Chess in the Geometry Classroom

The current interest in chess playing in this country prompts me to share a very effective technique I have used in introducing the study of formal geometry. Basically, it uses the chess game as an “advance organizer” of Euclidean geometry viewed as a deductive system. Of course, this is but one of several possible views of Euclidean geometry. For example, another view is that geometry is an abstraction of man's physical environment.

Download Full-text

Um Passeio pelo Labirinto da Lógica Matemática em companhia de Malba Tahan

Revista de Educação Matemática ◽

10.25090/remat25269062v15n192018p247-264 ◽

2018 ◽

Vol 15 (19) ◽

pp. 247-264

Author(s):

Inocêncio Fernandes Balieiro Filho

Keyword(s):

Mathematical Logic ◽

Euclidean Geometry ◽

History Of Mathematics ◽

Logical Structure ◽

Deductive System ◽

Axiomatic Method ◽

Concept Definition ◽

History Of ◽

Contemporary Approach

O presente artigo tem por objetivo discutir numa perspectiva contemporânea os conteúdos de Lógica, Matemática, Filosofia da Matemática e História da Matemática presentes no livro A Lógica na Matemática, escrito por Malba Tahan. Para isso, mediante o uso da historiografia, foram selecionados temas concernentes com os assuntos da pesquisa. Foram tratados os seguintes temas: a base lógica da Matemática, a definição de conceito, os princípios para se definir um objeto, as definições e a natureza dos axiomas em Matemática, o método axiomático e as diversas axiomáticas para a geometria euclidiana, a estrutura lógica de um sistema dedutivo, os métodos de demonstração em Matemática, a indução, analogia e dedução em Matemática. Palavras-chave: Lógica Matemática; História da Matemática; Filosofia da Matemática. A TOUR BY THE LABYRINTH OF MATHEMATICAL LOGIC IN THE COMPANY OF MALBA TAHAN Abstract In this paper we discuss the Mathematics, the Logic of Mathematics, the Philosophy and History of Mathematics that presents in the book A Lógica na Matemática of the Malba Tahan, in a contemporary approach. For that, we use the historiography to select matters in adherence with the research. Are treated this topics: the basis of the Logic of Mathematics; the concept definition; principles to define an object; definitions and nature of the axioms in Mathematics; the axiomatic method and the diverse axiomatic to the Euclidean Geometry; the logical structure of a deductive system; demonstration methods in mathematics; the induction, analogy and deduction in mathematics.

Download Full-text

The Teaching of Mathematics in the Elementary and the Secondary School

Mathematics Teacher ◽

10.5951/mt.1.4.0121 ◽

1909 ◽

Vol 1 (4) ◽

pp. 121-122

Author(s):

Arthur Sullivan Gale

Keyword(s):

Secondary School ◽

Experimental Work ◽

Euclidean Geometry ◽

Geometric Analysis ◽

Algebraic Equations ◽

Algebraic Expressions ◽

Formal Geometry ◽

Non Euclidean Geometry ◽

Teaching Of Mathematics ◽

Class Room

The chapter on geometry (pp. 257-291) is excellent. lt presents ideas on geometric analysis, concrete and formal geometry, methods for treating problems, modern geometry, and non-Euclidean geometry. Especially important is the discussion of problems which lead to algebraic equations and the construction of simple algebraic expressions. A timely plea is made for experimental work and the usc of models and apparatus. As an example of their value, a Rochester teacher exhibited a sextant before a class one morning. A pupil borrowed it for the noon hour and became so enthusiastic in its use that he “cut” his afternoon classes to do some rough surveying. Contrast the interest which the inc;trument developed with the lack of enthusiasm which causes so many absences from the mathematical class-room! The chapter closes with an analysis of trigonometry and suggestions as to where its various parts should be taught.

Download Full-text

PEDAGOGICAL CONDITIONS FOR DEVELOPING INTELLECTUAL ACTIVITY IN YOUNGER SCHOOLCHILDREN WHILE TEACHING THE GAME OF CHESS: THEORETICAL FOUNDATIONS

Scientific Review Series 2 Human sciences ◽

10.26653/2076-4685-2021-1-06 ◽

2021 ◽

pp. 65-75

Author(s):

Arcady B. Pankin ◽

◽

Arslang Y. Doglaev ◽

Keyword(s):

Fairy Tale ◽

Developing Strategy ◽

Main Concept ◽

Intellectual Activity ◽

Intellectual Elite ◽

Chess Game ◽

Theoretical Thinking ◽

Additional Education ◽

Theoretical Foundations ◽

Chess Playing

The article problematizes the development of intellectual elite as a guiding genesis of society and culture, the idea that chess school in the system of additional education is focused on the development of personality giftedness in extracurricular educational environment and is a structure called to create pedagogical conditions of support and accompaniment of development of creative abilities, stimulation of intellectual activity of the young schoolchildren according to the requirements of the chess game is substantiated. The authors analyze the ideas allowing to identify the theoretical and methodological foundations of the main concept of research — "intellectual activity" and related concepts; different contexts of the problem of intellectual activity in the process of learning chess playing are shown; the experience in teaching chess playing to junior schoolchildren is summarized and described; the pedagogical conditions of stimulating intellectual activity of junior schoolchildren in the practice of chess education are revealed. It is proved that the pedagogical conditions of development of intellectual activity of the junior schoolchildren while teaching them chess are the following: the establishment of the relations of collaboration and co-creativity in the pedagogical interaction; organization of classes taking into account the specificity of age with the use of fairy-tale plots, games, elements of dramatization in the context of creating the situation of success for everybody; realization of the developing strategy in teaching chess playing, which assumes forming theoretical thinking, mastering generalized ways of playing chess in the junior schoolchildren.

Download Full-text

The relations among navigation, object analysis, and magnitude perception in children: Evidence for a network of Euclidean geometry

10.31234/osf.io/wujhv ◽

2020 ◽

Author(s):

Vladislav Ayzenberg ◽

Stella F. Lourenco

Keyword(s):

Physical Environment ◽

Potential Role ◽

Euclidean Geometry ◽

Visual Analysis ◽

The Novel ◽

Geometric Properties ◽

Form Analysis ◽

Open Questions ◽

Potential Interactions

What is the relation between navigation and object analysis? Although researchers agree that such systems are, to some extent, dissociable, open questions remain about the potential interactions between them. Using an individual differences approach with young children (4-year-olds; N = 90), we examined the novel proposal that navigation and object analysis interact indirectly by way of magnitude perception. A system for magnitude perception may be ideally suited to support such interactions because it is present from early in development and is capable of discriminating geometric properties on the basis of spatial extent. Across two experiments, we tested for relations between children’s ability to reorient within an environment, analyze the geometric properties of object forms, and discriminate area. We found that area precision was correlated with both reorientation ability and geometric form analysis, though reorientation and form analysis did not correlate with one another. Importantly, we also found that the precision of children’s area discrimination supported an indirect link between their reorientation ability in a physical environment and visual analysis of object forms. Together, these results suggest a network for representing Euclidean geometry that includes both direct and indirect relations among navigation, object analysis, and magnitude perception. The potential role of magnitude perception within such a network is discussed.

Download Full-text

AEM analysis of crystallization in Ti-based amorphous alloys

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100075142 ◽

1983 ◽

Vol 41 ◽

pp. 270-271

Author(s):

A.R. Pelton ◽

A.F. Marshall ◽

Y.S. Lee

Keyword(s):

Magnetic Properties ◽

Amorphous Alloys ◽

Amorphous Materials ◽

Isothermal Annealing ◽

Amorphous Matrix ◽

Nucleation And Growth ◽

Current Interest ◽

Amorphous Films ◽

Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

Download Full-text

Lattice Imaging of Grain Boundaries in Ceramics

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100078171 ◽

1977 ◽

Vol 35 ◽

pp. 114-115

Author(s):

D. R. Clarke ◽

G. Thomas

Keyword(s):

Mechanical Properties ◽

High Temperature ◽

Silicon Nitride ◽

Grain Boundaries ◽

High Temperature Strength ◽

Current Interest ◽

Lattice Imaging ◽

Special Significance ◽

Structural Applications ◽

Refractory Ceramics

Grain boundaries have long held a special significance to ceramicists. In part, this has been because it has been impossible until now to actually observe the boundaries themselves. Just as important, however, is the fact that the grain boundaries and their environs have a determing influence on both the mechanisms by which powder compaction occurs during fabrication, and on the overall mechanical properties of the material. One area where the grain boundary plays a particularly important role is in the high temperature strength of hot-pressed ceramics. This is a subject of current interest as extensive efforts are being made to develop ceramics, such as silicon nitride alloys, for high temperature structural applications. In this presentation we describe how the techniques of lattice fringe imaging have made it possible to study the grain boundaries in a number of refractory ceramics, and illustrate some of the findings.

Download Full-text

Contrast and Resolution in Alfresco Microscopy and Thick Specimens

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100096102 ◽

1972 ◽

Vol 30 ◽

pp. 570-571

Author(s):

J. R. Sellar ◽

J. M. Cowley

Keyword(s):

Solid Material ◽

Cell Design ◽

Current Interest ◽

High Voltage Electron Microscopy ◽

Voltage Electron ◽

Transmission Electron ◽

Environmental Cell ◽

Scanning Transmission ◽

Conventional Transmission Electron Microscope ◽

Transmission Microscope

Current interest in high voltage electron microscopy, especially in the scanning mode, has prompted the development of a method for determining the contrast and resolution of images of specimens in controlled-atmosphere stages or open to the air, hydrated biological specimens being a good example. Such a method would be of use in the prediction of microscope performance and in the subsequent optimization of environmental cell design for given circumstances of accelerating voltage, cell gas pressure and constitution, and desired resolution.Fig. 1 depicts the alfresco cell of a focussed scanning transmission microscope with a layer of gas L (and possibly a thin window W) between the objective O and specimen T. Using the principle of reciprocity, it may be considered optically equivalent to a conventional transmission electron microscope, if the beams were reversed. The layer of gas or solid material after the specimen in the STEM or before the specimen in TEM has no great effect on resolution or contrast and so is ignored here.

Download Full-text

Novel Approaches to Low-Voltage Scanning Electron Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100180586 ◽

1990 ◽

Vol 48 (1) ◽

pp. 366-367

Author(s):

Arthur V. Jones

Keyword(s):

Electron Microscopy ◽

Scanning Electron Microscopy ◽

Low Voltage ◽

Electron Optics ◽

Current Interest ◽

New Concepts ◽

Opportune Time ◽

Novel Approaches ◽

Voltage Scanning ◽

Scanning Electron

In comparison with the developers of other forms of instrumentation, scanning electron microscope manufacturers are among the most conservative of people. New concepts usually must wait many years before being exploited commercially. The field emission gun, developed by Albert Crewe and his coworkers in 1968 is only now becoming widely available in commercial instruments, while the innovative lens designs of Mulvey are still waiting to be commercially exploited. The associated electronics is still in general based on operating procedures which have changed little since the original microscopes of Oatley and his co-workers.The current interest in low-voltage scanning electron microscopy will, if sub-nanometer resolution is to be obtained in a useable instrument, lead to fundamental changes in the design of the electron optics. Perhaps this is an opportune time to consider other fundamental changes in scanning electron microscopy instrumentation.

Download Full-text