University course timetabling is commonly evaluated through operational feasibility, while pedagogical considerations such as cognitive-demand distribution and class timing receive less explicit attention in optimization models. This study developed a spreadsheet-based Integer Linear Programming (ILP) model that integrates conventional hard constraints with pedagogically informed soft constraints to generate a feasible and more educationally responsive timetable. A mathematical modeling and computational simulation design was employed using synthetic data comprising five courses, four lecturers, two classrooms, five daily time slots, and two instructional days. The model incorporated classroom allocation, lecturer assignment and availability, room suitability and capacity, required course duration, and day-room consistency as hard constraints, while cognitive-demand distribution, post-lunch scheduling, and first-slot placement were treated as soft constraints. The formulation was implemented in Microsoft Excel and solved using OpenSolver with the COIN-OR CBC engine. The resulting timetable satisfied all hard constraints, confirming operational feasibility, while the three pedagogical penalty components produced values of 4, 0, and 1, respectively. These results indicate that excessive daily concentration of cognitively demanding courses could be fully avoided, although some temporal-placement penalties remained under the available scheduling conditions. The findings demonstrate that pedagogically informed preferences can be incorporated into an accessible ILP framework without compromising feasibility. This study therefore provides a transparent proof of concept for extending university timetabling beyond conflict avoidance toward multidimensional schedule quality that combines operational requirements with explicit cognitive and temporal considerations. The approach also preserves transparency for institutional inspection and adaptation.
The heat equation is a fundamental model for transient diffusion processes, yet its numerical treatment becomes increasingly demanding when spatial dimensionality, source placement, and matrix conditioning interact. This study investigates the direct application of the Method of Fundamental Solutions (MFS) to the homogeneous heat equation with Dirichlet boundary conditions in two-dimensional circular and three-dimensional spherical domains. A fixed-source, meshless formulation was implemented in MATLAB using equal numbers of source and collocation points, two fictitious-boundary radii (R=1.5 and R=2), and the Moore-Penrose pseudo-inverse to stabilize ill-conditioned systems. Numerical accuracy was evaluated against analytical solutions through pointwise absolute and relative errors. In two dimensions, relative errors remained small and stable, ranging from 7.5×10−3 to 8.1×10−3 as M=N increased from 100 to 250. In three dimensions, the main improvement occurred between M=N=10 and M=N=15, after which the relative error plateaued near 3.62×10−1 to 3.63×10−1 despite further increases in point number. Changing the fictitious-boundary radius produced only modest differences in both dimensions. These findings show that increasing the number of approximation points does not necessarily yield monotonic accuracy gains and that MFS performance is shaped by the combined effects of point configuration, source-boundary placement, spatial dimensionality, and numerical conditioning. The study provides controlled comparative evidence on the practical behavior of direct fixed-source MFS for transient heat conduction and clarifies the conditions under which additional discretization offers diminishing numerical returns within the tested computational configurations.
This study investigated the influence of three-dimensional visualization media on elementary school students’ spatial ability and creative thinking in geometry learning. The study was grounded in students’ persistent difficulties in interpreting solid figures, recognizing object viewpoints, and transforming spatial representations when geometry is taught through conventional explanation and static images. A quantitative quasi-experimental design with a non-equivalent control group pretest-posttest model was employed. The participants were 50 sixth-grade students from two public elementary schools in Karangpucung District, Cilacap Regency, Indonesia, consisting of 27 students in the experimental class and 23 students in the control class. The experimental class learned cubes, cuboids, and composite spatial structures using three-dimensional visualization media, whereas the control class received conventional instruction. Data were collected using spatial ability and creative thinking tests and analyzed through descriptive statistics, Shapiro-Wilk normality testing, Levene’s homogeneity testing, independent samples t-test, and Pearson correlation. The results showed that the experimental class achieved significantly higher outcomes than the control class in spatial ability, t(48) = 4.514, p < .001, and creative thinking, t(48) = 7.816, p < .001. A strong positive correlation was also found between spatial ability and creative thinking, r = .756, p < .001, indicating that students with stronger spatial reasoning tended to demonstrate stronger creative mathematical thinking. These findings suggest that three-dimensional visualization media function as cognitive scaffolds that support visual-spatial representation, mental transformation, and flexible idea generation. The study contributes to technology-enhanced geometry learning by demonstrating that interactive visual media can simultaneously strengthen spatial reasoning and creative thinking in elementary mathematics education.
Mathematical conceptual understanding is essential for enabling students to interpret mathematical ideas, connect concepts, and justify problem-solving procedures. However, many students still experience difficulty developing conceptual understanding because mathematics instruction often emphasizes procedural fluency rather than meaningful conceptual construction. The Heuristic Vee model has been proposed as an instructional approach that connects conceptual and methodological components of learning, yet empirical findings regarding its effectiveness remain fragmented across studies. This study conducted a group-contrast meta-analysis to examine the effect of the Heuristic Vee model on students’ mathematical conceptual understanding. Data were obtained from seven eligible empirical studies published between 2013 and 2023, selected through Publish or Perish from Google Scholar, Semantic Scholar, and Scopus. The included studies reported complete statistical information from experimental and control groups, including sample size, mean, and standard deviation. Effect sizes were calculated using standardized mean differences and analyzed with OpenMEE and JASP. The random-effects model produced a significant large pooled effect size of 2.529, indicating that Heuristic Vee-based instruction positively influenced students’ mathematical conceptual understanding. Subgroup analysis showed a significant large effect at the junior secondary level and in the Indonesian context, whereas higher education, Malaysia, and Turkey showed large numerical effects but limited statistical certainty due to small study representation. Heterogeneity was very high, and Egger’s test indicated possible funnel plot asymmetry, although fail-safe N and trim-and-fill analysis suggested relatively robust results. These findings indicate that Heuristic Vee is a promising instructional model for strengthening mathematical conceptual understanding when supported by appropriate scaffolding and contextual implementation.
Mathematical critical thinking is increasingly recognized as a fundamental competency in mathematics education because it enables students to analyze information, evaluate solution strategies, and justify mathematical conclusions. Despite the growing adoption of inquiry-oriented learning, limited empirical evidence explains how commonly available classroom technology can effectively support the cognitive processes underlying Discovery Learning to improve students' mathematical critical thinking. This study investigated the effect of PowerPoint-assisted Discovery Learning on eighth-grade students’ mathematical critical thinking in Systems of Linear Equations in Two Variables. A quantitative quasi-experimental approach employing a nonequivalent control group design was conducted with 50 students, consisting of 25 students in the experimental group and 25 in the control group. Data were collected using a validated essay-based mathematical critical thinking test comprising nine valid items with excellent reliability (Cronbach’s α = .9669) and analyzed using descriptive statistics and an independent-samples t-test after normality and homogeneity assumptions were satisfied. Students in the experimental group significantly outperformed those in the control group (87.40 vs. 58.08), t(48)=16.1489, p<.001., and the difference was statistically significant, t (48) = 16.1489, p < .001. These findings demonstrate that PowerPoint functions as an instructional scaffold supporting exploration, verification, and mathematical reasoning within Discovery Learning. The study contributes empirical evidence that integrating structured visual scaffolding with inquiry-oriented pedagogy provides an accessible and effective approach for strengthening mathematical critical thinking in lower-secondary mathematics education.
