Systematic Review: State of Knowledge on Learning Difficulties and Teaching Strategies in Linear Algebra Alethia Piñón Jiménez1 and Diana Margarita Córdova Esparza2 1 Facultad de Informática, Universidad Autónoma de Querétaro, Av. de las Ciencias S/N, Juriquilla, Querétaro 76230, México 2 Facultad de Informática, Universidad Autónoma de Querétaro, Av. de las Ciencias S/N, Juriquilla, Querétaro 76230, México Abstract This paper presents a systematic review focusing on diagnosing learning difficulties and implementing didactic strategies in linear algebra. We aim to deepen the understanding of this topic over the past decade. Our research, guided by four questions, analyzed 78 articles, ultimately including 37 in this review. We based our search strategy on the PRISM protocol and used specific indicators. Our findings indicate that most authors in this review primarily use the APOE theory and genetic decomposition for formal diagnosis of learning problems. This approach helps build knowledge frameworks, especially in vector spaces and linear transformations. A key finding is the prevalent use of digital technology in both the models and strategies proposed in these studies. This review highlights opportunities for future research in diagnosing learning problems and developing innovative, technology-integrated strategies in education. Keywords Education, didactic strategy, linear algebra 1 1. Introduction In the field of education, teaching and learning mathematics often presents significant challenges for teachers. These challenges include covering the subject's content within the allotted time and addressing the diverse learning difficulties that students face. Additionally, teachers must develop effective teaching strategies to enhance learning outcomes in mathematics. Each researcher in this field brings their unique perspective, knowledge, and experience to analyze the state of knowledge on teaching and learning mathematics. Despite these efforts, learning problems in linear algebra, especially in abstract topics like vector spaces and linear transformations, persist (31). This paper aims to conduct a systematic review to better understand how learning difficulties in linear algebra are formally diagnosed and what teaching strategies are being implemented. The importance of this review becomes evident when considering that linear algebra is a fundamental subject in science and engineering courses. It contributes significantly to developing students' logical, heuristic, and algorithmic thinking skills by using linear models to predict and control system behaviors. Therefore, this review will analyze current knowledge on diagnosing student learning problems in linear algebra and the recent implementation of didactic strategies to improve teaching and learning in this field. CISETC 2023: International Congress on Education and Technology in Sciences, December 04–06, 2023, Zacatecas, Mexico alethia463@gmail.com (A. Piñón-Jiménez); diana.cordova@uaq.mx (D. M. Córdova-Esparza) 0000-0003-0326-4741 (A. Piñón-Jiménez); 0000-0002-5657-7752 (D. M. Córdova-Esparza) © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) CEUR ht tp: // ceur -ws .or g Works hop I SSN1613- 0073 Pr oceedi ngs CEUR ceur-ws.org Workshop ISSN 1613-0073 Proceedings 2. Method Our search strategy used the PRISM (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) protocol as a reference and followed specific indicators. We guided our research with four key questions: 1. What are the main factors influencing learning problems in linear algebra? 2. Which learning theories have been applied to formally diagnose these learning problems and design teaching strategies for linear algebra? 3. What are the developed thematic strategies for linear algebra, and do they share any common characteristics? 4. What were the sizes of the groups used to validate the formal diagnoses or as pilot groups for implementing teaching strategies? To address our research questions and achieve the study's objective, we conducted a systematic literature review. This method is known for systematically integrating empirical results related to a specific research problem (34). We developed our research methodology in four distinct stages, which we detail in the subsequent paragraphs. Stage 1: Setting Inclusion and Exclusion Criteria for Research Studies In this first stage, we established specific criteria for including and excluding studies in our research. For inclusion, we focused on research articles, excluding other document types like theses and book chapters. We considered articles published from 2013 to 2022, ensuring the research was no more than 10 years old. Additionally, we included studies written in Spanish, English, or Portuguese. The final inclusion criterion was that the articles must be related to teaching or learning linear algebra; we excluded articles on topics outside this specific educational area. For exclusion, we omitted any articles that did not meet all our inclusion criteria. This also included articles that were duplicates in our study. Stage 2: Developing the Search Strategy In this stage, we executed our search strategy across various databases, yielding 71 articles for analysis. Our search criteria varied depending on the database to maximize results (see Figure 1). We selected databases that showed the highest number of relevant results for our topic. The databases and their respective search formulas were: 1. ERIC: Using the formula (“Education”) AND (“Linear Algebra”), we obtained 9 articles. 2. Scielo and DOAJ: We used (“Education”) AND (“Linear Algebra”) and (“Education”) AND (“Linear Algebra”), obtaining 8 and 30 articles, respectively. 3. Redalyc: With the formula (“Education”) AND (“Linear Algebra”), we found 8 articles. 4. Science Direct: We used (Teaching OR Learning) AND (“Linear Algebra”), leading to 9 articles. 5. Dialnet: The formulas (Teaching OR Learning) AND (“Linear Algebra”) and (Didactics) AND (“Linear Algebra”) resulted in 7 articles. Additionally, we identified 49 articles through references. After applying our exclusion criteria, 7 of these were ultimately included in our study. Figure 1: Overview of the Information Search and Data Collection Process. This flowchart details the search terms used across various databases, the number of articles retrieved from each, and the filtering process leading to the final selection of articles included in the review. It also outlines the exclusion criteria applied and the total number of articles analyzed. Stage 3: Information Purification In this stage, we conducted an initial review of the 78 articles gathered from the databases and references mentioned earlier. The purpose of this review was to assess each article's relevance to our research objectives. We rejected 34 articles during this process because they did not provide relevant data for our systematic review analysis or contribute to answering our research questions. Consequently, 37 articles were selected and included in our review. Stage 4: Data Coding and Analysis In this final stage, we analyzed the data based on specific categories. This structured approach helped us to thoroughly examine and understand the findings. The categories we focused on were: 1. Factors influencing learning problems in linear algebra. 2. Learning theories applied for diagnosing learning problems or implementing teaching strategies in linear algebra. 3. Thematic contents within the subject of linear algebra that were the focus of the research. 4. Strategies implemented in teaching linear algebra. 5. Sizes of the samples used for validation or implementation in pilot tests. This categorization facilitated a comprehensive analysis of the collected data, aligning it closely with our research objectives. 3. RESULTS Factors Influencing Linear Algebra Learning Problems The factors identified as influencing learning problems in linear algebra are varied, as observed in the systematic review of the research. Despite this diversity, there is a notable consistency in the findings. This is apparent when we see that several factors recur across multiple studies. In some instances, more than one factor is repeated between different investigations, as detailed in Table 1. This repetition underscores the commonalities in challenges faced by learners in linear algebra. Table 1 Influential Factors in Linear Algebra Learning Problems Identified in Scholarly Research This table compiles pivotal studies on linear algebra, listing the year of publication, authors, article title, and the predominant factor influencing learning difficulties as identified in each piece of research. Year of Authors Title article Predominant Factor publication Increasing Reality and Nishizawa et 2013 Educational Merits of a Virtual Abstract al. Game 2013 Parraguez The role of the body in the Abstract construction of the concept of Vector Space 2013 Rosso & A taxonomy of errors in Abstract, Language, Barros learning vector spaces Various representations University students’ solution Abstract, Prior 2014 Birinci et al. processes in systems of linear knowledge, Axiomatic equation Coordination of semiotic Ramírez- representation records in the 2014 Sandoval et Various representations use of linear transformations al. in the plane A teaching experience of Abstract Salgado & values, vectors and 2014 Trigueros eigenspaces based on APOE Gaisman theory Constructions and mental Abstract Trigueros mechanisms for learning the 2015 Gaisman et matrix theorem associated al. with linear transformation Berman & Definitions are important: the 2016 Formalism Shvartsman case of linear algebra Advanced mathematical Marins & thinking manifested in tasks 2016 Formalism, Abstract Pereira involving linear transformations A Teaching Proposal for the Abstract, Formalism 2017 Beltrán et al. Study of Eigenvectors and Eigenvalues. 2017 Costa & Teaching linear algebra in an Abstract, Without Rossignoli engineering school: connection with other Methodological and didactic subjects, Language aspects From Practical to Theorical 2018 Pierri Thinking: The Impact of the Abstract, Formalism Role-Play Activity. 2019 Álvarez- Teaching Linear Algebra in Epistemological Macea & engineering courses: an component, didactic Costa analysis of the process of schemes, Language mathematical modeling within the framework of the Anthropological theory of didactics Teaching Linear Algebra Abstract, procedures Aytekin & 2019 Supported by GeoGebra memorization, lack of Kiymaz Visualization Environment vinculation 2019 Gallo et al. Interpretation of linear Formalism, Language, transformations in the plane Various representations using GeoGebra Linear algebra learning García- 2019 focused on plausible reasoning Formalism Hurtado et al. in engineering programs Teaching-Learning of Matrices Abstract, Formalism, 2019 Xavier et al. in the civil Engineering Course Prior knowledge Construction of the meanings of vector space operations Abstract, Formalism 2020 Parraguez through linearly independent/dependent sets A Didactic Sequence for Teaching Linear Transformation: Unification of Concept application 2020 Pizarro Methods and Problems, conditions Modeling and Explanation of Learning 2021 Cárcamo et Hypothetical learning Abstract al. trajectories: an example in a linear algebra course 2021 Kariadinata Students Reflective Abstraction Abstract Ability on Linear Algebra Problem Solving and Relationship with Prerequisite Knowledge. 2021 Silva et al. Creation and uses of LineAlg Formalism application as a learning object in basic education 2021 Wibawa et al. Learning Effectiveness Abstract, Through Video Presentations Demonstrations, Large and WhatsApp Group (WAG) in number of operations the Pandemic Time Covid-19 between variables In our systematic review, we found that the most significant factor affecting linear algebra learning, as identified by various authors, is the subject's level of abstraction and formalism (see Figure 2). The high level of abstraction required by linear algebra itself poses a challenge for students, demanding a substantial degree of abstract thinking for proper understanding (37). As for formalism, it stems from the way linear algebra is presented, studied, and learned in the literature, which heavily relies on the formalism of mathematical language (25). Other key aspects impacting linear algebra learning difficulties include students' challenges in differentiating between a concept and its various representations (29) and the use of diverse languages when discussing vector spaces and linear transformations (8). Additionally, the connection to the teacher's training emerges as a notable factor. If a teacher has a background in mathematics or a related field, the issue often lies in not having the foundational structures in place. Conversely, for engineering educators, the challenge is often linking the relevance and applicability of linear algebra concepts to their specific field (28). Incident factors in learning Linear Algebra 18 16 14 12 10 8 6 4 2 0 ct ns ism ge ge rs k lin ra tio ua he d al le st No ng Ot ta rm w Ab en no Le Fo es rk pr io re Pr us rio Va Figure 2: Prevalence of Factors Impacting Learning in Linear Algebra This bar chart illustrates the frequency of various factors that influence the learning of linear algebra, as identified in the reviewed research, including abstraction, formalism, language, and others. Learning Theories Applied in Linear Algebra This section highlights the learning theories applied in diagnosing learning difficulties and implementing didactic strategies in linear algebra. It also covers the tools used in the various research projects analyzed. Additionally, we provide information about the countries where each study was conducted, as detailed in Table 2. Table 2 Learning Theories and Research Tools Utilized in Linear Algebra Studies. This table enumerates the studies included in the systematic review, outlining the applied learning theories, the research tools used, and the countries where the studies were conducted. Autor y Año País Applied learning theory Tool used (Parraguez, Chile APOE Semi-structured interview 2013) (Rosso & Theory of didactic Barros, Argentina Problems situations situations and constructivism 2013) (Parraguez Questionnaire and & Uzuriaga, Chile APOE interviews 2014) (Ramírez- Theory of semiotic Interview with sequence of Sandoval et México represetations 5 activities al., 2014) (Salgado & Trigueros Questionary and semi- México APOE Gaisman, structured interview 2014) (Trigueros Chile APOE Questionary and semi- Gaisman et structured interview al., 2015) (Murillo & Beltrán, Spain APOE RGB color system 2016) (González & Internalization of concrete Colombia APOE Roa, 2017) actions (Roa- Fuentes & Chile and APOE Questionary Parraguez, Colombia 2017) (Costa, Argentina Anthropological Theory of Study and research activity 2018) the Didactic (Karrer, Theory of semiotic Brazil Using GeoGebra 2018) represetations (Rodríguez Questionnaire and Chile APOE et al., 2018) interviews (Álvarez- Anthropological Theory of Macea & Colombia Study and research activity the Didactic Costa, 2019) (Gallo et al., Theory of semiotic Series of computer activities Argentina 2019) represetations using GeoGebra software (Parraguez, Chile APOE Written questionnaire 2020) (Fortuny & Guía escrita, archivos de Realistic mathematics Fuentealba, Spain audio y video, entrevistas education 2021) con algunos estudiantes. (Betancur et Questionary and semi- Colombia APOE al., 2022) structured interview In the systematic review, the APOE theory emerges as the most frequently applied learning theory in the research works analyzed (see Figure 3). This theory has been predominantly used to diagnose learning problems in linear algebra more accurately and deeply. It employs genetic decomposition to develop mental schemes or structures that aid students in constructing knowledge about specific concepts (30). Regarding the theory of semiotic representations, the reviewed studies have utilized it to support didactic strategies. These strategies involve varying representations of concepts, often enhanced by computational tools for better graphic representation (16). The anthropological theory of didacticism was applied to identify students' learning difficulties in linear algebra and to back didactic strategies using modeling, incorporating technology such as mobile devices and software (7). The theory of didactic situations was employed to categorize common errors in learning the topic of vector spaces (32). Additionally, the column labeled "others" in Figure 3 includes various theories like the theory of didactic proposal situations and realistic mathematical education (10). These theories have been instrumental in supporting didactic proposals for teaching linear algebra. Learning theories 12 10 8 6 4 2 0 APOE Semiotic representations Anthropology of the Others didactic Figure 3: Distribution of Learning Theories in Reviewed Research. This figure illustrates the prevalence of different learning theories as applied in the research works reviewed. The APOE theory leads in application, followed by semiotic representations, the anthropological theory of the didactic, and other various theories. In the systematic review, we noted the tools employed for conducting research. Prominent among these are questionnaires and interviews, particularly in studies implementing the APOE theory. The GeoGebra software stands out, along with the use of study guides on virtual platforms and a variety of activities grounded in learning theories. It is also worth noting the global reach of research in the field of linear algebra education. Chile emerges as a leader in research production within Latin America. However, countries outside the American continent, such as Spain, Turkey, and Indonesia, also contribute significantly. This underscores the universal relevance of the challenges in teaching and learning linear algebra, indicating that these difficulties are common in classrooms worldwide, irrespective of location. Regarding teaching strategies for linear algebra, the review also examined the specific subject topics that have been the focus of research and the sample sizes used in these studies (refer to Table 3). Table 3 Overview of Teaching Models or Strategies, Topics, and Sample Sizes in Linear Algebra Research. This table details the teaching models or strategies applied to linear algebra topics, specifying the topics addressed and the sample sizes involved in each study. Author and Model or year of Topics Sample size strategy publication (Nishizawa et Digital Vectors in 3D 40 students al., 2013) technology (Yildiz Ulus, Digital Eigenvectors and Not implementation 2013) technology eigenvalues (Salgado & Trigueros APOE-based 34 students on average Eigenvectors and eigenvalues Gaisman, activities per semester 2014) Matrices and determinants, (Petrov et Digital Vector spaces, Eigenvectors 37 students al., 2015) technology and eigenvalues (Gabriel Systems of linear equations, Digital 35 teachers and 5 Vergara et Matrices, Eigenvectors and technology students al., 2016) eigenvalues (Murillo & Digital Beltrán, Vector spaces Not implementation technology. 2016) Systems of linear equations, (Torres et al., Digital Vector spaces, Matrices, Linear Not implementation 2016) technology transformations, Eigenvectors and eigenvalues (Costa Digital &Rossignoli, Not specified Voluntaries 295 students technology 2017) (Meneu et Eigenvectors and Activities Not implementation al., 2017) eigenvalues Digital (Costa, 2018) Linear algebra with physics 50 students technology (Karrer, Digital Linear transformations 2 students 2018) technology (Kartika et Digital Vectors 3D 69 students al., 2018) technology Digital Systems of linear equations, (Pierri, 2018) 70 students technology Matrices, Vector spaces (Aytekin & Digital Kiymaz, Vector spaces 4 students technology 2019) (Gallo et al., Digital Linear transformations Not implementation 2019) technology (García- System of linear equations, Mathematical 36 students Hurtado et Matrices and determinants, modeling al., 2019) Vectors, Vector spaces (Villalobos & Digital Vector operations 40 students Ríos, 2019) technology (Xavier et al., Activities Matrices Not implementation 2019) Problem- 21 students and 21 (Nissa et al., Systems of linear equations, based control group 2020) Matrices learning Didactic engineering (Pizarro, 17 students and Linear transformations 2020) Mathematical modeling (Fortuny & Hypothetical 7 students Fuentealba, learning Vector spaces 2021) trajectories (Silva et al., Digital Matrices, systems of linear Not implementation 2021) technology equations (Wibawa et Digital Vector spaces 14 students al., 2021) technology This review reveals a strong emphasis on the use of digital technology in teaching the topics discussed, with the specific tools and elements varying according to the research aims (Figure 4). For instance, there is a focus on utilizing various mathematical software (35), knowledge management platforms (26), web-based learning tools (17), virtual games (21), and virtual evidence portfolios (36). The systematic and thorough diagnosis of mental structures that underpin the understanding of vector space concepts, linked to the design of proposed activities, was distinctly noted in the study by (33). However, a common thread across many studies is that topics of higher complexity and abstraction are most frequently addressed, both in diagnostic processes and in methodological proposals for teaching and learning. Notably, studies targeting instruction within the domain of engineering, particularly mathematical modeling, are prominent (13). This aligns with the practical application requirements characteristic of engineering curriculums. Teaching Model or Strategy 14 12 10 8 6 4 2 0 Digital technology Mathematical Activities Others modeling Figure 4: Frequency of Different Teaching Models or Strategies Used This bar graph illustrates the frequency with which various teaching models or strategies are applied in linear algebra education, showcasing a predominant use of digital technology, followed by mathematical modeling, diverse learning activities, and other strategies. Research Focus on Linear Algebra Topics The systematic review of research works revealed that most teaching strategies and diagnostic efforts in linear algebra are focused on the more abstract concepts. Vector spaces (24), linear transformations (30), and matrices are the topics most frequently addressed. Less commonly, but still noteworthy, are studies on systems of linear equations (6) and eigenvalues and eigenvectors (3). These findings align with the goal of the research: to develop tools that mitigate the factors impacting the teaching and learning of complex linear algebra topics (4). 4. Discussion and Conclusions from the Systematic Review The systematic review has led to several important conclusions regarding the factors that hinder students' learning of linear algebra. High levels of abstraction (23), unfamiliar formalism (18), language barriers (1), multiple representations of mathematical objects (12), lack of prior knowledge (40), and weak connections in learning (18) are significant challenges. Additionally, the complexity of new definitions, the quantity of operations between variables, and the subject's epistemological and axiomatic characteristics are noted as less frequent but still impactful factors. In terms of learning theories, the review underscores the APOE theory as the predominant framework for in-depth research on learning difficulties in linear algebra. The theory's popularity suggests it effectively uncovers and addresses students' mental structures during knowledge construction, as highlighted by Rodriguez et al. (31). Despite this, the APOE theory's main application is in diagnosis, with other theories more commonly used to explore the results of various teaching and learning strategies, except in the work of Salgado and Trigueros (33). This review reveals a gap: the direct link between systematic diagnosis and strategy application is often absent. This could be due to educational institutions' urgent need to produce quick results, relying on authors' experience and conceptual understanding to design their approaches. Digital technology's role is consistently significant in the research on teaching and learning strategies. Mathematical software applications (16), (20), (41), web-based learning tools— especially relevant during the COVID-19 pandemic for remote education (39), and virtual games (38) are some examples that reflect the growing, irreversible trend of digital integration in education. The main research focus in terms of content includes vector spaces (15) and linear transformations (14), likely due to their complex and abstract nature requiring a deep understanding. Regarding sample sizes for statistical analysis in the reviewed studies, they ranged from 2 to 295 participants, with variations in application time and students' nationalities. This indicates a need for further research with larger populations, leveraging digital technology for more extensive validation and evaluation. The reviewed research, regardless of its focus, often bases some methodological aspects on the authors' experiences, their conceptual understanding, and sometimes the influence of a research community. The effectiveness of proposed solutions is most significantly validated by the experiences of those who implement them. Therefore, future research should aim to enhance the authors' experiences and perspectives by developing methodologies that better connect with research communities and employing digital technology. This approach could allow a broader student population to engage with and benefit from the proposed methodologies in this review. Acknowledgements We would like to express our sincere gratitude to CONAHCYT for providing the scholarship that supported the graduate studies enabling this research. Their generous assistance was invaluable to the completion of this project. References [1] Álvarez-Macea, F., Costa, V. A. (2019). Enseñanza del Álgebra Lineal en Carreras de Ingeniería : Un Análisis del Proceso de la Modelización Matemática en el Marco de la Teoría Antropológica de lo Didáctico . Eco Matemático, 8231(2), 65–78. [2] Aytekin, C., y Kiymaz, Y. (2019). 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