Search Results (15)

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (${\mathcal{R}}_0$) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings. Full article

Digital learning environments create a dynamic and engaging learning and teaching context that promotes a deeper understanding of complex concepts, eases the teaching process and fosters a passion for learning. Moreover, integrating interactive materials into pilot courses can assist teachers in better assessing student learning and adjusting their teaching strategies accordingly. The teachers can also receive valuable insights into students’ strengths and weaknesses, allowing them to provide targeted support and intervention when needed. For students from the defence and security fields, digital learning environments can create realistic simulations and virtual training scenarios that allow students to practise their skills in a controlled and safe environment, develop hands-on experience, and enhance their decision-making abilities without the need for real-world training exercises. In this context, the purpose of this paper is to introduce an approach for solving mathematical problems embedded in technical scenarios within the defence and security fields with the aid of digital technology using different software environments such as Python, Matlab, or SolidWorks. In this way, students can visualise abstract concepts, experiment with different scenarios, and receive instant feedback on their understanding. At the same time, the use of didactic and interactive materials can increase the interest among students and teachers for utilising mathematical models and digital technologies in the educational process. This paper also helps to reinforce key concepts and enhance problem-solving skills, sparking curiosity and creativity, and encouraging active participation and collaboration. Throughout the development of this proposal, based on survey analysis, good practices are presented, and advice for improvement is collected while having a wide range of users giving feedback, and participating in discussions and testing (pilot) short-term learning/teaching/training activities. Full article

Inquiry and modelling are relevant processes of mathematical activity, and both are considered as important competencies for problem solving, since their subprocesses have a family resemblance that makes them mutually complementary and enriched. In this line, we report an investigation focused on the study of the synergies and relationships established between the inquiry and modelling processes emerging in a learning situation, based on a realistic and authentic archaeological context. To this end, we considered one theoretical cycle to explain each process, and then designed a learning situation contextualised in a realistic and authentic archaeological context which promoted the development of inquiry and modelling skills for its solving. Methodologically, we conducted a case study with students in a secondary school of Badalona (Catalonia, Spain), who solved the learning situation grouped into working teams. Through a thematic analysis, we identified the steps of both processes followed by the students when solving the learning situation, and thus we delved into the synergies and relationships established between inquiry and modelling processes. Finally, we reflect on how both processes mutually enrich when dealing with a problem-situation and propose an integrative cycle for the development of inquiry and modelling skills in the classroom. Full article

In this article, we chart the development of one of us—Sue Hough—from a teacher who wanted students to understand to one who gained new critical understandings of student thinking, pedagogy, and the very nature of mathematics. We comment on the role of research interventions and learning communities in this development, with a particular focus on Sue’s encounter with Realistic Mathematics Education and the connections it makes between informal and formal mathematics through the pedagogy of guided reinvention. Development towards teaching that enables all learners to make sense of mathematics requires fundamental changes in pedagogic practice and a reconceptualisation of progress. Bringing about such radical change relies on one further aspect of Sue’s story—the freedom to experiment and learn as a teacher. We note the remoteness of this possibility in a climate of performativity and marketised education, and we discuss the implications of Sue’s journey for our pedagogical responsibilities in professional development today. Full article

Spatial visualization ability (SVA) has been identified as a potential key factor for academic achievement and student retention in Science, Technology, Engineering, and Mathematics (STEM) in higher education, especially for engineering and related disciplines. Prior studies have shown that training using virtual reality (VR) has the potential to enhance learning through the use of more realistic and/or immersive experiences. The aim of this study was to investigate the effect of VR-based training using spatial visualization tasks on participant performance and mental workload using behavioral (i.e., time spent) and functional near infrared spectroscopy (fNIRS) brain-imaging-technology-derived measures. Data were collected from 10 first-year biomedical engineering students, who engaged with a custom-designed spatial visualization gaming application over a six-week training protocol consisting of tasks and procedures that varied in task load and spatial characteristics. Findings revealed significant small (Cohen’s d: 0.10) to large (Cohen’s d: 2.40) effects of task load and changes in the spatial characteristics of the task, such as orientation or position changes, on time spent and oxygenated hemoglobin (HbO) measures from all the prefrontal cortex (PFC) areas. Transfer had a large (d = 1.37) significant effect on time spent and HbO measures from right anterior medial PFC (AMPFC); while training had a moderate (d = 0.48) significant effect on time spent and HbR measures from left AMPFC. The findings from this study have important implications for VR training, research, and instructional design focusing on enhancing the learning, retention, and transfer of spatial skills within and across various VR-based training scenarios. Full article

This research describes the perspective of secondary school teachers on the selection, elaboration, and implementation of contextualized mathematical problems in secondary education in Costa Rica. An exploratory and descriptive quantitative study is carried out, applying a questionnaire to 67 in-service secondary school mathematics teachers. The dimensions of the sources used by teachers, the difficulties encountered and the learners’ background considered for problem selection and elaboration are analyzed. In addition, the contexts present in the problems that teachers propose are classified. The results indicate that teachers use textbooks for the selection of mathematical problems. In addition, they give a favorable opinion regarding the use of realistic contexts in the creation of new problems. However, several of the proposed problems present fictitious or mathematical contexts. The highest percentage of the participants points out the need for training for the elaboration of contextualized mathematical problems. Full article

Our society is immersed in the Fourth Industrial Revolution due to the fast evolution of the new technologies that are modifying the labor market. In the near future, technologies related to Industry 4.0 will produce totally new goods and services. Therefore, the educational systems should adapt their programs to the future needs of an uncertain labor market. In particular, mathematics will play a key role in future jobs and there is a strong need to connect its teaching methodologies to the new technological scene. This work uses the STEAM approach (science, technology, engineering, arts and mathematics) along with active methodologies and educational robotics with the aim of developing a new strategy for the application of mathematics and physics in an engineering degree. In particular, a transportation challenge is posed to tackle the teaching–learning process of the Bézier curves and their applications in physics. A pilot project is developed using a LEGO EV3 robot and an active methodology, where students become the center of the learning process. The experimental results of the pilot study indicate an increase in the motivation due to the use of robots and the realistic context of the challenge. Full article

This work aims to examine how the learning of mathematics in early childhood is developed in different policies, particularly within the processes of formal education and care in early years institutions. A comparative analysis of early mathematics education policies across countries must consider cultural differences, teaching practice, structural differences and institutional framework conditions, as well as the initial training and professional knowledge of teachers and educators. Extracted from the official country regulations, the following pages include some of the main characteristics of the national systems of early childhood education and care (ECEC) in Portugal, Slovenia and Spain, as well as a comparison of the ECEC guidelines concerning mathematics education between these three countries. There is an international consensus on an approach to early mathematics education inspired by realistic mathematics education (RME), i.e., on the importance of working mathematically in context, as well as on the idea of doing so through play, developing the language to communicate mathematical ideas. However, we found that these three aspects are reflected very differently in the official regulations of the three countries: while in Spain the development is very detailed and emphasizes the holistic approach and the role of mathematics in exploring the environment, the Portuguese curriculum emphasizes the role of mathematics as a form of language. The Slovenian curriculum, at last, focuses on the concepts and procedures associated with each mathematical sense. Furthermore, there are structural features concerning the regulation and type of ECEC system that have an influence in the implementation of the curriculum. Full article

Research suggests that integrated STEM activities can best support students in developing their mathematical and scientific understanding. On one hand, while science provides mathematics with real-life authentic problems to investigate, mathematics provides science powerful tools to explore those problems. In line with this call, in this study, we designed an integrated lesson at the cross-section of proportional reasoning and added sugar present in food products to explore how added sugar provides students with a meaningful context to engage in proportional reasoning and how proportional reasoning helps students identify the quantity of added sugar present in different food products and provides students with a platform to initiate a conversation around quality of food products. Developed on the theoretical framework of Realistic Mathematics Education (RME), this lesson was remotely implemented on three middle school students. The result section highlights the design principle of the lesson that provided students with an opportunity to construct an understanding of both the disciplines through a mutual interaction. Full article

A cube is one of the most fundamental shapes we can draw and can observe from a drawing. The two visualization methods most commonly applied in mathematics textbooks and education are the axonometric and the perspective representations. However, what we see in the drawing is really a cube or only a general cuboid (i.e., a polyhedron with different edge lengths). In this experimental study, 153 first-year ( 19–20-year-old) students, two-thirds of them being female, were asked to interactively adjust a cuboid figure until they believe what they see is really a cube. We were interested in how coherently people, who are actually students of arts studies and engineering with advanced spatial perception skills in most cases, evaluate these drawings. What we have experienced is that for most people there is a common visual understanding of seeing a cube (and not a general cuboid). Moreover, this common sense is surprisingly close to the conventions applied in axonometric drawings, and to the theoretical, geometric solution in the case of three-point perspective drawings, which is the most realistic visualization method. Full article

Clinical reasoning in multimorbidity conditions asks for the ability to anticipate the possible evolutions of the overall health state of a patient and to identify the interactions among the concurrent health issues and their treatments. The HIN (Health Issue Network) approach, as Petri Nets-based formal language, is introduced as capable of providing a novel perspective to facilitate the acquisition of such competencies, graphically representing the network among a set of health issues (HIs) that affect a person throughout their life, and describing how HIs evolve over time. The need to provide a more immediate user-oriented interface has led to the development of f-HIN (friendly HIN), a lighter version based on the same mathematical properties as HIN, from which stems in turn the f-HINe (friendly HIN extracted) model, used to represent networks related to either real patients’ clinical experiences extracted from electronic health records, or from teacher-designed realistic clinical histories. Such models have also been designed to be embedded in a software learning environment that allows drawing a f-HIN diagram, checking for its format correctness, as well as designing clinical exercises for the learners, including their computer-assisted assessment. The present paper aims at introducing and discussing the f-HIN/f-HINe models and their educational use. It also introduces the main features of the software learning environment it was built upon, pointing out its importance to: (i) help medical teachers in designing and representing the context of a learning outcome; and (ii) handle the complex history of a multimorbidity patient, to be conveyed in Case-Based Learning (CBL) exercises. Full article

Since the late 1960s, a reform in mathematics education, which is currently known under the name Realistic Mathematics Education (RME), has been taking place in the Netherlands. Characteristic for this approach to mathematics education is that mathematics is not seen as ready-made knowledge but as an activity of the learner. Although much has been written about the big ideas and intentions of RME, and multiple RME-oriented textbooks have been published, up to now the development of this approach to mathematics education has not been thoroughly investigated. In the research reported in this article, we traced how RME has evolved over the years. The focus in our study was on early addition and subtraction in primary school. For this, we studied RME core curriculum documents and analyzed RME-oriented textbooks that have been published between the onset of RME and the present. We found that the big ideas and teaching principles of RME were clearly reflected in the learning facilitators for learning early addition and subtraction and were steadily present in curriculum documents over the years, although some were made concrete in further detail. Furthermore, we found all RME learning facilitators also to be present in all RME-oriented textbooks, though in some cases in other ways than originally intended. Our research shows the complexity of a curriculum reform process and its implementation in textbooks. Full article

Software programming is one of the key abilities for the development of Computational Thinking (CT) skills in Science, Technology, Engineering and Mathematics (STEM). However, specific software tools to emulate realistic scenarios are required for effective teaching. Unfortunately, these tools have some limitations in educational environments due to the need of an adequate configuration and orchestration, which usually assumes an unaffordable work overload for teachers and is inaccessible for students outside the laboratories. To mitigate the aforementioned limitations, we rely on cloud solutions that automate the process of orchestration and configuration of software tools on top of cloud computing infrastructures. This way, the paper presents ACTaaS as a cloud-based educational resource that deploys and orchestrates a whole realistic software programming environment. ACTaaS provides a simple, fast and automatic way to set up a professional integrated environment without involving an overload to the teacher, and it provides an ubiquitous access to the environment. The solution has been tested in a pilot group of 28 students. Currently, there is no tool like ACTaaS that allows such a high grade of automation for the deployment of software production environments focused on educational activities supporting a wide range of cloud providers. Preliminary results through a pilot group predict its effectiveness due to the efficiency to set up a class environment in minutes without overloading the teachers, and providing ubiquitous access to students. In addition, the first student opinions about the experience were greatly positive. Full article

Creating mathematics tasks provide opportunities for students to develop their thinking, reasoning, communication, and creativity. This paper presents a study on teaching pre-service teachers to create realistic mathematics tasks in real contexts and amending them through an iterative process of analysis and refinement. The study was undertaken with pre-service teachers from two university training courses in Spain, undergraduate students from a primary teacher training course, and graduate students from an educational Master’s course. The students worked in groups to collaborate in the creation of the requested tasks and improvement of them based on critical thinking and creativity. The tasks were not only evaluated concerning their level of realism, but also regarding their level of authenticity, the cognitive domains involved, and their openness characteristic. These are the key characteristics related to environmental and sustainability aspects. The outcomes confirmed that the creation of realistic mathematics tasks was a challenge for future primary teachers; however, they were able to create tasks with high levels of cognitive domain, authenticity, and openness. This evidences, on the one hand, the difficulty that future teachers have in understanding the realism of a mathematics task, and, on the other, the possibilities offered by the task’s creation and the revision activity, which has educational implications and opens paths for future research. Full article

This paper presents several new techniques for volumetric cloud rendering using efficient algorithms and data structures based on ray-tracing methods for cumulus generation, achieving an optimum balance between realism and performance. These techniques target applications such as flight simulations, computer games, and educational software, even with conventional graphics hardware. The contours of clouds are defined by implicit mathematical expressions or triangulated structures inside which volumetric rendering is performed. Novel techniques are used to reproduce the asymmetrical nature of clouds and the effects of light-scattering, with low computing costs. The work includes a new method to create randomized fractal clouds using a recursive grammar. The graphical results are comparable to those produced by state-of-the-art, hyper-realistic algorithms. These methods provide real-time performance, and are superior to particle-based systems. These outcomes suggest that our methods offer a good balance between realism and performance, and are suitable for use in the standard graphics industry. Full article