Unveiling the complexities of ‘Abstract Algebra’ in University Mathematics Education (UME): fostering ‘Conceptualization and Understanding’ through advanced pedagogical approaches

Abstract

This scholarly inquiry critically examines the pedagogical aspects pertaining to the instruction and acquisition of Abstract Algebra within the realm of University Mathematics Education (UME). Drawing upon multiple lenses, including epistemological, cognitive, phenomenological, and institutional perspectives, this study investigates the formidable challenges encountered by students and delves into the intricate abstraction processes required to master Abstract Algebra concepts. Moreover, this article endeavors to shed light on the underlying rationales that underscore structural concepts while rigorously analyzing the predominant instructional practices adopted by educators in this domain. The study contends that Abstract Algebra presents distinctive hurdles by virtue of its characteristic dual movement of abstraction, namely idealization and thematization. As such, it necessitates the development and implementation of inquiry-based methodologies to foster optimal conceptualization and comprehension. However, despite ongoing research endeavors in this field, the need for further systematic investigations remains evident. Within this context, the research identifies several critical domains that demand deeper exploration. It underscores the imperative of unearthing the phenomenological foundations of Group Theory and extends this analysis to encompass Ring Theory and Field Theory. By comprehending these fundamental underpinnings, educators can derive valuable insights to inform effective instructional strategies tailored to the specific branches of Abstract Algebra.

Keywords:

• Mathematics pedagogy
• mathematics curriculum
• University Mathematics Education (UME)
• teaching and learning
• pedagogical practices
• Inquiry-Based approaches
• curriculum reform

Reviewing Editor:

• Vagn Lundsgaard Hansen, Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark

SUBJECT:

• History & Philosophy of Mathematics
• Mathematics Education
• Theories of Learning
• Teachers & Teacher Education
• Theory of Education
• Classroom Practice
• Curriculum Studies
• Education Policy & Politics
• Curriculum
• Pedagogy

Introduction

Abstract Algebra, also referred to as modern algebra (Albert, 2018) or structural algebra (Kieran, 2018), represents a distinguished branch of mathematical inquiry that distinguishes itself from classical algebra (Cooke, 2008) by virtue of its departure from the conventional preoccupation with the formal manipulation of abstract symbols (Landy & Goldstone, 2007) as a means of resolving equations. Instead, Abstract Algebra encapsulates the transformational paradigm of algebra that materialized at the dawn of the 20^th century. This novel manifestation of algebraic thought (Bell, 1996) found its initial manifestation within the annals of academia through the publication of the seminal textbook Moderne Algebra, penned by Waerden, (1930). This literary opus, heralded as an intellectual milestone, ingeniously inverted the established conceptual hierarchy of classical algebra, thrusting algebraic structures into the vanguard and elucidating the properties of numbers (Bancerek, 2013), polynomials (Barbeau, 2003), and analogous entities as corollaries (Rossi, 2011) that emerge from the knowledge of general structures, rather than adopting the conventional approach where the comprehension of these structures derives from the properties of specific elements.

The advent of Abstract Algebra engendered a profound paradigm shift within the mathematical landscape, representing a monumental departure from the time-honored tenets of classical algebra (Kleiner, 2007). While the traditional discipline focused on the manipulation of abstract symbols (Wagner & Parker, 1993) within the context of solving equations, Abstract Algebra embarked on a transformative journey that cast aside these long-standing conventions in favor of a more profound and comprehensive comprehension of mathematical structures. It unveiled an entirely novel perspective that elucidated the intrinsic relationships between various algebraic entities by situating them within a broader framework of algebraic structures (He & Kim, 2023), wherein their fundamental properties were revealed as logical consequences of the underlying organizational principles governing these structures. Consequently, Abstract Algebra liberated algebraic thought from the shackles of its former limitations, endowing it with unprecedented flexibility and facilitating a profound expansion of mathematical horizons.

Historically, the seminal work Moderne Algebra, written by the venerable mathematician Waerden, (1930), assumed a pivotal role in disseminating and solidifying the emergent principles and tenets of Abstract Algebra. This groundbreaking treatise constituted a watershed moment in the annals of mathematical scholarship, effectuating a dramatic rupture from the entrenched traditions of classical algebra. Its revolutionary approach challenged the prevailing orthodoxy by subverting the conventional emphasis on the properties of specific elements in favor of the primacy accorded to algebraic structures. In doing so, Waerden fundamentally altered the intellectual landscape, bestowing upon abstract algebraic structures a hitherto unseen prominence, while relegating the properties of numbers, polynomials, and similar mathematical entities to the status of derivative consequences arising from the broader understanding of these algebraic structures. The profound transformation instigated by Moderne Algebra can be succinctly encapsulated in the insightful assertion made by Corry (2016), who eloquently characterized this seminal work as an ‘inversion of the conceptual hierarchy of classical algebra’. This incisive appraisal aptly captures the radical shift in perspective that unfolded within the realm of algebraic inquiry (Sokolowski & Sokolowski, 2018). By espousing a fresh vision of mathematics, Moderne Algebra defied convention and repositioned algebraic structures (Downey, 2019) as the bedrock upon which the edifice of mathematical knowledge (Resnick, 1989) was constructed. It revolutionized the understanding of algebraic phenomena (Guo et al., 2013), fundamentally altering the manner in which mathematicians approached and apprehended the intricate interplay of numbers, polynomials, and related mathematical entities.

Abstract Algebra emerged as a formidable intellectual force that reshaped the landscape of mathematical discourse at the onset of the 21^st century. Through its bold departure from the classical precepts of algebra (Smarandache, 2022), it engendered a transformative shift in the understanding and exploration of mathematical structures (Corry, 2003). The seminal work Moderne Algebra, authored by Waerden, (1930), played a pivotal role in this intellectual revolution by inverting the conceptual hierarchy, elevating algebraic structures to the forefront, and unveiling the properties of numbers, polynomials, and analogous entities as natural consequences derived from the comprehension of these overarching structures. Thus, Abstract Algebra emerged as a dynamic and fertile discipline, breathing new life into the study of mathematics and engendering fresh avenues of inquiry that continue to invigorate and enrich the mathematical enterprise to this day. Within the realm of education, the term ‘Abstract Algebra’ commonly denotes a higher-level undergraduate course specifically tailored for mathematics majors. This course revolves around the intricate examination and analysis of various algebraic structures, namely groups (Halbeisen et al., 2007), rings (Rowen, 2018), and fields (Gouvêa, 2012). Typically, Abstract Algebra courses are mandatory for mathematics students and serve as a pivotal academic milestone in their educational journey. They are usually encountered subsequent to the completion of foundational courses in Linear Algebra (Wilkinson et al., 2013), as a means to build upon and expand the students’ mathematical repertoire.

These courses in Abstract Algebra are strategically positioned at the juncture between undergraduate and graduate studies, signifying a crucial transition in the students’ academic trajectory. While they serve as a bridge between these two academic realms, they also represent a distinct sphere of mathematical inquiry (Hunter, 2006), characterized by its emphasis on abstract structures and their inherent properties. The curriculum of Abstract Algebra courses varies in complexity and depth, ranging from introductory offerings to more advanced iterations that delve into specialized topics such as Category Theory (Awodey, 2010), Homological Algebra (Cartan & Eilenberg, 1999), Algebraic Geometry (Lang, 2019), Algebraic Number Theory (Neukirch, 2013), and even Galois Theory (Stewart, 2022) as a comprehensive framework for the study of diverse structures. As a foundational course in Abstract Algebra, the curriculum typically encompasses a comprehensive study of groups, which are mathematical structures endowed with a binary operation and governed by a set of defining axioms. Students explore the properties and characteristics of groups, investigating their internal composition and examining the intricate interplay between group elements and operations. This foundational knowledge sets the stage for a deeper understanding of algebraic structures and their myriad applications.

Moving beyond groups, the Abstract Algebra course delves into the study of rings, which are algebraic structures featuring two distinct operations: addition and multiplication. Students examine the intricate algebraic properties exhibited by rings, unraveling the relationship between these operations and the underlying structure of the ring. Emphasis is placed on the properties of ring elements (Veldsman, 1987), the existence of multiplicative identities (Karvellas, 1972), the concept of integral domains (Yih, 2016), and the exploration of polynomial rings (MacLane, 1936), among other topics of significance. The exploration of fields constitutes another fundamental pillar of Abstract Algebra education. Fields, which are algebraic structures possessing the properties of both a ring and a group, serve as fertile ground for further inquiry. Students delve into the study of field extensions, investigating the interconnections between distinct fields and analyzing the implications of these extensions on the structure and properties of the overarching algebraic system. Within the realm of Abstract Algebra education, students are exposed to a vast array of advanced topics, which extend beyond the foundational structures of groups, rings, and fields. In more advanced courses, students engage with Galois Theory, an elegant and powerful framework that explores the symmetries of algebraic equations and their roots. Galois Theory provides a profound understanding of the solutions to polynomial equations, employing sophisticated mathematical techniques to unravel their inherent symmetries and uncover the relationships between various algebraic objects.

Algebraic Number Theory (Fröhlich et al., 1991) represents another captivating avenue of exploration within Abstract Algebra. It delves into the intricate interplay between number theory and algebraic structures, investigating algebraic properties and phenomena that emerge in the realm of number theory (Everest & Ward, 2005). This subfield uncovers deep connections between algebraic structures and the properties of numbers, shedding light on profound arithmetic principles and providing a profound understanding of the underlying algebraic foundations. Abstract Algebra courses may also venture into the realm of Algebraic Geometry (Görtz & Wedhorn, 2010), where algebraic techniques and structures intertwine with the study of geometric objects and their properties. This interdisciplinary field explores the correspondence between geometric shapes (Alt & Guibas, 2000) and algebraic equations (Capraro & Joffrion, 2006), leveraging algebraic techniques to analyze geometric phenomena and uncover hidden algebraic structures within the realm of geometry. Homological Algebra (Gel′fand & Manin, 1994) serves as yet another captivating facet of Abstract Algebra. It investigates the intricate algebraic structures that arise from the study of chain complexes and their homology (Grigor’yan et al., 2020). Students delve into the study of exact sequences, homological functors (Weibel, 1994), and other advanced concepts, enabling them to analyze and comprehend the algebraic properties that underlie these complex chains. Abstract Algebra courses may even encompass Category Theory (Riehl, 2017), a general theory of structures that transcends the boundaries of specific algebraic domains. Category Theory provides a comprehensive framework for the study of diverse structures, emphasizing the relationships and transformations between them. This abstract and powerful mathematical framework equips students with a broader perspective on algebraic structures and their connections, fostering a deeper appreciation for the underlying unity and coherence of mathematics as a whole. Abstract Algebra courses within an educational context cater to upper division undergraduate students pursuing mathematics majors. They serve as pivotal courses that build upon the foundations of Linear Algebra, exploring the intricate structures of groups, rings, and fields. These courses act as gateways between undergraduate and graduate studies, with varying levels of complexity and specialization. The breadth and depth of Abstract Algebra courses enable students to engage with diverse and sophisticated mathematical concepts, fostering their intellectual growth and nurturing their mathematical intuition (Feferman, 2000).

Praxeologies

Drawing on the objects-structures dialectic, which elucidates the interplay between specific mathematical objects and their underlying structures, Hausberger (2018) highlights the dynamic relationship between the concrete and the abstract. Axiomatic structures, encompassing algebraic concepts and systems, are essential components of structuralist praxeologies. These structures offer a means of generalizing and simplifying mathematical phenomena by providing a coherent framework for understanding and analyzing various mathematical objects. By employing axiomatic structures, mathematicians can derive and demonstrate properties that hold true across a range of specific instances. This process of generalization allows for a deeper understanding of the underlying principles and structures that govern mathematical objects, fostering a more comprehensive comprehension of Abstract Algebra. Conversely, the objects-structures dialectic recognizes the reciprocal relationship between structures and objects. Through the examination of concrete examples and specific instances, mathematicians can subject formal general statements on structures to empirical verification. By testing these statements against known examples, mathematicians can ensure that the theoretical constructs accurately capture the essential properties of the mathematical objects they seek to describe. This interplay between structures and objects ensures a constant feedback loop, where the theoretical constructs are refined and verified through empirical evidence.

Hausberger’s praxeological models shed light on the challenges faced by students as they navigate the transition from the concrete to the abstract in the domain of Abstract Algebra. These models provide a valuable institutional perspective, taking into account the specific educational context in which students engage with Abstract Algebra. By examining the difficulties encountered by students at different academic levels, the models offer insights into the cognitive obstacles that impede the development of abstract mathematical thinking. This institutional perspective serves as a foundation for the design and implementation of innovative instructional approaches aimed at addressing these challenges. In this vein, inquiry-based instructional approaches are proposed as study and research paths that foster a deeper understanding of Abstract Algebra. These approaches empower students to engage in active exploration, investigation, and problem-solving, enabling them to construct their own mathematical knowledge and meaning. By immersing students in authentic mathematical experiences and encouraging them to pose questions, make conjectures, and seek solutions, inquiry-based approaches promote a deeper understanding of abstract algebraic concepts. These approaches also cultivate critical thinking skills, mathematical reasoning, and the ability to transfer mathematical knowledge to new contexts.

By employing praxeological models, educators can gain insights into the challenges faced by students during the transition from the concrete to the abstract. This understanding serves as a foundation for the development of inquiry-based instructional approaches that promote deep comprehension, critical thinking, and mathematical reasoning in the field of Abstract Algebra. In the work of Hausberger (2017), the newly devised structure draws upon phenomenological roots and exhibits resemblances to permutation theory, making it particularly suitable for fostering classroom dialogues centered on structuralist perspectives through the application of the meta-lever. The concept of the ‘banquet’ serves as a didactical engineering endeavor, meticulously designed to enhance students’ comprehension and assimilation of structuralist thinking. Through the introduction of this unique mathematical structure, Hausberger seeks to provide a practical framework that encapsulates essential aspects of structuralist thought and facilitates its internalization by students. By utilizing the banquet as a didactical tool, Hausberger aims to stimulate insightful discussions within the classroom, enabling students to explore and engage with the underlying principles of structuralist views.

The phenomenological underpinnings of the banquet structure emphasize the importance of grounding mathematical concepts in concrete experiences and perceptions. By anchoring abstract notions in the realm of tangible phenomena, students can develop a more intuitive understanding of the structuralist perspective. This phenomenological approach enables students to connect the banquet structure with their own experiential knowledge, enhancing their engagement and conceptual development. Moreover, the banquet structure exhibits intriguing parallels with permutation theory, a field of mathematics concerned with the arrangement and transformation of objects. The similarities between the banquet structure and permutation theory provide students with a bridge between familiar mathematical concepts and the abstract notions encapsulated within the structuralist framework. This connection allows students to draw upon their prior knowledge of permutation theory, facilitating a smoother transition and deeper exploration of structuralist concepts. A crucial aspect of the didactical engineering behind the banquet structure is its compatibility with the utilization of the meta-lever. The meta-lever functions as an instructional tool aimed at fostering meta-awareness and meta-cognition in students. By employing the meta-lever, educators can guide students in reflecting upon and analyzing the underlying assumptions, principles, and methodologies of structuralist thinking. Through these reflective processes, students develop a heightened awareness of the overarching structuralist framework and its implications for mathematical reasoning.

The banquet structure serves as a prime catalyst for classroom discussions centered on the objects-structures dialectic. By engaging with the banquet structure, students are encouraged to explore the interplay between specific objects and the broader structures that govern them. This dynamic dialogue allows students to examine how specific objects within the banquet structure relate to and interact with the overarching structural principles. The banquet structure thus provides a fertile ground for students to delve into the intricacies of the objects-structures dialectic and develop a more nuanced understanding of its implications. By incorporating semiotic, phenomenological, and didactical considerations, Hausberger’s work contributes to the pedagogical landscape, offering educators valuable tools and strategies to cultivate students’ engagement with abstract mathematical concepts. Through the utilization of the banquet structure and the meta-lever, students are guided towards a deeper appreciation of structuralist perspectives, paving the way for enhanced mathematical reasoning and proficiency in the field of Abstract Algebra.

Genetic decompositions, APOS framework, and group Theory (GT)

In the 1990s, Dubinsky et al. (1994) initiated an extensive and ongoing project in the United States that focused on the learning and teaching of undergraduate mathematics, specifically Group Theory (GT). The framework employed for this project was the APOS (Action, Process, Object, Schema) framework, which served as a valuable tool for investigating errors, misconceptions, and designing innovative pedagogical approaches. The researchers proposed ‘genetic decompositions’ of fundamental GT concepts, with particular emphasis on the examination of cosets in quotient groups. They also developed teaching sequences that revolved around computer-based experiences, subsequent discussions, and formal definitions. Through their research, Dubinsky and his team made significant strides in uncovering robust findings regarding the teaching and learning of GT. One notable discovery was the central role played by the encapsulation of the process of forming cosets within an object. This recognition highlighted the significance of effectively integrating the formation of cosets as an object of study in order to facilitate conceptual understanding. The underlying teaching methodology was based on the premise that the implementation of algorithms could enhance the construction of mathematical concepts within the learner’s cognitive framework.

To facilitate the implementation of their teaching approach, the researchers selected the ISETL programming language. This decision was motivated by the language’s close alignment with the mathematical constructs employed in the study, including sets, functions, and logical expressions. The choice of ISETL provided a means of integrating computational experiences into the learning process, offering learners a tangible platform for exploring and solidifying their understanding of GT concepts. The project launched by Dubinsky and his colleagues marked a significant advancement in the field of mathematics education, particularly in the context of undergraduate learning. By adopting the APOS framework and utilizing computer-based experiences, the researchers aimed to deepen their understanding of errors and misconceptions while simultaneously devising novel teaching approaches. The concept of ‘genetic decompositions’ played a pivotal role in the project, enabling the researchers to break down complex GT concepts into their fundamental components, thereby facilitating more targeted instruction and remediation. The teaching sequences developed by Dubinsky and his team followed a systematic structure, commencing with computer-based experiences that allowed learners to engage directly with the subject matter. These experiences served as a foundation for subsequent discussions, during which learners had the opportunity to articulate their observations, ask questions, and engage in collaborative sense-making activities. Through this iterative process of computer-based exploration followed by reflective discourse, learners were able to gradually build a coherent understanding of the GT concepts under investigation.

The incorporation of formal definitions within the teaching sequences added a crucial layer of precision and rigor to the learning process. By providing learners with explicit definitions, the researchers sought to establish a solid conceptual framework upon which subsequent learning could be built. The interplay between computer-based experiences, discussions, and formal definitions created a dynamic and comprehensive learning environment that aimed to address the various dimensions of conceptual understanding in GT. The project initiated by Dubinsky et al. (1994) represented a significant and ongoing endeavor in the field of undergraduate mathematics education. Their utilization of the APOS framework, coupled with the integration of computer-based experiences and systematic teaching sequences, allowed for a deeper exploration of errors, misconceptions, and effective pedagogical approaches in GT. The project’s findings shed light on the importance of encapsulating key processes within objects of study, and the selection of the ISETL programming language facilitated the computational exploration of GT concepts. This project, thus, contributed to the advancement of instructional methodologies in undergraduate mathematics education, providing valuable insights for both researchers and practitioners in the field.

Research-based inquiry to foster deep comprehension

In recent years, Larsen (2013) initiated the Teaching Abstract Algebra for Understanding project, which aimed to develop a curriculum for Abstract Algebra grounded in research-based inquiry and designed to foster deep comprehension. The instructional design of this project drew upon the Realistic Mathematics Education (RME) framework and placed particular emphasis on Group Theory (GT). The origins of GT, as perceived through a phenomenological lens following Freudenthal, were traced back to the study of symmetries exhibited by geometric figures. The resulting system served as a model for reasoning. The progression of activities culminated in the ‘reinvention’ of the definition of a group, wherein students independently arrived at a comprehensive understanding of the fundamental properties and structure of groups. An analogous process was employed to address quotient groups and the concept of isomorphism within the domain of Group Theory. Through carefully crafted activities and guided inquiries, students explored and analyzed the notion of quotient groups, investigated the relationship between these groups and the original group, and ultimately arrived at a deep comprehension of the isomorphism concept. By engaging in this process of exploration and discovery, students developed a profound understanding of abstract algebraic concepts within the context of GT. The curriculum design of the Teaching Abstract Algebra for Understanding project was firmly grounded in the RME framework. This framework advocates for the integration of real-world contexts and authentic problem-solving experiences to facilitate meaningful mathematical learning. By immersing students in the investigation of symmetries and algebraic structures present in geometric figures, the project aimed to provide a concrete and tangible foundation for the development of abstract algebraic reasoning.

The inquiry-oriented nature of the curriculum encouraged students to actively construct their own knowledge and engage in mathematical discourse with their peers. Through this collaborative exploration, students were able to develop mathematical reasoning skills, critically analyze mathematical phenomena, and draw connections between different mathematical concepts. The emphasis on inquiry-based learning fostered a deep understanding of abstract algebraic structures, enabling students to internalize key concepts and their interrelationships. This project exemplifies a contemporary approach to mathematics education that recognizes the importance of engaging students in meaningful mathematical experiences. By linking abstract algebraic concepts to concrete examples and utilizing the power of inquiry-based learning, the project aimed to promote conceptual understanding, foster mathematical reasoning, and cultivate a deep appreciation for the beauty and utility of Abstract Algebra. The incorporation of phenomenological roots, derived from Freudenthal’s insights on the symmetries of geometric figures, provided a rich and meaningful context for students to explore and discover the principles and structures inherent in abstract algebraic systems. Larsen’s project represents a valuable contribution to the field of mathematics education, offering a research-based and inquiry-oriented curriculum for teaching Abstract Algebra. By leveraging the RME framework and utilizing phenomenological connections, the curriculum aimed to enhance students’ understanding of abstract algebraic concepts, cultivate their mathematical thinking skills, and empower them to engage with the profound beauty and significance of Abstract Algebra. Through this approach, students were encouraged to become active participants in their own learning, developing a deep appreciation for the fundamental principles and structures that underpin abstract algebraic systems.

Epistemological and historical breakthroughs

As attested by historians, notably Wussing (2007), the profound significance of the group concept in permutation theory, pioneered by Galois, and its instrumental role in reconciling the diverse realms of geometry, championed by Klein, engendered a momentous advancement in mathematical thought prior to the dawn of the 20^th century. This pivotal epoch witnessed the emergence of the group structure as the primordial abstract mathematical construct, heralding a transformative shift in mathematical discourse. The assimilation and integration of disparate historical origins of groups were skillfully accomplished by Dyck, who introduced the abstract notion of groups through the lens of generating operations and their interrelations. Galois’ revolutionary insights illuminated the profound connection between the symmetries inherent in permutations and the fundamental structure of groups. By unveiling the intricate interplay between permutation theory (Kanji, 1976) and group theory (Rose, 1994), Galois laid the foundation for a more profound and abstract comprehension of mathematical structures, transcending the limitations imposed by the conventional paradigms of the time. Furthermore, the unification of the disparate branches of geometry, accomplished by Klein, constituted another pivotal milestone in the development of group theory. Klein’s visionary synthesis of different geometries, such as Euclidean (Ryan, 1986), hyperbolic (Cannon et al., 1997), and projective geometries (Cameron, 2020), under the overarching umbrella of group theory (Rose, 1994), illuminated the deep-seated symmetry and invariance exhibited by geometric objects. By harnessing the power of group theory, Klein unveiled the unifying principles that underpin these seemingly disparate geometric realms, underscoring the pervasive role of groups in shaping the fundamental fabric of mathematical structures.

The synthesis and consolidation of diverse historical sources of groups into a cohesive and unified framework were masterfully accomplished by Dyck. Recognizing the inherent generative nature of groups, Dyck introduced the concept of abstract groups by focusing on the operations that generate the elements of a group and their mutual relationships. This groundbreaking conceptualization transcended the limitations imposed by specific instances of groups and paved the way for a more general, abstract understanding of groups as mathematical structures governed by a set of axioms (Lane, 1996) and interrelations. Dyck’s seminal contributions formed the bedrock upon which the modern theory of abstract groups would be built. Hilbert’s systematic and rigorous approach to mathematics sought to establish a solid foundation through the articulation of precise axioms and logical deductions (Gentzen, 1964). This meticulous methodology not only bolstered the rigor of mathematical discourse (Moschkovich, 2007) but also provided the necessary framework to investigate and comprehend the abstract and relational nature of groups (Givant & Givant, 2017). By elevating the study of groups to a formal discipline guided by axiomatic principles, Hilbert facilitated a more comprehensive exploration of the intrinsic properties and interconnections of these mathematical structures. Hilbert’s promotion of the axiomatic method reinforced the formal, abstract, and relational perspective, providing a solid foundation for the study of groups. Collectively, these advancements paved the way for the establishment of group theory (Hamermesh, 2005) as a foundational discipline, laying the groundwork for the remarkable developments that would unfold in the field throughout the twentieth century and beyond. The overarching aim of Dedekind’s endeavors was to provide a generalization of the fundamental theorem of arithmetic, an undertaking that would profoundly impact the field of number theory. Finite fields, which play a vital role in modern mathematics, find their roots in the pioneering work of Galois and the theory of polynomial congruences developed by Gauss. These seminal contributions set the stage for the exploration and understanding of finite fields, laying the groundwork for subsequent investigations into their properties, structures, and applications.

The emergence of p-adic fields, attributed to the mathematician Hensel, constituted a further impetus for the study of fields. Hensel’s groundbreaking insights into p-adic numbers and their associated fields expanded the frontiers of mathematical inquiry, fostering a deeper comprehension of the structures and properties inherent in these fields. Inspired by these developments, Steinitz advanced the field of field theory in 1910 by formulating a comprehensive general theory of fields. Central to Steinitz’s framework was the classification of fields based on their characteristic, along with the pivotal notion of a prime field. This classification system offered a powerful tool for distinguishing and analyzing different types of fields, enabling mathematicians to investigate their distinct properties and characteristics in a more systematic manner. In contrast to fields, which had been formally introduced by Dedekind, the concept of rings was initially put forth by Hilbert in 1900. Hilbert’s pioneering work on rings primarily focused on the rings of integers associated with number fields. However, it is worth noting that at the time of Hilbert’s introduction of rings, he did not establish any explicit connections or relationships with groups or fields. Consequently, the concept of rings, as presented by Hilbert, existed as an independent and self-contained mathematical construct, separate from the overarching theories of groups and fields.

Hilbert’s notion of rings played a crucial role in the realm of number theory, providing a formal framework for the study of the rings of integers associated with number fields. By delving into the inherent properties and structures of these rings, mathematicians gained a deeper understanding of number theory and its intricate connections to algebraic structures. Nevertheless, the interconnections between rings, groups, and fields were not explicitly elucidated or explored by Hilbert in his seminal work. Dedekind’s introduction of fields in 1871 marked a significant breakthrough in mathematical theory. Concurrently, the study of finite fields, influenced by Galois and Gauss, garnered significant attention and became a fertile area of research. The advent of p-adic fields, pioneered by Hensel, further expanded the scope of field theory. Steinitz’s classification system, based on the characteristic and prime field, provided a framework for organizing and analyzing different types of fields. In contrast, Hilbert’s introduction of rings in 1900, while foundational in the context of number theory, was not explicitly linked to the theories of groups or fields, remaining an independent mathematical concept. These developments collectively contributed to the rich tapestry of algebraic structures and laid the groundwork for future advancements and investigations in the field of abstract algebra. Noether, a prominent mathematician of the 1920s, is credited with introducing the idea of algebraic structures as a unifying principle. Her seminal contributions propelled the field of Abstract Algebra beyond conventional notions of manipulating elements through basic operations, such as addition or multiplication in groups or rings. Noether’s revolutionary approach involved a paradigm shift, wherein structures were defined and understood in terms of meticulously chosen subsets, such as normal subgroups in Group Theory or ideals in Ring Theory, coupled with the exploration of homomorphisms. By reframing algebraic structures in this manner, Noether and her contemporaries transformed the landscape of mathematical theorems, focusing on comprehensive proofs that transcended specific calculations and emphasized the ‘most general and fundamental’ concepts. These overarching principles, according to structuralist viewpoints, inherently possess a certain elegance and simplicity. This mathematical reformation of algebra not only engendered a fresh perspective on the discipline but also paved the way for unprecedented mathematical constructs and insights.

The axiomatic method, employed to formulate and apply structuralist concepts, finds a comprehensive exposition in the influential manifesto authored by Bourbaki (1950). Drawing inspiration from the German algebraists, Bourbaki sought to extend the methodology to all branches of mathematics, promoting a unified and systematic approach rooted in structuralist thinking. Their concerted efforts were instrumental in shaping the course of mathematical inquiry across various domains. The impact of Noether’s structuralist approach extended beyond the realm of mathematics, influencing other scientific disciplines, including the human sciences of psychology and sociology. Furthermore, it catalyzed the New Math reform movement in the 1960s (Herrera & Owens, 2001), which sought to revolutionize the teaching of secondary education mathematics in countries such as the USA, France, and Germany. The reform entailed a dramatic shift in instructional methodologies, with a strong emphasis on mathematical formalism (Henderson, 2018) and elementary aspects of Set Theory (Hausdorff, 2021), symbolic logic (Lewis et al., 1959), and Abstract Algebra. By prioritizing the teaching of mathematical structures, this reform aimed to foster a deeper understanding of the abstraction processes (Mitchelmore & White, 1995) involved in conceptualizing these structures.

While Noether’s contributions and the subsequent advancements in Abstract Algebra have significantly impacted the mathematical landscape, there are still areas requiring further exploration. The notion of structure, being a vital organizing principle, demands continued investigation into its semiotic (Socas & Hernández, 2013), phenomenological (Hausberger, 2020), and didactical (D’Amore et al., 2012) aspects. Additionally, the persistent challenges in pedagogical practices and the incorporation of computational tools, such as Computer Algebra Systems (CAS) (Kramarski & Hirsch, 2003), warrant dedicated research efforts. Ultimately, a comprehensive reform of Abstract Algebra education, encompassing the reorganization of curricula at both undergraduate and graduate levels, appears essential to effectively address the enduring difficulties encountered by students. This transformative shift in perspective fundamentally altered the manner in which theorems were proven in algebra, with a newfound emphasis on general proofs that minimized cumbersome calculations and highlighted the ‘most general and fundamental’ concepts. By prioritizing these structuralist notions, Noether and her school facilitated a mathematical reconfiguration of algebra, offering a fresh vision of mathematics as a discipline and paving the way for the creation of unprecedented mathematical constructs. This mathematical re-foundation of algebra instigated a paradigm shift, propelling the discipline towards new horizons. No longer limited to the manipulation of symbols and operations on elements, algebra now embraced a broader conceptual framework, wherein structures and their interrelations assumed center stage. By elucidating the inherent relationships and properties of carefully chosen subsets and mapping structures, mathematicians were empowered to tackle complex problems and unveil profound insights into the nature of algebraic systems. This transformative approach, championed by Noether and her contemporaries, marked a departure from traditional algebraic methodologies and fostered the development of a more sophisticated and elegant algebraic discourse.

Axioms served as the bedrock upon which mathematical systems were constructed, providing a rigorous framework for the exploration of their properties and interrelationships. By systematically defining the axioms governing various mathematical domains, Bourbaki endeavored to establish a comprehensive and interconnected mathematical landscape, transcending the boundaries imposed by specific subfields. This ambitious undertaking sought to integrate disparate mathematical disciplines and foster a unified mathematical language, enabling mathematicians to communicate and collaborate across disciplinary boundaries. In adopting the structuralist perspective, Bourbaki espoused a holistic approach to mathematical inquiry, recognizing the inherent interconnectedness of mathematical structures. This approach enabled mathematicians to discern the underlying unity and coherence that permeates diverse branches of mathematics. By identifying and elucidating the fundamental concepts and principles that underpin mathematical structures, Bourbaki and their adherents aimed to forge a harmonious and all-encompassing mathematical framework. The concept of algebraic structures as a unifying principle emerged through the pioneering work of Noether and her contemporaries. By shifting the focus from operations on elements to the study of carefully selected subsets and homomorphisms, Noether revolutionized algebraic discourse, leading to a mathematical reconfiguration and the birth of unprecedented constructs. This structuralist approach, as outlined in the Bourbaki Manifesto, leveraged the power of the axiomatic method to foster a unified and interconnected mathematical landscape, transcending disciplinary boundaries and promoting a holistic understanding of mathematics.

Mathematical structures

Mathematical structuralism, with its profound insights and foundational principles, exerted a far-reaching influence not only within the realm of mathematics but also across various scientific disciplines. One notable consequence of mathematical structuralism was the New Math reform, which emerged during the 1960s and was implemented in countries such as Germany, France, and the USA. This reform brought about a significant transformation in the pedagogy of secondary education mathematics, heralding a departure from traditional instructional methods. Central to the New Math reform was a renewed emphasis on mathematical formalism, which encompassed elementary aspects of Abstract Algebra, symbolic logic, and Set Theory. Students were exposed to these advanced mathematical concepts at an earlier stage of their education, enabling them to develop a deeper understanding of the underlying structures that govern mathematical systems. This pedagogical approach aimed to cultivate students’ analytical and critical thinking skills, fostering a more profound appreciation for the abstract and foundational aspects of mathematics. The reformed curriculum prioritized the teaching of mathematical structures, prompting an upsurge in research endeavors focused on investigating the cognitive processes underlying the conceptualization and abstraction of these structures. Scholars delved into the intricate mechanisms through which individuals grasp and internalize abstract mathematical concepts, striving to unravel the cognitive pathways that facilitate the formation and manipulation of mathematical structures within the human mind.

The incorporation of structuralist principles into the teaching of mathematics heralded a new era of mathematical education, characterized by a heightened awareness of the fundamental role played by abstract structures in the field. Students were no longer confined to rote memorization (Skemp, 2012) or procedural calculations (Song & Ginsburg, 1987); instead, they were encouraged to explore the underlying principles and connections that unify diverse mathematical concepts. This pedagogical shift not only empowered students to engage with mathematics at a deeper level but also nurtured their capacity for critical thinking (Lai, 2011), problem-solving (Snyder & Snyder, 2008), and logical reasoning (Poole et al., 1987). However, despite the initial enthusiasm and widespread implementation of the New Math reform, it eventually faced significant challenges and criticism. Some detractors argued that the reform overemphasized abstract concepts at the expense of practical applications, leading to a perceived disconnect between the mathematics taught in schools and its real-world relevance. This criticism, coupled with concerns about the complexity and accessibility of the reformed curriculum, ultimately contributed to the decline and eventual abandonment of the reform in many educational systems. Nonetheless, the legacy of the New Math reform endures, leaving an indelible mark on the landscape of mathematical education. Its impact extended beyond the specific content taught in classrooms, instigating a broader conversation about the goals, methods, and pedagogical approaches in mathematics instruction. The reform sparked ongoing reflections on the balance between abstract and applied mathematics, the role of mathematical structures in fostering mathematical reasoning, and the development of cognitive skills essential for success in mathematical endeavors.

In the realm of University Mathematics Education (UME), particular attention is directed toward theoretical elaborations that revolve around the intricate concepts of structures and abstraction, particularly in relation to the domain of Abstract Algebra. Within this educational framework, emphasis is placed on the exploration and elucidation of mathematical structures. These structures serve as the foundational building blocks upon which the edifice of mathematical knowledge is erected. By comprehending the intricate interplay and interconnections between various elements within these structures, students are empowered to perceive the underlying coherence and organization of mathematical systems. Furthermore, the concept of abstraction assumes a paramount role in UME. Abstraction entails the process of distilling complex mathematical phenomena to their essential and generalizable components, allowing for the formulation of comprehensive and universal principles. By engaging in abstraction, students acquire the ability to transcend the specificities of individual instances and discern overarching patterns and principles that apply across a wide array of mathematical contexts.

Thematization and idealization

The notion of abstraction within the framework of structuralist thinking, as expounded by French philosopher Cavaillès, posits that it comprises two intertwined processes: idealization and thematization. Idealization involves the distillation of essential elements or properties from a given phenomenon, removing extraneous details to focus on its core conceptual framework. Thematization, on the other hand, entails the act of endowing these abstracted elements with a thematic structure, providing a coherent framework for their understanding and analysis. Benis-Sinaceur (2014) explores these twin processes, shedding light on their pivotal role in the development of structuralist thinking. Cavaillès’ philosophical dialectic between form and matter, as it pertains to algebraic structuralism, greatly influenced Piaget, a prominent psychologist. Piaget posited that mathematics as a whole could be comprehended through the lens of constructing structures via a process he termed ‘reflective abstraction’. In this process, forms or abstract concepts are encapsulated as new contents, enabling individuals to engage with and manipulate these structures. Piaget’s insights into reflective abstraction aligned closely with the principles of structuralist thinking, emphasizing the central role of abstract structures in mathematical cognition.

Building upon Piaget’s work, Dubinsky (1991) expounded on the encapsulation of processes into objects within the framework of APOS theory. According to Dubinsky, the APOS framework elucidates the cognitive progression from acting, to processes, to objects, and to schemas. Within this framework, the encapsulation of a process into an object signifies a pivotal step in the cognitive development of mathematical understanding. This conceptual encapsulation allows individuals to manipulate and reason with objects, providing a tangible foundation for engaging with abstract mathematical structures. Expanding upon the theoretical underpinnings of thematization in Abstract Algebra, Hausberger (2017) delved into its epistemological and didactical aspects. Thematization, as conceptualized by Hausberger, involves the creation of coherent thematic frameworks within which abstract algebraic concepts, such as homomorphism, can be comprehended. By contextualizing these concepts within well-defined themes and exploring their interconnections, students are afforded a deeper understanding of the fundamental principles and properties that underpin Abstract Algebra.

Hausberger’s research sheds light on the importance of thematization in Abstract Algebra education, highlighting the pedagogical significance of structuring mathematical concepts within meaningful contexts. By organizing abstract algebraic notions into coherent themes, educators can facilitate students’ comprehension and foster their ability to make connections and discern patterns within the abstract realm of algebraic structures. This didactical approach enhances students’ engagement and enables them to develop a more profound grasp of the intricate interplay between mathematical concepts. Cavaillès’ examination of abstraction as encompassing idealization and thematization provides a theoretical framework for understanding the cognitive processes involved in structuralist thinking. Piaget’s work on reflective abstraction and Dubinsky’s exploration of encapsulation in APOS theory further extend this understanding to the realm of mathematics. Hausberger’s investigation of thematization in Abstract Algebra elucidates the didactical and epistemological dimensions of organizing abstract concepts within coherent thematic frameworks. Together, these theoretical elaborations contribute to our comprehension of the intricate interplay between structures, abstraction, and cognition within the realm of mathematics.

Mathematical structures

Within the context of mathematical cognition, the process-object duality assumes a significant role, with structural aspects serving to characterize the object dimension. This duality is often referred to as the procedural-structural duality, highlighting the interplay between the processes involved in mathematical thinking and the structural properties exhibited by mathematical objects. Simpson and Stehlíková (2006) investigation sheds light on the cognitive processes involved in developing a deep understanding of abstract mathematical structures, specifically in the context of a commutative ring. They underscore the significance of perceptual and conceptual shifts required for students to grasp the underlying structural properties of mathematical objects. As students progress from working with concrete objects and operations to engaging with abstract structures, their attention must shift from the particulars of individual elements to discerning the intricate interrelationships that arise from the operations enacted upon these elements. The researchers Simpson and Stehlíková highlight the significance of this cognitive transition as students develop their structural sense, a heightened awareness of the underlying organization and coherence of abstract mathematical systems. By recognizing and understanding the interplay between objects within a mathematical structure, students are better equipped to reason abstractly and engage with the formal properties and relationships inherent in abstract algebraic settings.

The notion of structural sense, as explored by Simpson and Stehlíková, encompasses the capacity to discern and appreciate the intrinsic connections and dependencies that exist among the elements of a mathematical structure. It involves a profound understanding of how operations impact the relationships between objects, providing insights into the overarching properties and patterns that govern the structure as a whole. Developing structural sense entails a cognitive shift from perceiving mathematics solely in terms of isolated facts and procedures to recognizing the intricate web of connections that underlies mathematical systems. The findings of Simpson and Stehlíková (2006) align with the broader literature on mathematical cognition, which emphasizes the crucial role of abstraction and structural thinking in the development of mathematical understanding. The ability to shift attention from the specific to the general, from objects to relationships, is a hallmark of advanced mathematical thinking. It enables individuals to perceive the underlying coherence and organization of mathematical structures, paving the way for sophisticated reasoning and problem-solving abilities. The process-object duality, also known as the procedural-structural duality, plays a significant role in mathematical cognition. Simpson and Stehlíková (2006) investigation into the acquisition of structural sense highlights the cognitive shifts necessary for understanding abstract structures, particularly in the context of a commutative ring. This cognitive transition is crucial for students to perceive the underlying organization and coherence of mathematical structures, enabling them to reason abstractly and engage with the formal properties and relationships inherent in abstract algebraic settings.

In his phenomenological exploration of various types of structures—geometrical, topological, algebraic, and more—Freudenthal (1983) emphasizes the importance of situating these structures within a geometrical context. He argues that placing structures in a geometrical framework facilitates the development of intuitions and highlights the necessity and efficacy of spatial representations. From Freudenthal’s perspective, structures serve as tools for organizing the phenomena observed in the mental, social, and physical realms. Consequently, unraveling the process of idealization entails identifying the specific phenomena that a given structure aims to organize. Freudenthal’s phenomenological approach to structures highlights the significance of grounding mathematical abstractions in concrete spatial representations. By situating structures within a geometrical context, individuals can develop intuitive understandings that aid in their comprehension and manipulation of abstract mathematical concepts. This emphasis on spatial representations aligns with Freudenthal’s broader perspective that structures act as powerful tools for organizing and making sense of the diverse phenomena encountered in the physical, social, and mental realms. Furthermore, Freudenthal’s notion of idealization involves identifying the specific phenomena that a given structure seeks to capture and organize. Idealization, in this context, can be understood as the process of abstracting essential features and properties from observed phenomena, stripping away extraneous details to uncover the core conceptual framework that underlies the observed phenomena. By isolating and focusing on these fundamental aspects, mathematicians and scientists can construct mathematical structures that provide powerful tools for organizing and understanding complex phenomena.

Structures, as meta-concepts, not only organize the phenomena encountered in various domains but also guide the process of thematization—the vertical mathematization specific to the structuralist methodology. Thematization involves the systematic exploration and development of mathematical themes within a given structural context. By thematizing abstract concepts and discerning the underlying relationships and patterns within a structure, individuals gain a deeper understanding of the fundamental principles and properties that govern it. The concept of thematization, as discussed by Hausberger within the framework of Realistic Mathematics Education (RME), emphasizes the role of structures as organizing principles in mathematical abstraction. By organizing mathematical concepts into coherent themes, individuals can discern the intrinsic connections and relationships that exist within a structure. This thematized approach enhances understanding and promotes a more holistic comprehension of the abstract concepts and structures being studied. Freudenthal’s phenomenological approach underscores the importance of placing structures within a geometrical context to facilitate intuitive understanding and the development of spatial representations. He views structures as tools for organizing phenomena in the physical, social, and mental realms. By thematizing abstract concepts, individuals gain a deeper understanding of the underlying principles and relationships that govern the structure.

FUGS concepts and didactical consequences

Throughout history, several significant milestones have highlighted the formalizing, unifying, generalizing, and simplifying (FUGS) nature of concepts in Abstract Algebra. The recognition of the FUGS character of Abstract Algebra concepts underscores their significance in the development and advancement of mathematical knowledge. Formalizing refers to the process of establishing rigorous definitions, axioms, and logical frameworks to ensure precision and coherence within the field. Unifying involves finding common structures and patterns across diverse mathematical objects, leading to the discovery of overarching concepts and principles. Generalizing entails extending concepts and results to broader contexts and allowing for the application of abstract reasoning and techniques. Finally, simplifying involves distilling complex ideas into more manageable forms, stripping away unnecessary details while retaining the essential underlying structure. The epistemological nature of FUGS concepts presents challenges for teaching strategies rooted in traditional didactic tools. The Theory of Didactical Situations, which aims to create meaningful and engaging learning experiences by situating mathematical concepts in realistic contexts, may face difficulties in effectively addressing the FUGS nature of Abstract Algebra. Dorier (1995) argues that finding a single ‘fundamental situation’ that encompasses the central concepts of Linear Algebra or similar abstract concepts is problematic due to the wide range of mathematical objects and the diversity of relationships between them. This challenge calls for a shift in approach and the incorporation of meta-aspects in the teaching and learning process.

To address these difficulties, Dorier suggests employing a ‘meta-lever’ approach. This approach involves emphasizing meta-level considerations, such as the axiomatic method and the structuralist methodology, in the teaching and learning of Abstract Algebra. By focusing on the underlying principles, structures, and relationships that govern the abstract concepts, students can develop a deeper understanding and appreciation of the subject matter. The meta-lever approach encourages students to engage with the fundamental ideas and principles that transcend specific instances or examples, promoting a more comprehensive grasp of the abstract concepts in Abstract Algebra. The search for fundamental situations that effectively introduce FUGS concepts remains a topic of debate within the UME community. This perspective suggests that exploring the symmetries of geometric shapes can provide a concrete and accessible entry point into the abstract concepts of groups. However, the identification of fundamental situations for other FUGS concepts, such as those found in other branches of Abstract Algebra, continues to be a subject of ongoing investigation and scholarly discourse. The historical landmarks in the development of Abstract Algebra highlight its FUGS character, characterized by formalizing, unifying, generalizing, and simplifying concepts. The epistemological nature of FUGS concepts presents challenges for traditional didactic tools, leading to the exploration of alternative approaches such as the meta-lever approach. The search for fundamental situations that effectively introduce FUGS concepts continues to be a topic of debate, with proposals suggesting the use of counting problems and symmetries as potential entry points. Understanding and addressing the epistemological nature of Abstract Algebra concepts are crucial for enhancing the learning and teaching experience in University Mathematics Education.

FUGS, which stands for ‘Formalizing, Unifying, Generalizing, and Simplifying’, is a problem-solving strategy or a set of principles often used in mathematics and science to tackle complex problems and concepts. It is a systematic approach that involves the following four steps:

1. Formalizing: Formalizing involves expressing a problem or concept in a precise, mathematical, or systematic manner. It is about creating a clear and unambiguous representation of the problem. This step often includes defining terms and variables, establishing relationships, and using mathematical notation when appropriate. Formalizing helps remove ambiguity and lays the foundation for rigorous analysis.

2. Unifying: Unifying is the process of identifying common patterns or principles that exist within the problem or concept. It involves looking for connections, relationships, or similarities among different elements or aspects of the problem. By unifying various components, you can simplify the problem by focusing on its essential characteristics. Unifying often leads to insights and a deeper understanding of the problem.

3. Generalizing: Generalizing means identifying broader principles or rules that apply not only to the specific problem at hand but also to a broader class of problems. It involves finding a more abstract and generalized representation of the problem or concept. Generalization allows one to apply the same solution or framework to a range of related problems, making problem-solving more efficient and insightful.

4. Simplifying: Simplifying involves the process of reducing complexity. After formalizing, unifying, and generalizing, one should be able to simplify the problem or concept by stripping away unnecessary details or complexities. This can make the problem more manageable and easier to understand. Simplifying does not necessarily mean making the problem trivial; it means focusing on the core elements and key principles.

FUGS is a problem-solving methodology that helps individuals approach complex problems or abstract concepts in a systematic and structured manner. It is commonly used in mathematics, physics, and engineering, but it can also be applied in various other fields. By following these steps, problem solvers can make complex issues more understandable and develop elegant and insightful solutions. This approach is particularly useful when dealing with abstract and challenging problems where a systematic strategy can lead to more effective problem-solving and deeper insights. To understand it better, let us take a mathematical problem in abstract algebra:

Problem Statement: Consider the field extension ℚ(√2) over ℚ (the rational numbers). Determine the minimal polynomial of √2 over ℚ.

FUGS approach

• Step 1. Formalizing: The first step is to formalize the problem by defining key terms and notations. In this case:

• Field Extension: ℚ(√2) over ℚ signifies that we are extending the field of rational numbers (ℚ) to include the square root of 2 (ℚ(√2)).

• Minimal Polynomial: We need to determine the polynomial that √2 satisfies in the extended field, i.e. a polynomial with coefficients in ℚ that evaluates to 0 when √2 is substituted.

• Step 2. Unifying: In the unifying step, we seek to identify patterns, relationships, or connections within the problem.

• Unification: We observe that the problem involves the extension of a field (ℚ) by adding an element (√2) and finding a polynomial that this element satisfies.

• Connection to Field Extensions: This is a common pattern in field extensions, where we seek minimal polynomials for elements added to a base field.

• Step 3. Generalizing: In the generalizing step, we aim to find a broader, abstract principle that applies to similar problems.

• Generalization: We aim to derive a general principle for finding minimal polynomials of elements added to fields, applicable to various field extensions.

• Applicability: The approach used to find the minimal polynomial for √2 can be applied to other algebraic elements added to fields, such as ∛3 to ℚ, ∛2 to ℚ(√2), and so on.

• Step 4. Simplifying: Simplifying involves reducing the problem’s complexity by focusing on essential elements.

• Simplification: We aim to simplify the problem by breaking it into smaller, more manageable steps.

• Core Focus: The core of the problem involves finding a polynomial for √2, which can be achieved by verifying that the polynomial x² − 2 satisfies the criteria.

Now, let us proceed with the detailed solution, applying each step of the FUGS approach:

• Step 1: Formalizing: We have already formalized the problem in the problem statement. We are dealing with the field extension ℚ(√2) over ℚ, and we need to find the minimal polynomial of √2 over ℚ.

• Step 2: Unifying: In this step, we recognize the pattern of extending a field by adding an element and determining the minimal polynomial for that element. This process is not unique to this problem but is a general approach in field theory.

• Step 3: Generalizing: Now, let us extract the general principle for finding minimal polynomials for elements added to fields:

General Principle: To find the minimal polynomial of an element (denoted as α) added to a field F, follow these steps:

1. Consider the field extension F(α) over F.

2. Construct a polynomial in F[x] of the form f(x) = (x − α).

3. If f(x) is irreducible over F, it is the minimal polynomial of α over F.

This general principle applies to various field extensions, not just ℚ(√2) over ℚ.

• Step 4: Simplifying: We have simplified the problem by breaking it into steps. Now, let us proceed with solving it.

Solution: We want to determine the minimal polynomial of √2 over ℚ. Following the general principle, we derived earlier:

1. Consider the field extension ℚ(√2) over ℚ.

2. Construct a polynomial in ℚ[x] of the form f(x) = (x − √2).

We have $\mathrm{f}(\mathrm{x})=\mathrm{x}-\surd 2.$

Now, we need to show that this polynomial is irreducible over ℚ. In other words, we want to prove that there are no non-constant polynomials g(x) in ℚ[x] that divide f(x) without leaving a non-constant remainder. To prove that x − √2 is irreducible over ℚ, we use a proof by contradiction. We assume that there exists a non-constant polynomial g(x) in ℚ[x] such that: g(x) divides f(x) without leaving a non-constant remainder. Mathematically, this means: $$\mathrm{f}\left(\mathrm{x}\right)=g\left(\mathrm{x}\right)*h\left(\mathrm{x}\right)$$ where g(x) and h(x) are polynomials in ℚ[x], and h(x) is not a constant polynomial.

Now, let us substitute f(x) with x − √2 in the equation: $$\mathrm{x}-\surd 2\text{=}g\left(\mathrm{x}\right)*h\left(\mathrm{x}\right)$$

Since √2 is an irrational number, it cannot be expressed as a simple fraction in ℚ. Therefore, it is clear that x − √2 is not a rational number. However, the left-hand side of the equation, x − √2, is a rational number when x is a rational number (since x belongs to ℚ). This is a contradiction. Therefore, the assumption that g(x) exists is false. There is no non-constant polynomial in ℚ[x] that can divide x − √2 without leaving a non-constant remainder. Hence, x − √2 is irreducible over ℚ. Since x − √2 is irreducible over ℚ and is the polynomial that √2 satisfies, it is the minimal polynomial of √2 over ℚ. In this solution, we formalized the problem, unified it with the general approach of finding minimal polynomials in field extensions, generalized the approach to a broader principle, and simplified the problem by focusing on proving the irreducibility of x − √2. The FUGS approach allowed us to systematically tackle a complex abstract algebra problem in a structured and insightful manner.

Instructional approaches

In a comprehensive study conducted by Fukawa-Connelly et al. (2016), a remarkable phenomenon of steadfast resistance to change was observed among Abstract Algebra instructors in the United States. The researchers discovered that these educators displayed a notable inclination towards adhering to traditional pedagogical methods, with lectures remaining the predominant mode of instruction. Surprisingly, despite the extensive dissemination efforts and the availability of innovative reform materials generated by University Mathematics Education (UME) research, the utilization of such resources by Abstract Algebra instructors was found to be meager. This observation of limited adoption and incorporation of new pedagogical approaches and existing reform materials within the field of Abstract Algebra is indicative of a potent inertia prevailing among instructors. The reluctance to embrace novel teaching methodologies and to fully embrace the fruits of UME research implies a resistance to change deeply rooted within the community. The study’s findings shed light on the prevailing educational landscape in the United States, suggesting that the identified patterns of instructional practices in Abstract Algebra may extend beyond national borders and resonate globally.

It is essential to delve into the potential reasons underlying this resistance to pedagogical change and the underutilization of reform materials. Several factors could contribute to this inertia among Abstract Algebra instructors. First and foremost, the deeply entrenched tradition of the lecture-based instructional model may perpetuate a sense of familiarity and comfort for both instructors and students. Traditional lecture formats have been the hallmark of mathematical education for generations, and the perpetuation of this approach could stem from a desire to maintain continuity and stability in the teaching-learning process. Additionally, the discrepancy between research and practice might contribute to the observed inertia. Despite the efforts to disseminate reform materials and promote innovative teaching practices, there could be a disconnection between UME research and the realities of the classroom. Instructors might face challenges in effectively accessing, interpreting, and implementing the reform materials due to various constraints, including time limitations, lack of awareness, or perceived difficulties in aligning the materials with existing curricula. Furthermore, instructors’ beliefs and attitudes towards teaching and learning might play a pivotal role in perpetuating the inertia. Long-standing beliefs and pedagogical philosophies, coupled with the demands of traditional educational systems, can influence instructors’ perceptions of their roles and responsibilities. A resistance to change could stem from a perceived conflict between innovative teaching methods and deeply held convictions regarding the nature of mathematics education and the most effective instructional strategies.

The findings of Fukawa-Connelly et al. (2016) indicate a need for further exploration and analysis of the prevailing instructional practices in Abstract Algebra across different educational contexts worldwide. While the study focused on the situation in the United States, it suggests that similar patterns may be present in other countries as well. By investigating the reasons underlying the resistance to change, it may be possible to develop strategies and interventions that bridge the gap between research and practice, encouraging the adoption of pedagogical innovations and reform materials in Abstract Algebra instruction. The study conducted by Fukawa-Connelly et al. (2016) thus provides valuable insights into the prevailing instructional practices in Abstract Algebra, revealing a notable inertia among instructors in the United States. The study highlights the limited adoption of new pedagogical approaches and the underutilization of existing reform materials generated by UME research. While the specific factors contributing to this resistance to change require further investigation, the findings suggest that similar patterns may exist globally. Understanding and addressing the underlying reasons for this inertia is crucial for promoting innovative teaching practices and bridging the gap between research and practice in Abstract Algebra instruction.

Implications for teachers

Teaching abstract algebra effectively requires a well-thought-out approach that takes into consideration the nature of the subject and the needs of the students. Abstract algebra can be challenging because it deals with abstract concepts and structures, so a clear and structured teaching strategy is essential. Here are some steps to consider:

1. Prerequisite Knowledge: Ensure that students have a strong foundation in elementary algebra, number theory, and mathematical proof techniques. It is important that they understand the fundamental concepts of algebra, including groups, rings, and fields.

2. Start with Concrete Examples: Begin with concrete and familiar examples of algebraic structures, such as integers under addition, modular arithmetic, or matrices. These examples can serve as a bridge to abstract algebraic concepts.

3. Motivate the Abstract: Help students understand why abstract algebra is important and how it is applicable in various areas of mathematics and science. Show real-world examples of abstract algebra in action, such as cryptography or coding theory.

4. Emphasize Definitions and Terminology: Clear and precise definitions are critical in abstract algebra. Ensure that students have a solid grasp of terms like groups, rings, fields, subgroups, cosets, homomorphisms, and isomorphisms. Encourage them to actively use this terminology in their discussions.

5. Provide Proofs: Abstract algebra relies heavily on proof-based reasoning. Teach students how to write and understand mathematical proofs. Start with simpler proofs and gradually progress to more complex ones as the course advances.

6. Exercises and Problem Solving: Assign a variety of exercises and problems to reinforce the material. Include both computational and theoretical problems. Encourage students to work in groups, discuss solutions, and seek help when needed.

7. Visual Aids: Use visual aids, diagrams, and examples to illustrate abstract concepts. This can help students develop a more intuitive understanding of algebraic structures.

8. Historical Context: Introduce students to the history of abstract algebra and the mathematicians who contributed to its development. Understanding the historical context can make the subject more engaging.

9. Relate to Other Mathematics: Show connections between abstract algebra and other branches of mathematics, such as linear algebra, number theory, and geometry. Demonstrating how abstract algebra fits into the broader mathematical landscape can make it more accessible.

10. Encourage Exploration: Allow students to explore topics that interest them within abstract algebra. Provide opportunities for independent research or projects to delve deeper into specific areas.

11. Use Technology: Utilize mathematical software, computer algebra systems, and graphing tools to illustrate abstract algebraic concepts and perform computations. This can help students experiment and visualize algebraic structures.

12. Assess Progress: Regularly assess students’ understanding through quizzes, assignments, and exams. Provide constructive feedback and offer additional support to struggling students.

13. Foster Discussion: Encourage classroom discussions and questions. Create an environment where students feel comfortable asking for clarification and sharing their insights.

14. Real-Life Applications: Show practical applications of abstract algebra in fields such as computer science, cryptography, coding theory, and physics to demonstrate its relevance beyond pure mathematics.

Teaching abstract algebra can be challenging, but with a structured approach, a strong emphasis on fundamental concepts, and opportunities for exploration and application, a teacher can help students develop a deep understanding of this abstract and fascinating field of mathematics. Abstract algebra courses at universities typically cover a range of topics. Here are seven core topics that are commonly taught in abstract algebra courses:

1. Group Theory: Group theory is often the starting point in abstract algebra. Students learn about groups, which are algebraic structures with an operation that satisfies closure, associativity, identity element, and inverses. They study subgroups, cyclic groups, permutation groups, and group isomorphisms.

2. Ring Theory: Ring theory introduces students to rings, which are algebraic structures that combine addition and multiplication operations. Topics include integral domains, fields, polynomial rings, and the properties of ideals.

3. Field Theory: Field theory explores the properties and structure of fields, which are algebraic structures that extend the concept of rings. Students study field extensions, algebraic and transcendental elements, and the construction of splitting fields.

4. Vector Spaces: Vector spaces are a fundamental concept in linear algebra, and they are often covered in an abstract algebra course. Students learn about vector space axioms, linear independence, bases, and dimension.

5. Linear Transformations: Linear transformations and their properties are studied, connecting abstract algebra with linear algebra. Topics include linear operators, kernels, images, and matrix representations.

6. Galois Theory: Galois theory is a more advanced topic that builds on field theory. Students learn about the Galois group of a polynomial, solvability by radicals, and the Fundamental Theorem of Galois Theory, which relates field extensions to group theory.

7. Modules: Modules extend the concepts of vector spaces to rings. Students explore module axioms, module homomorphisms, and properties of modules over various rings, including integral domains.

These topics form the core of most abstract algebra courses, but the specific content and depth of coverage may vary depending on the level of the course (undergraduate or graduate) and the goals of the curriculum. Additionally, some courses may include additional topics or variations on these foundational concepts. To understand it better, let us work through an advanced problem in Group Theory, with a typical step-by-step approach and providing detailed explanations throughout. This problem focuses on a specific group and its properties.

Problem: Consider the set of all 3x3 invertible matrices with real entries (denoted as GL(3, ℝ)), and the operation of matrix multiplication. Prove that GL(3, ℝ) is a group under this operation.

Solution: To demonstrate that GL(3, ℝ) forms a group under matrix multiplication, we need to verify the four group axioms: closure, associativity, identity element, and inverses.

Step 1. Closure: To show that GL(3, ℝ) is closed under matrix multiplication, we need to prove that the product of any two invertible 3x3 matrices results in another invertible 3x3 matrix.

Let A and B be two invertible matrices in GL(3, ℝ). We want to show that AB is also in GL(3, ℝ).

Proof:

Since A and B are invertible, they have inverses, denoted as A ^ (−1) and B^(−1), respectively. We need to prove that (AB) ^ (−1) exists. We know that (AB) ^ (−1) = B ^ (−1)A ^ (−1) (This is a standard property of matrix inverses). Now, B ^ (−1) and A ^ (−1) are also matrices, and the product of two invertible matrices is another invertible matrix. Therefore, B^(−1)A ^ (−1) is invertible. This proves closure, as the product of two invertible 3x3 matrices, AB, results in another invertible 3x3 matrix.

Step 2. Associativity: Matrix multiplication is known to be associative. That is, for any three matrices A, B, and C, (AB)C = A(BC). Therefore, associativity is satisfied.

Step 3. Identity Element: A group must have an identity element, which is an element such that, when combined with any other element, leaves the other element unchanged. In this case, the identity element is the 3x3 identity matrix, denoted as I. We need to show that for any matrix A in GL(3, ℝ), AI = IA = A.

Proof:

Let A be any matrix in GL(3, ℝ). We need to prove that AI = A. We know that I is the identity matrix, and the product of any matrix with the identity matrix results in the same matrix. Therefore, AI = A, which is the left identity. Similarly, we can prove that IA = A, which is the right identity. Hence, the identity element is indeed the 3x3 identity matrix I.

Step 4. Inverses: To demonstrate that GL(3, ℝ) has inverses, we need to show that for any matrix A in GL(3, ℝ), there exists a matrix B in GL(3, ℝ) such that AB = BA = I.

Proof:

Let A be any matrix in GL(3, ℝ). Since A is invertible, it has an inverse, denoted as A^(−1). We need to show that AA^(−1) = A^(−1)A = I. Now, we know that A^(−1) is the inverse of A, and by definition, the product of a matrix and its inverse is the identity matrix. Therefore, AA^(−1) = I. Similarly, A^(−1)A = I. This proves that GL(3, ℝ) has inverses for all its elements.

Here, we have shown that GL(3, ℝ) satisfies all four group axioms: closure, associativity, identity element, and inverses. Therefore, GL(3, ℝ) is indeed a group under matrix multiplication. This detailed solution walks through the proof of GL(3, ℝ) being a group, emphasizing the importance of each group axiom and the properties of matrix algebra. When teaching this concept effectively, it is essential to break down each step and provide clear explanations, as demonstrated here, to help students grasp the abstract algebraic concepts involved.

Let us take another example (say Field Theory) to understand it well. Teaching Field Theory effectively in a university mathematics class requires a carefully designed plan that should incorporate the following 14 points. Field Theory is an abstract and advanced topic in abstract algebra, and it is crucial to approach it in a structured and engaging manner to help students grasp its intricacies.

1. Prerequisite Knowledge: Before delving into Field Theory, it is important to ensure that students have a solid foundation in abstract algebra, including knowledge of groups, rings, and integral domains. Students should also have a strong background in mathematical proof techniques, as these will be essential in Field Theory.

2. Start with Concrete Examples: Begin the course with concrete and familiar examples of fields. For instance, the professor can start with the field of real numbers and the field of complex numbers, both of which are well-known fields. Show students how these fields exhibit the basic properties of fields, such as addition, multiplication, and inverses.

3. Motivate the Abstract: Help students understand why Field Theory is important and how it plays a crucial role in various mathematical areas. Emphasize that fields are fundamental structures in mathematics and are used in diverse fields like cryptography, coding theory, algebraic geometry, and more.

4. Emphasize Definitions and Terminology: Field Theory is built upon a precise set of definitions and terminology. Ensure that students have a clear understanding of terms like fields, subfields, field extensions, and the concept of field isomorphisms. Regularly reinforce these definitions throughout the course.

5. Provide Proofs: Field Theory, like all abstract algebra, relies heavily on proof-based reasoning. Teach students how to write and understand mathematical proofs. Start with simpler proofs and gradually introduce more complex ones as the course progresses.

6. Exercises and Problem Solving: Assign a wide range of exercises and problems to reinforce the material. Include both computational problems and theoretical proofs. Encourage students to work in groups, discuss solutions, and seek help when needed.

7. Visual Aids: Use visual aids, diagrams, and examples to illustrate abstract concepts in Field Theory. Visual representations of field structures, subfields, and field extensions can help students develop an intuitive understanding.

8. Historical Context: Introduce students to the historical context of Field Theory and the mathematicians who contributed to its development. Understanding the history of the field can make it more engaging and relatable.

9. Relate to other Mathematics: Show connections between Field Theory and other branches of mathematics. Explain how field extensions relate to concepts in linear algebra, algebraic geometry, and number theory. Demonstrating these connections can make the subject more accessible and meaningful.

10. Encourage Exploration: Give students the opportunity to explore specific topics or questions within Field Theory that pique their interest. Allow for independent research projects or presentations to delve deeper into areas like Galois Theory or finite fields.

11. Use Technology: Incorporate mathematical software and computer algebra systems to illustrate abstract algebraic concepts, especially when dealing with complex field extensions. Technology can help students experiment and visualize algebraic structures and operations.

12. Assess Progress: Regularly assess students’ understanding through quizzes, assignments, and exams. Provide constructive feedback and offer additional support to those who may be struggling with the material. Periodic assessments will help track progress and identify areas that need more attention.

13. Foster Discussion: Encourage classroom discussions and questions. Create an environment where students feel comfortable asking for clarification and sharing their insights. Field Theory can be challenging, and discussion can provide a valuable platform for collective learning.

14. Real-Life Applications: Highlight practical applications of Field Theory in various fields, such as cryptography, coding theory, and algebraic geometry. Show how the abstract concepts and structures students are learning have real-world relevance.

Thus, teaching Field Theory effectively involves a well-structured plan that considers the prerequisites, motivates students, emphasizes definitions and proofs, and encourages exploration. By providing a rich learning environment with clear explanations and real-life applications, students can develop a deep understanding of this abstract yet powerful area of mathematics. Remember to adapt your teaching strategies to the needs and pace of your students, ensuring that they are engaged and can apply the knowledge gained in various mathematical contexts. Let us explore it further with a mathematical problem. The problem that we are discussing here focuses on field extensions and the concept of minimal polynomials, which is a fundamental topic in Field Theory.

Problem: Consider the field extension ℚ(√2) over ℚ (the rational numbers). Determine the minimal polynomial of √2 over ℚ.

Solution:

• Step 1. Prerequisite Knowledge: Before diving into the problem, students should have a strong foundation in the definitions of fields, field extensions, and the basic properties of rational numbers and real numbers.

• Step 2. Start with Concrete Examples: Begin by introducing the concept of field extensions through a simple example. Explain that we are extending the field of rational numbers (ℚ) to include the square root of 2 (ℚ(√2)).

• Step 3. Motivate the Abstract: Motivate the need for studying field extensions by discussing that not all numbers can be expressed using rational numbers. √2 is an example of such a number, and to understand it better, we extend the field.

• Step 4. Emphasize Definitions and Terminology: Define the terms field extension, rational numbers, and square root of 2. Clarify the properties of fields and subfields, which are essential in field theory.

• Step 5. Provide Proofs: Explain that in Field Theory, we often use proofs to establish facts. For this problem, we will focus on finding the minimal polynomial of √2.

• Step 6. Exercises and Problem Solving: Assign problems related to field extensions, subfields, and minimal polynomials to reinforce understanding.

• Step 7. Visual Aids: Use visual aids to represent the field extension ℚ(√2) and its relationship to ℚ. Draw diagrams to show the inclusion of √2 in the extended field.

• Step 8. Historical Context: Briefly mention that Field Theory, especially the study of field extensions, was foundational to the work of mathematicians like Galois in the 19^th century.

• Step 9. Relate to other Mathematics: Explain that Field Theory connects with concepts in number theory, algebraic geometry, and even cryptography, showcasing the broader relevance of this abstract field.

• Step 10. Encourage Exploration: Encourage students to explore other field extensions, such as ℚ(∛3), and investigate the properties of minimal polynomials in these cases.

• Step 11. Use Technology: Demonstrate how to use mathematical software or calculators to calculate minimal polynomials and perform algebraic operations.

• Step 12. Assess Progress: After discussing the minimal polynomial of √2, assign exercises to test students’ understanding of the topic. Provide feedback and clarification as needed.

• Step 13. Foster Discussion: Encourage students to ask questions and discuss their thought processes when solving problems related to field extensions and minimal polynomials.

• Step 14. Real-Life Applications: Show that field extensions and minimal polynomials are used in coding theory, error-correcting codes, and cryptography.

Now, proceed with solving the problem:

To determine the minimal polynomial of √2 over ℚ, we will follow these steps:

• Step 1: Start with the field extension ℚ(√2) over ℚ. We want to find a polynomial that √2 satisfies.

• Step 2: Consider the polynomial f(x) = x² − 2. This polynomial is in ℚ[x] (the polynomial ring with coefficients in ℚ), and it evaluates to f(√2) = (√2)² − 2 = 2 − 2 = 0.

• Step 3: Since f(√2) = 0, the polynomial f(x) = x² − 2 has √2 as a root.

• Step 4: Now, we need to show that this polynomial is the minimal polynomial of √2. This means that there should be no other polynomial g(x) in ℚ[x] with a smaller degree such that g(√2) = 0.

• Step 5: To prove this, assume there exists a polynomial g(x) in ℚ[x] of degree less than 2 such that g(√2) = 0.

• Step 6: The only constant polynomials in ℚ[x] are of the form c, where c is a rational number. Since the degree of g(x) should be less than 2, the only other possibility is a linear polynomial, g(x) = ax + b, where a and b are rational numbers.

• Step 7: Now, we have g(√2) = a√2 + b = 0. If a or b is nonzero, this equation is impossible because the square root of 2 is irrational. Therefore, both a and b must be zero, which means g(x) = 0.

• Step 8: Since we have shown that any polynomial of degree less than 2 cannot have √2 as a root, the polynomial x² − 2 is the minimal polynomial of √2 over ℚ.

In this detailed solution, we have approached the problem of finding the minimal polynomial of √2 over ℚ with a focus on step-by-step explanation, proof, and clarification of abstract concepts. Field Theory, as demonstrated through this problem, involves a precise understanding of field extensions and their applications in abstract algebra and beyond. This approach aligns with the principles for effective teaching of abstract algebra concepts in a university mathematics class.

Conclusion

The teaching and learning of Abstract Algebra within the realm of University Mathematics Education (UME) has been the subject of investigation from various perspectives, encompassing epistemological, cognitive, phenomenological, and institutional lenses. Researchers have focused on elucidating the challenges students encounter and the processes of abstraction necessary for grasping Abstract Algebra’s intricacies. Additionally, investigations have delved into the rationale underlying specific structural concepts and analyzed the entrenched practices of instructors, aiming to shed light on the inertia often observed in their pedagogical approaches. Encompassing both thematization and idealization, Abstract Algebra presents researchers with a plethora of challenges, prompting the development of inquiry-based methodologies aimed at fostering robust conceptualization and comprehension. Despite ongoing scholarly efforts, research in the field of Abstract Algebra education remains relatively limited, and several crucial issues warrant further systematic investigation. For instance, while the phenomenological roots of Group Theory (GT) have been identified in the realm of symmetries, it becomes pertinent to inquire into the corresponding phenomenological foundations of Ring Theory and Field Theory. Understanding the phenomenological underpinnings of these branches of Abstract Algebra could offer valuable insights into effective teaching and learning strategies tailored to their unique characteristics.

In order to advance the field, it is imperative to chart sustainable study and research paths. These paths should encompass a comprehensive exploration of the challenges students face and the abstraction processes that prove instrumental in their mastery of Abstract Algebra. By pursuing such avenues, researchers can identify innovative instructional approaches that encourage deeper conceptualization and foster a more profound understanding of the subject matter. The integration of Computer Algebra System (CAS) technologies constitutes another pivotal area worthy of investigation. Understanding how to leverage the potential of CAS technologies in Abstract Algebra instruction holds promise for enhancing students’ engagement and proficiency. Exploring the optimal usage of these tools within the learning environment could open new avenues for inquiry-based learning and provide valuable insights into the impact of technology on students’ mathematical comprehension. A fundamental question that necessitates careful consideration revolves around the potential reform of Abstract Algebra education. By reevaluating the existing syllabi and implementing targeted reforms, educators can create a conducive learning environment that promotes greater mastery of Abstract Algebra concepts. Moreover, it is crucial to delve into the enduring difficulties students encounter in their journey through Abstract Algebra. Identifying the specific areas of struggle and investigating the root causes of these challenges can inform the design of targeted interventions and pedagogical strategies. By addressing these persistent difficulties, educators can facilitate more effective learning experiences and equip students with the necessary tools to navigate the complexities of Abstract Algebra. The research in the teaching and learning of Abstract Algebra within UME encompasses a range of perspectives, aiming to uncover the underlying epistemological, cognitive, phenomenological, and institutional aspects of this field. While progress has been made, the research landscape remains relatively limited, necessitating further investigation. Key areas for exploration include understanding the phenomenological foundations of different branches of Abstract Algebra, charting sustainable research paths, exploring the integration of CAS technologies, and contemplating comprehensive reforms to address persistent difficulties. By addressing these issues, researchers can contribute to the advancement of Abstract Algebra education and facilitate students’ mastery of its intricate concepts.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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No potential competing interest was reported by the authors.

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Notes on contributors

Ashraf Alam

Ashraf Alam is a Ph.D. Scholar at IIT Kharagpur. He holds a master’s degree in education and a bachelor’s degree in computer science and engineering from the University of Delhi. Over the course of Ashraf’s academic journey, his research and teaching interests have inclined towards the philosophy and sociology of education and in areas of research ethics, educational psychology, and educational technology. He works at the crossroads of research and action for sustainable development, focusing on policies that impact the vulnerable. Currently, he is researching the different facets of ‘Positive Education’ at the Rekhi Centre of Excellence for the Science of Happiness at IIT Kharagpur under the esteemed guidance and patronage of Prof. Atasi Mohanty.

Atasi Mohanty

Atasi Mohanty, PhD, is an Assistant Professor at the Rekhi Centre of Excellence for the Science of Happiness, Indian Institute of Technology Kharagpur, India. Her research interests include positive youth development, social psychology, youth mental health, the science of happiness, prosocial behavior, happiness at the workplace, sustainable health and wellbeing, positive psychology, organizational behavior, Indian psychology, and sustainable entrepreneurship. She is currently guiding 14 Ph.D. research scholars.