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Vol.33 No.2
Abstract
Analogical reasoning helps extend mathematical concepts and can be used as a heuristic for solving mathematical problems. In this study, identities of the Horadam sequence were conjectured through conceptual analogy, and the conjectured identities were proved through methodological analogy. In addition, further methods for proving the identities were presented using matrix powers, determinants, and the Cayley–Hamilton theorem. This study is expected to help preservice and inservice teachers experience the process of formulating and justifying mathematical conjectures through analogical reasoning, and to serve as instructional material for providing gifted students with opportunities to gradually expand their mathematical knowledge by using analogical reasoning.
Abstract
This study examined representations of Aristarchus’ measurement method and the educational value of GeoGebra-based modeling. Textbook analysis and a case study revealed that fixed values constrained critical thinking, whereas modeling tasks led students to derive 89.85° and experience cognitive conflict. Through technological refinement, students recognized the model’s validity and engaged in deeper validation. The study highlights the role of technology in fostering data evaluation and mathematical understanding.
Abstract
In this paper, we examine the process of mathematization from the instructor’s perspective, based on lessons related to the nets of a cube and a regular octahedron for elementary gifted students in an advanced program. Specifically, we investigate how the number of nets considered by students, based on the adjacency and parallelism of faces in the nets (represented respectively by the concepts of trees and partitions of sets), leads to a process of mathematization by the instructor using the concept of matchings in graphs to determine the number of nets for a rectangular prism and a square prism.
Abstract
This study aimed to investigate a generalized method for finding one integer solution to a linear equation in two variables of the form r_1 x + r_2 y = c. In school mathematics, solving linear equations in two variables is an important topic, and in number theory this problem is closely related to linear Diophantine equations. However, textbooks do not usually present a systematic method for obtaining a particular integer solution. To address this limitation, this study focused on the case where (r_1,r_2) = g and g | c, and explored a generalized formula based on the simple continued fraction expansion of r1/r2. First, cases with n = 0,1,2,3 were analyzed inductively to identify a recurring pattern. Then, this pattern was generalized into a recursive formula for generating one integer solution and justified by mathematical induction. The study also showed that the procedure can be explained structurally through matrix representation. The findings indicate that one integer solution can be obtained systematically from the coefficients of the continued fraction, and that the method can be applied to both middle school textbook problems and university-level number theory examples. Thus, this study provides a meaningful link between school mathematics and number theory.
Abstract
We develop a classroom module that introduces the perceptron through the XOR gate, a compact example that links secondary-school mathematics (coordinate geometry and systems of inequalities) to the idea that composing simple linear separators can create non-linear decision boundaries. The module is organized as three lessons and includes activity outlines, student-facing proof sketches for the limitation of single-layer perceptrons on XOR, and a minimal two-hidden-unit construction that realizes XOR. An optional computational lab visualizes a learned decision boundary of a small multilayer perceptron trained on XOR.
Abstract
This paper presents an elementary derivation of the oriented area formula Area(D)= {1} over {2} int _{a} ^{b} {(x(t)y prime (t)−y(t)x prime (t))} dt for a parametrized simple closed curve. Instead of taking Green’s theorem as a starting point, we construct inscribed polygons from a partition of the parameter interval, apply the Shoelace formula to each polygon, and pass to the limit through a Riemann-sum argument. This approach makes the geometric origin of the area integrand explicit and provides a bridge between polygonal area computation and line-integral area formulas. As an application, we compare two similar four-petal-type curves and verify that they have the same enclosed area while having slightly different perimeters.