Transcultural Networks and Connectivities: The Circulation of Mathematical Ideas between India and England in the Nineteenth Century

The nineteenth century has been characterised as a period in which mathematics proper acquired a disciplinary and institutional autonomy. This article explores the intertwining of three intersecting worlds of the history of mathematics inasmuch as it engages with historicising the pursuit of novel mathematics, the history of disciplines and, more specifically, that of the British Indological writings on Indian mathematics, and finally, the history of mathematics education in nineteenth century India. But, more importantly, the article is concerned with a class of science and mathematics teaching problems that are taken up by researchers—in other words, science and mathematics teaching problems that lead to scientific and mathematical research. The article argues that over a period of 50 years, a network of scholars crystallised around a discussion on mathematics proper, the history of mathematics and education. This discussion spanned not just nineteenth-century England but India as well, involving scholars from both worlds. This network included Scottish mathematicians, East India Company officials and administrators who went on to constitute the first generation of British Indologists, a group of mathematicians in England referred to as the Analytics, and traditional Indian scholars and mathematics teachers. The focus will be on the concerns and genealogies of investigation that forged this network and sustained it for over half a century.

Domestication of Mathematical Pathologies

Certain mathematical objects bear the name “pathological” (or “paradoxical”). They either occur as unexpected and (temporarily) unwilling in mathematical research practice, or are constructed deliberately, for instance in order to delimit the scope of application of a theorem. I discuss examples of mathematical pathologies and the circumstances of their emergence. I focus my attention on the creative role of pathologies in the development of mathematics. Finally, I propose a few reflections concerning the degree of cognitive accessibility of mathematical objects. I believe that the problems discussed in the paper may attract the attention of philosophers interested in concept formation and the development of mathematical ideas.

POLYPHONIC GENRES IN PIANO CREATIVITY OF CHINESE COMPOSERS

Statement of the problem. The twentieth century marked an increased interest in polyphonic music. The geography of polyphonic works for piano expanded significantly and a creative development of many Chinese composers, writing polyphonic piano pieces, took place. Today, polyphonic pieces make up a significant part of the piano repertoire in China, but they are little studied by musicologists and performers. The objective of this study – to reveal the contribution of Chinese composers to the creation of polyphonic piano repertoire of the XX – early XXI century. Analysis of the research and publications on the theme. А large number of modern authors study polyphony from the point of physical and mathematical research methods (Igarashi, Yu. & Ito, Masashi & Ito, Akinori, 2013; Weiwei, Zhang & Zhe, Chen, & Fuliang, Yin, 2016; Li, Xiaoquan et al. others, 2018). This approach does not reveal the factual musical component of polyphonic genres. In the 20th century, musicologists explored polyphony in musical folklore (Wiant, 1936; Fan Zuyin, 2004; Li Hong, 2015) and in professional Chinese composing (Sun Wei-bo, 2006, Winzenburg, 2018). The scientific novelty. This article studies the role of Chinese composers in the development of the world polyphonic piano repertoire of the XX – early XXI century. The methodological basis for the analysis of polyphonic works was the theoretical concepts of P. Hindemith, Peng Cheng, Fang Zuin, Li Hong, Sun Wei-bo. The results of the study. The research outcomes demonstrate the evolutionary development of the genre diversity of Chinese piano polyphony as well as those composers who created magnificent musical pieces. Conclusions. Chinese composers have fully mastered the art of modern counterpoint, represented by the genres of polyphonic program pieces (He Lu Ting), invention (Xiao Shu Xian, Du Qian, Sun Yun Yin, Chen Chen Quang), polyphonic suite (Ma Gui), large polyphonic cycle ( He Shao, Chen Hua Do, Xiao Shu Xian), fugue (Li Jun Yong, Yu Su Yan, Chen Gang, Tian Lei Lei, Duan Ping Tai, Zheng Zhong, Xiao Shu Xian) and small cycle “Prelude and Fugue” (Ding Shan Te, Chen Zhi Ming, Wang Li Shan). Creatively assimilating and rethinking the experience of Western polyphonists, Chinese composers have filled their polyphonic works with national features, firmly linking them with the origins of Chinese traditional and folk music. The polyphonic way of transmitting musical material becomes the most expressive at the moments of profound creativity and musical dramatization.

Structure of Mathematical Research

In this paper, we discuss about the structure of various components of mathematical research. We mainly focus on the structure of PhD proposal, PhD thesis and research article. We describe in detail about the standard structure of these components in the context of mathematics.

INVESTIGAÇÃO MATEMÁTICA: UMA ALTERNATIVA METODOLÓGICA PARA O ENSINO DA GEOMETRIA COM ALUNOS DO 5º ANO DO ENSINO FUNDAMENTAL / MATHEMATICAL RESEARCH: A METHODOLOGICAL ALTERNATIVE FOR TEACHING GEOMETRY WITH STUDENTS OF THE 5th YEAR OF FUNDAMENTAL EDUCATION

Neste trabalho, teve-se por objetivo, analisar estratégias que os alunos de uma turma de 5º ano do Ensino Fundamental, utilizam ao realizar tarefas investigativas, envolvendo o cálculo de áreas e perímetros de figuras planas. Ademais procurou-se, investigar as conjecturas elaboradas por estes alunos para comparar figuras de mesma área, mas com valores de perímetros diferentes e vice-versa. Foram utilizadas as etapas propostas por Ponte, Brocardo e Oliveira (2006), para desenvolver duas tarefas envolvendo a Investigação Matemática. Como instrumentos de coletas de dados foram utilizados diários de campo, resolução de tarefas, observações, questionários, gravação de voz e filmagens. Para a análise dos dados, optou-se pela análise descritiva, que consiste na descrição de características de determinados fenômenos. Para a resolução das tarefas investigativas propostas os alunos usaram o material concreto e o desenho. Percebeu-se que o trabalho em grupo foi produtivo, para elaboração das conjecturas e compreensão dos conceitos geométricos.