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Search Results (1,006)

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12 pages, 261 KB  
Article
Differences in the Performance of Physical Education Teacher Education Students in the National Diagnostic Assessment: A Comparative Analysis by Themes and Type of Institution
by Francisco Gallardo-Fuentes, Bastian Carter-Thuillier, Johan Rivas-Valenzuela, Sebastián Peña-Troncoso, Jorge Gallardo-Fuentes and Luis Añazco-Martínez
Educ. Sci. 2026, 16(4), 609; https://doi.org/10.3390/educsci16040609 (registering DOI) - 10 Apr 2026
Abstract
A system of initial teacher education must incorporate instruments capable of capturing the complexity of professional learning. In this context, national diagnostic assessments have become central mechanisms for monitoring outcomes in initial teacher education. This study examines student performance in Initial Teacher Education [...] Read more.
A system of initial teacher education must incorporate instruments capable of capturing the complexity of professional learning. In this context, national diagnostic assessments have become central mechanisms for monitoring outcomes in initial teacher education. This study examines student performance in Initial Teacher Education in Physical Education (ITEPE) programs using the themes assessed by the National Diagnostic Assessment (NDA) 2024, comparing achievement levels and analyzing differences according to the type of higher education institution in Chile. A quantitative, cross-sectional, and comparative design was employed, using official data from 1102 students enrolled in all Chilean universities offering the program. Descriptive and nonparametric inferential analyses were conducted to examine differences by sex and type of institution. The results show relatively homogeneous performance across standards, with higher percentages of achievement in dimensions related to didactic organization and assessment, and lower results in the standard associated with understanding student characteristics. Although statistically significant differences were identified according to administrative dependency, the effect sizes were small. Consequently, the NDA is positioned as a formative input to guide contextualized curricular improvements rather than as a mechanism for institutional ranking. Full article
(This article belongs to the Section Higher Education)
17 pages, 1272 KB  
Article
Integrating Bioclimatology into Environmental Education Through ICT: Implications for Sustainable Land Management and Climate Change Mitigation
by Ana Cano-Ortiz, Juan Peña-Martínez and Jose Daniel Sánchez-Martínez
Sustainability 2026, 18(8), 3727; https://doi.org/10.3390/su18083727 - 9 Apr 2026
Abstract
This study proposes an interdisciplinary approach that integrates bioclimatology, agronomy, environmental education, and information and communication technologies (ICT) to analyze their potential to support sustainable land management in the context of climate change. The research focuses on the application of bioclimatic indices, the [...] Read more.
This study proposes an interdisciplinary approach that integrates bioclimatology, agronomy, environmental education, and information and communication technologies (ICT) to analyze their potential to support sustainable land management in the context of climate change. The research focuses on the application of bioclimatic indices, the continentality index (Ic), the ombrothermic index (Io), and the thermicity index (It/Itc), combined with the use of a virtual herbarium as a didactic resource for interpreting ecological indicators associated with vegetation. The study was conducted using a pretest–posttest design aimed at assessing students’ self-reported understanding of ecological concepts, bioclimatology, geobotany, and the use of digital tools for learning plant species. The results show a significant improvement in students’ perceived understanding following the educational intervention, with the mean questionnaire score increasing from 21.99 (SD = 5.03) in the pretest to 31.33 (SD = 5.06) in the posttest (t(69) = 37.13, p < 0.001). The normalized gain (g = 0.42) indicates a moderate improvement in students’ self-reported comprehension of bioclimatic and ecological concepts. These findings highlight the potential of ICT to strengthen environmental education and to foster the development of competencies related to sustainable agricultural and forest land management. Full article
7 pages, 473 KB  
Proceeding Paper
Visual Teaching, Accessibility, and Hybridization: At the Intersection of Visual Education, Artificial Intelligence, and Universal Design for Learning
by Pierangelo Berardi and Carmela Paladino
Proceedings 2026, 139(1), 5; https://doi.org/10.3390/proceedings2026139005 - 8 Apr 2026
Abstract
Positioned at the intersection of instructional mediation, Visual Education, and Universal Design for Learning (UDL), this research aims to ascertain whether the use of Artificial Intelligence (AI) enhances accessibility for students with sensory disabilities. The study involved 137 pre-service teachers attending the “Special [...] Read more.
Positioned at the intersection of instructional mediation, Visual Education, and Universal Design for Learning (UDL), this research aims to ascertain whether the use of Artificial Intelligence (AI) enhances accessibility for students with sensory disabilities. The study involved 137 pre-service teachers attending the “Special Didactics and Learning for Sensory Disabilities” course within the teacher specialization program (TFA) at the University of Foggia. Although the hybridization of AI, UDL, and Visual Education was favourably received, its application remains sporadic, highlighting the challenge of balancing the need for simplification with requisite conceptual accuracy. This underscores the necessity of integrating more structured and continuous training pathways into teacher education, grounded in visual education and featuring micro-modules dedicated to specific skills such as writing alternative text, subtitling, and verifying color contrast according to recognized standards. Full article
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18 pages, 1363 KB  
Article
The Spectrum of Teaching Styles in Physical Education: A Feasibility Study on Children’s Physical Fitness and Self-Perception
by Domenico Monacis, Giacomo Pascali, Angel-Alex Haisan, Matteo Bibba and Dario Colella
Educ. Sci. 2026, 16(4), 576; https://doi.org/10.3390/educsci16040576 - 4 Apr 2026
Viewed by 148
Abstract
This pilot study investigates the feasibility and preliminary outcomes of an experimental intervention based on practice variability and teaching styles’ variation, gathering preliminary data on changes in physical fitness and self-perception in primary school children. The sample consists of 142 children (boys = [...] Read more.
This pilot study investigates the feasibility and preliminary outcomes of an experimental intervention based on practice variability and teaching styles’ variation, gathering preliminary data on changes in physical fitness and self-perception in primary school children. The sample consists of 142 children (boys = 76, age = 8.97 ± 0.82; girls = 66, age = 9.03 ± 0.77) recruited from one school participating in the SBAM! Project. Physical fitness components were assessed before (t0) and after (t1) a 4-month intervention protocol using the following physical fitness measures: BMI cutoff, standing long jump, 4 × 10 m shuttle run, medicine ball throw (1 kg), and 20 m slalom. Self-perception was assessed with a validated questionnaire. The pre-experimental didactic intervention involved motor tasks based on practice variability to promote inclusion, following the principles of non-linear pedagogy. Pre-and post-intervention changes were analyzed using multivariate analysis accounting for gender and BMI cutoff. The results showed statistically significant differences (p < 0.05) for all included variables, regardless of group. Preliminary findings suggest that this intervention is feasible in fostering positive changes and improvements in physical fitness and self-perception. Further investigations are needed to extend the generalizability of the results through more robust research designs. Full article
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14 pages, 245 KB  
Article
Exploring Strategies to Detect and Mitigate Bias in AI in Education: Students’ Perceptions and Didactic Approaches
by María Ribes-Lafoz, Borja Navarro-Colorado and José Rovira-Collado
Trends High. Educ. 2026, 5(2), 33; https://doi.org/10.3390/higheredu5020033 - 3 Apr 2026
Viewed by 206
Abstract
The increasing integration of Generative AI (GenAI) into higher education, particularly in the domain of language teaching, presents both opportunities and challenges. While AI-powered tools such as ChatGPT-5 can support language learning by generating personalised content which enables real-time interaction and feedback, they [...] Read more.
The increasing integration of Generative AI (GenAI) into higher education, particularly in the domain of language teaching, presents both opportunities and challenges. While AI-powered tools such as ChatGPT-5 can support language learning by generating personalised content which enables real-time interaction and feedback, they also risk perpetuating biases embedded in training data. These biases can appear in linguistic, cultural or socio-political forms, reinforcing stereotypes and influencing language norms. Therefore, equipping students and educators with strategies to critically assess AI outputs is essential for ethical and responsible AI use in language education. While recent research highlights the risks of algorithmic bias, less attention has been given to the perceptions and attitudes of pre-service teachers, whose future practice will shape classroom uses of these technologies. This exploratory pilot study adopts a survey-based approach to examine pre-service teachers’ baseline awareness of bias in artificial intelligence, with particular attention to linguistic and cultural dimensions Data were collected through an online questionnaire administered to 65 undergraduate students enrolled in Primary Education degree programmes. The study documents baseline perceptions prior to any instructional intervention and provides preliminary empirical evidence to inform the future design of pedagogical strategies aimed at developing critical AI literacy in teacher education. Full article
22 pages, 395 KB  
Article
Shifting Models of Early Childhood Education: A Study of Curriculum Ambivalence in English Preschool Mathematics
by Paul Andrews and Pernille Bødtker Sunde
Educ. Sci. 2026, 16(4), 509; https://doi.org/10.3390/educsci16040509 - 25 Mar 2026
Viewed by 388
Abstract
In this paper, by means of a comprehensive analysis of the statutory and non-statutory documents that govern its preschool provision, we examine how early childhood education and care (ECEC), particularly in relation to mathematics, is conceptualised by the English educational authorities. Situated within [...] Read more.
In this paper, by means of a comprehensive analysis of the statutory and non-statutory documents that govern its preschool provision, we examine how early childhood education and care (ECEC), particularly in relation to mathematics, is conceptualised by the English educational authorities. Situated within international debates about economic (school-readiness, accountability-driven) versus social (holistic, play-based, rights-oriented) models of ECEC, the study explores how curriculum expectations, assessment practices and didactical guidance collectively frame young children’s learning opportunities. Drawing on a document-based analytic approach, and guided by six literature-derived questions, the analysis reveals significant inconsistencies both within and between documents, including conflicting messages about the purpose of preschool, an uneven emphasis on school readiness, and ambivalent statements regarding the role of play, instruction and practitioner agency, as well as contradictory and shifting expectations surrounding the scope, status and pedagogical treatment of early mathematics. While statutory materials frequently privilege school readiness and narrowly defined number outcomes, non-statutory guidance promotes broader mathematical thinking, exploratory play and child-initiated reasoning. Overall, the findings demonstrate limited coherence across the English authorities’ ECEC expectations and highlight the interpretive and professional challenges faced by practitioners expected to implement this fragmented early years mathematics policy landscape. Full article
(This article belongs to the Section Early Childhood Education)
17 pages, 330 KB  
Article
The Influence of Heritage Language Teachers’ Educational Backgrounds on Their Reasoning About Special Educational Needs in Multilingual Students
by Christa Roux Sparreskog
Educ. Sci. 2026, 16(3), 468; https://doi.org/10.3390/educsci16030468 - 19 Mar 2026
Viewed by 237
Abstract
Swedish heritage language teachers have diverse educational backgrounds, and collaboration with mainstream and special needs teachers often fails. Consequently, many heritage language teachers must independently assess and support students with special educational needs. This study explores how their educational background shapes their reasoning [...] Read more.
Swedish heritage language teachers have diverse educational backgrounds, and collaboration with mainstream and special needs teachers often fails. Consequently, many heritage language teachers must independently assess and support students with special educational needs. This study explores how their educational background shapes their reasoning about such needs in multilingual students. All heritage language teachers in one Swedish municipality completed a questionnaire analyzed through content analysis and from a communicative-relational perspective. Findings show that teachers with Swedish teaching degrees describe students’ needs in nuanced ways, using specialized language while considering individual as well as didactic factors. Those with non-didactic degrees more often generalize groups who need support and attribute their needs mainly to individual factors. Full article
(This article belongs to the Special Issue Research, Innovation, and Practice in Bilingual Education)
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16 pages, 18259 KB  
Article
Pedagogy in Built Form: A Diachronic Reading of the UPAT
by Guiomar Martín Domínguez
Architecture 2026, 6(1), 47; https://doi.org/10.3390/architecture6010047 - 18 Mar 2026
Viewed by 212
Abstract
This article examines the Unité Pédagogique d’Architecture in Toulouse (UPAT) as a paradigmatic example of the palimpsestic architectures that characterize many contemporary university campuses. Conceived in the immediate aftermath of May 1968, the school emerged at a moment when pedagogical reform, political commitment, [...] Read more.
This article examines the Unité Pédagogique d’Architecture in Toulouse (UPAT) as a paradigmatic example of the palimpsestic architectures that characterize many contemporary university campuses. Conceived in the immediate aftermath of May 1968, the school emerged at a moment when pedagogical reform, political commitment, and architectural experimentation became closely intertwined. These conditions gave rise to a singular spatial organization based on a combinatory grid, intended to give architectural form to a democratic ideal of education grounded in openness, flexibility, and collective agency. The study adopts a historical–critical methodology based on the systematic analysis of primary and secondary sources, complemented by original graphic interpretations. This approach makes it possible to read the UPAT simultaneously as a didactic instrument and as an ideological manifesto, one whose ambitions were inherently marked by internal tensions and contradictions. A diachronic examination of subsequent extensions and transformations reveals how these founding intentions were progressively reinterpreted, constrained, or displaced in response to changing institutional, social, and cultural conditions. Taken as a whole, the evolving trajectory of this “manifesto school” illuminates the ways in which architectural ideals—particularly the pursuit of openness—are negotiated over time, offering a critical perspective on the reciprocal shaping of architecture, pedagogy, and institutional identity within the history of university buildings. Full article
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29 pages, 1565 KB  
Article
Integer Intelligence: A Reproducible Path from Training to FPGA
by Manjusha Shanker and Tee Hui Teo
Electronics 2026, 15(5), 1117; https://doi.org/10.3390/electronics15051117 - 8 Mar 2026
Viewed by 303
Abstract
A transparent, end-to-end pathway from learning-level training to deployable fixed-point hardware is presented and framed as gradients to gates. A didactic XOR convolutional network is first employed so that backpropagation, post-training quantization in INT8, and fixed-point arithmetic can be made concrete and verified [...] Read more.
A transparent, end-to-end pathway from learning-level training to deployable fixed-point hardware is presented and framed as gradients to gates. A didactic XOR convolutional network is first employed so that backpropagation, post-training quantization in INT8, and fixed-point arithmetic can be made concrete and verified with exact checks. The same methodology was applied to a compact LeNet-5 case study. On the software side, the training-to-export flow was formalized, and a bit-accurate Python reference was constructed for the quantized network. On the hardware side, a synthesizable INT8 datapath was implemented in Verilog, including multiply–accumulate units, sigmoid activation stages, and per-layer requantization with rounding and saturation. Test benches are provided so that the exported weights and activations can be ingested, and layer-wise matches can be reported. A co-simulation harness was used to coordinate framework inference, quantization, file conversion, HDL simulation, and regression checks, which enabled deterministic comparisons of the activations, partial sums and outputs. The complete loop was mapped to Artix-7 on the CMOD A7 development board, and the resource usage, maximum clock frequency, inference latency, and throughput were determined. The approach aligns with an educational HDL-to-Caffe pipeline by using reusable parameterized Verilog primitives for convolution, pooling, activation, and fully connected layers, training in Colab with AccDNN, Caffe, quantization, and an automated bit-for-bit verification regime before FPGA synthesis. Methodological contributions are provided, including a minimal and auditable XOR CNN that exposes scales, shifts, and saturation; a practical quantization recipe with INT32 accumulation and unit tests that guarantee agreement within one least significant bit between RTL and the INT8 reference; and a scalable mapping to LeNet-5 using a row-stationary and line-buffered dataflow on an Artix-7 FPGA. Empirical evidence shows feasibility at 100 MHz with representative utilization, millisecond-scale latency and zero mismatches across large test sets, which validates the quantization configuration and the verification strategy. Full article
(This article belongs to the Special Issue Recent Advances in AI Hardware Design)
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24 pages, 1291 KB  
Article
Integration of Adapted Podcasts and Digital Media into English Language Teaching for Primary School Children: Developing Creative Speech Skills
by Sholpan Kalbergenova, Larissa Lebedeva, Larissa Ageyeva, Jesus Garcia Laborda, Elmira Uaidullakyzy and Mahfuza Gafurova
Educ. Sci. 2026, 16(3), 405; https://doi.org/10.3390/educsci16030405 - 6 Mar 2026
Viewed by 379
Abstract
The accelerating digitalization of education has heightened the need for instructional approaches that are developmentally appropriate for young learners and capable of supporting both linguistic growth and creative speech production. This study investigates the pedagogical potential of integrating adapted English-language podcasts and media-based [...] Read more.
The accelerating digitalization of education has heightened the need for instructional approaches that are developmentally appropriate for young learners and capable of supporting both linguistic growth and creative speech production. This study investigates the pedagogical potential of integrating adapted English-language podcasts and media-based tasks into primary English instruction to foster originality, contextual coherence, expressive flexibility, and emotional richness in learners’ productive speech. The research employed a quasi-experimental pretest–posttest design implemented in intact classroom groups and involved 233 third- and fourth-grade students (experimental group, n = 116; control group, n = 117). Over one academic semester (January–May 2025), the experimental group participated in a structured programme embedded within regular lessons that combined short podcast episodes with dialogic and narrative tasks, while the control group followed the standard curriculum without podcast integration or comparable multimedia enrichment. Data analysis combined quantitative comparison of pre- and post-intervention speech outcomes with qualitative evaluation of learners’ oral and written products, supplemented by student and teacher feedback. The results showed statistically significant improvements in the experimental group across key indicators of creative speech performance. Qualitative evidence further indicated a shift toward more independent, personally meaningful language use, with learners demonstrating greater willingness to experiment with narrative transformation, evaluative retelling, and expressive variation. Taken together, the findings suggest that adapted podcasts, when systematically integrated into routine classroom practice, can serve as an effective and feasible tool for strengthening creative speech development in primary English language education and for enriching contemporary media pedagogy in digitally evolving learning environments. Full article
(This article belongs to the Section Teacher Education)
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15 pages, 1478 KB  
Article
SSR-Based Genetic Diversity Assessment Among Varieties Conserved in a Romanian Grapevine (Vitis vinifera L.) Collection
by Monica Hârţa, Doina Clapa, Gabriella De Lorenzis, Lucia Cintia Colibaba and Liliana Rotaru
Agriculture 2026, 16(5), 605; https://doi.org/10.3390/agriculture16050605 - 5 Mar 2026
Viewed by 484
Abstract
The present study highlights the genetic relationships among 52 grapevine varieties, including traditional and newly developed Romanian varieties and international reference cultivars, held in a small didactic ampelographic collection in Iaşi, Romania, and their genetic diversity, assessed using multivariate analysis. Twelve nuclear simple [...] Read more.
The present study highlights the genetic relationships among 52 grapevine varieties, including traditional and newly developed Romanian varieties and international reference cultivars, held in a small didactic ampelographic collection in Iaşi, Romania, and their genetic diversity, assessed using multivariate analysis. Twelve nuclear simple sequence repeat (SSR) markers, including nine OIV-standard descriptors, were used for genetic profiling. A total of 102 alleles were identified, with an average of 8.5 alleles per locus. The mean polymorphic information content (PIC) of 0.779 confirmed the high discriminatory power of the chosen markers. NJ dendrogram and PCoA yielded mostly similar results but did not clearly differentiate genotypes based on the selected criteria for genotype comparison (usage or historical status). STRUCTURE analysis assigned genotypes to SSR-group 1 (23.07%) and SSR-group 2 (34.61%) under K = 2 and a Q-value threshold of 0.85. The high proportion of admixed genotypes (42.32%) may reflect complex pedigrees and the migration of grapevine varieties across a wider territory surrounding Romania. The present research may serve as a starting point for future studies in Romania on the genetic structure and parental analysis of grapevine varieties held in small didactic collections, aiming to characterize and hold valuable grapevine varieties under secure conditions for future generations. Full article
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26 pages, 1201 KB  
Article
The Perceptions of Early Childhood Education Teachers About Folklore as an Educational Resource
by Miriam López-Santos, Alejandro Rodríguez-García, Pablo Alonso Diez and Claudia Mónica Rolando
Educ. Sci. 2026, 16(3), 396; https://doi.org/10.3390/educsci16030396 - 5 Mar 2026
Viewed by 422
Abstract
Folklore, as an expression of intangible cultural heritage, plays a key role in the transmission of collective memory, values, and cultural identity and has been widely recognized for its educational potential in early childhood. This study analyzes the perceptions of Early Childhood Education [...] Read more.
Folklore, as an expression of intangible cultural heritage, plays a key role in the transmission of collective memory, values, and cultural identity and has been widely recognized for its educational potential in early childhood. This study analyzes the perceptions of Early Childhood Education teachers in the province of León (Spain) regarding the pedagogical value of folklore and its use in classroom practice. A quantitative, cross-sectional, non-experimental design was employed using a validated questionnaire administered to a voluntary sample of 100 teachers from schools offering the second cycle of Early Childhood Education. The instrument assessed three dimensions: knowledge of folklore, perceived relevance of folk literature, and application in educational practice. The results indicate that teachers hold very positive attitudes toward folklore as a cultural and educational resource, particularly for children’s social, emotional, and cultural development. However, a clear gap emerges between this strong theoretical appreciation and its occasional classroom use. Significant differences were identified according to school environment, type of school, and teaching experience, indicating that contextual and professional factors influence folklore integration. Insufficient training and low self-confidence were identified as the main barriers. This study highlights the need to strengthen initial and in-service teacher education in cultural heritage and in the didactics of literature and music to promote a more meaningful and sustainable use of folklore in early childhood education. Full article
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27 pages, 2093 KB  
Article
Enhancing GreenComp Sustainability Skills in STEM Disciplines: A Didactic Proposal for Extreme Weather Preparedness in Secondary Education
by José Luis del Río-Rodríguez, Sergio Campos Fernández and María Calero Llinares
Sustainability 2026, 18(5), 2487; https://doi.org/10.3390/su18052487 - 4 Mar 2026
Viewed by 321
Abstract
This study addresses the growing vulnerability of societies to extreme weather events intensified by climate change and explores how Secondary Education can foster sustainability competences aligned with the European GreenComp framework. A mixed-methods design was used, combining a content analysis of 279 curricular [...] Read more.
This study addresses the growing vulnerability of societies to extreme weather events intensified by climate change and explores how Secondary Education can foster sustainability competences aligned with the European GreenComp framework. A mixed-methods design was used, combining a content analysis of 279 curricular units from educational legislation and STEM subjects in Compulsory Secondary Education and Baccalaureate, a questionnaire administered to 190 students, and the design and classroom implementation of a GreenComp-based teaching intervention. The curricular analysis revealed uneven integration of sustainability competences across STEM disciplines, with stronger presence in Biology, Geology and Technology, and limited representation in Mathematics and Physics and Chemistry. Student perceptions showed fragmented understandings of extreme weather events, their causes and consequences, and limited awareness of global frameworks such as the SDGs and COP meetings. The implemented teaching sequence improved students’ knowledge of extreme events, strengthened their recognition of links with climate change, and increased awareness of mitigation, adaptation, and the role of education and political action. Overall, the findings highlight both opportunities and gaps in current curricula and demonstrate the potential of contextualized, inquiry-based STEM approaches to develop sustainability competences and better prepare students to face extreme weather events. Full article
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24 pages, 1717 KB  
Article
Linguistic Landscape as a Resource in EGAP Courses: A Case Study
by Maria Yelenevskaya
Educ. Sci. 2026, 16(3), 359; https://doi.org/10.3390/educsci16030359 - 25 Feb 2026
Viewed by 370
Abstract
This article explores the incorporation of linguistic landscape (LL) studies into English for General Academic Purposes (EGAP) courses, emphasizing its potential to enhance language learning through real-world engagement. This study highlights the growing interest in LL as a sociolinguistic phenomenon that reflects urban [...] Read more.
This article explores the incorporation of linguistic landscape (LL) studies into English for General Academic Purposes (EGAP) courses, emphasizing its potential to enhance language learning through real-world engagement. This study highlights the growing interest in LL as a sociolinguistic phenomenon that reflects urban multilingualism and cultural dynamics. The goal of this article is to analyze pedagogical benefits of integrating LL into language education, such as fostering critical thinking, pragmatic competence, intercultural awareness among students, and creating situations in which the target language is used in natural communication. Through a case study conducted at the Guangdong Technion–Israel Institute of Technology, the author presents specific classroom activities and reports on how they can be combined with fieldwork conducted by students. The goal of the tasks was to let students analyze language use in public spaces, classifying the surrounding signs into top-down and bottom-up, and informative and regulatory, and discuss how social prestige of languages is reflected in multilingual signs. In documenting written language in public places, creating their own signs and assessing their peers’ work, students were practicing both receptive and productive skills. Most of the work was done in small groups, which contributed to the students’ ability to collaborate with peers. The findings suggest that LL projects can effectively bridge classroom learning with lived language experiences, although challenges remain in implementation due to time constraints and pedagogical ideologies. Full article
(This article belongs to the Special Issue Innovation and Design in Multilingual Education)
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34 pages, 476 KB  
Article
Discrete Quantization on Spherical Geometries: Explicit Models, Computations, and Didactic Exposition
by Mrinal Kanti Roychowdhury
Mathematics 2026, 14(5), 750; https://doi.org/10.3390/math14050750 - 24 Feb 2026
Viewed by 298
Abstract
This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere S2, we investigate three explicit models of discrete uniform distributions and [...] Read more.
This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere S2, we investigate three explicit models of discrete uniform distributions and derive closed-form expressions for their optimal quantizers and corresponding mean square quantization errors. (I) For N equally spaced points on the equator, we obtain exact error formulas for both divisible and non-divisible cases nN, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes ±ϕ0, each with M longitudes, we prove a no-cross-circle Voronoi phenomenon, establish symmetry-preserving optimality, and derive finite-sum error formulas together with sharp curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude ϕ0, we obtain analogous exact error formulas and show that curvature reduces distortion by a factor of cos2ϕ0, while preserving the n2 decay rate. Across all models, we rigorously establish the “block midpoint principle”: optimal Voronoi cells on a circle are contiguous azimuthal blocks, and their optimal representatives are the corresponding azimuthal midpoints. Numerical tables and illustrative figures highlight curvature effects and compare divisible and non-divisible cases. An algorithmic appendix provides pseudocode and a small, commented Python implementation to facilitate reproducibility. Written with didactic clarity while maintaining full mathematical rigor, this work bridges geometric intuition and analytic precision, providing explicit benchmark models that illuminate curvature effects and support further developments in quantization on curved manifolds. Full article
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