a}^{-1}$ | ||||

□ | ⋯${a}^{-1}\circ b$ | ||||

□ | ⋯${a}^{-1}\circ {b}^{-1}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 5: One can show that $a\star b:=a+b-5$ defines an operation on $\mathbb{Z}$ such that $(\mathbb{Z},\star )$ is a group. The neutral element of this operation is ⋯ | |||||

□ | ⋯ 5 | ||||

□ | ⋯ 0 | ||||

□ | ⋯$-5$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 6: One can show that $a\u2022b:=\frac{ab}{7}$ defines an operation on $\mathbb{Q}\backslash \left\{0\right\}$ such that $(\mathbb{Q}\backslash \{0\},\u2022)$ is a group. The inverse of $x\in \mathbb{Q}\backslash \left\{0\right\}$ is given by ⋯ | |||||

□ | ⋯$\frac{49}{x}$ | ||||

□ | ⋯$\frac{49}{{x}^{2}}$ | ||||

□ | ⋯$\frac{7}{x}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 7: Let $f\left(x\right)=\frac{2}{x-1}$ and $g\left(x\right)={e}^{x+1}$, then ⋯ | |||||

□ | ⋯$(g\circ f)\left(x\right)={e}^{\frac{x+1}{x-1}}$ | ||||

□ | ⋯$(g\circ f)\left(x\right)=\frac{2}{{e}^{x-1}-1}$ | ||||

□ | ⋯$(f\circ g)\left(x\right)={e}^{\frac{2}{x}}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 8: In the group ${D}_{4}$ the equation ${s}_{1}\circ (x\circ {s}_{1})={s}_{1}$ is solved by ⋯ | |||||

□ | ⋯$x={s}_{1}$ | ||||

□ | ⋯$x={r}_{90}$ | ||||

□ | ⋯$x=id$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 9: The notion isomorphic means that ⋯ | |||||

□ | ⋯ the groups are indifferentiable from a mathematical point of view. | ||||

□ | ⋯ the Cayley tables are identical. | ||||

□ | ⋯ the groups are identical. | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 10: The operation ⊕ within the group $({\mathbb{Z}}_{4},\oplus )$ has been altered to ∘ such that $\left[0\right]$ is no longer necessarily the neutral element. Find the neutral element with the help of the Cayley table. | |||||

$\begin{array}{ccccc}\circ & \left[0\right]& \left[1\right]& \left[2\right]& \left[3\right]\\ \left[0\right]& \left[1\right]& \left[3\right]& \left[0\right]& \left[2\right]\\ \left[1\right]& \left[3\right]& \left[2\right]& \left[1\right]& \left[0\right]\\ \left[2\right]& \left[0\right]& \left[1\right]& \left[2\right]& \left[3\right]\\ \left[3\right]& \left[2\right]& \left[0\right]& \left[3\right]& \left[1\right]\end{array}$ | |||||

□ | $\left[2\right]$ | ||||

□ | $\left[1\right]$ | ||||

□ | $\left[3\right]$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 11: A group structure is to be established on the set $\{0,\pi ,55\}$ where the following Cayley table is given. Which element must be at ⋆? | |||||

$\begin{array}{cccc}\circ & 0& \pi & 55\\ 0& \star & & 0\\ \pi & & & \pi \\ 55& 0& \pi & 55\end{array}$ | |||||

□ | $\star =\pi $ | ||||

□ | $\star =0$ | ||||

□ | $\star =55$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 12: The set $\{a,t,w,z\}$ has been equipped with a group structure by the following Cayley table. What is the inverse of z? | |||||

$\begin{array}{ccccc}\circ & a& t& w& z\\ a& a& w& a& t\\ t& w& z& t& a\\ w& a& t& w& z\\ z& t& a& z& w\end{array}$ | |||||

□ | ${z}^{-1}=z$ | ||||

□ | ${z}^{-1}=t$ | ||||

□ | ${z}^{-1}=w$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 13: What is the symmetry group of the figure? | |||||

□ | ${\mathbb{Z}}_{2}$ | ||||

□ | ${D}_{2}$ | ||||

□ | ${D}_{4}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 14: What is the symmetry group of the figure? | |||||

□ | ${\mathbb{Z}}_{3}$ | ||||

□ | ${D}_{3}$ | ||||

□ | ${\mathbb{Z}}_{6}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 15: What is the symmetry group of the figure? | |||||

□ | ${D}_{5}$ | ||||

□ | ${\mathbb{Z}}_{5}$ | ||||

□ | ${\mathbb{Z}}_{10}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 16: Which two of the following figures have an isomorphic symmetry group? | |||||

□ | The first and the third. | ||||

□ | The first and the second. | ||||

□ | The second and the third. | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 17: If a Group is commutative the Cayley tyble is ⋯ | |||||

□ | ⋯ axially symmetric to the diagonal. | ||||

□ | ⋯ point symmetric to the entry in the middle. | ||||

□ | ⋯ axially symmetric to the anti diagonal (top left to bottom right). | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 18: Which of the following sets is a subgroup of $({\mathbb{Z}}_{10},\oplus )$ if equipped with ⊕? | |||||

□ | $\left\{\right[0],[2],[4],[6],[8\left]\right\}$ | ||||

□ | $\left\{\right[0],[1],[2],[5\left]\right\}$ | ||||

□ | $\left\{\right[0],[1],[3],[5],[7],[9\left]\right\}$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 19: Which of the following permutations does not describe an isometry of the square? | |||||

□ | $\pi =\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 3& 2& 4\end{array}\right)$ | ||||

□ | $\sigma =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right)$ | ||||

□ | $\tau =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 1& 4& 3\end{array}\right)$ | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

Item 20: Which of the following tables is a Cayley table? | |||||

$\begin{array}{cccc}\circ & a& b& c\\ a& a& c& b\\ b& c& a& b\\ c& b& b& a\end{array}$ $\begin{array}{cccc}\circ & a& b& c\\ a& a& c& b\\ b& c& a& b\\ c& b& b& a\end{array}$ $\begin{array}{cccc}\circ & a& b& c\\ a& a& c& b\\ b& c& a& b\\ c& b& b& a\end{array}$ | |||||

□ | The third. | ||||

□ | The first. | ||||

□ | The second. | ||||

□ Very sure | □ Sure | □ Undecided | □ Unsure | □ Guessed |

## References

- Wasserman, N.H. Introducing Algebraic Structures through Solving Equations: Vertical Content Knowledge for K-12 Mathematics Teachers. Primus
**2014**, 24, 191–214. [Google Scholar] [CrossRef] - Wasserman, N.H. Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction. Can. J. Sci. Math. Technol. Educ.
**2016**, 16, 28–47. [Google Scholar] [CrossRef] - Wasserman, N.H. Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure. Math. Think. Learn.
**2017**, 19, 181–201. [Google Scholar] [CrossRef] - Wasserman, H.H. Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers, 1st ed.; Springer: Basel, Switzerland, 2018. [Google Scholar]
- Melhuish, K. The Design and Validation of a Group Theory Concept Inventory. Ph.D. Thesis, Portland State University, Portland, OR, USA, 2015. [Google Scholar]
- Melhuish, K.; Fagan, J. Connecting the Group Theory Concept Assessment to Core Concepts at the Secondary Level. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers; Springer: Basel, Switzerland, 2018; pp. 19–45. [Google Scholar]
- Melhuish, K. The Group Theory Concept Assessment: A Tool for Measuring Conceptual Understanding in Introductory Group Theory. Int. J. Res. Undergrad. Math. Educ.
**2019**, 5, 359–393. [Google Scholar] [CrossRef] - Veith, J.M.; Bitzenbauer, P. What Group Theory Can Do for You: From Magmas to Abstract Thinking in School Mathematics. Mathematics
**2022**, 10, 703. [Google Scholar] [CrossRef] - Larsen, S. Struggling to Disentangle the Associative and Commutative Properties. Learn. Math.
**2010**, 30, 37–42. [Google Scholar] - Zaslavsky, O.; Peled, I. Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operation. J. Res. Math. Educ.
**1996**, 27, 67–78. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Even, R. The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioners’ views. ZDM Math. Educ.
**2011**, 43, 941–950. [Google Scholar] [CrossRef] - Veith, J.M.; Bitzenbauer, P.; Girnat, B. Assessing Learners’ Conceptual Understanding of Introductory Group Theory Using the CI
^{2}GT: Development and Analysis of a Concept Inventory. Educ. Sci.**2022**, 12, 376. [Google Scholar] [CrossRef] - Chick, H.L.; Harris, K. Grade 5/6 Teachers’ Perceptions of Algebra in the Primary School Curriculum. In Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Seoul, Korea, 8–13 July 2007; pp. 127–134. [Google Scholar]
- Common Core State Standards in Mathematics (CCSS-M). 2010. Available online: http://www.corestandards.org/Math/Content/mathematics-glossary (accessed on 4 July 2022).
- Branco, N.; Ponte, J.P. Analysis of Teaching and Learning Situations in Algebra in Prospective Teacher Education. J. Educ.
**2013**, 1, 182–213. [Google Scholar] - Shamash, J.; Barabash, M.; Even, R. From Equations to Structures: Modes of Relevance of Abstract Algebra to School Mathematics as Viewed by Teacher Educators and Teachers. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers; Springer: Basel, Switzerland, 2018; pp. 241–262. [Google Scholar]
- Burn, R. What Are the Fundamental Concepts of Group Theory? Educ. Stud. Math.
**1996**, 31, 371–377. [Google Scholar] [CrossRef] - Baldinger, E.E. Learning Mathematical Practices to Connect Abstract Algebra to High School Algebra. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers; Springer: Cham, Switzerland, 2018; pp. 211–239. [Google Scholar]
- Shimizu, J.K. The Nature of Secondary Mathematics Teachers’ Efforts to Make Ideas of School Algebra Accessible. Ph.D. Thesis, The Pennsylvania State University, University Park, PA, USA, 2013. [Google Scholar]
- Zbiek, R.M.; Heid, M.K. Making Connections from the Secondary Classroom to the Abstract Algebra Course: A Mathematical Activity Approach. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers; Springer: Cham, Switzerland, 2018; pp. 189–209. [Google Scholar]
- Leron, U.; Dubinsky, E. An abstract algebra story. Am. Math. Mon.
**1995**, 102, 227–242. [Google Scholar] [CrossRef] - Hake, R.R. Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. Am. J. Phys.
**1998**, 66, 64. [Google Scholar] [CrossRef] - Hasan, S.; Bagayoko, D.; Kelley, E.L. Misconceptions and the certainty of response index. Phys. Educ.
**1999**, 34, 294–299. [Google Scholar] [CrossRef] - Rappaport, D. The New Math and Its Aftermath. Sch. Sci. Math.
**1976**, 7, 563–570. [Google Scholar] [CrossRef] - Larsen, S.P. A local instructional theory for the guided reinvention of the group and isomorphism concepts. J. Math. Behav.
**2013**, 32, 712–725. [Google Scholar] [CrossRef] - Larsen, S.P.; Lockwood, E. A local instructional theory for the guided reinvention of the quotient group concept. J. Math. Behav.
**2013**, 32, 726–742. [Google Scholar] [CrossRef] - Leppig, M. Beispiele zum Rechnen in endlichen Gruppen. Der Mathematikunterricht
**1966**, 2, 39–49. [Google Scholar] - Machi, A. Groups. An Instruction to Ideas and Methods of the Theory of Groups, 1st ed.; Springer: Milano, Italy, 2012. [Google Scholar]
- Lienert, G.A.; Raatz, U. Testaufbau und Testanalyse, 6th ed.; Verlagsgruppe Beltz: Weinheim, Germany, 1998. [Google Scholar]
- Bauer, A. Cronbachs α im Kontext des Grundmodells der Klassischen Testtheorie und darüber Hinaus. Master’s Thesis, Ludwig-Maximilian-University, Munich, Germany, 2015. [Google Scholar]
- Coletta, V.P.; Steinert, J.J. Why normalized gain should continue to be used in analyzing preinstruction and postinstruction scores on concept inventories. Phys. Rev. Phys. Educ. Res.
**2020**, 16, 010108. [Google Scholar] [CrossRef] - Finkenberg, F. Flipped Classroom im Physikunterricht; Logos Verlag Berlin: Berlin, Germany, 2018. [Google Scholar]
- Bitzenbauer, P. Quantenoptik an Schulen. Studie im Mixed-Methods Design zur Evaluation des Erlanger Unterrichtskonzepts zur Quantenoptik, 1st ed.; Logos Verlag Berlin: Berlin, Germany, 2020. [Google Scholar]
- Wilcoxon, F. Individual comparisons by ranking methods. Biometrics
**1945**, 1, 80–83. [Google Scholar] [CrossRef] - Kruskal, W.H.; Wallis, W.A. Use of Ranks in One-Criterion Variance Analysis. J. Am. Stat. Assoc.
**1952**, 47, 260. [Google Scholar] [CrossRef] - Douglas, C.E.; Michael, F.A. On Distribution-Free Multiple Comparisons in the One-Way Analysis of Variance. Commun. Stat.—Theory Methods
**1991**, 20, 127–139. [Google Scholar] [CrossRef] - Rost, J. Allgemeine Standards für die Evaluationsforschung. In Handbuch Evaluation Psychologischer Interventionsmaßnahmen; Verlag Hans Huber: Bern, Switzerland, 2000; pp. 129–140. [Google Scholar]
- Zenger, T.; Bitzenbauer, P. Exploring German Secondary School Students’ Conceptual Knowledge of Density. Sci. Educ. Int.
**2022**, 33, 86–92. [Google Scholar] [CrossRef] - Tirosh, D.; Hadass, R.; Movshovitz-Hadar, N. Overcoming overgeneralizations: The case of commutativity and associativity. In Proceedings of the Fifteenth Annual Conference of the International Group for Psychology of Mathematics Education, Assisi, Italy, 29 June–4 July 1991; pp. 310–315. [Google Scholar]
- Ubben, M.S.; Heusler, S. Gestalt and Functionality as Independent Dimensions of Mental Models in Science. Res. Sci. Educ.
**2021**, 51, 1349–1363. [Google Scholar] [CrossRef] - Ubben, M.S.; Bitzenbauer, P. Two Cognitive Dimensions of Students’ Mental Models in Science: Fidelity of Gestalt and Functional Fidelty. Educ. Sci.
**2022**, 12, 163. [Google Scholar] [CrossRef] - Rupnow, R.; Sassman, P. Sameness in algebra: Views of isomorphism and homomorphism. Educ. Stud. Math.
**2022**. [Google Scholar] [CrossRef]

**Figure 1.**Bar Chart for the first round of the expert survey showing what percentage of the experts’ votes account for each domain.

**Figure 2.**Bar Chart for the second round of the expert survey showing how much percent of the experts’ votes account for each domain.

**Figure 3.**Boxplots of the normalized gains for each of the three domains as well as the total learning gain.

**Figure 4.**Mean values of the normalized gains for each of the three domains as well as the total learning gain. The error bars indicate the $95\%$ confidence intervals. The asterisks indicate statistical significance (⋆) and high statistical significance ($\star \star $).

**Figure 5.**The figures of items 13 (

**left**) and 14 (

**right**) of the CI${}^{2}$GT. The task lies in classifying their symmetry group.

**Table 1.**Sub-domains of group theory and the respective items of the CI${}^{2}$GT, as well as their internal consistencies, expressed by Cronbach’s $\alpha $ and the adjusted Cronbach’s ${\alpha}^{\star}$ as explained above.

Domain | Items | Length | Description | ${\mathit{\alpha}}^{\star}$ | $\mathit{\alpha}$ |
---|---|---|---|---|---|

1 | 1,2,3,7 | 4 | Definitional Fundamentals | 0.41 | 0.28 |

2 | 4,5,6,10,11,12,17,20 | 8 | Beginner Concepts | 0.48 | 0.54 |

3 | 8,9,13,14,15,16,18,19 | 8 | Intermediate Concepts | 0.48 | 0.50 |

**Table 2.**Decision matrix based on combinations of correct or wrong answer and low or high CRI adapted from [24].

Low CRI ($\le 3$) | High CRI ($>3$) | |
---|---|---|

Correct Answer | Correct answer and low CRI Uncertainty of Knowledge | Correct answer and high CRI Knowledge of scientific concept |

Wrong Answer | Wrong answer and low CRI Lack of knowledge | Wrong answer and high CRI Misconceptions |

**Table 3.**Mean value $\mu $ and standard derivation $\sigma $ for the pre- and post-test scores as well as the test statistics of a Wilcoxon Signed-Rank Test.

$\mathit{\mu}$ | $\mathit{\sigma}$ | Wilcoxon Signed-Rank Test | |
---|---|---|---|

pretest | $1.28$ | $2.74$ | $Z\left(136\right)=147$, $p<0.001$; $r=0.965$ |

post-test | $8.99$ | $3.54$ |

**Table 4.**Mean value $\mu $ and standard derivation $\sigma $ for the normalized gain g for each of the three domains as well as the total gain ${g}_{\mathrm{tot}}$.

Domain | Normalized Gain | $\mathit{\mu}$ | $\mathit{\sigma}$ |
---|---|---|---|

Domain 1: Definitional Fundamentals | ${g}_{1}$ | 0.37 | 0.26 |

(i.e., Naive set theory, binary operations, associativity, commutativity) | |||

Domain 2: Beginner Concepts | ${g}_{2}$ | 0.44 | 0.26 |

(i.e., Neutral element, inverses, Cayley Tables) | |||

Domain 3: Intermediate Concepts | ${g}_{3}$ | 0.47 | 0.26 |

(i.e., Dihedral groups, cyclical groups, isomorphisms) | |||

Overall CI${}^{2}$GT | ${g}_{\mathrm{tot}}$ | 0.40 | 0.21 |

(i.e., all domains) |

**Table 5.**Number of responses regarding the wrong answer options 2 and 3 chosen confidently or very confidently by our study participants for each item. The first column provides the total number of responses (tot. #), the second column provides the relative number of responses (rel. #) and the third column provides the average CRI.

Option 2 | Option 3 | |||||
---|---|---|---|---|---|---|

Item | tot. # | rel. # | $\langle CRI\rangle $ | tot. # | rel. # | $\langle CRI\rangle $ |

1 | 71 | 50% | 3.86 | 4 | 3% | 3.50 |

2 | 10 | 7% | 3.48 | 0 | 0% | 2.83 |

3 | 19 | 13% | 3.77 | 15 | 10% | 3.48 |

4 | 7 | 5% | 3.50 | 54 | 38% | 3.44 |

5 | 15 | 10% | 3.15 | 7 | 5% | 3.10 |

6 | 13 | 9% | 2.00 | 1 | 0% | 2.54 |

7 | 25 | 17% | 3.37 | 15 | 10% | 3.02 |

8 | 7 | 5% | 2.58 | 2 | 1% | 3.00 |

9 | 25 | 17% | 4.13 | 39 | 27% | 4.15 |

10 | 13 | 10% | 3.13 | 2 | 1% | 2.75 |

11 | 17 | 12% | 3.74 | 15 | 10% | 3.56 |

12 | 5 | 3% | 3.00 | 36 | 25% | 3.48 |

13 | 25 | 17% | 2.90 | 13 | 9% | 3.25 |

14 | 26 | 18% | 3.11 | 0 | 0% | 1.86 |

15 | 8 | 6% | 2.66 | 5 | 3% | 3.27 |

16 | 10 | 7% | 3.50 | 5 | 3% | 3.27 |

17 | 3 | 2% | 2.92 | 5 | 3% | 2.69 |

18 | 13 | 9% | 3.00 | 13 | 9% | 2.82 |

19 | 15 | 10% | 3.58 | 15 | 10% | 3.67 |

20 | 9 | 6% | 3.46 | 20 | 14% | 3.56 |

**Table 6.**An overview of the three recurring themes in learning difficulties. The percentages describe how many of the participants selected the respective answer option confidently (CRI = 4) or very confidently (CRI = 5).

Domain | Domain Description | Learning Difficulty | Item | Option 2 | Option 3 |
---|---|---|---|---|---|

1 | Definitional Fundamentals (Naive set theory, associativity, commutativity, etc.) | Problems with associativity | 1 | 50% | – |

3 | 13% | 10% | |||

4 | – | 38% | |||

7 | 17% | 10% | |||

2 | Beginner Concepts (Neutral element, inverses, etc.) | Problems with inverses and the neutral element | 5 | 10% | – |

10 | 10% | – | |||

12 | – | 25% | |||

20 | – | 14% | |||

3 | Intermediate Concepts (Dihedral and cyclic groups, isomorphism, etc.) | Problems with visualizing abstract notions | 9 | 17% | 27% |

13 | 17% | – | |||

14 | 18% | – |

**Table 7.**The Cayley Table from item 12 of the CI${}^{2}$GT: The task lies in finding the inverse of z with respect to ∘.

∘ | a | t | w | z |

a | z | w | a | t |

t | w | z | t | a |

w | a | t | w | z |

z | t | a | z | w |

**Table 8.**The tables from item 20 of the CI${}^{2}$GT: The task lies in identifying which of the three presented tables is a Cayley Table.

∘ | a | b | c | ∘ | a | b | c | ∘ | a | b | c | ||

a | a | c | b | a | c | a | b | a | c | a | b | ||

b | c | a | b | b | b | c | a | b | a | b | c | ||

c | b | b | a | c | a | b | c | c | b | c | a |

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## Share and Cite

**MDPI and ACS Style**

Veith, J.M.; Bitzenbauer, P.; Girnat, B.
Exploring Learning Difficulties in Abstract Algebra: The Case of Group Theory. *Educ. Sci.* **2022**, *12*, 516.
https://doi.org/10.3390/educsci12080516

**AMA Style**

Veith JM, Bitzenbauer P, Girnat B.
Exploring Learning Difficulties in Abstract Algebra: The Case of Group Theory. *Education Sciences*. 2022; 12(8):516.
https://doi.org/10.3390/educsci12080516

**Chicago/Turabian Style**

Veith, Joaquin M., Philipp Bitzenbauer, and Boris Girnat.
2022. "Exploring Learning Difficulties in Abstract Algebra: The Case of Group Theory" *Education Sciences* 12, no. 8: 516.
https://doi.org/10.3390/educsci12080516