Mathematical Context Research Articles

Problem Posing as a Way of Promoting Individual Mathematical Thinking in STEM Contexts – The Case of Climate Change

AbstractProblems encountered in Science, Technology, Engineering and Mathematics (STEM) contexts cannot be adequately described or solved with the knowledge of a single discipline. Instead, a high level of inter- and transdisciplinary knowledge and methods is required to overcome them. These help to pose problems about the complex challenges and solve them in a creative way. The better the knowledge within one and of different disciplines is interlinked, the more targeted questions can be formulated and answered. Mathematical concepts serve as the foundation for many of the pressing problems of our time. At the same time, these problems offer a wide range of opportunities for individualized exploration. They are equally suitable for students with different interests and levels of ability, as everyone can identify and work on an individual problem within the given context. However, numerous studies have shown that posing adequate mathematical problems must be learnt as well as knowledge from different disciplines is not automatically linked or transferred to other situations. The ability to grasp the formal structure of a problem, recognize problems, and find connecting problems are characteristics of mathematically gifted children and young people that need to be promoted. In our theory-based contribution, we use a concrete context to illustrate which possible mathematically rich problems can be posed by students of different ages and abilities. This approach facilitates the development of their individual abilities according to their interests and potential.

Understanding the Flows of Signals and Gradients: A Tutorial on Algorithms Needed to Implement a Deep Neural Network from Scratch

Theano, TensorFlow, Keras, Torch, PyTorch, and other software frameworks have remarkably stimulated the popularity of deep learning (DL). Apart from all the good they achieve, the danger of such frameworks is that they unintentionally spur a black-box attitude. Some practitioners play around with building blocks offered by frameworks and rely on them, having a superficial understanding of the internal mechanics. This paper constitutes a concise tutorial that elucidates the flows of signals and gradients in deep neural networks, enabling readers to successfully implement a deep network from scratch. By “from scratch”, we mean with access to a programming language and numerical libraries but without any components that hide DL computations underneath. To achieve this goal, the following five topics need to be well understood: (1) automatic differentiation, (2) the initialization of weights, (3) learning algorithms, (4) regularization, and (5) the organization of computations. We cover all of these topics in the paper. From a tutorial perspective, the key contributions include the following: (a) proposition of R and S operators for tensors—rashape and stack, respectively—that facilitate algebraic notation of computations involved in convolutional, pooling, and flattening layers; (b) a Python project named hmdl (“home-made deep learning”); and (c) consistent notation across all mathematical contexts involved. The hmdl project serves as a practical example of implementation and a reference. It was built using NumPy and Numba modules with JIT and CUDA amenities applied. In the experimental section, we compare hmdl implementation to Keras (backed with TensorFlow). Finally, we point out the consistency of the two in terms of convergence and accuracy, and we observe the superiority of the latter in terms of efficiency.

LEARNING MATHEMATICS IN TELUKDALAM MARKET: CALCULATING PRICES AND MONEY IN LOCAL TRADE

This study aims to develop a contextual mathematics learning method using the market as a learning medium in Telukdalam. The main focus of the research is to enhance students' abilities in calculating prices and money within local trade. Through this approach, students not only learn basic mathematical concepts but also relate them to real-life situations they encounter in their daily lives. A literature review method is employed to examine relevant literature on contextual mathematics learning and the importance of social interaction in the learning process. The results indicate that learning experiences in the market can enhance students' motivation and understanding of mathematical material. Additionally, this learning approach enriches local social and cultural values, leading students to appreciate their environment and traditions more. It is hoped that the findings of this study will contribute to the development of a more relevant and contextual mathematics curriculum.

The More the Better? A Systematic Review and Meta-Analysis of the Benefits of More than Two External Representations in STEM Education

Over the last decades, a multitude of results in educational and psychological research have shown that the implementation of multiple external representations (MERs) in educational contexts represents a valuable tool for fostering learning and problem-solving skills. The context of science, technology, engineering, and mathematics (STEM) education has received great attention because it necessitates using various symbolic (e.g., text and formula) and graphical representations (e.g., pictures and graphs) to convey subject content. Research has mainly explored effects of combining two representations, but the potential benefits of integrating more than two representations on students’ learning remain underexplored. This gap limits our understanding of promising educational practices and restricts the development of effective teaching strategies catering to students’ cognitive needs. To close this gap, we conducted a systematic review of 46 studies and a meta-analysis that included 132 effect sizes to evaluate the effectiveness of using more than two representations in STEM education and to identify moderating factors influencing learning and problem-solving. A network diagram analysis revealed that the advantages of learning and problem-solving with MERs are also applicable to more than two representations. A subsequent meta-analysis revealed that the learning with more than two representations in STEM can have advantageous effects on students cognitive load (Hedges′g=0.324,p<.001,95%CI[0.164,0.484]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Hedges}}{^\\prime}g =0.324,~p<.001,~95\\%~\ ext{CI}~[0.164, 0.484]$$\\end{document}) and performance (Hedges′g=0.118,p<.001,95%CI[0.050,0.185]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Hedges}}{^\\prime}g =0.118,~p<.001,~95\\%~\ ext{CI}~[0.050, 0.185]$$\\end{document}) compared to learning with two representations without notable differences in learning time. The analysis of moderating factors revealed that benefits of learning with more than two representations primarily depend on the provision of appropriate support.

Some Mathematical Examples of Emergent Intuitive Local Time Flow

After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds.

Developing a GeoGebra-Based Teaching Module on Quadrilateral Area and Perimeter to Enhance Seventh-Grade Students' Critical Thinking Skills

Objective: Seventh-grade students often need help to grasp the concepts of area and perimeter, particularly when applied to real-world problems requiring critical thinking. This difficulty highlights a need for engaging and effective teaching resources beyond traditional methods. Method: This study employed a Research and Development (R&amp;D) approach using the ADDIE model to design, develop, and evaluate a GeoGebra-based teaching module explicitly targeting the area and perimeter of quadrilaterals. The module, designed to foster critical thinking, underwent rigorous validation by media and material experts before being tested for practicality and effectiveness with seventh-grade students. Result: The GeoGebra-based teaching module was valid and practical, receiving high scores from expert evaluations and user feedback. More importantly, the module's implementation positively impacted students' critical thinking skills related to area and perimeter, as evidenced by significant improvement between their pre-intervention and post-intervention assessments. Novelty: This study provides valuable evidence for the efficacy of GeoGebra-based teaching modules in significantly improving critical thinking skills within a specific mathematical context. It addresses a critical gap in existing educational resources by offering a validated, practical, and effective tool that can be adapted to elevate mathematical understanding and cognitive skills in middle school education

Self-Regulation Profiles of Pre-Service Mathematics Teachers for Primary Education in Mathematical Problem-Solving Contexts

Self-regulation in mathematical problem solving is of particular importance in the initial training of Primary Education teachers as it can contribute to improving their own competence in mathematics and at the same time enable them to teach effectively and in a personalized way. With this idea in mind, this study identifies self-regulation profiles present in university students in mathematical problem-solving contexts based on data from the application of a validated scale on a sample of 402 pre-service teachers of primary education at University of the Basque Country. The cluster analysis carried out made it possible to identify and characterize three profiles, labelled according to the level of self-regulation represented by each of them (low, medium, high). In view of the results obtained, we perceive a need to design didactic proposals that allow these profiles to evolve from the lowest to the highest levels.

Domain effects on interpretations of general conditionals: The case of mathematics

Mathematics is often thought to have a unique association with certainty. The present study investigated a possible consequence of this association, namely that general conditionals are interpreted more deterministically in math than in other domains. To test this hypothesis, in two studies (Ns = 146 and 117), adults were presented general conditionals involving fictional categories in math and science and were asked to judge whether the conditionals were compatible with various frequencies of exceptions to them. Participants indicated that even rare exceptions (e.g., 1 exception per 99 confirming cases) would falsify a conditional (Studies 1 and 2), that a conditional could not be true and rare exceptions to it at the same time exist (Study 1), and that the truth of a conditional precluded the existence of even rare exceptions (Study 2), more when the conditionals involved math than science. The findings are consistent with the hypothesis that mathematical context is particularly likely to elicit deterministic interpretations of general conditionals. Implications of the findings for theories of conditional reasoning, and for individual differences in conditional reasoning, are discussed.

Mathematical formulation of the physical properties of Hydrocarbon Gases and the determination of these parameters in a Gas-Condensate field in Albania

Abstract The physic-chemical properties of fluids that saturate a bed have been followed throughout the course of resource exploitation through different analyses. Fluids released from the formation over time of exploitation change due to the change in the phase state of the fluids, their material balance and energy. Determining the type of reservoir fluid and its properties, including physical properties and chemical properties, play a key role in reservoir engineering for many decisions made in the development of an oil and gas field. In this manuscript, the properties of gases are analyzed step by step in both their physical and mathematical contexts, as well as that of gas. In this context, a gas-condensate field located in Albania was taken into consideration, determining its properties through the relevant correlations and the use of different graphs both for the measured value of the initial pressure of the formation and for the different values of the pressure taken into consideration.

Proses Berpikir Numerik dalam Menyelesaikan Soal Cerita Materi FPB dan KPK Kelas IV SD Berbasis Kearifan Lokal

The aim of this research is to analyze students' numerical thinking processes in solving FPB and KPK story problems based on local wisdom, as well as to identify effective learning strategies to improve students' numerical abilities. The research method used is a qualitative method with a descriptive approach. Data collection was carried out through numerical thinking process tests, interviews with students and teachers, and observation of learning implementation. The research subjects consisted of 29 class IV students at SDN Wonokromo III Surabaya who were grouped based on high, medium and low abilities. Data was analyzed through reduction, presentation and verification. The research results show that students with high ability (MY) can solve questions well, showing strong and consistent numerical abilities. Students with moderate abilities (RP) are able to solve several questions, but still need improvement in understanding the KPK concept. Low ability (MM) students experienced difficulty in all questions, indicating the need for deeper and more focused learning strategies. The integration of local context in mathematics learning has proven effective in increasing student engagement and understanding.

Short Note: Exploring Ideals in Graph Theory

We are re-evaluating the novel concept of 'ideal' within the context of graph theory. The concept of an 'ideal' is well-established in the fields of topology and algebra, having been extensively researched by numerous scholars due to its mathematical significance (see references) [2-6]. Tree-width is a well-known graph width parameter. In our discussion, we explore the relationship between tree-width and ideal. Overall, this exploration contributes to a deeper understanding of these concepts and their relevance in various mathematical and logical contexts.

Undergraduate Students’ Logical Consistency in Mathematical Thinking: Implications for Teaching and Learning

This paper introduces the concept of logical consistency in students’ thinking in mathematical contexts. We present the Logical in-Consistency (LinC) instrument as a valuable assessment tool designed to examine the prevalence and types of logical inconsistencies among undergraduate students’ evaluation of mathematical statements and accompanying arguments. Beyond its utility as an assessment tool, this paper also underscores the potential of the LinC instrument as an instructional aid, effective for identifying students who may inadvertently overlook logical contradictions within their mathematical assertions. Furthermore, this paper outlines a practical classroom activity based on the LinC instrument format. This activity is tailored to enhance students’ capacity for reasoning with logical consistency, specifically in proof-oriented mathematics courses. We hope this activity helps instructors address and mitigate logical inconsistencies in students’ thinking throughout their classroom activities.

Cultural Transitions in Mathematical Discourse: Unveiling Mathematical Writing Hurdles during the Rite of Passage

This qualitative case study delves into the intricate landscape of mathematical writing challenges faced by first-year university students undergoing the critical transition from school-level to university-level mathematical discourse. Conducted at a prominent South African university, the research, employing a purposive sampling technique, engaged thirty first-year mathematics students. Guided by the community of practice theory by Lave and Wenger, alongside van Gennep’s rite of passage analytical lens, the study sought answers to the question: What are the mathematical writing challenges encountered by first-year university students during their rite of passage period? Thematic analysis, informed by Adu's (2019) coding framework, was utilized to systematically examine common themes and patterns within the qualitative data. The findings illuminated key hurdles during this transitional phase, prominently including the inconsistency in working with mathematical notations, erroneous use of universal and existential quantifiers, and a notable confusion between the acts of illustrating and proving in mathematical contexts. In response to these challenges, the study advocates for the explicit incorporation of mathematical writing instruction to scaffold students during this rite of passage. Furthermore, it recommends a shift in the emphasis of first-year mathematics courses—suggesting a redirection from a content-centric approach to one that prioritizes the cultivation of students' new identities. This entails focused attention on teaching the customs, traditions, and adept ways of constructing and articulating mathematical proofs in the university context. The implications of this study extend beyond the immediate challenges identified, offering actionable recommendations to enhance the pedagogical strategies employed in the crucial transition period for first-year mathematics students.

What is the room for guessing in metacognition? Findings in mathematics problem solving based on gender differences

Introduction. Understanding the interaction of metacognitive strategies with guessing in problem-solving is a focus point in educational psychology, especially in mathematics education, especially with gender differences. These provide different nuances in understanding students' cognitive dynamics in guessing and metacognitive strategies. These interactions must be explained in depth to understand the unique phenomenon of second-guessing that sometimes arises in metacognitive activities based on students' gender. This research aims to reveal the process of metacognition and students' guessing thinking in solving mathematical problems. This research seeks to contribute to an empirical of metacognitive processes and their practical implications for educators and curriculum designers by uncovering these activities. Study participants and methods. The participants in this study were (30) 12th-grade high school students, consisting of (16) female students and (14) male students who had studied mathematics with statistical material. The primary method in this research is a case study with a qualitative descriptive approach. The indepth interview technique using the MAI protocol is based on the results of mathematical problem-solving to explore information about students' natural metacognitive processes according to what they think when working on problems. Semi-structured interviews used the Indonesian language to eliminate the influence of differences in regional language proficiency levels as much as possible. Results. The results of interviews on mathematical problem-solving tasks show a 'guessing' strategy in students' metacognitive activities. However, there is a striking difference between female and male students' use of the 'guessing' strategy when solving mathematical problems. Female students use the 'guessing' method more often than male students; even the 'guessing' method used by male students cannot solve the problem, thus changing the guessing strategy to estimation. This means that most of the students' answers used the guessing method. These findings underscore the dynamic nature of the cognitive processes used by students, revealing diverse interactions between metacognitive strategies and guessing during problem-solving efforts. Conclusion. This research shows that rather than standing in isolation, guessing has a place within metacognitive processes, with metacognitive regulation guiding and shaping the deliberate application of guessing in mathematical problem-solving contexts. This can be a factor for reconsidering the perceived dichotomy between precision-driven methodologies and intuitive guessing.

Level of Students' Mathematical Literacy Based on Cognitive Conceptual Tempo (Reflective-Impulsive)

This research aims to describe students' mathematical literacy abilities based on their cognitive conceptual tempo. The research method used in this study is a qualitative descriptive approach. The research was conducted at SMKN 1 Ngasem in class XI-2 Design, Modeling, and Building Information. Data collection techniques in this research involved the use of tests and interviews. The tests were divided into two parts: the Matching Familiar Figures Test (MFFT) to gather data on students' cognitive styles (conceptual tempo) and a mathematical literacy test to assess the level of mathematical literacy abilities of the students. The results of this research indicate that reflective subjects were able to achieve a mathematical literacy level of 3 for S1 and level 4 for S2. Reflective students fall within the medium scale. Meanwhile, impulsive subjects achieved mathematical literacy indicators at level 1 and level 4 for S3, and level 1, level 3, and level 4 for S4. Impulsive students also fall within the medium scale. Furthermore, future researchers can explore more extensive mathematical literacy content and contexts.

Improved algebraic inverter modeling for four-wire power flow optimization

This paper discusses the modeling of inverters used in distributed energy resources in steady state. Modeling the interaction between distribution grids and inverter-based resources is crucial to understand the consequences for the network’s operational and planning processes. This work highlights the limitations of existing models and emphasizes the need for better representations of inverters and their control laws in decision-making contexts. Improved steady-state grid-following and grid-forming inverter models are presented, including both three-leg and four-leg converter variants. The advantages of these improved models in mathematical optimization contexts are showcased by investigating the power quality improvement capabilities of the inverters. Numerical studies integrating the proposed inverter models in a four-wire unbalanced optimal power flow engine are presented, and trade-offs between modeling detail and computational intensity are illustrated.

A Critical Thinking Profile Based on Sex Differences: A Case Study on High School Students in Palu City, Indonesia

This study aims to describe students' critical thinking in solving derivative problems in terms of gender. The subjects in this study were male students, S1 and female students, S2 who had high mathematical abilities. This type of research is a descriptiv research with a qualitative approach. The research instrument consists of the main instrument (researchers themselves) and supporting instruments including the provision of written assignments and interviews. The results of this study indicate that S1 and S2 in solving derivative problems according to the criteria for critical thinking starting from the indicators of focus, reason, inference, situation, clarity and overview. However, S1 and S2 have different thinking activities, S1 completed the task with a more concise completion in a more logical way of thinking, while S2 completed the task with a longer completion with his verbal abilities with precision. and careful thinking. This shows that even though S1 and S2 have equal mathematical abilities, there are differences in thinking activities in solving problems. These findings are supported by written answer data and interview results. This research provides important insights into the influence of gender on critical thinking in mathematics contexts, which can be used as a basis for developing more effective and inclusive teaching strategies.

Fixed-Point Results of Generalized (ϕ,Ψ)-Contractive Mappings in Partially Ordered Controlled Metric Spaces with an Application to a System of Integral Equations

In this manuscript, we prove numerous results concerning fixed points, common fixed points, coincidence points, coupled coincidence points, and coupled common fixed points for (ϕ,Ψ)-contractive mappings in the framework of partially ordered controlled metric spaces. Our findings introduce a novel perspective on this mathematical context, and we illustrate the uniqueness of our findings through various explanatory examples. Also, we apply the main result to find the existence and uniqueness of the solution of the system of integral equations as an application.

Hidden dual mathematical symmetry in the genetic code

We describe the genetic code in terms of numbers that help us to find several dual symmetries. Our formulation can even be rewritten regarding the up-down and right-left dual concepts. We argue that our work may bring many topological tools to studying the DNA molecule, including the Grassmann-Plücker coordinates, which are important in mathematical and physical contexts.

Cryptographic Application of Elliptic Curve Generated through Centered Hexadecagonal Numbers

Background/Objectives: Elliptic Curve Cryptography (ECC) is a public-key encryption method that is similar to RSA. ECC uses the mathematical concept of elliptic curves to achieve the same level of security with significantly smaller keys, whereas RSA's security depends on large prime numbers. Elliptic curves and their applications in cryptography will be discussed in this paper. The elliptic curve is formed by the extension of a Diophantine pair of Centered Hexadecagonal numbers to a Diophantine triple with property D(8). Method: The Diffie–Hellman key exchange, named for Whitfield Diffie and Martin Hellman, was developed by Ralph Merkle and is a mathematical technique for safely transferring cryptographic keys over a public channel. Based on the Diffie–Hellman key exchange, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography. The generation of keys, encryption and decryption are the three main operations of the ElGamal cryptosystem. Findings: Given the relative modesty of our objectives, the fundamental algebraic and geometric characteristics of elliptic curves shall be delineated. Then the behaviour of elliptic curves modulo p: ultimately, there is a fairly strong analogy between the structure of the points on an elliptic curve modulo p and the integers modulo n will be studied. In the end, elliptic curve ElGamal encryption analogues of Diffie–Hellman key exchange will be created. Novelty: Elliptic curves are encountered in a multitude of mathematical contexts and have a varied and fascinating history. Elliptic curves are very significant in number theory and are a focus of much recent work. The earlier research works in Elliptic Curve Cryptography has concentrated on computer algorithms and pairing – based algorithms. In this paper, the concept of polygonal numbers and its extension from Diophantine pair to triples is encountered, thus forming an elliptic curve and perform the encryption-decryption process. MSC Classification Number: 11D09, 11D99,11T71,11G05. Keywords: Elliptic curves, Cryptography, Encryption, Decryption, Centered polygonal numbers