scholarly journals The challenge of computer mathematics

2005 ◽  
Vol 363 (1835) ◽  
pp. 2351-2375 ◽  
Author(s):  
Henk Barendregt ◽  
Freek Wiedijk
Keyword(s):  
Computer Systems ◽  
Foundations Of Mathematics ◽  
Mathematical Concepts ◽  
Mathematical Objects

Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This is not in spite of, but partially based on the famous results of Gödel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, and computations are dealing with objects and proofs. Interactive computer systems for a full integration of defining, computing and proving are based on this. The human defines concepts, constructs algorithms and provides proofs, while the machine checks that the definitions are well formed and the proofs and computations are correct. Results formalized so far demonstrate the feasibility of this ‘computer mathematics’. Also there are very good applications. The challenge is to make the systems more mathematician-friendly, by building libraries and tools. The eventual goal is to help humans to learn, develop, communicate, referee and apply mathematics.

scholarly journals Digital Representations of Mathematical Objects in the Context of Various Forms of Representation of Mathematical Knowledge

2020 ◽  
pp. 58-86
Author(s):  
Semjon F. Adlaj ◽  
◽  
Sergey N. Pozdniakov ◽  
Keyword(s):  
Comparative Analysis ◽  
Digital Media ◽  
Mathematical Knowledge ◽  
Information Environment ◽  
Teaching Mathematics ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Computer Tools ◽  
Digital Representations ◽  
The Impact

This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.

THE DEVELOPMENT OF MATHEMATICAL SPEECH OF FUTURE TEACHERS IN THE STUDY COURSE METHODOLOGY OF MATHEMATICS

2017 ◽  
Vol 1 ◽  
pp. 364
Author(s):  
Larisa Sergeeva ◽  
Alina Ledovaya
Keyword(s):  
Cultural Development ◽  
Professional Activity ◽  
Mathematical Language ◽  
Urgent Problem ◽  
Future Teachers ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Specialized Language

The article is devoted to setting forth of the urgent problem of the cultural development of mathematical speech of future teachers. The article shows the study of mathematical concepts by students taking into account specialized language of the mathematics as one of the aspects of this problem. The proposed method of development of mathematical speech is based on the constructed hierarchy of mathematical concepts consideration the specifics of mathematical objects’ nature and characteristics of the used mathematical language. The conducted research allowed to establish the influence of the created active studying for developing students' methodological speech, cognitive motives, interest in future professional activity.

Connecting Research to Teaching: Visual and Analytic Thinking in Calculus

2009 ◽  
Vol 103 (2) ◽  
pp. 140-145
Author(s):  
Erhan Selcuk Haciomeroglu ◽  
Leslie Aspinwall ◽  
Norma C. Presmeg
Keyword(s):  
Mathematics Education ◽  
Multiple Representations ◽  
Reform Movement ◽  
National Council ◽  
Analytic Thinking ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Ways Of Thinking ◽  
Calculus Reform

A frequent message in mathematics education focuses on the benefits of multiple representations of mathematical concepts (Aspinwall and Shaw 2002). The National Council of Teachers of Mathematics, for instance, claims that “different representations support different ways of thinking about and manipulating mathematical objects” (NCTM 2000, p. 360). A recommendation conveyed in the ongoing calculus reform movement is that students should use multiple representations and make connections among them so that they can develop deeper and more robust understanding of the concepts.

scholarly journals AVICENNA ON GRASPING MATHEMATICAL CONCEPTS

2021 ◽  
Vol 31 (1) ◽  
pp. 95-126
Author(s):  
Mohammad Saleh Zarepour
Keyword(s):  
Mathematical Object ◽  
Physical Object ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Concept Empiricism

AbstractAccording to Avicenna, some of the objects of mathematics exist and some do not. Every existing mathematical object is a non-sensible connotational attribute of a physical object and can be perceived by the faculty of estimation. Non-existing mathematical objects can be represented and perceived by the faculty of imagination through separating and combining parts of the images of existing mathematical objects that are previously perceived by estimation. In any case, even non-existing mathematical objects should be considered as properties of material entities. They can never be grasped as fully immaterial entities. Avicenna believes that we cannot grasp any mathematical concepts unless we first have some specific perceptual experiences. It is only through the ineliminable and irreplaceable operation of the faculties of estimation and imagination upon some sensible data that we can grasp mathematical concepts. This shows that Avicenna endorses some sort of concept empiricism about mathematics.

Connecting Research to Teaching: When Visualization Is a Barrier to Mathematical Understanding

2002 ◽  
Vol 95 (9) ◽  
pp. 714-717
Author(s):  
Leslie Aspinwall ◽  
Kenneth L. Shaw
Keyword(s):  
Mathematics Teachers ◽  
Mathematics Learning ◽  
Mathematical Understanding ◽  
Mathematical Representation ◽  
National Council ◽  
Mathematical Task ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Graphic Representations ◽  
Ways Of Thinking

AS MATHEMATICS TEACHERS, OUR INTUITION TELLS us that students benefit from being able to understand a variety of representations for mathematical concepts and being able to select and apply a representation that is suited to a particular mathematical task. Students develop mathematical power as they learn to operate on mathematical objects and to translate from one mathematical representation to another when necessary. For example, graphic representations convey mathematical information visually, whereas expressions represented symbolically may be easier to manipulate, analyze, or transform. The National Council of Teachers of Mathematics (NCTM) reinforces our intuition: “Different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses” (NCTM 2000, p. 360). Notwithstanding our intuition and experience with students' representations, we teachers tend to think that students' graphic and visual abilities are always an advantage in mathematics learning. Indeed, we accept on pedagogical faith that students' conceptual understandings are enhanced whenever they use visualization.

scholarly journals Second-Order Logic and Foundations of Mathematics

2001 ◽  
Vol 7 (4) ◽  
pp. 504-520 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
Foundations Of Mathematics ◽  
Mathematical Concepts ◽  
Informal Reasoning ◽  
First Order ◽  
Second Order Logic ◽  
Order Set

AbstractWe discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

scholarly journals Rules, understanding and language games in mathematics

2021 ◽  
pp. 35-45
Author(s):  
V. V. Tselishchev
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Language Game ◽  
Language Games ◽  
Logical Necessity ◽  
Foundations Of Mathematics ◽  
Mathematical Concepts ◽  
Philosophical Investigations ◽  
Form Of Life ◽  
Inherent Ambiguity

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games.

Application of Information Visualization Techniques to the Design of a Mathematical Mindtool: A Usability Study

2003 ◽  
Vol 2 (3) ◽  
pp. 142-159 ◽  
Author(s):  
Kamran Sedig ◽  
Sonja Rowhani ◽  
Jim Morey ◽  
Hai-Ning Liang
Keyword(s):  
Information Visualization ◽  
Usability Study ◽  
Visual Aids ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Learning Tasks ◽  
Cognitive Aids ◽  
Fertile Ground ◽  
Human Thinking ◽  
Visualization Techniques

One of the goals of information visualization is to support human thinking through the use of external visual aids. Mathematical mindtools can act as visual cognitive aids to enhance thinking and reasoning about mathematical objects and concepts. Although some mathematical mindtools incorporate information visualization techniques, the systematic use of these techniques in the design of these tools and their effect on users’ thinking and reasoning need to be investigated. A mathematical mindtool called PARSE (Platonic-Archimedean Solids Explorer) is presented in this paper. PARSE is intended to support the exploration and learning of a subset of geometric shapes. The mathematical objects and concepts embedded in PARSE have been enhanced using information visualization techniques. A usability study of PARSE and its information visualization techniques have been conducted and reported. The study shows that information visualization techniques enhance and support learning and exploration of mathematical concepts. The findings reported in this paper suggest that mathematical mindtools provide a fertile ground for investigating different information visualization techniques and their effectiveness in supporting learning tasks.

scholarly journals Digital Representations of Mathematical Objects in the Context of Various Forms of Representation of Mathematical Knowledge

2020 ◽  
pp. 58-86
Author(s):  
Semjon F. Adlaj ◽  
◽  
Sergey N. Pozdnyakov ◽  
Keyword(s):  
Comparative Analysis ◽  
Digital Media ◽  
Mathematical Knowledge ◽  
Information Environment ◽  
Teaching Mathematics ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Computer Tools ◽  
Digital Representations ◽  
The Impact

This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.

scholarly journals Digital Representations of Mathematical Objects in the Context of Various Forms of Representation of Mathematical Knowledge

2020 ◽  
pp. 58-86
Author(s):  
Semjon F. Adlaj ◽  
◽  
Sergey N. Pozdniakov ◽  
Keyword(s):  
Comparative Analysis ◽  
Digital Media ◽  
Mathematical Knowledge ◽  
Information Environment ◽  
Teaching Mathematics ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Computer Tools ◽  
Digital Representations ◽  
The Impact

This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.

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