Journal of Mathematics and Statistics Studies
Journal of Mathematics and Statistics Studies (ISSN: 2709-4200) is a double-blind peer-reviewed, open-access monthly journal published by Al-Kindi Center for Research and Development. The journal is led by expert international editorial board members who take an objective approach to peer review, ensuring each research paper is reviewed and evaluated on its own scholarly merits and research integrity. Publishing your work in the Journal of Mathematics and Statistics Studiesmeans you will receive prompt and reliable publishing and a global platform for your research to reach its full potential. The journal is published in both online and print versions.
Al-Kindi Center for Research and Development.
Analysis of the Value of Production of Milkfish
Milkfish is important in the Philippine aquaculture, yet limited research has explored the economic value of its production and the factors that influence it. This study analyzes the trend in milkfish production value from 2008 to 2022 and examines its relationships with volume production, retail pricing, and export production. Using regression analysis, the results show a significant upward trend in production value over 15 years, with volume and retail price identified as primary drivers. Although export production showed a positive relationship in simple analysis, it was not significant in the combined model. The findings highlight that strengthening local production and retail strategies may be more effective than relying on exports to enhance industry value. Recommendations include targeted support for enhancing production efficiency, stabilizing pricing, and developing the domestic market to promote sustainable growth within the milkfish industry.
Copyright (c) 2026 Journal of Mathematics and Statistics Studies
Atom-Bond Connectivity Index on a Graph of Finite Free Semilattices
Let SLn be the finite free semilattice on Xn={1,2,..,n} and Γn be the zero-divisor graph of SLn . In this paper, we find atom-bond connectivity index, Randic index and harmonic index of Γn for n≥3
Finite free semilattice, ABC index, Randić index, Harmonic index
Stability and Numerical Simulation of 2D Fractional Pseudo-Hyperbolic Partial Differential Equations
This study investigates the numerical solution and stability analysis of two-dimensional fractional pseudo-hyperbolic partial differential equations. Recent advancements in fractional calculus have demonstrated its efficacy in modelling complex physical phenomena, particularly through generalized derivatives that capture memory and non-local effects. The primary contribution of this work is the development and analysis of finite difference schemes for solving initial boundary value problems associated with two dimensional fractional pseudo-hyperbolic equations. We propose first-order accurate difference schemes that incorporate Caputo fractional derivatives with order 1 . Comprehensive error analysis is conducted through numerical simulations, and Von Neumann stability analysis establishes the conditional stability of the proposed numerical method. Experimental validation confirms the accuracy and reliability of the proposed schemes against exact solutions.
Pseudo hyperbolic partial differential equation
Jackson-Steklov Approximation Theory in Weighted Sobolev Spaces with Operations Research Applications
This paper studies the theoretical framework that brings together the Jackson-Steklov theory of approximation in weighted Sobolev spaces and optimization techniques used in Operations Research (OR). In this work, several new direct Jackson-type theorems are established, and a detailed analysis of Steklov eigenvalues in weighted spaces is carried out, along with error bounds and conditioning results for polynomial approximations. The results of this paper are intended for use in Operations Research, where stochastic programming with heavy-tailed distributions, PDE constrained optimization with irregular domains, and regularized inverse problems are considered. In this paper, key inequalities are established, which relate the approximation error with the modulus of continuity, as well as equivalence results for the K-functionals and the continuity module, and the role of Steklov eigenvalues in conditioning optimization problems discretized in weighted spaces. Numerical examples on standard problems in Operations Research support the results of this paper and show significant improvements in accuracy, conditioning, and efficiency over existing methods. This paper brings together abstract approximation theory and Operations Research optimization, which offers a firm foundation for the development of sophisticated computational methods in weighted function spaces.
Weighted Sobolev spaces; operations research; Jackson theorem; Steklov eigenvalues; optimization; polynomial approximation
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Some Topological Indices on a Graph of Finite Free Semilattices
Let SLn be the finite free semilattice on Xn={1,2,...,n} and Γn be the zero-divisor graph of SLn. In this paper, we find the first Zagreb index, second Zagreb index, and Sombor index of Γn for n≥3
Finite free semilattice, Zagreb index, Sombor index.
Forecasting Theobroma Cacao Value of Production in the Philippines Using Time Series Modeling Approach
Cacao production contributes significantly to the Philippine agricultural sector, and having reliable forecasts of its value is necessary to guide policy decisions, investments, and industry strategies. This study aimed to analyze historical trends in the cacao value of production from 2000 to 2025 and forecast production value for 2027 by applying time series models. Using data from the Philippine Statistics Authority, several modeling techniques were tested, including linear, exponential, polynomial (quadratic to sextic), moving average, exponential smoothing, and autoregression. The models were then assessed using the coefficient of determination (R²) and standard error (SE) to determine which produced the most reliable forecasts. The quintic model emerged as the best fit since it provided the most accurate forecasted values and showed strong explanatory power. For 2027, the projected value is 5,070.934 million pesos. These findings give policymakers, agricultural agencies, and those in the cacao industry a reliable reference for anticipating production trends. Based on the results, the study recommends using the quintic model for planning, resource allocation, and program development to help sustain growth and improve the resilience of the cacao sector.
agricultural forecasting; cacao production; mathematical modeling; nonlinear trends; time series analysis; Philippines
Line Graphs of Zero-Divisor Graphs of Finite Free Semilattices
Let SLn be the finite free semilattice on xn={1,2,...,n} and Γn be the zero-divisor graph of SLn. Moreover, let L(Γn) be the line graph of Γn. In this paper, we find the cardinality of the vertex set of L(Γn), diameter, maximum degree, minimum degree, girth, clique number, chromatic number, and degrees of all vertices in L(Γn)for n≥3.
Finite free semilattice, line graph, diameter, clique number
Forecasting CO₂ Emissions Per Capita in the Philippines Using Time Series Analysis
Forecasting carbon dioxide (CO₂) emissions per capita in the Philippines is crucial for evaluating individual contributions and informing targeted policies. It also helps anticipate potential environmental and social impacts while facilitating data-driven decision-making. This study analyzes the trends in CO₂ emissions per capita in the Philippines from 1979 to 2024. Various time-series models were constructed to select the best-fit model and to predict per capita emissions for 2027. The data were obtained from the website of Our World in Data. The results revealed a generally upward trend throughout the study period, despite short-term fluctuations and temporary decreases. Moreover, the first-order autoregressive (AR(1)) model emerged as the best fit, projecting an increase in per capita emissions to approximately 1.689 tonnes per person by 2027. These findings provide insight into potential future emission trajectories at the individual level and underscore the need for targeted interventions to effectively manage per capita emissions.
Orthogonal Jordan Triple Higher Reverse Left (resp.Right) Centralizer on Semiprime T-Rings
The goal of this job offer the entrance of orthogonal Jordan triple higher reverse left (resp.right) centralizers a -ring Ň (mention -ring Ň by Ň ) & examine this concept
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Enhancing Students’ Self-Efficacy and Mathematics Achievement Through Competency Sub-Tasking
This study assessed Grade 11 students’ self-efficacy and Mathematics achievement when exposed to competency sub-tasking at two identified public high schools in Cebu during the 2022-2023 school year, as the basis for the learning enhancement plan. A descriptive-correlational method was employed, with 154 Grade 11 students chosen as respondents. The researchers used a survey questionnaire consisting of three parts to gather information. Part I provided information on students’ profiles. Part II questions focused on students' self-efficacy in learning mathematics when exposed to competency sub-tasking. Part III consists of individual students’ assessment scores. The data were analyzed using percentages, frequencies, means, standard deviations, and Pearson's r. Based on the findings, most students were 17 years old, most parents were high school graduates, and most students had low combined family monthly incomes. As revealed by the study, the student respondents' performance on the mathematics achievement test using the competency sub-tasking strategy was very satisfactory (at a proficient level). Regarding students’ self-efficacy, the respondents agreed, indicating that most of the time the performance indicators were true. It has been found that there is a significant relationship between students’ self-efficacy and academic performance when they are exposed to competency sub-tasking. It is recommended that the learning enhancement plan be implemented to enhance achievement in general mathematics competencies in modular-based instruction.
Teaching Mathematics, students’ self-efficacy, competency sub-tasking performance, descriptive-correlation, mathematics, Cebu
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Strategies and Challenges in Implementing Differentiated Instruction Among High School Mathematics Teachers
Differentiated Instruction (DI) is widely recognized as an effective approach for addressing learner diversity in Mathematics classrooms; however, its consistent implementation remains challenging for many teachers. This study examined the extent to which high school Mathematics teachers practice differentiated instruction strategies and the challenges they encounter in implementing them. Quantitative findings revealed that teachers demonstrated a high extent of practice in differentiated instruction across content, process, product, and learning environment. Despite this, challenges related to time constraints, large class sizes, limited instructional resources, diverse learner needs, and workload demands were also reported. Statistical analysis indicated no significant relationship between the extent of differentiated instruction practice and the challenges encountered. The findings suggest that while teachers actively employ differentiated instruction strategies, systemic and contextual barriers continue to influence their classroom practices. These results highlight the need for teachers to employ various strategies to address these difficulties, including flexible grouping, formative assessment, peer collaboration, contextualized activities, and continuous monitoring, sustained instructional support and professional development to strengthen the effective implementation of differentiated instruction in high school Mathematics classrooms.
Differentiated Instruction; Diverse Learners; Mathematics Education; Teaching Strategies
Nonparametric Density Estimation in Survey Sampling
Nonparametric methods for estimating probability densities are popular because they provide flexible tools for exploratory analysis, model checking, and inference when little is known about the underlying distributional form. In the context of sample surveys where data arise from complex designs involving stratification, clustering, and unequal inclusion probabilities, naive application of standard nonparametric estimators can, however, produce biased and inconsistent results. This paper reviews foundations of nonparametric density estimation and use of kernel and local polynomial methods and discusses their adaptation to design-based and model-based survey frameworks. Practical implementation issues involving bandwidth selection, boundary correction, and computational considerations are made. Throughout, emphasis is placed on methods that respect survey design information, and on trade-offs between design-based validity and model-based efficiency. The paper concludes with recommendations for practice and directions for future research.
nonparametric density estimation, kernel density estimator, survey sampling, design-based inference, weighted estimators, variance estimation, bandwidth selection.
A Least Squares Hyperbolic Transformation to Moderate School Assessments
This research develops a new method to mathematically moderate school assessments to make them comparable across schools in a large-scale public examination system. In many systems the school assessment is partly used to determine the final mark in a course. A hyperbola was chosen for this method as it has two important features. First, if chosen from the second or fourth quadrants of the Cartesian plane it is always monotonic increasing. Second, it gives a curvilinear transformation which helps to overcome differences in skew between a school’s raw assessments and the school’s exam marks. If the skew difference is large it can disadvantage the top students in a linear moderation, giving them moderated assessments below their exam marks. A hyperbola is closely fitted to the moderation criterion (the sorted exam marks) using a least squares process. The moderation equation itself is simple, with three constants, but their calculation is more complex. However a central education authority should have the computing resources to calculate these constants for each school group. This new method provides another option for school systems to consider in a high-stakes large-scale public examinations environment.
school assessments, moderation method, least squares, external examinations, high stakes, large scale
NEW MULTIPLE SOLUTIONS FOR SOME PERIODIC BOUNDARY VALUE PROBLEMS WITH ψ-LAPLACIAN
We study the existence of multiple solutions of the quasilinear equation (ψ(u'(t)))'= f(t,u(t),u'(t)), t∈[0,T] submitted to periodic boundary conditions, where ψ:]-a,a[→R, with 0<a < +∞, is an increasing homeomorphism such that ψ(0)=0. Combining some sign conditions and lower and upper solutions method, we obtain existence of two or Three solutions.
ψ- Laplacian; L^1-Carathéodory function; nonlinear Neumann-Steklov problem; Periodic problem; Lower and upper-solutions; sign conditions.
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Influence of Administrative and Professional Support on Teaching Attitudes in Mathematics Instruction
This study investigated the influence of administrative and professional support on the teaching attitudes of junior high school mathematics teachers. The inquiry was undertaken in response to persistent challenges in mathematics performance and the varied implementation of assessment initiatives, particularly the Rapid Mathematics Assessment (RMA). Employing a descriptive-correlational research design, quantitative data were collected from junior high school mathematics teachers through a researcher-made questionnaire. Descriptive statistical tools, including frequency, percentage, weighted mean, and standard deviation, were used to determine the level of administrative support, professional support, and teaching attitudes. The Pearson Product-Moment Correlation Coefficient was utilized to examine relationships among the key variables. Results revealed that the level of administrative support and professional support was generally high, while teaching attitudes toward mathematics instruction were positive. Further analysis showed a significant relationship between administrative support and teaching attitudes, as well as between professional support and teaching attitudes. Teachers who reported receiving regular guidance, constructive feedback, and sufficient instructional resources demonstrated more favorable attitudes toward mathematics teaching. In addition, professional development activities that were relevant, sustained, and aligned with classroom realities were associated with increased confidence and openness to assessment innovations such as RMA. The study concludes that both administrative and professional support significantly influence the teaching attitudes of junior high school mathematics teachers. Strengthening leadership practices and professional learning systems is recommended to enhance instructional effectiveness and improve mathematics learning outcomes.
Mathematics instruction, teacher attitudes, administrative support, professional development
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Forecasting Philippine Rice Prices: Comparison of Traditional Time Series and Machine Learning Models
The Philippine rice market is characterized by high volatility and a critical impact on national food security, necessitating the use of accurate forecasting tools for price monitoring and risk mitigation. The primary objective of this study was to determine the optimal forecasting methodology for the nominal wholesale price of regular-milled rice. The analysis utilized 430 monthly observations (January 1990 – October 2025), partitioned into an 80% training set (344 data points) and a 20% out-of-sample test set (86 data points). Eight time series models including traditional methods: Seasonal Naïve, ETS, ARIMA, TBATS, Theta, and Prophet, and machine learning algorithms: Random Forest and XGBoost were evaluated. Performance was assessed using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE), supplemented by Ljung–Box and Lilliefors tests for residual diagnostics. Evaluation revealed that the Random Forest model achieved the best predictive accuracy, confirming the superior capability of non-linear models to capture the volatile patterns present in the wholesale price data. Residual diagnostics indicated a fundamental trade-off between structural adequacy and predictive accuracy. The final projection forecasts that wholesale prices will stabilize within the Php 34.00/kg to Php 35.50/kg range through 2027. The general price stability predicted for the next two years suggests policy focus may prioritize long-term supply-side improvements rather than short-term demand controls, unless external shocks occur.
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The Susceptible-Infected-Recovered (SIR) model, first introduced by Kermack and McKendrick in 1927, represents a fundamental framework for understanding epidemic dynamics and has evolved into one of the most influential mathematical models in epidemiology. This article examines the foundations of the SIR model, including its mathematical formulation, underlying assumptions, and parameter estimation techniques. The review also explores diverse applications of the SIR model across multiple domains, including traditional epidemiological studies, economic modeling, social network analysis, and emerging areas such as cybersecurity and marketing. Through systematic analysis of recent literature, this work demonstrates the model's adaptability and continued relevance in addressing contemporary challenges, from COVID-19 pandemic modeling to understanding information diffusion in digital networks. The findings highlight both the strengths and limitations of the SIR framework while identifying future research directions for model enhancement and application.
SIR model, epidemiology, mathematical modeling, infection dynamics, parameter estimation, COVID-19, stochastic models
Time Series Analysis to Evaluate the Impact of Foreign Direct Investment on Economic Growth in Sudan
The study aimed to evaluate how foreign direct investment (FDI) impacts economic growth in Sudan. A standard model was designed to analyze the relationship between the study variables from 2000 to 2022. The primary problem of the study is whether Sudan can offer the required standards, incentives, and privileges to attract foreign investors. In the theoretical aspect, the study utilized a descriptive analytical approach, but in the practical aspect, it employed a statistical method grounded on econometric tools in the analysis, applying the Autoregressive Distributed Lag (ARDL) model. The relevant data were acquired from the Sudan Central Bureau of Statistics for the study period. The study hypothesized that there is a statistically significant correlation between FDI and economic growth in Sudan. The findings determined a statistically significant long-term association between FDI and economic growth. The study recommended enhancement of an environment favorable to both foreign and domestic investment.
Economic growth, foreign direct investment, capital formation, GDP, trade openness.
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Mathematical Modeling and Control of Drone Swarm Coordination Using PD Approach
This research paper introduces a mathematical model and control framework for coordinating multiple drones (a swarm) using a Proportional–Derivative (PD) control approach grounded in graph theory. Each drone functions as an independent agent, communicating only with its nearby neighbors. This local interaction creates a decentralized and scalable system that eliminates the need for a central controller. The inter-drone connectivity and interactions are represented through the Laplacian matrix, while eigenvalue analysis helps in designing the control gains to ensure smooth and stable motion.The model is formulated using Newton’s equations of motion, expressed through differential equations that describe each drone’s dynamic behavior. The proposed approach is tested on a five-drone ring formation, demonstrating that the drones can maintain formation, avoid collisions, and reach the target position with stable convergence.
Mathematical Modeling; Control of Drone Swarm Coordination; PD Approach
Mathematical and AI-Blockchain Integrated Framework for Strengthening Cybersecurity in National Critical Infrastructure
This article examines the intersection between mathematical modeling, artificial intelligence (AI), and blockchain technology as a way of strengthening cybersecurity in national critical infrastructures (NCI). The increasing frequency and sophistication of cyber threats against vital infrastructures, such as electrical grids, healthcare information networks, and transportation infrastructures, creates the need to develop some innovative protective mechanisms. To solve these concerns, in the article the authors propose a hybrid framework that combines the AI-driven predictive analysis and the decentralized ledger capabilities of blockchain technology. Decentralized ledger technology (DLT) forms the backbone of secure and tamper-proof data management and machine learning algorithms are used to identify and predict emerging threats in real-time. By combining the immutability of the blockchain technology and the adaptive analytical capabilities of AI, this framework aims to improve the integrity of the data, maintain the privacy of it, and provide a fast-responding mechanism in the NCI environments. The paper outlines possible applications, lists the benefits that come with such applications, and addresses challenges inherent in the implementation of such an integrated system in order to provide a blueprint for future scholarly investigation and practical implementation.
AI, Blockchain, Cybersecurity, Critical Infrastructure, Mathematical Framework, Decentralized Ledger Technology, Machine Learning
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A Comparative Statistical Analysis of Machine Learning Regression Models for Economic Indicator Forecasting
Accurate forecasting of economic indicators is essential for informed policy-making and strategic financial planning. This study presents a comprehensive comparative statistical analysis of multiple machine learning regression models—including linear regression, ridge regression, lasso regression, support vector regression, and random forest regression—for predicting key economic indicators such as GDP growth, unemployment rate, and inflation. Using a simulated dataset of 1000 samples with multiple economic features, each model’s performance was evaluated using mean squared error (MSE), mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination ($R^2$). Results indicate that ensemble methods such as random forest provide superior predictive accuracy, while regularized models outperform ordinary linear regression. Visualizations of predicted versus actual values, residual analyses, and statistical comparisons are provided to support the findings. Implications for model selection and economic forecasting are discussed.
A Bazilevic function related to the class of meromorphic functions
The goal of this article is to further geometric function theory by focusing on meromorphic functions and relying on the Bazilevic function, which represents the interactions of analytic functions. It focuses on presenting and investigating a differential operator for meromorphic functions. Using this operator, create meromorphic Bazilevic functions, a new subclass in the punctured unit disk This text also greatly expands knowledge and understanding of the meromorphic function to fall into this new subclass.
Bazilevic functions, Meromorphic functions, Diﬀerential operator
Leveraging Artificial Intelligence to Recommend Personalized Learning Pathways: A Case Study on Secondary Education Students
This study aims to develop an explainable artificial intelligence (XAI)–based educational recommendation system tailored for secondary school students. The system integrates classification and regression models to suggest personalized learning paths while estimating the confidence level of each recommendation. The dataset used was obtained from EdNet, comprising data for over 50,000 students including academic performance, educational level, interests, and learning pathways. The study followed an applied quantitative methodology involving data preprocessing, feature encoding, TF-IDF for text processing, model building, and the development of a user interface using Gradio. Four classification algorithms were evaluated: Decision Tree, Random Forest, KNN, and XGBoost. XGBoost outperformed others with an accuracy of 89%. For regression, LightGBM achieved the highest performance (R² = 0.9997, MAE = 0.000008). SHAP was utilized to interpret model outputs and provide visual explanations of the recommendation logic. The results demonstrated that incorporating explainability into educational recommender systems significantly increases users’ trust and engagement. The study recommends enriching the input data with behavioral indicators, testing the model in diverse learning contexts, and deploying the system as an interactive platform aligned with Saudi Vision 2030’s digital education goals.
Recommender systems, Explainable AI, Machine Learning, Adaptive Learning, XGBoost, SHAP, EdNet
Impact of Academic Stress on The Performance of The Grade 9 Students in Mathematics
One of the most frequent stressors that students encounter is academic stress. The purpose of this study was to determine whether there is a significant relationship between the academic performance of grade 9 students at Campagao High School in Campagao, Bilar, Bohol, and their level of academic stress. The study employed the descriptive-correlational survey research design and the complete enumeration technique was used to obtain 63 respondents. The survey instrument was based on a standardized questionnaire and the data gathered were analyzed using the Pearson Product Moment Coefficient of Correlation at 0.05 alpha level. The major finding of the study indicated that there is a significant relationship between the level of academic stress and the level of mathematics performance. Furthermore, it was revealed that most of the respondents were 14 years old and were female. Majority of the respondents’ female parents were high school level while the highest educational attainment of the male parents was either high school level or elementary level with a combined family monthly income of P10 000 and below. Moreover, the level of academic stress experienced by the respondents in learning mathematics was moderate while the level of the students’ academic performance in math was developing. In line with the findings and conclusion of the study, the researcher recommends that the proposed math performance enhancement plan can be used and monitored.
Academic Stress, Mathematics Performance, High School Education, Stress and Learning
Relationship Between Academic Stress and Math Scores in Grade 9 Students
This study investigated the relationship between academic stress and math scores in Grade 9 students. Additionally, it sought to examine the relationship between students' academic stress and their academic performance in the subject of Mathematics. It utilized the descriptive correlational method of research with the use standardized questionnaire. There were 210 respondents in the study and it used a simple random sampling technique. The respondents were Grade 9 students from Jagobiao National High School, one of the public high schools in the city of Mandaue. There were four statistical treatments employed in this study, namely, frequency count, simple percentage, weighted mean, and Pearson’s r. It was revealed that most of the respondents were 14 years old and were female. The level of academic stress experienced by the respondents in learning mathematics was moderate while the level of the students’ academic performance in math was approaching proficiency. The study found no statistically significant correlation between academic stress and math performance. However, the researcher suggests that the recommended approach for improving math performance should be implemented and monitored.
Comparing three recently developed techniques and using them to address the non-Newtonian fluid flow and heat transfer issue on a turbine disk
This study is investigating three distinct semi-analytical methods (SAGPM, q-HALPM, and PYRDTM) to solve nonlinear equations in the context of turbine cooling during heat transfer and fluid flow. These methods include Shehu, Laplace, and Young transformations and are improved by Padé approximation. The analysis not only explores how these methods deal with the complicated problems of cooling turbine systems but also remove to test efficiency, accuracy, and convergence of each of them. A comprehensive comparison of approximate solution methods to identify the most effective approach for modeling nonlinear thermal systems is conducted. The findings provide critical insights in to efficient analytical estimations for applications of the above recent methods. Ultimately, the results contribute to the advancement of accurate and computationally efficient solutions for complicated fluid dynamics problems, with broad applicability to various flow-thermal challenges. Moreover, the results showed that the methods are effective and powerful and give an indication that q-HALPM has superior convergence and high accuracy compared to PYRDTM and SAGPM
Fluid dynamics, SAGPM, q-HALPM, PYRDTM, Nonlinear thermal systems, Padé approximations, Computational efficiency, convergence, accuracy
The Impact of Effective Teaching Practices and Student Engagement on Grade 10 Students’ Mathematics Performance
This study explored the connection between teaching effectiveness, student engagement, and the academic performance of Grade Ten students in Mathematics at Don Vicente Rama Memorial National High School during the school year twenty twenty-four to twenty twenty-five. A correlational research design was used to examine how these variables are related within actual classroom settings. Teaching effectiveness was assessed through indicators such as clarity of explanations, provision of feedback, use of interactive strategies, and the creation of a supportive learning environment. Student engagement was examined across behavioral, cognitive, and emotional aspects, while academic performance was evaluated based on the students’ Mathematics grades. Data were gathered using a structured survey and analyzed through descriptive and inferential statistics to identify patterns and relationships. Findings revealed that teaching effectiveness was perceived at an excellent level, showing that teachers consistently applied high-quality instructional practices. Student engagement was rated as high, with particularly strong responses in active participation, asking questions, and enjoying interactive lessons. The academic performance of the students was also commendable, with the majority reaching outstanding and very satisfactory levels. Results further indicated a moderate positive relationship between teaching effectiveness and academic performance and a weak but meaningful link between student engagement and academic performance. These outcomes highlight the importance of maintaining effective, student-centered teaching strategies and fostering engaging classroom experiences to enhance motivation, active participation, and achievement in Mathematics.
Teaching effectiveness, student engagement, academic performance, Mathematics education, instructional strategies
Investigating the Effectiveness of Gamification Teaching Method on Self-directed Learning in Mathematics among High School Students of Herat City
Math education faces an ongoing challenge - many students struggle with fundamental concepts and lack the confidence to direct their own learning. Traditional teacher-centered approaches often leave learners dependent on instructors, hesitant to tackle difficult problems. Yet fostering independent, curious students who take ownership of their math education remains a critical goal. This study explored how gamification - using game elements in teaching - could help high school students develop self-directed math skills like planning, self-monitoring, and problem-solving. The study conducted on 194 tenth-grade high school students of Herat City, dividing them into experimental (gamification) and control (traditional teaching) groups. Over eight lessons, the experimental group learned challenging math concepts through gamified activities, while the control group received conventional instruction. Using a validated self-assessment questionnaire (α = 0.89 overall), we measured four key skills: motivation, planning, self-monitoring, and communication. The results of the data analysis showed that teaching by gamification has an effect on promoting the independence and responsibility of students' learning in mathematics and on the three components of self-directed learning in mathematics. In other words, teaching by gamification had a positive effect on students' self-directed learning by creating a feeling of need to learn and accepting their role and responsibility in it. The study showed that teaching by gamification had a positive effect on the students’ mathematical performance by increasing the level of interactive learning and creating independence in students. Teachers, authors of textbooks, planners, and educational experts can benefit from the results of this research to improve and modify educational methods.
Teaching method, gamification, self-directed learning, Herat City.
Applications of pre-open & α-open sets in bitopological spaces
Using the concept of pre-open set & α-open set in bitopological spaces, we introduce and study topological properties of pre-limit points, pre-derived sets, pre-interior and pre-closure of a set, pre-interior points, α-limit points, α-derived sets, -interiαor and α-closure of a set and α-interior points in a bitopological space.
ij-p-limit points, ij-p-derived set, ij-p-interior, ij-p-closure, ij-α-limit point, ij-α-derived, ij-α-interior, ij-α-closure
Methods of Calculating and Reducing Sampling Error
Sampling error is a significant factor in research, denoting the variance between sample statistics and actual population values. This study examines techniques for quantifying and mitigating sampling error to improve the reliability and accuracy of research findings. Essential methods for determining sampling error, such as the standard error of the mean, confidence intervals, proportional error estimates, and bootstrapping, are examined comprehensively. Strategies to mitigate sampling error, including augmenting sample size, using stratified sampling, utilizing systematic sampling, implementing weighted adjustments, and enhancing sampling frames, are examined. The results underscore the significance of rigorous sampling techniques in reducing error, guaranteeing representativeness, and improving the validity of outcomes. The research emphasizes the significance of sophisticated statistical methodologies and pilot studies in mitigating constraints in sampling methods. This study offers pragmatic insights and methodological directives for academics, policymakers, and practitioners in several fields. It also delineates avenues for further investigation, including the use of sophisticated computational techniques and context-specific sampling methodologies, to further reduce sample error and enhance study quality.
sampling error, bootstrapping, actual population values, statistics, mean.