scholarly journals Remarks on the Principle of Permanence of Forms

2020 ◽  
Vol 11 ◽  
pp. 127-141
Author(s):  
Jerzy Pogonowski
Keyword(s):  
Abstract Algebra ◽  
Formal Logic ◽  
Mathematical Education ◽  
Heuristic Principle

We discuss the role of a heuristic principle known as the Principle of Permanence of Forms in the development of mathematics, especially in abstract algebra. We try to find some analogies in the development of modern formal logic. Finally, we add a few remarks on the use of the principle in question in mathematical education.

scholarly journals On the links between mathematics education and democracy: A literature review

Pythagoras ◽  
2012 ◽  
Vol 33 (2) ◽  
Author(s):  
Mario Sánchez Aguilar ◽  
Juan Gabriel Molina Zavaleta
Keyword(s):  
Mathematics Education ◽  
Literature Review ◽  
Democratic Education ◽  
Research Area ◽  
Mathematical Education ◽  
Mathematics Students ◽  
The World ◽  
Education For Democracy ◽  
Further Development

This article reports the results of a literature review focused on identifying the links between mathematics education and democracy. The review is based on the analysis of a collection of manuscripts produced in different regions of the world. The analysis of these articles focuses on six aspects, namely, (1) definitions of democracy used in these texts, (2) identified links between mathematics education and democracy, (3) suggested strategies to foster a democratic competence in mathematics students (4) tensions and difficulties inherent in mathematical education for democracy, (5) the fundamental role of the teacher in the implementation of democratic education and (6) selected criticisms of mathematical education for democracy. The main contributions of this article are to provide the reader with an overview of the literature related to mathematics education and democracy, and to highlight some of the theoretical and empirical topics that are necessary to further development within this research area.

Lightweight tension structures – an aesthetic integration of geometry and mechanics. Part 1. The role of minimal surfaces and soap films. Part 2. Finding the form of a minimal surface, by W. J. Lewis, Mathematics Today 35 (1) pp. 10–16, (3) pp. 80–84, 1999. - Magic squares indeed!, by Arthur T. Benjamin and Kan Yasuda, American Mathematical Monthly 106 (2), pp. 152–156, 1999. - Pierre-Simon de Laplace: 1749–1827, by Roger Cook, Mathematical Spectrum 31 (3), pp. 49–51, 1998/9. - Unifying threads in Alfred Tarski’s work, by Steven Givant, The Mathematical Intelligencer 21 (1), pp. 47–58, 1999. - André Weil and algebraic topology, by Armand Borel, pp. 422–427. - André Weil as I knew him, by Goro Shimura, pp. 428–433. - André Weil: A prologue, by Anthony W. Knapp, pp. 434–439. - André Weil (1906-1998), by Armand Borel, Pierre Carrier, Komaravolu Chandrasekharan, Shiing-Shen Chern and Shokichi Iyanaga, pp. 440–447. - The apprenticeship of a mathematician – autobiography of André Weil, reviewed by V. S. Varadarajan, pp. 448–456. - Introduction to metric preserving functions, by Paul Corazza, The American Mathematical Monthly 106 (4) pp. 309–323, 1999. - Visual aspects of understanding group theory, by D. F. Almeida, Int. J. of Mathematical Education in Science and Technology 30 (2) pp. 159–166, 1999 - Marriage, Magic, and Solitaire, by David B. Leep and Gerry Myerson, The American Mathematical Monthly 106 (5) pp. 419–429, 1999. - Professional Development of Mathematics Teachers, by H. Wu, Notices of the American Mathematical Society 46 (5) pp. 535–541, 1999. - Mathematics Today 35 (4) pp. 118–122, 1999 contains three short articles under the heading Mathematics and Dyslexia. - Difficulties in Knowledge Integration: Revisiting Zeno’s Paradox with Irrational Numbers, by Irit Peled and Sara Hershkovitz, International Journal of Mathematical Education in Science and Technology 30 (1) pp. 39–46, 1999.

1999 ◽  
Vol 83 (498) ◽  
pp. 529-532
Author(s):  
Anne C. Baker ◽  
G. Jackson
Keyword(s):  
Algebraic Topology ◽  
Minimal Surfaces ◽  
Knowledge Integration ◽  
Science And Technology ◽  
American Mathematical Society ◽  
Mathematical Education ◽  
Magic Squares ◽  
Tension Structures ◽  
Zeno's Paradox

scholarly journals Euathlus and Crocodile paradoxes: dialectic solution’s advantages

2020 ◽  
Vol 164 ◽  
pp. 11022
Author(s):  
Anton Zamorev ◽  
Alexander Fedyukovsky
Keyword(s):  
Formal Logic ◽  
Central Problem ◽  
Legal Cases ◽  
Universal Approach ◽  
Uniform Solution ◽  
Consistent Application

The paper is devoted to two ancient legal cases which, to date, have had no uniform solution: The Euathlus paradox and The Crocodile paradox. The aim of this work is not only searching logically faultless solution of both problems, but also developing the general approach to solving any similar cases without involving principles other than formal logic and the primary contract between litigants. The central problem of the research is that of incompleteness of this problem provisions, resulting in a set of various treatments the same questions. In the paper the following problems are solved: four exhaustive approaches to the problem of legal cases, which are called formal, authoritative, liberal and dialectic, are specified; the solution of the Euathlus paradox, which is inevitable with all the four approaches in the condition of their consistent application, is obtained; the solution of the Crocodile paradox, which is true with the dialectic approach, but impossible with three others, is obtained; it is proved that the dialectic approach not only combines the advantages of the first three approaches, but it is without their disadvantages that makes it a unique worthy applicant for the role of the universal approach.

FEATURES OF STUDYING THE SECTION "MULTIPLIERS. THE ELEMENTS OF LOGIC" IN THE COURSE OF PRIMARY SCHOOL MATHEMATICS ON THE UPDATED CONTENT OF EDUCATION

2021 ◽  
Vol 73 (1) ◽  
pp. 42-49
Author(s):  
M.Zh. Мynzhasarova ◽  
◽  
A.B. Akpaeva ◽  
L.A. Lebedeva ◽  
◽  
...  
Keyword(s):  
Primary School ◽  
School Mathematics ◽  
Educational Goals ◽  
Mathematical Education ◽  
Special Role ◽  
Logical Thinking ◽  
School Students ◽  
Elementary School Mathematics ◽  
Independent Work

This article discusses the features of studying the topics of the section "Set. Elements of Logic" in the course of elementary school mathematics. The analysis of the content of the program and its implementation in the textbook "Mathematics" is given. The article describes the features of the updated content of mathematical education in primary classes. The role of the "Set. Elements of logic" in the development of logical thinking of primary school students. A feature of studying the section "Set. Elements of Logic" is that a system of exercises has been developed that implement the formation of thinking techniques. The analysis of the proposed system of exercises in the implementation of the educational goals of this section is carried out. When developing the system of exercises, the age-specific features of the students ' thinking development were taken into account. In the organization of work with the system of exercises, the ways of increasing the mental activity of students, the development of independent work were considered. A special role in the development of logical thinking of students is also occupied by logical tasks presented in the category "You are a researcher".

Issues concerning Evidence

1992 ◽  
Vol 23 (4) ◽  
pp. 341-344
Keyword(s):  
Education Research ◽  
Mathematics Education ◽  
Mathematics Education Research ◽  
Mathematical Education ◽  
National Council ◽  
Research Traditions ◽  
Quantitative Studies ◽  
Education Community ◽  
To Come

As noted in Bishop (1992), at the initial meeting of the International Congress on Mathematical Education (ICME) in 1969, only one presentation directly addressed the role of inquiry in mathematics education research. If ICME 1988 is an indicator, then such discussion will be a hallmark of the international interchange to be held in Quebec in August 1992. In 1994, the presses will release the 25th volume of the Journal for Research in Mathematics Education (JRME). At the Research Presession of the Annual Meeting of the National Council of Teachers of Mathematics, Thomas Carpenter, outgoing editor of the JRME, remarked that whereas over 70% of volume 1 of the JRME reported purely quantitative studies, nearly 50% of the 1991 volume presented qualitative works (1992). Mathematics education research traditions still are evolving, in comparison to the more established research traditions in some disciplines, but the field is beginning to come of age. At this time it is reasonable for the mathematics education community to examine the varying approaches and traditions that characterize mathematics education research as well as the nature of evidence within these approaches and traditions.

scholarly journals The Concept of Theoretical-Empirical Dualism in Teaching Math

2020 ◽  
Vol 29 (4) ◽  
pp. 85-95
Author(s):  
G. D. Gefan
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Research Work ◽  
Cognitive Activity ◽  
Teaching Mathematics ◽  
Mathematical Education ◽  
A Posteriori ◽  
Research Activities ◽  
New Knowledge

Among the problems of mathematical education, the article highlights: (1) insufficient attention paid to the fundamental, structure-forming role of mathematics; (2) speculative learning, its isolation from practice. The concept of theoretical-empirical dualism in teaching is formulated as the unity of the abstract-theoretical and experimental-empirical cognitive activity of students. According to the author, a priori and a posteriori mathematical knowledge should be distinguished. A priori knowledge either seems to an individual to be completely obvious, indisputable, or he assimilates it uncritically, “on faith”. A posteriori mathematical knowledge subjectively arises in the process of student’s intense theoretical and practical activity, and is being actively and comprehensively verified experimentally – either using mathematical applications, or through mathematical experiments. The empirical component of teaching mathematics implies a variety of forms and methods of active (including computer) and professionally oriented learning, giving experience in independent formulation of problems, joint search for ways to solve them, interaction and teamwork. Particular attention is paid to the use of mathematical experiments in those frequent cases when it is necessary to replace or supplement complex evidence, illustrate new knowledge, and give research skills. Monte Carlo mathematical experiments are demonstrated, which serve, in particular, as a bright, figurative, and convincing form of reinforcing theoretical knowledge in the field of stochastic branches of mathematics. The research work of students is considered as the highest stage of the students’ theoretical-empirical activity. The article proposes subjects of research activities of students in the process or upon completion of the study of probabilistic and statistical disciplines.

On the role of implication in formal logic

2000 ◽  
Vol 65 (3) ◽  
pp. 1076-1114 ◽  
Author(s):  
Jonathan P. Seldin
Keyword(s):  
Higher Order ◽  
Formal Logic ◽  
Combinatory Logic ◽  
Cut Elimination ◽  
Classical Version ◽  
Logical Constants ◽  
Logical Strength ◽  
Special Case

AbstractEvidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a “classical” version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic.

scholarly journals Counting with Alice

2020 ◽  
Vol 10 (2) ◽  
pp. 58-68
Author(s):  
Amirouche Moktefi
Keyword(s):  
Personal Identity ◽  
Literary Texts ◽  
Mathematical Education ◽  
Mathematical Culture ◽  
Mental Calculation ◽  
Mathematical Ideas ◽  
Alice's Adventures In Wonderland ◽  
Alice’S Adventures In Wonderland ◽  
Looking Glass

There are many mathematical references in Lewis Carroll’s two tales for children: Alice’s Adventures in Wonderland (1865) and Through the Looking-Glass (1872). Many critics suggested that Carroll inserted hidden meanings in those passages. We rather consider them as part of the story’s setting and narrative. Yet, those passages may be interpreted and used as convenient to illustrate mathematical ideas. In this paper, we consider two passages from the Alice tales that relate to arithmetic, and we discuss them in relation to issues of personal identity, mathematical certainty, the role of notations and the processes of composition and decomposition in mental calculation. Hence, we show how literary texts can be used to convey ideas related to mathematics, mathematical culture and mathematical education. We conclude on the importance of mathematical writings as literary texts.

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