Search Results (49)

Kirchhoff’s current law was originally derived for systems such as telegraphs that switch in 0.1 s. It is used widely today to design circuits in computers that switch in ~0.1 nanoseconds, one billion times faster. Current behaves differently in one second and one-tenth of a nanosecond. A derivation of a current law from the fundamental equations of electrodynamics—the Maxwell equations—is needed. Here is a derivation in one line: $\mathbf{d}\mathbf{i}\mathbf{v} \mathbf{c}\mathbf{u}\mathbf{r}\mathbf{l}\mathbf{B}/{\mathsf{\mu }}_0=0=\mathbf{d}\mathbf{i}\mathbf{v}\left(\mathbf{J}+({\mathsf{\epsilon }}_{\mathit{r}}-1){\mathsf{\epsilon }}_0\mathbf{\partial }\mathbf{E}/\mathbf{\partial }\mathbf{t}+{\mathsf{\epsilon }}_0\mathbf{\partial }\mathbf{E}/\mathbf{\partial }\mathbf{t}\right)=\mathbf{d}\mathbf{i}\mathbf{v}{\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$. Maxwell’s ‘true’ current is defined as ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$. The universal displacement current found everywhere is ${\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$. The conduction current $\mathbf{J}$ is carried by any charge with mass, no matter how small, brief, or transient, driven by any source, e.g., diffusion. The second term $({\mathsf{\epsilon }}_r-1){\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$ is the usual approximation to the polarization currents of ideal dielectrics. The dielectric constant ${\mathsf{\epsilon }}_r$ is a dimensionless real number. Real dielectrics can be very complicated. They require a complete theory of polarization to replace the $({\mathsf{\epsilon }}_r-1){\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t}$ term. The Maxwell current law $\mathrm{d}\mathrm{i}\mathrm{v}{\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}=0$ defines the solenoidal field of total current that has zero divergence, typically characterized in two dimensions by streamlines that end where they begin, flowing in loops that form circuits. Note that the conduction current $\mathbf{J}$ is not solenoidal. Conduction current $\mathbf{J}$ accumulates significantly in many chemical and biological applications. Total current ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ does not accumulate in any time interval or in any circumstance where the Maxwell equations are valid. ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ does not accumulate during the transitions of electrons from orbital to orbital within a chemical reaction, for example. ${\mathbf{J}}_{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$ should be included in chemical reaction kinetics. The classical Kirchhoff current law $\mathrm{d}\mathrm{i}\mathrm{v} \mathbf{J}=0$ is an approximation used to analyze idealized topological circuits found in textbooks. The classical Kirchhoff current law is shown here by mathematics to be valid only when $\mathbf{J}\gg {\mathsf{\epsilon }}_0\partial \mathbf{E}/\partial \mathrm{t},$ typically in the steady state. The Kirchhoff current law is often extended to much shorter times to help topological circuits approximate some of the displacement currents not found in the classical Kirchhoff current law. The original circuit is modified. Circuit elements—invented or redefined—are added to the topological circuit for that purpose. Full article

This study is focused on the analysis of the level of geometric thinking of 15-year-old Slovak pupils in relation to the difficulty of geometric problems, their gender, and their assessment in mathematics. The main aim of this study was to determine the level of geometric thinking of 15-year-old Slovak pupils, to examine the relationship between their mathematics assessment and the level of geometric thinking, and to find out gender differences in relation to the different levels of geometric thinking. The van Hiele test was adapted and applied to a representative sample of 15-year-old Slovak pupils to determine the level of geometric thinking. We used reliability/item analysis. The reliability of the knowledge test (after adaptation) was assessed using Cronbach’s alpha (0.64). The validity of the test was demonstrated by the correlation of the Usiskin test results with pupils’ mathematics grades (Goodman–Kruskal’s gamma, p < 0.05). Statistical analysis showed that 15-year-old Slovak pupils achieve different levels of geometric thinking depending on the difficulty of the tasks. Pupil achievement declined significantly as task difficulty increased. Pupils had the greatest difficulty with tasks classified as the fifth (rigorous) and partly the fourth (deductive) van Hiele level, which require a deep understanding of geometric systems and the ability to prove logically. The lower-level tasks (visualization, analysis, and abstraction) were able to differentiate students according to different levels of geometric thinking. The results showed a significant positive relationship (Goodman–Kruskal’s gamma, p < 0.05) between the pupils’ overall mathematics scores (expressed as a grade) and their level of geometric thinking as detected by the van Hiele test. The analysis of gender differences (Duncan’s test, p < 0.05) showed that in the less challenging tasks, corresponding to the first three van Hiele levels (visualization, analysis, abstraction), girls performed statistically significantly better than boys. In the more challenging tasks, classified as the fourth (deductive) and fifth (rigorous) levels of geometric thinking, there were no statistically significant differences between boys and girls. In the more challenging tasks, the performances of both genders were comparable. The presented study identifies significant deficits in the development of higher levels of geometric thinking among 15-year-old Slovak pupils. These findings strongly imply the necessity for the transformation of the curriculum, textbooks, and didactic approaches with the aim of systematically developing deductive and rigorous reasoning, while it is essential to account for the demonstrated gender differences in performance. 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

The point-kernel (PK) technique has a long history in applied radiation shielding, originating from the early days of digital computers. The PK technique applied to gamma-ray attenuation is one of many successful applications, based on the linear superposition principle applied to distributed radiation sources. Mathematically speaking, the distributed source will produce a detector response equivalent to the numerical integration of the radiation received from an equivalent number of point sources. In this treatment, there is no interference between individual point sources, while inherent limitations of the PK method are its inability to simulate gamma scattering in shields and the usage of simple boundary conditions. The PK method generally works for gamma-ray shielding with corrective B-factor for scattering and only specifically for fast neutron attenuation in a hydrogenous medium with the definition of cross section removal. This paper presents theoretical and programming aspects of the PK program developed for a distributed source of photons (line, disc, plane, sphere, slab volume, etc.) and slab shields. The derived flux solutions go beyond classical textbooks as they include the analytical integration of Taylor B-factor, obtaining a closed form readily suitable for programming. The specific computational modules are unified with a graphical user interface (GUI), assisting users with input/output data and visualization, developed for the fast radiological characterization of simple shielding problems. Numerical results of the selected PK test cases are presented and verified with the CADIS hybrid shielding methodology of the MAVRIC/SCALE6.1.3 code package from the ORNL. Full article

This paper presents research on the origin, scope, evolution, and rationale of knowledge about polynomials in high school mathematics. Within the framework of the Anthropological Theory of the Didactic, Croatian high school curricula and textbooks were analyzed, and four models of knowledge to be taught were identified in the period following the formal abandonment of New Math principles. None of the identified models provides a unified discourse that integrates knowledge about polynomials transposed from scholarly domains of algebra and mathematical analysis. In relation to other curricular content the knowledge about polynomials has two-fold importance: (1) contributing to the development of various techniques related to high school algebra and calculus; (2) serving as a fundamental example in the formation of the notion of a function. Thus, the observed reduction in polynomial content over the analyzed period affects both practical and theoretical knowledge. The findings suggest that curricular changes have primarily focused on the selection of knowledge, with scarce adaptations of knowledge to be taught compared to the knowledge before each curricular change. This has led to a persistent gap between algebraic and analytical approaches to polynomials, potentially influencing the learned knowledge even among the highest-achieving students. Despite polynomials’ epistemological and didactical potential to bridge high school algebra and calculus, their restriction to specific forms of algebraic expressions and linear and quadratic functions contributes more to the fragmentation of high school mathematics. Full article

Readability formulas are mathematical functions that assess the ‘difficulty’ level of a given text. They play a crucial role in aligning educational texts with student reading abilities; however, existing models are often not tailored to specific linguistic or regional contexts. This study aims to develop and evaluate two novel readability formulas specifically designed for the Mexican Spanish language, targeting elementary education levels. The formulas were trained on a corpus of 540 texts drawn from official elementary-level textbooks issued by the Mexican public education system. The first formula was constructed using multiple linear regression, emulating the structure of traditional readability models. The second was derived through genetic programming (GP), a machine learning technique that evolves symbolic expressions based on training data. Both approaches prioritize interpretability and use standard textual features, such as sentence length, word length, and lexical and syntactic complexity. Experimental results show that the proposed formulas outperform several well-established Spanish and non-Spanish readability formulas in distinguishing between grade levels, particularly for early and intermediate stages of elementary education. The GP-based formula achieved the highest alignment with target grade levels while maintaining a clear analytical form. These findings underscore the potential of combining machine learning with interpretable modeling techniques and highlight the importance of linguistic and curricular adaptation in readability assessment tools. Full article

Digital platforms are increasingly prevalent among young students in K-12 education, offering significant opportunities but also raising concerns about their effects on self-assessment and academic performance. This study investigates the effectiveness of Kahoot! compared to traditional instructional methods in enhancing mathematics achievement and its impact on multiple screen addiction (MSA) among Greek students aged 9 to 12 during a STEM summer camp. A quasi-experimental design was employed with a purposefully selected sample of one hundred and ten (n = 110) students, who were non-randomly divided into two groups: (a) an experimental group of fifty-five students (n = 55) who engaged with Kahoot! (using dynamic visual aids and interactive content) and (b) a control group of fifty-five students (n = 55) who received traditional instruction (using digital textbooks and PowerPoint slides with multimedia content) on laptops and tablets. The findings revealed a statistically significant difference in MSA scores, with the experimental group exhibiting lower MSA scores compared to their counterparts, indicating a positive impact on reducing screen addiction levels. While Kahoot! led to lower MSA levels, it significantly improved overall mathematical achievement, with a substantial effect size, suggesting a strong positive impact on learning outcomes. The current study highlights the importance of aligning educational tools with the intended outcomes and recommends further research to explore the broader impact of gamified learning on student engagement, screen addiction, and learning outcomes. Full article

This quasi-experimental study investigated the impact of computational learning activities on high school students’ computational thinking (CT) and computational modeling (CM) skills. High school students (n = 90) aged 16 to 19 engaged in activities using computer models versus textbook-based models in mathematics and social science. The results indicated that students using computer models showed significant improvements in CT and CM skills compared to their peers in conventional learning settings. However, a potential ceiling effect in the CT assessments suggests that the test may not fully capture the extent of skill development. These findings highlight the importance of integrating computational learning activities in education, as they enhance students’ abilities to apply these skills beyond the classroom. Full article

Algebra as a school subject is ill defined. Students experience algebra quite differently depending on the perspective of algebra taken by authors of the textbooks from which they learn. Through a content analysis of problems (n = 63,174) in the narrative and homework sections of six high school mathematics textbook series published in the U.S., we acquired systematic and reliable information about the algebra strand (i.e., symbolic algebra and functions) of each textbook series. We introduce plots to show the density, distribution, and sequencing of content, and present analyses of data for cognitive behavior, real-world context, technology, and manipulatives. Feedback on this study from an author of each textbook series is shared, and findings are discussed in terms of students’ opportunities to learn. Full article

The traditional educational system, in certain aspects, limits personalized learning. This is mainly evident in the fact that average students, who do not have any learning difficulties, are required to solve the same tasks from the same textbook in the same order. Artificial intelligence and other smart learning tools present great opportunities for implementing a personalized learning system. Our previous surveys and literature reviews also show that educators see the greatest potential in personalized education for the assimilation of artificial intelligence into education. In this context, we have developed educational software called “Learn with M.E. as Math Educator”, which facilitates more personalized teaching of basic mathematical operations. This study presents the structure and operation of this application. We tested the usability of the software in several institutions. Our research target group consists of elementary school students, specifically those aged 11–15. This article provides a detailed overview of the accuracy and educational outcomes of the completed application. We evaluated the application and its effectiveness using both qualitative and quantitative methods. Our research design combined elements of educational technology development and effectiveness assessment. To evaluate student performance, we employed a control group methodology. Data were analyzed by comparing test results between students using the software and those receiving traditional instruction. We examined user satisfaction through survey questionnaires. Teachers’ opinions were gathered through structured interviews, and their responses were categorized using a SWOT analysis. The findings indicated that the use of the software significantly improved students’ mathematics performance compared to the control group. Students provided positive feedback on the software’s user interface, describing it as user-friendly and motivating. Teachers regarded the software as an effective educational tool, facilitating differentiated instruction and enhancing student engagement. The results suggest that digital educational tools, such as the developed software, can provide substantial added value in education. Full article

The design of end-of-shaft angle sensing magnetic positioning systems (MPS) requires accurate field computations in the sensing area for the magnetostatic inversion procedure. Highly resolved field computations on a 3-D domain make FEM simulations unfeasible and favour analytical solutions. Analytical textbook field solutions of a number of standard magnet shapes are however numerically unstable along symmetry axes, body edges as well as in the far field. For the particular application of an end-of-shaft system, only a particular instability close to the symmetry axis, ρ → 0 plays a detrimental role. We stabilize the field equation by mathematical reformulation of naturally occurring numerically unstable combinations of elliptic integrals in the derivation. The resulting formulas or even their ready-to-use implementation in the freely available Python package Magpylib can be used without limitations for end-of-shaft MPS designs. Full article

Faculty members in science, technology, engineering, and mathematics (STEM) fields are accustomed to presenting their research findings through journal publications, conference presentations, textbooks, and other academic mediums. However, the audience for these traditional forms of communication are other researchers, which raises concerns about how science research and knowledge are communicated to audiences who have less expertise on these topics. We sought to understand how faculty members develop their identities through collaborative professional development opportunities aimed at growing communication skills to communicate with audiences less familiar with research through interdisciplinary science, technology, engineering, arts, and mathematics (STEAM) activities. We conducted a qualitative, longitudinal study with sixteen STEAM faculty members to explore their identity trajectories as their interdisciplinary cohorts participated in various collaborations to engage with public audiences about their research. Through our analysis, we found that each faculty member’s dominant identity played a significant role in their identity trajectory through their professional development. We observed a significant growth in faculty members’ communication skills, such as learning new presentation techniques to engage others in their research areas of expertise and in their understanding of interdisciplinary STEAM collaboration. Our results provide insights into the identity trajectories of faculty members and how their identity development through these interdisciplinary STEAM collaborations will impact their formal education roles as researchers and teachers moving forward. Full article

This paper aims to discuss the revitalization of education for sustainable development (ESD) in mathematics education, particularly in relation to mathematics curricula for grades 1–10, using the example of the new Norwegian curriculum, LK20, which came into effect at the beginning of the 2020 school year. Several studies in the past two decades have identified disengagement of sustainability learning (SL) within mathematics education and called for a change in the philosophy of mathematics education to integrate sustainability into the teaching and learning of mathematics. Using the qualitative content analysis method, we examined three types of documents: the core curriculum, the mathematics curriculum, and one Norwegian mathematics textbook series called Matemagisk. We find that sustainable development (SD) is one of the interdisciplinary issues addressed in LK20. Even though the mathematics curriculum does not explicitly incorporate terms such as ‘sustainability’ or ‘sustainable development’, indicating that mathematics and sustainability are unconnected, its six core elements—exploration and problem-solving, modeling and applications, reasoning and argumentation, representation and communication, abstraction and generalization, and mathematical fields of knowledge—provide opportunities for integrating sustainability learning (SL). On the other hand, looking at one of the mathematics textbook series, Matemagisk, for grades 4, 7, and 10, it appears that tasks that use sustainability contexts were included implicitly or explicitly, indicating another opportunity to facilitate sustainable learning. By including sustainability contexts in tasks and mathematical activities, it is possible to boost the process of embedding ESD in mathematics education without compromising content. Full article

We argue that a clear view of quantum mechanics is obtained by considering that the unicity of the macroscopic world is a fundamental postulate of physics, rather than an issue that must be mathematically justified or demonstrated. This postulate allows for a framework in which quantum mechanics can be constructed in a complete mathematically consistent way. This is made possible by using general operator algebras to extend the mathematical description of the physical world toward macroscopic systems. Such an approach goes beyond the usual type-I operator algebras used in standard textbook quantum mechanics. This avoids a major pitfall, which is the temptation to make the usual type-I formalism ’universal’. This may also provide a meta-framework for both classical and quantum physics, shedding new light on ancient conceptual antagonisms and clarifying the status of quantum objects. Beyond exploring remote corners of quantum physics, we expect these ideas to be helpful to better understand and develop quantum technologies. Full article

Literature and most textbooks around the world describe Silver-Meal in such a way that periods with zero demand make Silver-Meal suggest a higher frequency of order replenishments than necessary and therefore higher total costs than necessary. Silver-Meal, still the best-known technique, is therefore inferior to other lesser-known techniques when the time interval in the calculations presently is days and not months. The purpose of this article is to show that another mathematical formulation of Silver-Meal avoids this trouble. We also point to characteristics such as Silver-Meal, Least Unit Cost, Part-Period Balancing, and lot-sizing techniques that are available in many textbooks for operations and supply chain management. We illustrate the techniques with different examples of periods without demand, declining demand, and varying demand. We point out possible problems with the different techniques. Literature mostly does not consider periods of zero demand, which was not so important before. Lot-sizing methods must cope with the important performance indicator “Days of inventory”. Numerous practical situations with zero demand periods exist where a lot of sizing techniques help for efficient operations. It is necessary knowledge and a tool for students (future users, performers, and managers). “Lägsta periodkostnad” is a restored and reformulated Silver-Meal, with Silver-Meal’s characteristics already presented in literature, except those difficulties with zero demand periods disappear. Full article