Acquisition of the counting principles during the subset-knower stages: Insights from children’s errors

Studies on children’s understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn’s (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children’s understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel’s (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children’s acquisition of the counting principles, and that learning to coordinate all three principles represents an additional step beyond learning them individually.

The Relationship Between Confidence and Conformity in a Non-routine Counting Task With Young Children: Dedicated to the Memory of Purificación Rodríguez

Counting is a complex cognitive process that is paramount to arithmetical development at school. The improvement of counting skills of children depends on their understanding of the logical and conventional rules involved. While the logical rules are mandatory and related to one-to-one correspondence, stable order, and cardinal principles, conventional rules are optional and associated with social customs. This study contributes to unravel the conceptual understanding of counting rules of children. It explores, with a developmental approach, the performance of children on non-routine counting detection tasks, their confidence in their answers (metacognitive monitoring skills), and their ability to change a wrong answer by deferring to the opinion of a unanimous majority who justified or did not justify their claims. Hundred and forty nine children aged from 5 to 8 years were randomized to one of the experimental conditions of the testimony of teachers: with (n = 74) or without justification (n = 75). Participants judged the correctness of different types of counting procedures presented by a computerized detection task, such as (a) pseudoerrors that are correct counts where conventional rules are violated (e.g., first counting six footballs, followed by other six basketballs that were interspersed along the row), and (b) compensation errors that are incorrect counts where logical rules were broken twice (e.g., skipping the third element of the row and then labeling the sixth element with two number words, 5 and 6). Afterwards, children rated their confidence in their detection answer with a 5-point scale. Subsequently, they listened to the testimony of the teachers and showed either conformity or non-conformity. The participants considered both compensation errors and pseudoerrors as incorrect counts in the detection task. The analysis of the confidence of children in their responses suggested that they were not sensitive to their incorrect performance. Finally, children tended to conform more often after hearing a justification of the testimony than after hearing only the testimonies of the teachers. It can be concluded that the age range of the evaluated children failed to recognize the optional nature of conventional counting rules and were unaware of their misconceptions. Nevertheless, the reasoned justifications of the testimony, offered by a unanimous majority, promoted considerable improvement in the tendency of the children to revise those misconceptions.

Half Step Numerical Method for Solution of Second Order Initial Value Problems

This paper presents a half step numerical method for solving directly general second order initial value problems. The scheme is developed via collocation and interpolation technique invoked on power series polynomial. The proposed method is consistent, zero stable, order four and three. This method can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over some existing schemes of same and higher order.

Sociology as a Science of Reality: A Logical Foundation for the System of Sociology

The Centre for Fundamental Sociology (HSE University, Moscow) and Vladimir Dal Publishing House (St. Petersburg) have initiated the Russian translation and publication of Sociology as a Science of Reality: A Logical Foundation for the System of Sociology (1930), a key work of the famous German philosopher and sociologist, Hans Freyer. In the early 1920s, Freyer, who became the first full professor of sociology in Germany, published several seminal works covering a wide range of topics in social science and political philosophy. The Introduction to the thinker’s first work on sociology in its proper meaning, published here, has the characteristics of a program manifesto outlining the basic principles for comprehending the discipline and its subject matter as a social and historical phenomenon. Freyer argues that sociology as a scholarly discipline emerges in a society that is being detached from the state; now, instead of an obvious and stable order, an insecure, precarious and unpredictable society arises, becoming a problem for itself. Consequently, alongside the formation of sociology, its object emerges; it is a heterogeneous “society” that has gained autonomy from the state while sharply divergent from that same society regarding the principles of the organization of social life. Meanwhile, the distinctive feature of European sociology is not simply its embeddedness in history, but its immediate substantial connection with the preceding philosophical tradition. This enables Freyer to raise the question of the philosophical basis of sociology as a scientific system. He also formulates the task of defining the forms of this system and outlining its primary lines. The structural and methodological comparison between the European sociology version and the American version of the discipline is particularly interesting from the perspective of the academic history.

Connotation of “Belonging-Identity” and contemporary appeal under political confucianism

The politicization of Confucianism has always been an important dimension to the practice of Confucian realistic values. Strengthening the concept of national identity with Confucianism can maintain the stable order of the country, then realize the enlightenment of social ethics and promote democratic political reform. Yet, Confucianism does not have a specific way to interpret and realize national identity in its cultural connotation. Instead, it shows some meanings of “belonging-identity” in the relationship among individuals, society, and the country. The key point of Confucian “belonging-identity” is its moral nature. We should take the “Supreme goodness” of Confucianism into actualization, socialization, politicization to re-examine and construct an effective mechanism of national identity. Also, Confucianism belongs not only to China. We can apply the Confucian “belonging-identity” system into other Asian countries with similar cultures to realize harmonious relations among individuals, families, and countries. National identity is more a kind of sense, but if we use it in rational political philosophy, such as Confucianism, it can promote the development of democracy of a country.

Beckett's Broken Circle: A Literary Fractal

This essay proposes that the picture of a broken circle encountered by Watt during the second part of his tale marks a crucial collision point between Beckett's literary and mathematical interests and triggers a process of fractal scaling self-similarity. Building on recent interest concerning the role of the mathematics and mathematical forms found in Beckett's work, I argue that the broken circle depicted in the picture from Watt is a geometric form which (re)appears within at least three interlocking scales throughout Beckett's novel-length prose: (i) its moment of arrival in the picture from Watt, (ii) a macroscopic reinscription in the names of the protagonists populating the five novels spanning Watt through to The Unnamable and (iii) buried within the narratological depths of How It Is. As a structural principle, the interminable irregularity of fractals offered Beckett a viable solution for what he considered the defining task of the modern artist: ‘to find a form to accommodate the mess’. Moreover, the specific shape selected for his fractal is shown to contain within its geometry one of Beckett's most universal and pressing concerns: the inevitable insufficiency of language. Therefore, although this essay restricts itself to examining Beckett's novel-length prose, the idea of a broken circle fractal promises to provide a valuable heuristic through which to reassess the author's other generic avenues. Fractals thus offer a means through which one can bind together the length and breadth of Beckett's oeuvre without ever reducing dynamic chaos to stable order.

In Search of Order: Portrayal of Communists in Cold War Italy

This article analyzes the battle between Italian Communists and anti-Communists during the early Cold War. Without denying the critical role of high politics or the rivalry between Washington and Moscow, the article places the home front at the center of its analysis, using a bottom-up perspective. Drawing on the rhetoric, remembrances, and visual propaganda produced by Communists and anti-Communists alike, the article shifts the focus of attention away from governments and diplomacy and toward imagination and culture as agents of historical change. Looking beyond the institutional-administrative sphere of politics, the article explores the aspirations, emotions, expectations, and hopes formulated by Italians under conditions of existential uncertainty. The article concludes that the Cold War in Italy was, above all else, an internal contest between two parties over what the recent past meant and what would ensure a just and stable order for society.

Hybrid Wars and Conflict Management Experience on the Example of Democratic Republic of the Congo

The article is devoted to the issue of hybrid warfare and the experience of postwar management on the example of Democratic Republic of the Congo.It is revealed the weakness of the state that is not able to control the territory, the risks of democratic transformation and the influence of external forces. The author examines the tools of the hybrid war, analyzes the current situation in the country, the inability of the central government and the factors of external influence. The urgency of the topic is causes by importance of the planning postwar managementinUkraine, which is negotiating about peacekeeping mission on the territories of Donetsk and Luhansk regions. Ukraine needs to know the possible consequences of this decision, studieng the experience of the countries with similar problems. An example of DR Congo demonstrates the loss of capital sovereignty over economically important regions, which can not be returned without external assistance. Moreover, even in liberated territories it is difficult to establish a stable order. The processes of unconventional conflicts lead to the weakening of sovereignty and the reduction or even loss of subjectivity. Accordingly, the methods of solving these issues with the involvement of international organizations often freeze the situation, leaving the subjectivity of the governments recognized in the world fragile. Such a development scenario also threatens Ukraine in the case of introduction the peacekeeping forces. The author comes to the conclusion that hybrid conflicts lead to a weakening of sovereignty, which is deepened by the influence of external forces. Intervention of the international community does not give the expected effect. Keywords: hybridwar, conflict management, peacemaking operation.

The Settlement of 1815

This chapter looks at the order building in the settlement of 1815. The peace settlement that ended the Napoleonic wars in 1815 gave Europe the most elaborately organized political order yet. Led by Great Britain, the European states mounted a sustained effort to find a mutually agreeable, comprehensive, and stable order; this effort culminated in the celebrated Congress of Vienna. The Vienna settlement departed from earlier postwar settlements in the way the leading state attempted to use institutions to manage relations among the great powers. There were, however, sharp limits to the binding character of these institutions. The British proposal for general security guarantee failed primarily because of the inability of the states involved to make binding commitments. There are some traces of constitutional order, but the 1815 case reveals the limits to which nondemocratic states can create binding institutions.

BEAUTY IN MATHEMATICAL EDUCATION: SYNERGETIC WORLDVIEW

Introduction. The most important concepts underlying beauty are the mathematical concepts of symmetry and fractality. These categories are fundamental for modern mathematics, science and culture in general. However, in mathematical education and pedagogical literature, the ratio of these main categories has not been considered yet. Of special interest is the fact that the concepts of fractals, fractality and fractal geometry and fractal graphics are not included in the vast majority of high school programmes, although they have become commonly used among mathematicians and graphic designers.The aims of the article were the following: to demonstrate intersectionality and correlations of the basic concepts of symmetry and fractals from the point of view of synergetics, to establish the relevance of studying these concepts in the course of mathematics for aesthetic education of students and development of their worldview.Methodology and research methods. A significant role in the study is given to post-non-classical methodology based on synergetic worldview. The author employed the provisions of trinitarian methodology: in addition to two binary oppositions, the third element is necessary to solve the problem of contradiction of theseoppositions and integration into one coherent whole as the onditions of their coexistence. In the course of the research, analysis and generalisation of pedagogical and methodical literature, methods of comparative, historical and logical types of analysis were used.Results and scientific novelty.For centuries, beauty has been understood as a stable order and symmetry. The synergetrics as a general scientific theory about self-organisation of complex systems allows us to give another interpretation of beauty – as a kind of attractor, the result of self-organisation of nature or theflight of human thought. In the most general view, symmetry can be considered as transformation of similarity, which is also the core of another concept – fractality. On the one hand, fractality can be considered as one of the manifestations of symmetry in the broad sense. On the other hand, symmetry can be considered as a manifestation of fractality with a finite number of iterations. Thus, the concepts of symmetry and fractality are closely interrelated. Symmetry and fractality are two opposites, mutually complementing each other, aesthetically and mathematically mutually passing into each other. Symmetry reveals the beauty of a sustainable order and fractality reflects the beauty of the result of self-organisation of the chaos of nature or the freedom of the human mind. Therefore, symmetry and fractals are the most important concepts for the disclosure of the beauty of the universe, which determines their importance for mathematical learning and for aesthetic education of students.Practical significance. Taking into account the fact that the concepts of symmetry and fractals are directly related to each other, they should be jointly-taught. This will contribute to the development concept of mathematics education: to increase motivation for mathematical studies, to develop cognitive interests and activities, to narrow the gap between education and research processes, to overcome the problems with aesthetic education of students.