This article describes the design of a model-eliciting activity (MEA) that integrates GeoGebra drawing within the models and modeling perspective (MMP) and the action, process, object, and schema (APOS) theory to foster understanding of eigenvalues and eigenvectors. Insights from two pilot tests with linear algebra students led to refinement of the design to ensure alignment with the six principles of models and modeling theory, prior to its implementation with a group of university students in an introductory linear algebra course. Student work related to MEA generated during the implementation is analyzed using elements of genetic decomposition and modeling cycles. The reflections highlight the richness of the MEA, its potential to generate new situations and questions for further exploration of eigenvalues and eigenvectors, and its relevance for developing a more profound understanding of other fundamental concepts in linear algebra.

This study presents the outcomes of a pedagogical intervention conducted at the Keiser University during the 2024 − 2025 academic cycle, focusing on the integration of academic posters as an active learning strategy within higher education mathematics instruction. Grounded in constructivist theories and didactic approaches that foster active learner engagement, the intervention aimed to deepen students’ understanding of algebraic concepts through the contextualization of mathematical theory in socially and environmentally relevant phenomena. Empirical data reveal a 15% improvement in algebraic achievement alongside a 30% reduction in mathematics anxiety, thereby demonstrating the effectiveness of the approach in facilitating the internalization of both abstract and procedural mathematical knowledge. The incorporation of contextualized content enabled a significant shift from rote memorization towards reasoning, mathematical argumentation, and modeling. These pedagogical shifts promoted the development of critical skills, mathematical communication, and cooperative competencies-attributes essential for contemporary learning environments. The employment of academic posters as multimodal visual artifacts enhances the externalization of algebraic representations, supporting the construction of mathematically meaningful knowledge applicable to real-world contexts and strengthening the link between theoretical modeling and practical application. These findings emphasize the potential of innovative didactic strategies to bolster formative processes oriented toward systems thinking and social responsibility, positioning mathematics as a visual, argumentative, and propositional discipline. The research validates academic posters as effective tools for situated instruction, enhancing mathematical literacy, curricular innovation, and socio-cognitive skills in higher education.

The development of knowledge during the Abbasid Caliphate marked the emergence of the Islamic Golden Age, supported by political and institutional backing for intellectual activities. In this context, mathematics flourished through processes of translation, synthesis, and innovation, reaching its peak in the works of Al-Khawarizmi. This study examines the historical dynamics of the development of mathematics during the Abbasid Caliphate, focusing its analysis on Al-Khawarizmi’s intellectual contributions. The research employs the historical method with a descriptive-analytical approach through literature-based data collection techniques. The research procedures include heuristics, interpretation, and synthesis of various sources discussing the development of algebra, arithmetic, and algorithms during that period. The findings indicate that Al-Khawarizmi made significant contributions to the systematic formulation of algebraic concepts, the introduction of the Hindu-Arabic numeral system, and the development of algorithmic methods that later became foundational to modern computer science. With the support of the Abbasid caliphs, particularly through the House of Wisdom in Baghdad, mathematics advanced rapidly and had a profound impact on Islamic civilization as well as the wider world.

Misconceptions are common among students when solving mathematics problems and, if left unaddressed, may hinder their ability to understand the material. This study seeks to design and develop diagnostic test instruments—comprising questions and questionnaires—on algebra topics using the Appsheet application. The objectives are to assess the development process, feasibility, and effectiveness of the instruments, as well as to identify challenges faced during implementation. The research employed a Design Research approach with a development studies type, consisting of two stages: preliminary design and formative evaluation. The instruments underwent a one-to-one trial with three students, a small-group trial with six students, and a field test involving 33 eighth-grade students at SMP Negeri 1 Margaasih. Expert validation rated the instruments as “feasible.” Data were gathered through documentation, tests, and interviews. The instruments included 10 three-tier multiple-choice questions, a closed-ended questionnaire with 13 items, and interviews with both teachers and students. Analysis revealed that 3.9% of students understood the concepts, 8.2% exhibited positive misconceptions, 8.4% negative misconceptions, 22.1% general misconceptions, while 57.3% showed no conceptual understanding. These findings demonstrate that the instruments effectively identified algebraic misconceptions. Challenges encountered included limited school facilities for mobile device usage and occasional errors in the application when accessed simultaneously by many users, indicating the need for updates. Nevertheless, the Appsheet-assisted three-tier diagnostic test provides a feasible and promising tool for detecting misconceptions in junior high school algebra learning

This study seeks to enhance mathematics teaching and learning by facilitating students’ transition from secondary school to university. It provides an in-depth analysis of the mathematics curriculum from primary education through the first year of university, with particular emphasis on equations and related algebraic concepts. The research adopts a mixed-methods design combining several complementary approaches: (1) a longitudinal analysis of school curricula to examine the progression of mathematical concepts across educational levels; (2) questionnaires administered to university instructors to explore their perceptions of students’ learning difficulties and their pedagogical practices, particularly regarding the integration of Information and Communication Technologies (ICT); (3) a student questionnaire aimed at identifying perceived learning obstacles; and (4) a diagnostic test focused on equations to assess first-year students’ prerequisite knowledge and competency levels. The originality of this study lies in its systematic articulation of curricular content across educational stages, revealing discontinuities and conceptual gaps in the progression of mathematical knowledge. To our knowledge, this is the first study conducted in Morocco that integrates curriculum analysis, teachers’ and students’ perceptions, and an equation-based diagnostic assessment within a unified analytical framework. This comprehensive approach contributes to a deeper understanding of the structural causes underlying students’ difficulties in mathematics and proposes concrete strategies to enhance academic achievement. The findings provide valuable insights into critical learning bottlenecks and offer evidence-based recommendations for improving mathematics instruction and strengthening foundational competencies essential for sustained academic success. Keywords: mathematics education; transition to university; curriculum analysis; learning difficulties; diagnostic assessment; information and communication technologies (ICT).

Teaching algebra in Middle School and High School presents challenges related to the transition of arithmetic thinking to algebraic thinking, marked by abstraction and symbolic mathematics. Faced with this context, this article analyses the contributions of the use of factual materials in the process of teaching-learning algebra. The research is characterized as a mixed-methods study, both descriptive and explorative nature, based in literature review and empirical investigation with students and teachers of public and private schools. Materials such as algebraic blocks, algebraic balance, cards and mathematical games were used, integrated into the classes with proper planning. The data collected through questionnaires, interviews, observations and evaluations' analysis show increase in the academic performance, enhancing the participation and comprehension of algebraic concepts. The results demonstrate that factual materials act as knowledge mediators, favoring significant learning and the progressive construction of algebraic thinking, as long as they are associated with an adequate pedagogical mediation.

We discuss the geometry behind the classical Heisenberg model at the level suitable for third- or fourth-year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for the classical Heisenberg model, starting from the two-sphere geometry, thereby demonstrating the relevance of standard canonical procedures in the case of the Heisenberg model.

This study introduces and investigates intuitionistic fuzzy Hom-subgroups, extending the concept of fuzzy Hom-subgroups in the context of Hom-group theory. This approach integrates Atanassov’s intuitionistic fuzzy sets with Hom-group structures, allowing each element to be represented through both membership and non-membership degrees, thus offering a richer description of uncertainty. Fundamental properties and characterizations are established using level subsets, and the behavior of these subgroups under Hom-group homomorphisms is investigated. Several illustrative examples, including constructions on the real-line Hom-group, are provided to demonstrate the developed concepts. The proposed framework extends existing results on fuzzy Hom-groups and illustrates the developed algebraic concepts within the intuitionistic fuzzy setting.

This study explores the impact of the Compare and Discuss Multiple Strategies (CDMS) framework for teaching multiple solution methods for factoring trinomials, a foundational algebraic skill serving as a gateway to college-level STEM coursework and frequently identified as a conceptual hurdle for students transitioning from arithmetic to algebraic thinking. Using a mixed-methods pre–post quasi-experimental design, 39 undergraduate students enrolled in developmental mathematics participated in a CDMS-based intervention focusing on multiple solution methods for factoring trinomials. The research investigated performance changes across trinomial types, method usage patterns, and student experiences. Results revealed large effect sizes significantly exceeding typical educational interventions, with substantial improvements across all trinomial types. Method diversity increased from two to four approaches, with students shifting from single-method dependency to flexible use of multiple strategies, particularly visual methods like Box and Tic-Tac-Toe methods. Qualitative findings indicated increased student confidence and strategic thinking, with nearly half expressing appreciation for multiple solution methods. The intervention was consistently effective across different algebraic structures and complexity levels. These findings provide the first empirical evidence for implementing CDMS in developmental mathematics at the college level, demonstrating that multiple solution methods can produce meaningful learning gains among undergraduate students who traditionally struggle with algebraic concepts.

The connections between the Cube puzzle, group theory, and Cayley graphs is explored. It provides a comprehensive introduction to these concepts and their relation. The Cube puzzle is a popular puzzle, and the collection of all its possible states constitutes a group under face rotation operations. Focusing on the Cube puzzle group, we delve into the mathematical structure underlying this puzzle's structure. A key object of this exploration is the Cayley graph, which visually represents the Cube puzzle group. This graph connects the Cube puzzle with many other topics and lead to many applications. By modeling the Cube puzzle as a random walk on its corresponding Cayley graph, we gain insights into scrambling and solving processes. Another surprising application is its connection to musical patterns through modulation graphs. This paper aims to bridge the gap between abstract algebraic concepts and their tangible applications in the Cube puzzle.

We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series of rank one, and discuss connections to other concepts in tropical algebra and combinatorial algebraic geometry.

Vertex $F$-algebras are a deformation of the concept of an ordinary vertex algebra in which the additive formal group law is replaced by an arbitrary formal group law $F$. The main theorem of this paper constructs a Lie algebra from a vertex $F$-algebra - for the additive formal group law, this extends Borcherds' well-known construction for ordinary vertex algebras. Our construction involves the new concept of an $F$-residue and some other new algebraic concepts, which are deformations of familiar concepts for the special case of an additive formal group law.

The concepts of regular Noetherianity and regular coherence, which extend the classical notions of Noetherian and coherent rings, have been fundamental in the study of algebraic structures. In this paper, we aim to expand these notions to the realm of module theory. Specifically, we introduce and explore weak versions of injective, flat, and projective modules, which we term as reg-injective, reg-flat, and reg-projective modules. We provide analogues of classical results and establish their properties, offering examples to illustrate modules that meet these new criteria but not their classical counterparts. Additionally, we define and study regularly Noetherian and regularly coherent modules, characterizing their properties and examining their stability under various ring constructions. Our results contribute new examples and broaden the current understanding of these algebraic concepts.

In this paper, a monad-based denotational model is introduced and shown adequate for the Proto-Quipper family of calculi, themselves being idealized versions of the Quipper programming language. The use of a monadic approach allows us to separate the value to which a term reduces from the circuit that the term itself produces as a side effect. In turn, this enables the denotational interpretation and validation of rich type systems in which the size of the produced circuit can be controlled. Notably, the proposed semantic framework, through the novel concept of circuit algebra, suggests forms of effect typing guaranteeing quantitative properties about the resulting circuit, even in presence of optimizations.

Purity is one of the important concepts in Algebra, which was studied by many others in Acts, modules, and groups. An K-K-subact B of K-act is SP-sub-act, if every a ∈ A and a ∉ B, there is a pure K-subact P of A that contains B and a ∉ P. Now we describe that an SPsub- act L in K-act A, if and only if, Lj are pure K-subacts of A, for each j ∈ J, and L = ∩j∈J Lj. SP-subact as a generalization of purity. We discuss many Algebraic properties. We found the relationship between SP-sub-act idempotent subact, and multiplication.

Peer tutoring in recent times has garnered significant academic attention within education as a cornerstone of cooperative learning and a pivotal instructional method for inclusive education. This study specifically investigated the impact of peer tutoring on mathematics academic performance among student-teachers. Grounded in positivist philosophy, the study employed a pre-test, post-test quasi-experimental design with two intact classes: fifty student-teachers in the control group (receiving traditional instruction) and sixty in the experimental group (engaged in peer tutoring). Academic achievement was evaluated using the Linear Equation in Algebra Test (LEAT), a 30-item multiple-choice objective test designed to measure proficiency in algebraic concepts. The results indicated that student-teachers in the peer tutoring group achieved higher scores on the LEAT compared to those in the traditional instruction group. This outcome underscored the efficacy of peer tutoring in enhancing academic performance in mathematics. Based on these findings, it is recommended that teachers adopt peer tutoring instructional strategies in their classrooms to improve student-teachers' academic outcomes. Peer tutoring not only facilitates deeper learning through collaborative interaction but also promotes a supportive learning environment conducive to academic growth and achievement. This approach aligns with inclusive education principles by fostering peer support and active student engagement in the learning process.

Abstract This tutorial reviews the mathematical foundations of single-antenna radio polarimetry with the aim of fostering a conceptual understanding of the relationships between a physical description of signal propagation (gain, delay, reflection, down-conversion, etc.), the corresponding transformations of the electric field vector, and the equivalent operations on the Stokes parameters. The adopted framework is based on the work of Britton and Hamaker and applied to analyze the signal path described by Hamaker et al. with additional corrections for phase convention and reflection. Some objective criteria for selecting a model of the instrumental response are introduced and discussed, along with some practical guidelines that facilitate polarimetric calibration. Further relevant background material and lengthier mathematical proofs are included in the appendix, which introduces the vector, matrix, and tensor notation and concepts of linear algebra used in this work. The appendix also reviews some of the basics of analog and digital signal processing that are relevant to radio astronomy, and discusses some numerical instabilities that arise when modeling observations.

A new version of digital signature scheme (DSS) has been proposed in this work. This version has used a new concept of linear algebra which is called the integer matrix of Fibonacci numbers with size 2×2 (IFM2×2). The main point is to use the IFM2×2 with its right power (RPIFM2×2) for creating the digital signature. The verification is done of this signature using the computations of the RPIFM2×2 as well. New implemented results on the proposed DSS-RPIFM2×2 have been done using Python programming. The issue of the security has been determined based on the randomization for generating the IFM2×2 of the elements over a field Fp of prime numbers secretly. The DSS- PRIFM2×2 version is proposed to be a more secure for sending the signed messages and to avoid forging the signature of them. Thus, the proposed version of DSS- RPIFM2×2 considers as a bright point for more authenticity, integrity and non-repudiation in compare to the previous communication schemes.

The Algebra Concept Inventory (ACI) is the first large-scale instrument validated to measure the foundational algebraic conceptual understanding of college students. This study uses ACI scores to conduct the first quantitative analysis on the relationship between algebraic conceptual understanding and college outcomes, thus exploring the predictive validity of the ACI. Specifically, we investigate whether ACI scores predict: (1) math course grades; (2) subsequent completion of math courses required for STEM (science, technology, engineering, and mathematics) majors; (3) completion of STEM versus non-STEM degrees; and (4) the extent that differences in these outcomes by race/ethnicity or gender are explained by ACI scores. Results indicate that ACI scores significantly predict math course outcomes and STEM versus non-STEM degree completion, as well as significant proportions of differences in these outcomes by race/ethnicity and gender. This illustrates the importance of providing every student instruction that supports development of the kinds of foundational algebraic conceptual understanding measured by the ACI.

The Indian knowledge system in mathematics is one of the world's most ancient and profound knowledge systems. In addition to introducing the concept of zero, Indian mathematicians also made significant contributions to the study of other fields, including geometry, arithmetic, binary mathematics, the concept of negative numbers, algebra, trigonometry, and calculus. The decimal place value system used worldwide today was first developed in India. This comprehensive exploration of the Indian knowledge system in mathematics provides a detailed analysis of its historical development, significant contributions, and underlying philosophical principles. The paper aims to highlight the works of ancient Indian mathematicians, methods of knowledge dissemination, and the far-reaching influence of Indian mathematics on global mathematical thought.