Abstraction and embodiment: exploring the process of grasping a general

This paper reports from a case study which explores kindergarten children’s mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes “a point of view from which the concrete can be seen as meaningfully related” (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children’s mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys’ awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys’ gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children’s concrete embodied activities.

Predicting the Mathematical Abstraction Processes Using the Revised Bloom’s Taxonomy: Secondary School 7th Graders

In this study, it was investigated whether it would be possible to observe abstraction processes of secondary school 7th graders using the Revised Bloom’s Taxonomy. For this purpose, eight students participated in the study. The study was conducted at a state secondary school in Turkey. Purposeful sampling method was used in the selection of students and different students were examined by their achievement levels. The research was modeled as a case study and the data were obtained through interviews. Therefore, the data were collected through an interview form developed by the researchers. The collected data were analyzed according to descriptive analysis method. The findings show that the abstraction process differs according to the dimensions of the taxonomy. Accordingly, it was determined that a student who abstracts information should behave at least at the application level in the cognitive level and at least at the conceptual knowledge level in the knowledge dimension. It was also considered that the Revised Bloom’s Taxonomy categorized the cognitive mechanisms required by abstraction processes thoroughly. Supporting this study with quantitative data is suggested so that the findings may become more significant. Keywords: mathematical abstraction, mathematics education, equations, Revised Bloom’s Taxonomy, APOS theory

Psychology of Dwelling and Visual Appropriations—An Anthropological Application

We find ourselves situated within a world that can be experienced visually, for the first time, in its wholeness. Using conceptual analysis, we intend to show that notions born within the practice of habitation, such as the sense of place, place attachment, and hearth, can help us evaluate the psychological implications of the images of Earth taken from space. We chose a phenomenological approach to human habitation because it allows concepts pertaining to connected and inherently interdisciplinary fields, for instance environmental psychology or human geography, to be reunited under the umbrella of an anthropological interpretation. The sensory and imaginary connotations of the notion of place may be noticed starting from the distinction between space as mathematical abstraction and concrete places being experienced directly. An analysis of the nature of this connection leads to the finding that we actively imagine and reimagine the surrounding world as an unfolding space in which we are constantly attempting to dwell. What is of particular interest for us is the manner in which technologically-mediated visual experience may inspire cognitive representations or may generate profound emotions, such as the attachment to a particular place. Therefore, the value of imagination for the anthropology of habitation is not rendered by its compensatory role, but by its link to ontogenesis. Familiar places, which continue to attract us, are capable of triggering unique imaginary processes, reveries which refer us to the primordial steps of ontogenesis with outmost intensity. The process of subjective appropriation of the world begins with that privileged space of origin specific to each of us, the space which we identify with most intensely. Thus, the psychological impact of the image of Earth from space: we become intensely aware that this planet is our Place within a hostile universe.

PENINGKATAN KEMAMPUAN ABSTRAKSI MATEMATIS SISWA SMPN 16 BANDUNG MELALUI PEMBELAJARAN DENGAN MODEL ELICITING ACTIVITIES (MEAS)

Abstraction ability in mathematics is very important because it can be used to describe mathematical concepts in a mathematical problem. One way to help students develop mathematical abstraction skills is to apply the Eliciting Activities Model (MEAs). The purpose of this study was to determine the increase in mathematical abstraction skills between students who received mathematics learning with MEAs compared to the expository model, and to determine students' attitudes towards mathematics learning with MEAs. The method used in this research is quasi-experimental. The population in this study were students of class VIII SMP Negeri 16 Bandung with a sample of two classes from all available class VIII. The instruments used are mathematical abstraction ability test instruments, questionnaires, observation sheets, and students' daily journals. The results showed that the improvement of mathematical abstraction skills with MEAs was better than students who received mathematics learning using the expository model. In addition, students gave a positive response to learning mathematics with MEAs. ABSTRAKKemampuan abstraksi dalam matematika sangat penting karena dapat digunakan untuk menggambarkan konsep matematis dalam sebuah permasalahan matematis. Salah satu cara untuk membantu siswa menumbuh kembangkan kemampuan abstraksi matematis adalah dengan menerapkan Model Eliciting Activities (MEAs). Tujuan dari penelitian ini adalah untuk mengetahui peningkatan kemampuan abstraksi matematis antara siswa yang mendapatkan pembelajaran matematika dengan MEAs dibandingkan dengan model ekspositori, dan mengetahui sikap siswa terhadap pembelajaran matematika dengan MEAs. Metode yang digunakan pada penelitian ini adalah kuasi eksperimen. Populasi pada penelitian ini adalah siswa kelas VIII SMP Negeri 16 Bandung dengan sampel dua kelas dari keseluruhan kelas VIII yang tersedia. Instrumen yang digunakan adalah instrumen tes kemampuan abstraksi matematis, angket, lembar observasi, dan jurnal harian siswa. Hasil penelitian menunjukkan bahwa peningkatan kemampuan abstraksi matematis dengan MEAs lebih baik dari pada siswa yang mendapatkan pembelajaran matematika dengan model ekspositori,. Selain itu, siswa memberikan respons yang positif terhadap pembelajaran matematika dengan MEAs.

Science vs. Sophistry—A historical debate on bipolar fuzzy sets and equilibrium-based mathematics for AI&QI

The road from bipolar fuzzy sets to equilibrium-based mathematical abstraction is surveyed. A continuing historical debate on bipolarity and isomorphism is outlined. Related literatures are critically reviewed to counter plagiarism, distortion, renaming, and sophistry. Based on the debate, the term “isomorphistry” is coined. It is concluded that if isomorphism is used correctly it can be helpful in mathematics. If abused it may become isomorphistry—a kind of historical, socially constructed, entrenched, and “noble” hypocrisy hindering major scientific advances. It is shown that isomorphistry can be motivated by “denying” the originality of bipolar fuzzy sets and aimed at “justifying” plagiarism and distortion. Thus, isomorphistry is sophistry on isomorphism . Some (-,+)-bipolar isomorphistry behaviors are critiqued. YinYang vs. YangYin are distinguished. The geometrical and logical basis of equilibrium-based AI&QI computing machinery is introduced as a new computing paradigm with logically definable causality for mind-body unity. A philosophical joke on sophistry is appended.

ANALYSIS OF MATHEMATICAL ABSTRACTION ABILITY OF GRADE VIII STUDENTS IN CARTESIAN COORDINATE MATERIAL

Mathematics is one of the most important branches of science because mathematics is indispensable for everyday life and is the basis for other branches of science. Therefore, students' abstraction ability is very important. Each student has abstraction ability in solving problems in mathematics lessons in solving different problems according to the students' level of thinking ability and intelligence. Therefore, the purpose of this research is to describe the mathematical abstraction ability of grade VIII students in cartesian coordinate material at SMPN 1 Cikedal. The type of research is a qualitative descriptive approach. The research subjects are 6 students of grade VIII A. The research instruments used are test sheets and interview guidelines containing 14 questions. The results showed that all the research subjects had low mathematical abstraction skills, as none of the students met all the levels and indicators used in this study, namely the level of recognition, representation, and structural abstraction.

On Entropy, Information, and Conservation of Information

The term entropy is used in different meanings in different contexts, sometimes in contradictory ways, resulting in misunderstandings and confusion. The root cause of the problem is the close resemblance of the defining mathematical expressions of entropy in statistical thermodynamics and information in the communications field, also called entropy, differing only by a constant factor with the unit ‘J/K’ in thermodynamics and ‘bits’ in the information theory. The thermodynamic property entropy is closely associated with the physical quantities of thermal energy and temperature, while the entropy used in the communications field is a mathematical abstraction based on probabilities of messages. The terms information and entropy are often used interchangeably in several branches of sciences. This practice gives rise to the phrase conservation of entropy in the sense of conservation of information, which is in contradiction to the fundamental increase of entropy principle in thermodynamics as an expression of the second law. The aim of this paper is to clarify matters and eliminate confusion by putting things into their rightful places within their domains. The notion of conservation of information is also put into a proper perspective.

A Tale of Two Maps: A Study of Western Cartographic Practices and Indigenous Pictorial Representation of Varanasi

This paper argues that western cartography notions and practices have rejected mainly the Indian way of understanding the space and its distinct representational approaches. The map-making tradition in India visualises space as organic, complex and connected arrangements, exercises for mapping Varanasi city carried out by the British scholar James Prinsep in 1822, in contrast, relied on mathematical abstraction and land centric ideologies. The consequences of such contrasting styles and methodologies for visualising space, I argue, was made most acutely perceptible in the manner in which the river Gaṅgā was understood and positioned within the respective frameworks. While in western cartographic reckoning Varanasi was considered to be a dry space that was abutting the flowing Ganga river, in the indigenous representational formats the very same space was characterised as being a region where land and water met and interwove a continuum between the fluvial and the terrestrial.