Typical errors in presenting concepts in textbooks for the first four grades of primary school

This paper offers a systematization of typical errors in presenting scientific concepts in textbooks for the first four grades of primary school. The subject of our analysis and systematization were presentations of concepts which deviate from or violate the internal cognitive and logical nature of a scientific concept, thus representing a source of potential difficulties for students in understanding scientific knowledge. Starting from Vygotsky's theory of the development of scientific concepts, as well as the general standards of textbook quality and a review of studies analyzing textbooks in this field, we have made a systematization of typical errors in the presentation of scientific terms. Five typical errors are explained and elucidated: a simple description of a phenomenon or the statement of its function, use or usefulness; a simple establishment of connections between a concept (word) and an object (image); offering ready-made phrases and scientific statements without relating them to a system of concepts; providing only typical examples or providing examples that lack variety, and presenting important and unimportant facts on the same level, without pointing out the differences. Every typical error is explained using examples from textbooks in which scientific concepts relevant to grades 1-4 are introduced (settlement, village, city, plants, relief, historical figure and birds). In the absence of scientific principles in presenting concepts in textbooks, their authors rely on implicit assumptions about concepts as phenomenal or factual kinds of knowledge. Due to the importance of acquiring scientific concepts for the cognitive development of the individual, the practical implications of the findings are that in textbook design but also in teacher education particular attention must be devoted to the area of teaching and learning scientific concepts.

Epistemology and rationality of intuition and imagination in the field of mathematics

This paper aims to contribute to the clarification of the role of mathematical intuition and imagination in the constitution of mathematical knowledge, evidencing its epistemological and procedural characteristics. For that, an "epistemology of intuition and imagination" in the field of mathematics is outlined (suggested) emphasizing the need to adopt a dynamic conception of mathematics. In this context, intuition and imagination present themselves as forms of mathematical experience that give access, through paths that are not purely logical, to mathematical knowledge. Its epistemological and rationality characteristics, a rational of a non-logical nature, are highlighted by several examples, resources for moving the ideas involved. The epistemological study of intuition and imagination also allows highlighting its ontology, constituted of more relations than objects. From the pedagogical point of view, we discuss the formative character of philosophical studies involving intuition and imagination, mainly related to creativity in mathematics. Keywords: Mathematical knowledge; Mathematical experience; Epistemology of intuition and imagination; Creativity in mathematics.

THE SYSTEM OF CATEGORIES OF PSYCHOLOGY AND THE CONCEPTS OF THE INDIVIDUAL, SUBJECT OF ACTIVITY AND PERSONALITY: SOME LESSONS OF THE CONCEPTS OF KANT, FICHTE, HEGEL AND SARTRE

The categories of psychology are understood as the limit generalizations of various classes of psychical phenomena, including both the corresponding groups of particular concepts and the phenomena not yet subjected to final conceptual identification. They can be understood as the basic determinants of the psychic, having a socio-cultural nature (at the level of the subject of activity and personality) or bio-logical nature (at the level of the individual). The individual, the subject of activity and the personality, not being forms of “psychic”, should nev-ertheless possess special levers of regulation (or self-regulation) of such basic determinants of psyche, as motive, image, communication and action (that indirectly assumes also regulation of all vital activity of the individual subject of life). Not limited to the formula “personality as a transformed individual”, the author reveals the genetic continuity of the levels of the individual, the subject of activity and personality in the aspect of increasing of the degree of manifestation of the generic essence and “essential forces” of man in his individual exis-tence. At the same time, analyzing multilevel human activity from the point of view of “spatial” paradigm (in the aspect of human integration into different spheres of living space), the author finds the key to the relative independence of these levels and to the constant transitions from one of these levels of life activ-ity to another. The justification of the proposed provisions is given on the complex of the corresponding ideas of Kant, Fichte, Hegel and Sartre, taking into account the continuity between them.

Design Complexity for Objective Function Points

This paper investigates correlating the basic elements of Unified Modeling Language and Cyclomatic Complexity with Function Point Analysis (FPA) principles to develop an automated software functional sizing tool. This concept has been difficult to achieve due to the logical nature of the FPA sizing methodology versus the physical nature of source lines of code (SLOC). In this approach, we examine software complexity from design and maintainability perspectives in order to understand relationships in physical code. Our hypothesis is that this method will “simulate” FPA principles and produce an objective sizing method. This would provide the foundation for an automated tool that scans physical software code to derive “Objective Function Points”(OFPs) functional size measure.

Second-Order Abstraction Before and After Russell’s Paradox

In this article, I discuss certain aspects of Frege’s paradigms of second-order abstraction principles, Hume’s Principle and Basic Law V, with special emphasis on the latter. I begin by arguing that, contrary to a widespread view, Frege did not express any dissatisfaction with Basic Law V before 1902. In particular, he did not raise any doubt about its assumed logical nature. I then show why Frege nonetheless fails to justify Basic Law V as a primitive logical truth along the lines of the semantic justification that he provides for the other axioms of his system. In subsequent sections, I argue (a) that Frege could not have chosen Hume’s Principle as a logical axiom, neither before 1902 nor after 1902; (b) that even if in the wake of Russell’s Paradox Frege had accepted Hume’s Principle as a logical axiom, such an axiom could not have replaced Basic Law V which was designed to introduce logical objects of a fundamental and irreducible kind and to afford us the right cognitive access to them; (c) that Frege most likely held that the two sides of Basic Law V express different thoughts; (d) that for Basic Law V or for any other Fregean abstraction principle that is laid down as an axiom of a theory, the case in which both real epistemic value and self-evidence are given their due is ruled out. I make a proposal as to how Frege might have escaped this epistemic dilemma.

The new big picture for linguistics: Complex systems

In the history of linguistics there have been crucial moments when those of us interested in language have essentially changed the way we study our subject. We stand now at such a moment. In this presentation I will review the history of linguistics in order to highlight some past important changes in the field, and then turn to where we stand now. Some things that we thought we knew have turned out not to be true, like the systematic, logical nature of languages. Other things that we had not suspected, like a universal underlying emergent pattern for all the features of a language, are now evident. This emergent pattern is fractal, that is, we can observe the same distributional pattern in frequency profiles for linguistic variants at every level of scale in our analysis. We also have hints that time, as the persistence of a preference for particular variants of features, is a much more important part of our language than we had previously believed. We need to explore the new realities of language as we now understand them, chief among them the idea that patterned variation, not logical system, is the central factor in human speech. In order to account for what we now understand, we need to get used to new methods of study and presentation, and place new emphasis on different communities and groups of speakers. Because the underlying pattern of language is fractal, we need to examine the habits of every group of speakers at every location for themselves, as opposed to our previous emphasis on overall grammars. We need to make our studies much more local, as opposed to global. We do still want to make grammars and to understand language in global terms, but such generalizations need to follow from what we can now see as the pattern of language as it is actually used.

Logicism

The term ‘logicism’ refers to the doctrine that mathematics is a part of (deductive) logic. It is often said that Gottlob Frege and Bertrand Russell were the first proponents of such a view; this is inaccurate, in that Frege did not make such a claim for all of mathematics. On the other hand, Richard Dedekind deserves to be mentioned among those who first expressed the conviction that arithmetic is a branch of logic. The logicist claim has two parts: that our knowledge of mathematical theorems is grounded fully in logical demonstrations from basic truths of logic; and that the concepts involved in such theorems, and the objects whose existence they imply, are of a purely logical nature. Thus Frege maintained that arithmetic requires no assumptions besides those of logic; that the concept of number is a concept of pure logic; and that numbers themselves are, as he put it, logical objects. This view of mathematics would not have been possible without a profound transformation of logic that occurred in the late nineteenth century – most especially through the work of Frege. Before that time, actual mathematical reasoning could not be carried out under the recognized logical forms of argument: this circumstance lent considerable plausibility to Immanuel Kant’s teaching that mathematical reasoning is not ‘purely discursive’, but relies upon ‘constructions’ grounded in intuition. The new logic, however, made it possible to represent standard mathematical reasoning in the form of purely logical derivations – as Frege, on the one hand, and Russell, in collaboration with Whitehead, on the other, undertook to show in detail. It is now generally held that logicism has been undermined by two developments: first, the discovery that the principles assumed in Frege’s major work are inconsistent, and the more or less unsatisfying character (or so it is claimed) of the systems devised to remedy this defect; second, the epoch-making discovery by Kurt Gödel that the ‘logic’ that would be required for derivability of all mathematical truths can in principle not be ‘formalized’. Whether these considerations ‘refute’ logicism will be considered further below.

The Transcendental Foundations of Science

It is a philosophical naiveté to believe that the object of science is some ready-made world out there that the scientist, free of any preconceptions, simply stumbles upon. Of course, there is a world out there, given to us through the senses, but that must be intentionally elaborated to become a world for us and a possible object of scientific inquiry. The intentional constitution of the world of science supports and “justifies” a priori conceptions about the empirical world, even those of a logical nature, that are, then, properly transcendental rather than metaphysical. My goal here is to investigate what these presuppositions are and on what they are based.