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Glushchenko, Alexandra; Glushchenko, Alexander; Glushchenko, Eugenia – European Journal of Physics Education, 2020
The cosine theorem is used in solving triangulation problems and in physics when solving problems of addition of unidirectional oscillations. However, this theorem is used only for the analytical calculation of triangles or when solving problems of adding two oscillations. Here we propose a generalization of the cosine theorem for the case of…
Descriptors: Light, Radiation, Physics, Geometry
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Lingefjärd, Thomas; Hatami, Russell – Policy Futures in Education, 2020
This is an article about abstraction, generalization, and the beauty of mathematics. We claim that abstraction and generalization in of itself may very well be a beauty of the human mind. The fact that we humans continue to explore and expand mathematics is truly beautiful and remarkable. Many years ago, our ancestors understood that seven stones,…
Descriptors: Abstract Reasoning, Aesthetics, Mathematics, Mathematical Concepts
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Suzuka, Kara; Venenciano, Linda – Mathematics Teacher, 2019
Fragile understanding is where new learning begins. Students' understanding of new concepts is often shaky at first, when they have only had limited experiences with or single viewpoints on an idea. This is not inherently bad. Despite teachers' best efforts, students' tenuous grasp of mathematics concepts often falters with time or when presented…
Descriptors: Mathematics Instruction, Mathematical Concepts, Concept Formation, Misconceptions
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Wares, Arsalan – International Journal of Mathematical Education in Science and Technology, 2020
The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.
Descriptors: Generalization, Competition, Mathematics, Problem Solving
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Stupel, Moshe; Sigler, Avi; Tal, Idan – International Journal for Technology in Mathematics Education, 2019
We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect…
Descriptors: Mathematics Instruction, Teaching Methods, Validity, Mathematical Logic
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de Villiers, Michael – International Journal of Mathematical Education in Science and Technology, 2017
This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of…
Descriptors: Geometry, Mathematical Logic, Validity, Transformations (Mathematics)
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Kovács, Zoltán; Recio, Tomás; Vélez, M. Pilar – International Journal for Technology in Mathematics Education, 2018
This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some…
Descriptors: Geometry, Mathematics Instruction, Teaching Methods, Comparative Analysis
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Lo, Jane-Jane; Cox, Dana C. – Mathematics Teacher: Learning and Teaching PK-12, 2020
The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement…
Descriptors: Thinking Skills, Mathematics Instruction, Geometry, Standards
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Roberts, Sally K.; Borum, Viveka O. – Mathematics Teaching in the Middle School, 2012
Students often view mathematics as a set of unrelated facts and procedures and fail to make the connections between and among related topics. One role of a teacher is to help students understand that mathematics is an interrelated discipline. Another role is to assist students in the scaffolding of their knowledge so that they can make connections…
Descriptors: State Standards, Teaching Methods, Mathematics Instruction, Middle Schools
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Mercer, Peter R. – International Journal of Mathematical Education in Science and Technology, 2004
The location of the number "c" arising from Cauchy's Average Value Theorem is described when the size of the interval is small. This article discusses various generalizations of theorem 1, to the context of Cauchy?s Average Value Theorem--but without appealing to theorem 1. Obviously, hypotheses involving the functions "f" and "g" will be…
Descriptors: Geometry, Generalization, Classroom Techniques, Mathematics
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Fay, Temple H. – International Journal of Mathematical Education in Science and Technology, 2002
Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given…
Descriptors: Geometric Concepts, Mathematics Instruction, Geometry, Generalization
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DeVilliers, Michael – International Journal of Mathematical Education in Science & Technology, 2006
A heuristic description is given of the rediscovery with "Sketchpad" of a less-well-known, but beautiful, generalization of the nine-point circle to a nine-point conic, as well as an associated generalization of the Euler line. The author's initial analytic geometry proofs, which made use of the symbolic algebra facility of the TI-92 calculator,…
Descriptors: Geometry, Mathematical Logic, Algebra, Mathematics Education
Leung, Allen – International Group for the Psychology of Mathematics Education, 2003
In this paper, the theory of variation in the tradition of phenomenographic research approach is placed in the context of Dynamic Geometry Environment (DGE). Central concepts of discernment, variation, simultaneity and space of learning in the theory of variation are discussed for simple dragging episodes in DGE to illustrate the potential…
Descriptors: Research Methodology, Geometry, Geometric Concepts, Phenomenology