Journal of Hunan University Natural Sciences

Thinking about the introduction of algebra in the first years of school implies, among other aspects, a transformation of the school mathematics curriculum and a change in the limited and traditional conceptions that are held about its teaching. This article characterizes the conceptions of a group of primary school teachers from public institutions of the Buenaventura District about the introduction of algebra from an early age based on the perspective of curricular change called early algebra. The research was of a qualitative nature through the hermeneutic phenomenological approach with which the teachers’ conceptions were described from their classroom and personal experiences in teaching algebra. The analysis of the teachers’ conceptions provided fundamental didactic elements about the advantages of reflecting on the importance of transforming the traditional practices of algebra teaching at school. It was found that the conception that one has regarding the teaching of mathematics and the disciplinary training that teachers have play an essential role in the design and implementation of tasks that promote the development of algebraic thinking in students from the first years of school.

Keywords: conceptions, teachers, primary education, early algebra.

https://doi.org/10.55463/issn.1674-2974.50.7.25

This study is situated within the framework of realistic mathematics education (RME) and presents a classroom proposal aimed at illuminating and modeling everyday periodic phenomena. Emphasis is placed on recognizing periodic functions as fundamental tools for characterizing the sine and cosine trigonometric functions. The investigation uses the concept of mathematization levels to delineate the transition from informal to formal knowledge through real-world problem-solving using the GeoGebra platform. This research was conducted with tenth-grade students at the San Vicente Educational Institution in the Buenaventura District. The outcomes reveal that students engaged in activities, and subsequent analyses demonstrate a nuanced understanding of periodic phenomena within realistic contexts. This comprehension encompasses proficiency in various representation systems, including tabular, graphical, and algebraic methods. Moreover, the process of progressing from informal to formal knowledge is marked by incremental levels of mathematization, encompassing both horizontal and vertical processes. The main focus of this research is to characterize the process of mathematization within a realistic context, particularly concerning the sine and cosine trigonometric functions, and evaluate the strengths and limitations of this approach. This study underscores the potential of RME for elucidating intricate mathematical modeling processes within real-world contexts. These insights underscore the value of integrating authentic scenarios to foster a comprehensive understanding of mathematical concepts. Challenges are also acknowledged, particularly in achieving the formalization of generalized models. Thus, this research contributes to discussions on effective mathematics education by showcasing the transformative potential of contextualized pedagogical strategies.

Keywords: realistic mathematics education, trigonometric functions, levels of mathematization, context, levels of understanding.

https://doi.org/10.55463/issn.1674-2974.50.7.24

The main objective of this paper is to present an approximate numerical solution of the Klein-Gordon equation. For this purpose, the basic concepts associated with the numerical method used to obtain the approximate solution will be presented initially. In our case, we considered as a hypothesis the fact that using the finite element method and in particular the Galerkin method, it is possible to perform a superb numerical approximation of the solution of the Klein-Gordon equation. For this purpose, we will present some of the solutions that have been obtained by employing the modified Adomian decomposition method, via a transformation and Exp-function method and comparison with Adomian's, and using the variational method. This will allow better understanding of the effectiveness of the finite element method because the solutions will be represented in space and time. The novelty results from the fact of being able to take the solutions known as initial and boundary conditions and observe that effectively at great lengths the solution stabilizes at zero and that over time the approximate solutions stabilize. Finally, the simulations obtained by the finite element method indicate that the Klein-Gordon equation is damped nonlinear, its solution stabilizes at  and  and for short times, the function remains close to zero.

Keywords: approximate solution, finite element method, nonlinear equation, the Klein-Gordon equation, the Galerkin solution.

https://doi.org/10.55463/issn.1674-2974.50.7.23

In this study, we enter the field of nonlinear partial differential equations (PDEs) and propose a novel approach to their solution. We employ the iterative Adomian decomposition method (ADM) to address a distinct subclass of nonlinear PDEs. The main objective of this paper is to evaluate the efficiency and reliability of the ADM in handling complex mathematical models. To this end, we explore the convergence characteristics of the method when applied to the aforementioned class of equations. Our analysis sheds light on the intricate interplay between the iterative steps of the method and the underlying nonlinear dynamics, providing valuable insights into the behavior of the solution process. Through this investigation, we find noise terms in the context of inhomogeneous equations. We reveal that the presence of noise is intrinsically linked to the inherent heterogeneity of the equations, and its impact on convergence behavior is systematically elucidated. To validate the robustness and accuracy of the proposed method, we present a series of numerical results. These results not only underscore the reliability of the method in producing accurate solutions but also highlight its potential to overcome computational challenges associated with nonlinear PDEs. In summary, our study contributes to the broader picture of nonlinear PDE solvers by showing the potential of the Adomian decomposition method, especially in models involving noise terms and inhomogeneous equations.  The efficiency of this method is illustrated by investigating the convergence results for this equation. We show that the noise terms are conditional for nonhomogeneous equations, and the numerical results demonstrate the reliability and accuracy of the ADM.

Keywords: the Adomian method, noise terms, iterative method.

https://doi.org/10.55463/issn.1674-2974.50.7.22

Risk management is frequently cited as one of the most important focuses for executives working in the financial industry. As a result, a great number of books and articles on risk management have been written. Regarding keeping track of a company’s risk profile and identifying opportunities to improve risk management practices, the importance of internal auditing in today’s rapidly evolving business climate cannot be underestimated. During the COVID-19 pandemic, this study investigated the impact of internal auditing and risk management on commercial banks listed on the Amman Stock Exchange. Only 198 of the 315 questionnaires that were sent out through email to various financial managers and accountants could be considered legitimate for the study. In addition, the data were evaluated using partial least square structural equation modeling (PLS-SEM). The findings indicated that an internal audit, an internal auditor, and the additional value of an internal audit are statistically substantially associated with risk management. This finding is consistent with the theory and our expectations. On the basis of these findings, the study recommended that more attention be paid to internal audit because of its importance in supporting the function of risk management through relevant specialized training courses and the provision of support to internal audit management in terms of trained and qualified human resources, both academically and practically, particularly during the COVID-19 pandemic. This recommendation was made because of the importance of internal audit in supporting the function of risk management through relevant specialized training courses.

Keywords: commercial banks, COVID-19 pandemic, partial least square structural equation modeling, risk management.

https://doi.org/10.55463/issn.1674-2974.50.9.12