Search Results (81)

This study explores how to support Turkish–American secondary school students to co-develop covariational reasoning (CR) and representational fluency (RF) in solving contextually based quadratic function tasks in an after-school community center learning setting. We conducted a teaching experiment (n = 8) at a community center. Ongoing and retrospective analyses of classroom interaction and video transcripts revealed a culturally inclusive mathematics learning environment framework with several intertwined components: co-developing CR and RF and community-based practices. This study provides evidence that students coordinate symbolic, tabular, and graphical representations, which not only deepen their understanding of how quantities change in relation to one another but also enable them to interpret and construct representations in increasingly flexible ways. This reciprocal process of co-developing CR and RF allowed students to recognize and express quantitative relationships as meaningful functional relationships, demonstrating a dynamic interplay between reasoning about change and fluency across representations. This study situates learning within culturally inclusive learning environments and acknowledges the reflexive positionality of the teacher–researcher in relation to students. We highlight how shared community-based practices can enhance mathematics teaching and learning. Full article

Probabilistic language is a main component in the teaching and learning of probability; however, research analyzing probabilistic language in textbooks, which are fundamental didactic tools, is scarce. Consequently, in this research, we studied the various probabilistic languages used in Spanish secondary school textbooks. We performed a detailed content analysis of two complete series (grades 1 to 4; the last with two options) of Spanish prestigious editorials published after the last curricular guidelines in 2022; 10 books in total. We researched the verbal, symbolic, tabular, and graphical language in each textbook. Results suggest differences in the way each editorial introduces its everyday and probabilistic language. Although the number of new symbols is small, some of them are complex or used inconsistently. There is scarce use of tables and graphs, except for tree diagrams and two-way tables, in the study of conditional and compound probability. We conclude with recommendations to improve probabilistic language in textbooks and facilitate the learning of probability in secondary education in this way. Full article

Children’s graphic development reflects cognitive, emotional, motor and social processes, and is an indicator of the infant’s integral development. This study analyzed the validity of Viktor Lowenfeld’s developmental model in children aged 3 to 5 years, assessing whether the stages proposed by the author correspond to the patterns of graphic representation currently observed. A mixed design with a descriptive–comparative approach was used. The sample consisted of 218 drawings collected in educational centers in Madrid. Each drawing was evaluated according to criteria derived from Lowenfeld’s model (stroke, motor control, symbolism, use of color, spatial organization) and classified into one of the five developmental stages. The analysis included descriptive statistics, chi-squared tests (χ² = 104.92, gl = 10, p < 0.0001), and Spearman correlations (ρ = 0.661, p < 0.0001), in addition to qualitative interpretation by inter-rater consensus. The results confirm that the general graphic sequence follows Lowenfeld’s scheme, but with notable variations in the age of onset of each stage. Pre-schematic manifestations were observed in children as young as 3 years old and persistence of scribbling in some 5-year-olds. These findings point to the influence of individual, sociocultural, and technological factors on the rate of graphic development. Lowenfeld’s model remains a valid framework for observing children’s drawing, but its application should be flexible. Interindividual differences and contemporary visual culture require a pedagogical adaptation that addresses the diversity of developmental rhythms. Its integration with inclusive methodologies and future cross-cultural studies is suggested. Full article

Clifford algebras are an active area of mathematical research having numerous applications in mathematical physics and computer graphics, among many others. This paper demonstrates algorithms for the computation of characteristic polynomials, inverses, and minimal polynomials of general multivectors residing in a non-degenerate Clifford algebra of an arbitrary dimension. The characteristic polynomial and inverse computation are achieved by a translation of the classical Faddeev–LeVerrier–Souriau (FVS) algorithm in the language of Clifford algebra. The demonstrated algorithms are implemented in the Clifford package of the open source computer algebra system Maxima. Symbolic and numerical examples residing in different Clifford algebras are presented. Full article

DNA origami is a powerful technique for constructing nanoscale structures by folding a single-stranded DNA scaffold with short staple strands. While traditional models assume staples bind to a fixed side of the scaffold, we introduce a side-aware DNA origami framework that incorporates the directional binding of staples to either the left or right side. The graphical representation of DNA origami is described using rectangular basic modules of scaffolds and staples, which we refer to as symbols in side-aware DNA origami words. We further define the concatenation of these symbols to represent side-aware DNA origami words. A set of rewriting rules is introduced to define equivalent words that correspond to the same graphical structure. Finally, we compute the number of possible structures by determining the equivalence classes of these words. Full article

A model transformation method based on state refinement and semantic mapping is proposed to address the challenges of high modeling complexity and resource consumption in symbolic validation of industrial software requirements. First, a rule-based semantic mapping system is constructed through the explicit definition of element correspondence between statechart components and verification models, coupled with a composite state-level refinement strategy to structurally optimize model hierarchy. Second, an automated transformation algorithm is developed to bridge graphical modeling tools with formal verification environments, supported by quantitative evaluation metrics for mapping validity. To demonstrate its practical applicability, the methodology is systematically applied to railway infrastructure safety—specifically the railroad turnout control system—as a critical case study. The experimental implementation converts operational statecharts of turnout control logic into optimized NuSMV models. Not only did the models remain intact, but the state space was also effectively reduced through the optimization of the hierarchical structure. In the validation phase, the converted model is tested for robustness using the fault injection method, and boundary condition anomalies that are not explicitly stated in the requirement specification are successfully detected. The experimental results show that the validation model generated by this method has improved validation efficiency in the NuSMV tool, which is significantly better than the traditional conversion method. Full article

Buddhist cosmology is prominently reflected in artistic expressions across various regions, particularly through its extensive representation in Buddhist art. From the 12th century onward, depictions of Buddhist cosmology became increasingly significant in Theravāda Buddhist nations in Southeast Asia, underscoring the enduring appeal of these cosmological concepts within local cultures. This paper examines the formation, evolution, and iconography of Buddhist cosmological paintings, with a specific focus on Traiphum representations in central Siam. By drawing on visual and textual evidence from key Theravāda Buddhist centers, including Pagan, Ayutthaya, Thonburi, and Bangkok, the study analyzes the graphic characteristics and symbolic nuances of cosmological imagery through a comparative investigation of murals, illustrated manuscripts, and related texts. Situating Traiphum paintings within the broader framework of religious art and sociopolitical structures, this research highlights their role as both doctrinal expressions and mediums of cultural and ideological integration. The findings contribute to a deeper understanding of how Buddhist cosmological theories were visually articulated, transmitted, and localized in pre-modern Southeast Asia. Full article

Two new methods for handling a system of nonlinear fractional differential equations are presented in this investigation. Based on the characteristics of fractional calculus, the Caputo fractional partial derivative provides an easy way to determine the approximate solution for systems of nonlinear fractional differential equations. These methods provide a convergent series solution by using simple steps and symbolic computation. Several graphical representations and tables provide numerical simulations of the results, which demonstrate the effectiveness and dependability of the current schemes in locating the numerical solutions of coupled systems of fractional nonlinear differential equations. By comparing the numerical solutions of the systems under study with the accurate results in situations when a known solution exists, the viability and dependability of the suggested methodologies are clearly depicted. Additionally, we compared our results with those of the homotopy decomposition method, the natural decomposition method, and the modified Mittag-Leffler function method. It is clear from the comparison that our techniques yield better results than other approaches. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. We demonstrated that our methods for fractional models are straightforward and accurate, and researchers can apply these methods to tackle a range of issues. These methods also make clear how to use fractal calculus in real life. Furthermore, the results of this study support the value and significance of fractional operators in real-world applications. Full article

In the development of rural revitalization, the art intervention in rural cultural tourism has become a hot topic recently. With the advent of the new media era, it is particularly important to study the factors that affect the artistic value of environmental graphic design. This study proposes six dimensions of environmental graphic design for rural cultural tourism through literature analysis and experience summary, which are refined into 21 impact indicators. It is based on the case of the environmental graphic publicity of cultural tourism in Chongqing Nanshan Cattle Village, using the SBE beauty evaluation method to evaluate the influencing factors, while using the analytic hierarchy process (AHP) to measure each indicator. The consistency of the two evaluation methods verifies the feasibility of the influencing factors, and this study found the influence of visuality, cultural nature, artistic nature, interactivity, spatiality, and communicativeness on aesthetic value. Moreover, the core of environmental graphic design lies in information transmission, in which artistic nature occupies an important position, while cultural nature, although it has a relatively small weight in aesthetic value, cannot be ignored in social media communication. In terms of visual design, a reasonable layout of equipment significantly enhances visual appeal, while spatial adaptability is considered a key factor. This paper emphasizes the integration of art principles into design to promote sustainable development, while also pointing out that the identification of rural cultural symbols can enhance visitors’ cultural experience. In terms of interactivity, it is crucial to encourage visitors to actively participate and share their experiences, while communication emphasizes the important impact of exposure to rural scenes on brand communication. This research provides systematic theoretical support and practical guidance for the environmental graphic design of rural cultural tourism. Full article

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. Full article

This document introduces methods for describing long texts as groups of repeating symbols or patterns. The process converts a series of real-number values into texts. Developed tailored algorithms for identifying repeated sequences in the text are applied to decompose the text into nested tree-like structures of repeating symbols and is called the Nested Repeated Sequence Decomposition Model (NRSDM). The NRSDM is especially valuable for extracting repetitive behaviors in oscillatory but non-periodic and chaotic processes where the classical Fourier transform has limited application. The NRSDM along with the two graphical representations proposed here form a promising tool for characterizing long texts configured to represent the behavior of unidimensional processes. Full article

In the field of visual graphic design, complexity plays a crucial role in visual information processing, and it is assumed to be an absolute quantity based on the number of the presenting features and components. However, it remains unclear whether the visual order of the constituent elements in graphical symbol complexity affects cognitive processing, especially memory processing. Our research innovatively generated four groups of novel, meaningless graphical symbols (complex and ordered, complex and disordered, simple and ordered, and simple and disordered) and experimentally manipulated the level of complexity and order in these stimuli. Before the formal experiment, a five-point scale was used to further rule out differences between objective and subjective definitions of these graphical symbols on ratings of complexity, order, concreteness, and familiarity. Then, we used a cue-recall task to compare subjects’ memory performance of those four graphical symbol groups. The analytical results showed a significant interaction between visual order and graphical symbol complexity, with the complexity effect appearing only when the stimuli were in disordered condition and disappearing once the stimuli were ordered. In addition, this study conducted a practical application validation to confirm that increasing the level of visual order is an effective way to improve user experience while maintaining the same level of complexity. The findings can serve as a reference for graphical symbol design, graphic design, and visual communication design. Full article

The traditional interpretations of the Ramayana have been critiqued for preserving and promoting patriarchal gender structures by emphasising masculine heroism and often portraying female characters as unidimensional symbols of selflessness, purity, and honour. This paper analyses how Samhita Arni and Chitrakar’s graphic novel Sita’s Ramayana offers a retelling that foregrounds Sita’s perspective to question and reinterpret the social constructs. By analysing the text through a feminist literary lens, this paper examines how the novel adapts the traditional narrative to provide centre stage to Sita’s various encounters with instances of oppression. The findings reveal how Arni’s retelling employs unique aesthetics that combine texts and Chitrakar’s patua art illustrations to question the traditional male-centred versions, making this novel a part of a broader structure of feminist reinterpretations that aim to highlight female agency in cultural canons. This paper examines Sita’s stance against societal expectations for women, such as self-sacrifice, while also tracking her personal growth, which is symbolically represented by her reunion with Mother Earth. The novel contributes to the ongoing tradition of literary revisionism by offering a nuanced critique of the patriarchal foundations within classical myths. This is underscored by the novel’s reinterpretation of the epic in a way that points out the plasticity of the Ramayana, which can be reshaped to support more progressive views, encouraging discourse on existing gender norms present in contemporary societies. Full article

This article aims to introduce a set of hybrid matrix polynomials associated with $\lambda$-polynomials and explore their properties using a symbolic approach. The main outcomes of this study include the derivation of generating functions, series definitions, and differential equations for the newly introduced two-variable Hermite $\lambda$-matrix polynomials. Furthermore, we establish the quasi-monomiality property of these polynomials, derive summation formulae and integral representations, and examine the graphical representation and symmetric structure of their approximate zeros using computer-aided programs. Finally, this article concludes by introducing the idea of 1-variable Hermite $\lambda$ matrix polynomials and their structure of zeros using a computer-aided program. Full article

Human–animal relations have been a fruitful research topic worldwide. The importance of deer in hunter–gatherer societies is undeniable, with cervids being commonly found in archaeological and past artistic records, with a notable amount of information recovered in the Iberian Peninsula. This relevance continues during Late Prehistory, but the attempt to discuss cervids under broader perspectives and based on different types of data is not as common. We intend to approach human–deer relations in Central and Southern Portuguese Late Prehistory by considering the zooarchaeological records, both deer abundance in faunal spectra and their presence in “meaningful” assemblages and structured depositions, as well as the use of deer and deer body parts in other socio–cultural and ideological practices. The synthesis of available data shows that human–deer relations changed through time and space, with different abundances related to hunting depending on chronology and geography. The use of deer or their body parts as a resource of symbolic nature also varied, being included in food-sharing events, offerings, structured depositions, and graphic representations. Changeability is part of the different relationships, ontologies, and cosmogonies that humans and deer developed in the Late Prehistoric relational world. Full article