Representation Of Mathematical Knowledge Research Articles

Three‐component mathematics for students

AbstractThere has been a long‐standing debate on situational and symbolic mathematics, which is associated with how to design and execute mathematics education for all students. Brain studies can give some clues for how to deal with the debate. There are situational, verbalized, and visuospatial brain networks and the connectivity among the networks, which suggest the representation of mathematical knowledge should be situational, verbalized, and symbolic. Moreover, the representation should be mapped onto one another. Previous studies have shown the importance of situational and verbalized mathematics, but they are typically treated as preliminary and auxiliary introductions for formal symbolic mathematics. Each of the three mathematics types and their mapping would be the core learning goal in mathematics education.

Neural network model of mathematical knowledge and development of information and educational environment for mathematical training of engineers

The paper proposes a new model of mathematical knowledge representation based on the hierarchical neural network structure of the links between various elements of knowledge. The concept of the level of knowledge and the minimum element of knowledge-«the quantum of knowledge» are introduced. On the base of the developed model, the structuring of mathematical knowledge corresponding to the basic mathematical training of engineers was carried out. The information and educational environment Nomotex was developed for mathematical training of engineers based on the proposed model of knowledge representation and a unique collection of 3D animation of all basic mathematical knowledge, as well as examples of the use of knowledge in engineering. The information and educational environment allows implementing individual educational trajectories in which the educational process is automatically formed from the neural network structure of knowledge according to the specified output competences of the engineer. A specialized software for visualizing educational trajectories was developed. Informational and educational environment allows to realize the multilevel and continuity of mathematical preparation, taking into account the nature of the activity of engineers in their mathematical preparation, taking into account the specifics of engineering training in mathematical courses. The information and educational environment allows implementing distance education, creating online courses, video lectures and automated testing. The introduction of the information and educational environment provides an increase in the attractiveness of the learning process through the extensive use of 3D visualization and animation, as well as computer game elements in the process of mathematical learning. Information and educational environment NOMOTEX, provides the possibility of implementing a systematic approach in the study of mathematical knowledge. The results of the experimental implementation of the information and educational environment in the educational process of engineers at the Bauman MSTU are presented.

IMPLEMENTATION OF THE SHELL AND KNOWLEDGE PORTAL FOR MATHEMATICAL PROOFS VERIFICATION ON THE IACPaaS PLATFORM

The paper presents a conceptual architecture of the shell for interactive systems for mathematical proofs verification and an evolutionary thematic knowledge portal created with its help. The process of implementation of all software and information components of the shell on the IACPaaS cloud platform with the use of its technologies and tools of their support is described. The process of development of the initial state of the knowledge portal for mathematical proofs verification using shell tools is considered. The way to use the knowledge portal by members of the mathematical community as well as mechanisms for changing the state of the portal by its administrator is discussed. The initial state of the knowledge portal includes: the ontology model of the mathematical knowledge base, including the specification of the initial state of the language for mathematical knowledge representation, the editor for the ontology model of the mathematical knowledge base, the mathematical knowledge base editor, the base of methods of reasoning editor, and the problem solver of the shell that implements the process of constructing proofs in terms of ontology model of proofs. The initial state of the knowledge portal also includes the initial state of the mathematical knowledge base and the initial state of the base of the methods of reasoning. The evolvement of the knowledge portal consists in the development of the three listed information components. All interested members of the mathematical community can participate in this process. It is implemented with the use of personal cabinets of the IACPaaS platform, where each user can independently develop his personal copy of the current state of the common knowledge portal. The transfer of new results to the common knowledge portal is controlled by its administrator.

Mathematical knowledge representation: semantic models and formalisms

The paper provides a survey of semantic methods for solution of fundamental tasks in mathematical knowledge management. Ontological models and formalisms are discussed. We propose an ontology of mathematical knowledge, covering wide range of fields of mathematics. We demonstrate applications of this representation in mathematical formula search, and learning.

Bilingual learning and mathematical knowledge representation in the brain

Bilingual learning and mathematical knowledge representation in the brain

What Is a “Semiotic Perspective”, and What Could It Be? Some Comments on the Contributions to This Special Issue

This comment attempts to identify different “semiotic perspectives” proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to “semiotics in mathematics education”.

Representation of Mathematical Knowledge in National Knowledge Infrastructure

The representation of mathematical knowledge is an important aspect of knowledge representation. It is the foundation for knowledge-based automated theorem proving, mathematical knowledge retrieval and intelligent tutoring systems, etc. According to the problems that are encountered in designing the mathematical knowledge representation language in NKI (national knowledge infrastructure) and after the discussion of ontological assumptions for mathematical objects, two kinds of formalisms for the representation of mathematical knowledge are provided. One is a description logic in which the range of an attribute can be a formula in some logical language; and another is a first order logic in which an ontology represented by the description logic is a part of the logical language. In the former representation, if no restrictions are imposed on formulas, then there is no algorithm to realize the reasoning in the resulted knowledge base; in the latter representation, the reasoning in the ontology represented by the description logic is decidable, while in general, for mathematical knowledge described by the first order logic which contains the ontology represented by the description logic, there is no algorithm to realize its reasoning. Hence, in the representation of mathematical knowledge, it is necessary to distinguish conceptual knowledge (knowledge in an ontology) and non-conceptual knowledge (knowledge represented by a language containing the ontology). Frames and description logics can represent and reason effectively about conceptual knowledge, but the addition of non-conceptual knowledge to frames or knowledge bases may make the reasoning in the resulted knowledge bases not decidable and there is even no algorithm to reason about the knowledge bases. Therefore, it is suggested that in representing mathematical knowledge, frames or description logics are used to

Consequences of Skill: The Case of Abacus Training in Taiwan

A general orientation toward the study of skills and their development is outlined, in which analyses of representation, transfer, and context are used to explore the consequences of developing a specific skill. This general approach is then applied to the study of abacus training and its implications for school achievement and cognitive development. Previous studies of abacus skill are reviewed, and two new studies are reported. Previous research showed abacus training to result in qualitative changes in children's representations of mental calculation through development of a "mental abacus." In Study 1, mental abacus skill was found to develop primarily as a result of practice rather than of selection factors such as socioeconomic status, ability, and previous mathematical knowledge. Abacus skill did, however, have positive effects on future achievement. Study 2 clarified the mechanisms underlying the functional relationships discovered in Study 1. Abacus training was found to affect both calculation skills and conceptual knowledge of the numeration system. The consequences of abacus training are both varied and limited. Consideration of effects of abacus training on the mental representation of mathematical knowledge, transfer of abacus skill to other tasks, and the contexts in which abacus skill develops demonstrates the multiple ways in which specific skills can contribute to cognitive development.