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Vol.32 No.2
Abstract
This study explores how the initial terms of the series expansions of exponential, logarithmic, and trigonometric functions can be derived using substitution and various algebraic techniques, without relying on differentiation. Based on these findings, the study discusses several educational implications, particularly in the context of high school mathematics instruction.
Abstract
Problem-making has been continuously studied in mathematics education and problem-solving. In particular, the deductive problem-making method has potential applicability to various mathematical topics. By analyzing the problem solving process into several substructures and creating problems backward by altering elements within those substructures, this method becomes meaningfully relevant to the reflection phase of problem-solving. This study utilizes the deductive problem making method for a problem on combination with repetition and examines several pedagogical characteristics.
Abstract
This study examines historical approaches to deriving conic sections-ellipse, hyperbola, and parabola-as intersections of a cone and aplane. Focusing on two early 20th-century textbooks from the Real Gymnasium, it analyzes the mathematical knowledge and proof methods used in each. By comparing different approaches and tools, the study provides a systematic understanding of how conic sections were logically derived. The findings offer insights into teaching analytic geometry in secondary education through historically grounded methods.
Abstract
Mathematising is emphasized in school mathematics and is considered a core competency for preservice teachers as they prepare to instruct future students. To enhance their mathematising skills, preservice mathematics teachers must engage in meaningful mathematising experiences. One such experience involves transforming thephainomenon of the Fibonacci sequence into nooumenon through the processes of generalization and abstraction. In the present study, three novel identities and twelve corollaries related to the generalized Fibonacci sequence are presented, contributing to the mathematical foundation necessary for fostering mathematising abilities.
Abstract
This paper investigates a part of Faulhaber formula studied in the <Algebra> curriculum. Specifically, it examines the formula for sum_{k=1}^n k^{alpha} (alpha = 1, 2, 3). Based on textbook analysis, this study attempts to derive a general method of inductive justification for the formula when α=1, 2, 3), and further extends it to the cases where α=4, 5. First, through an analysis of previous research, two consistent methods of inductive justification were identified. One is the 'geometric rotational symmetry' method, and the other uses ratios. Second, usingthese two methods, the formula for sum_{k=1}^n k^{alpha} was inductively derived for α=4, 5.