Mathematical Problem Research Articles

Causal Inference Thinking of Elementary School Teacher Professional Education Program Students in Determining Digital Strategies to Solve Mathematical Problems

This research aims to examine the causal inference thinking of PPG PGSD students in solving mathematical problems. This research uses a qualitative approach with a case study type. The subjects of this research were four pre-service PPG students from PGSD UM. The instruments for this research include researchers, indicator rubrics, validation sheets, researcher notes, mathematical problems, internet networks, laptops, and handphones. Research subjects as data sources produce data that is analyzed in the form of written answers and recorded interviews. The qualitative data analysis technique used in this research is the interactive model. The results of this research show that fast causal inference thinking is characterized by expressing familiarity with mathematical problem solving, expressing familiarity with accessing digital platforms to solve mathematical problems, accessing digital platforms with keywords, correlating the characteristics of information with specific digital platforms, and Accessing specific digital platforms spontaneously. Slow Causal Inference Thinking is characterized by accessing more than one digital platform to understand a piece of information and read the math problem text repeatedly. The digital platforms that the subjects considered to be a solution were the Google and YouTube platforms.

AI ChatBots’ solutions to mathematical problems in interactive e-textbooks: Affordances and constraints from the eyes of students and teachers

AI ChatBots’ solutions to mathematical problems in interactive e-textbooks: Affordances and constraints from the eyes of students and teachers

Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

Exploring nanoparticle dynamics in binary chemical reactions within magnetized porous media: a computational analysis

Artificial Neural Networks are incredibly efficient at handling complicated and nonlinear mathematical problems, making them very useful for tackling these challenges. Artificial neural networks offer a special computational architecture that is extremely valuable in disciplines like biotechnology, biological computing, and computational fluid dynamics. The present work investigates the applicability of back-propagation artificial neural networks in conjunction with the Levenberg-Marquardt algorithm for evaluating heat transmission in hybrid nanofluids. This work focuses on the computational analysis of a MgO + GO/EG hybrid nanofluid’s steady mixed convection flow over an exponentially stretched sheet, considering multiple slip boundary conditions, thermal conductivity, heat generation, and thermal radiation. A nonlinear system of ordinary differential equations is produced from the basic associated partial differential system by performing the proper exponential similarities modifications. For generating benchmark datasets, the resulting ordinary differential equations are processed employing the bvp4c method. Considering benchmark datasets set aside for training (70%), testing (15%), and validation (15%), the Levenberg-Marquardt algorithm, which employs back-propagation in artificial neural networks, is implemented. The accuracy of the suggested strategy for handling nonlinear problems is verified utilizing mean squared error, error histograms, and regression analysis, which are all used to evaluate the methodology. Outstanding agreement is seen when ANN outputs are compared to numerical results. The flow properties, including temperature, velocity, and concentration profiles, are shown graphically and numerically. For practical purposes, it is therefore essential to analyze the flow and heat transfer in hybrid nanofluids over exponentially extending and shrinking surfaces under mixed convection and heat source scenarios. Hybrid nanofluid problems have a wide range of practical and industrial applications, such as medication delivery, manufacturing, microelectronics, nuclear plant cooling, and marine engineering.

Large language models can help boost food production, but be mindful of their risks

Coverage of ChatGPT-style large language models (LLMs) in the media has focused on their eye-catching achievements, including solving advanced mathematical problems and reaching expert proficiency in medical examinations. But the gradual adoption of LLMs in agriculture, an industry which touches every human life, has received much less public scrutiny. In this short perspective, we examine risks and opportunities related to more widespread adoption of language models in food production systems. While LLMs can potentially enhance agricultural efficiency, drive innovation, and inform better policies, challenges like agricultural misinformation, collection of vast amounts of farmer data, and threats to agricultural jobs are important concerns. The rapid evolution of the LLM landscape underscores the need for agricultural policymakers to think carefully about frameworks and guidelines that ensure the responsible use of LLMs in food production before these technologies become so ingrained that policy intervention becomes challenging.

Exploring Sources of Bias to Improve the Universal Design of Mathematical Problem-Solving Tests

ABSTRACT Advancement of testing of mathematical problem-solving skills calls for open-ended, realistic tasks particularly susceptible to bias, compromising the score validity and fairness of tests. Informed by universal design principles, this study framed 360 prototype items developed for the Problem-solving Measures Grades 6–8 Computer Adaptive Tests as a bounded case to illuminate sources of bias in middle-school mathematics assessment. Experts’ shared understandings of potential sources of bias were collected using an iterative Delphi technique. Four themes in sources emerged from a thematic analysis of 201 sources of bias identified: assumed lived experiences, language use, item presentation, and stereotypes. These sources potentially placed subgroups of students from certain sociocultural backgrounds (e.g. culture, gender, race/ethnicity, socioeconomic status) at a disadvantage by challenging them in ways unrelated to mathematical problem solving. Recommendations for minimizing potential sources of bias in test development to achieve equitable K-12 mathematics problem-solving tests are discussed.

APOS Framework Didactize Equivalence Linear Simultaneous Equations in Senior High School Mathematics

Many students followed rote-learned procedural rules without thinking about the meaning of quadratic equations, and the Chief examiners’ reports attributed the trend to poor teaching methods. In certain instances, candidates cannot use even the conventional methods to factorize quadratics equations due to a lack of understanding of the zero-product property in graphing, factorizing, and completing the squares or quadratic formula. In the worst circumstances, the candidates failed to scaffold higher-order quadratic equations using the Conjugale and Equivalence Linear Simultaneous Equation methods. However, the action, process, object, and schema theory which came out of the constructivist learning theory and created by Dubinsky can be applied to teach mathematics. With the aid of the APOS framework, this study sought to digitize the Equivalence Linear Simultaneous Equations in senior high schools. In this mixed-method embedded design, there were 286 first-year students selected from one school. All the students received four phases of the APOS framework. The four phases were collected based on the Actions for Factorization, Processes for Quadratic Formula, Object for Conjugale, and Schema for ELSE method. The data, which was both qualitatively and quantitatively, was analyzed using IBM SPSS Statistics (Version 26) deterministic analytics software. The data collection covered four contact periods of a total duration of four hours. The results showed that students in the Actions and Processes were not statistically significant. However, the results were statistically significant at the Object and Schema phases. It was concluded that students’ learning through the APOS framework improved their academic performance. The positive effect was the triumphant mental constructions involving encapsulation and interiorization in the conjugale and ELSE methods. It was, therefore, recommended that the framework be promoted to find solutions to many other complex mathematics problems.

A clash of epistemic tools: computer programming and paper-and-pencil in secondary school mathematics problem solving

A clash of epistemic tools: computer programming and paper-and-pencil in secondary school mathematics problem solving

A new conceptual and analytic framework for collective engagement in mathematics

This paper proposes a new social construct and conceptual-analytic framework for the study of group engagement in mathematics, which we call collective engagement structures. This new framework is offered to address research and practice-identified needs to better understand how individual student dispositions and preferences might interact with group processes to inform student engagement and learning. We demonstrate the collective engagement structures framework using a case study analysis of three middle school students working in a group on a complex mathematics problem. Our analysis details how three students’ individual engagement structures and positional influence interact to form two collective engagement structures: Follow The Leader and Let's Figure This Out. Our findings indicate that students’ collective engagement may present itself in identifiable and patterned ways, providing fresh theoretical and practical insight into how to better structure group-centered instruction.

Symbolic equation solving via reinforcement learning

Machine-learning methods are rapidly being adopted in a wide variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). These rules have usually needed to be discovered and subsequently programmed by humans. Efforts to automate this process by machine-learning approaches are faced with challenges like the singular nature of solutions to mathematical problems, when approximations are unacceptable, as well as hallucination effects leading to flawed reasoning. We propose a novel deep-learning interface involving a reinforcement-learning agent that operates a symbolic stack calculator to explore mathematical relations. By construction, this system is capable of exact transformations and immune to hallucination. Using the paradigmatic example of solving linear equations in symbolic form, we demonstrate how our reinforcement-learning agent autonomously discovers elementary transformation rules and step-by-step solutions.

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

A New Approximation for the Perimeter of an Ellipse

We consider the problem of approximating the perimeter of an ellipse, for which there is no known finite formula, in the context of high-precision performance. Ellipses are broadly used in many fields, like astronomy, manufacturing, medical imaging, and geophysics. They are applied on large and nanoscales, and while numerical integration can be used to obtain precision measurements, having a finite formula can be used for modeling. We propose an iterative symbolic regression approach, utilizing the pioneering work of Ramanujan’s second approximation introduced in 1914 and a known Padé approximation, leading to good results for both low and high eccentricities. Our proposed model is also compared with a very comprehensive historical collection of different approximations collated by Stanislav Sýkora. Compared with the best-known approximations in this centuries-old mathematical problem, our proposed model performs at both extremities while remaining consistent in mid-range eccentricities, whereas existing models excel only at one extremity.

Progress and Applications of Fully Homomorphic Encryption

In the era of digital transformation, marked by the widespread adoption of cloud-based technologies such as cloud computing and storage, ensuring data security and protecting user privacy have emerged as critical industry concerns. Fully Homomorphic Encryption (FHE) addresses these issues by allowing various computations to be performed on ciphertexts without the need for decryption, thus safeguarding data security and privacy. This paper explores the mathematical and algorithmic foundations essential for understanding FHE. It provides a detailed analysis of the advancements in FHE schemes over the past decade, focusing on various mathematical problems, including ideal lattices and integers. The discussion extends to three practical applications: cloud computing, machine learning, and electronic voting, highlighting the progress and exploring potential future research avenues and developmental strategies in the field of FHE. This comprehensive review not only underscores the significance of FHE in enhancing data security but also charts a path for its future exploration and integration into emerging technologies.

Проблеми вивчення математики в межах реалізації концепції Нової Української Школи

In this publication, we pay attention to the problems of learning mathematics in the context of the implementation of the concept of the New Ukrainian School. During preparation for publication, the following work was done. Conversations with mathematics teachers and their questionnaires were conducted. A survey of students was conducted. Sources of information in terms of informing teachers about the implementation of the concept of the New Ukrainian School and methods of teaching mathematics at the National University of Ukraine were analyzed. A comparative analysis of educational achievements of seventh-grade students who studied in experimental and control classes was made. The tasks of the final exams of high school students abroad and our graduates were analyzed.

Реализация познавательного и воспитательного потенциала рассказов М.М. Зощенко на уроках литературного чтения в начальных классах

The authors consider the possibilities of using M.M.Zoshchenko's stories in the process of forming the life experience of primary school students, reveal the essence of an integrated approach to the organization of ideological and thematic analysis of the text, methodological features of composing a mathematical problem based on the storyline of a work of art.

Evolution of the Four Fours Problem

Summary Besides being popular and entertaining, recreational mathematics problems are often of historical interest. We highlight the origin and evolution of one such simply-stated yet deep problem. For nearly three centuries the four fours problem has been generalized and expanded, providing hours of mathematical activity for enthusiasts as well as inspiring some intriguing discoveries.

<span><strong>Enhancing students’ mathematical competence in learning spatial geometry in grade 11<sup>th</sup> through practical and experiential activities</strong></span>

This study embarks on enriching 11th-grade students' mathematical competencies, particularly in spatial geometry, through experiential learning. Specifically, the study focused on the practical concepts of perpendicular and parallel relationships in space. Following the qualitative approach, we collected and analyzed the data of the students' behaviors observed within a teaching scenario, which was specifically designed for them to apply their knowledge to the task "determining the classroom's center". The scenario involved three phases: defining the "classroom center," problem-solving in the classroom, and group presentations analyzing each other's methods. The results showed that all student groups successfully solved the task, which was implemented in a serious way using their mathematical understanding in a real-life context. Thus, students enhanced their two mathematical competencies, including mathematical problem handling competency (posing and solving mathematical problems) and mathematical communication competency (communicating in, with and about mathematics). This study underscores the effectiveness of practical and experiential activities in enhancing mathematical competencies among high school students, specifically within spatial geometry. It highlights the benefits of applying mathematical concepts to practical situations, promoting more profound understanding and student engagement.

<span>Enhancing student's mathematical competency in learning the graph and properties of the sine function by Google Sheets app on smartphone</span>

This study explores the integration of the Google Sheets app on smartphones to enhance students' mathematical competency in understanding the graph and properties of the sine function. The paper was an exploratory study involving 32 11th-grade students. The teaching intervention was structured into four phases: familiarizing with the Google Sheets app, creating the sine function value table, sketching the function graph, and analyzing the graph to deduce the sine function's properties. Data were collected through worksheets and audio -video recordings of group work, focusing on manifestations of mathematical competencies. The findings indicate that students completed the tasks and demonstrated significant competencies in mathematical reasoning and using digital tools. Students were able to identify and solve mathematical problems, utilize Google Sheets effectively, and use logical reasoning to enhance their understanding of the sine function. The study concludes that integrating Google Sheets into mathematics education not only aids in visualizing and comprehending mathematical concepts but also promotes interactive and collaborative learning environments, ultimately fostering a deeper understanding and proficiency in mathematics.

PENINGKATAN KEMAMPUAN PEMECAHAN MASALAH MATEMATIS SISWA KELAS III DENGAN MEDIA POWER POINT INTERAKTIF MACAN (MATERI PECAHAN)

This research aims to improve the low level of mathematical problem solving regarding fractions using Interactive Power Point media for class III students at SD 2 Panjunan. Data collection techniques in this research used observation, interviews and documentation techniques, tests. The instruments of this research are interview sheets, observation sheets, and test question sheets. The method in this research uses the classroom action research method. The subjects in this research were class III students at SD 2 Panjunan with a total of 26 students. This research uses the Kemmis and Mc Taggart model research design, where each cycle or round consists of several stages which include, planning, action, observation, and reflection. The research results in Cycle I showed an average of 42% where the level of learning success was still low, and in cycle II after using interactive power point media it showed an average of 88% with a very good category, this shows that the application of interactive Power point in the learning process with fraction material can improve mathematics learning outcomes for class III students at SD 2 Panjunan. The conclusion of this research is that Interactive Power Point media can improve students’ mathematical problem solving abilities in fraction material.

Impact of induced electric field on the motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time

The classical and quantum motion of a charged particle in a uniform constant magnetic field is widely studied in the scientific literature since it is a problem that allows an exact mathematical analytical solution. On the other hand the cases of a non-uniform, time-dependent or both non-uniform and time-dependent magnetic field lead to mathematical problems that do not have general exact analytical solutions except for a few very specific cases. We present in this work, the results for the two-dimensional motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time for a certain finite interval and then stabilizes to a constant value afterwards. The work draws attention to certain interesting features observed for the two-dimensional motion of the charged particle as a result of the interplay between the effects of an induced electric field and the time varying magnetic field.