On Painter’s Paradox: Contextual and Mathematical Approaches to Infinity

Wijeratne, Chanakya; Zazkis, Rina

doi:10.1007/s40753-015-0004-z

Cited by 12 publications

(4 citation statements)

References 16 publications

(20 reference statements)

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“…To create Gabriel's horn [8,9], one constructs the graph of 1 z . By considering the domain z ≥ z 0 , the graph is rotated about the z-axis in three dimensions to form pseudosphere.…”

Section: Proof Of the Complexity-volumementioning

confidence: 99%

Dark Matter on the Closed Region Connected With Black Hole by Wormhole

Bousder¹

2021

Preprint

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In this letter, instead of choosing the Einstein Rosen bridge between two black holes as in ER=EPR, we consider a wormhole between a black hole and a closed edge of the wormhole. We assume that information in a black hole travels through a wormhole, turns to mass (dark matter) in the closed region. This study is in contradiction with the existence of the white hole in our Universe. We replace the notion of the white hole with the massive closed region. We prove the metric of the closed region by the Hopf fibration, this new metric generalizes the $AdS_{5}$\ metric.

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“…To create Gabriel's horn [8,9], one constructs the graph of 1 z . By considering the domain z ≥ z 0 , the graph is rotated about the z-axis in three dimensions to form pseudosphere.…”

Section: Proof Of the Complexity-volumementioning

confidence: 99%

Dark Matter on the Closed Region Connected With Black Hole by Wormhole

Bousder¹

2021

Preprint

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“…The main obstacle of learning is the lack of experience. This claim is also relevant to the learning and understanding of infinity properties (Denbel, 2014;Tsamir & Tirosh, 2007;Wijeratne & Zazkis, 2015). Tall's (2008) mentioned in his transition-in-thinking framework that students lacked infinite experience and hence encountered difficulties when learning in a context of infinite sets.…”

Section: Introductionmentioning

confidence: 97%

“…Another type is reasoning derived from an inadequate understanding of infinity. Lacking a proper thinking tool or schema to deal with an infinite task, students could generate improper reasoning and hence the thought of infinity as singularity and incomparability (Mamolo & Zazkis, 2008;Wijeratne & Zazkis, 2015). Having been introduced to the formal concept of infinite sets comparison, whose traditional approach involves the Cantorian Set Theory, some students still struggled as they inconsistently used several criteria to compare the infinite sets without awareness of a contradiction (Tsamir, 1999).…”

Section: Introductionmentioning

confidence: 99%

Capturing Conceptual Development through the Embodied-Based Experience in Infinite Sets Comparison

Cherdsak¹,

Nokkaew²,

Wongkia³

2019

INT J INSTRUCTION

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Designing pedagogical experience to serve as groundwork on which to build an understanding of abstract concepts is a challenging mission for educators. Much research has found that embodied activities could facilitate conceptual metaphor for students to understand such concepts. This study has captured the trajectory of reasoning occurred during the embodied-based experience designed for the infinity concept. The participants, 42 secondary students who had never been exposed to the formal concept of infinity, passed through a designed series of making-sense tasks, so called the Embodied-Based Activity of Infinite Sets Comparison (EBIC). The EBIC intervention was conducted in 2 phases; (i) careful observation of 2 voluntary students, and (ii) integration for 40-classroom practice. The first phase was to examine the students' conceptual development through their utterances, actions and inscriptions. Open-ended questions were used to capture the students' insights and reasoning in the second phase. The qualitative data was collected and mined to associate pairability and Cantor's metaphor. The finding showed the reasoning trajectory of some participants and the shift from improper reasoning to 1-1 correspondence pairing of countably infinite sets. This mathematical thinking shift was a consequence of conceptual embodiment, outlining the analysis of the embodied-based experience to develop abstract reasoning.

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“…The literature refers to studies that raise the most common difficulties in understanding infinity among students and in-service or prospective teachers. To name a few examples: its abstract nature (Manfreda Kolar and Čadež, 2012), its double dichotomy both actual and potential (Dubinsky et al, 2005a(Dubinsky et al, , 2005bMonaghan, 2001), the lack of a pictorial or mental image of what is being represented (Ángeles-Navarro and Pérez-Carreras, 2010), large finite numbers deemed as infinites (Manfreda Kolar and Čadež, 2012;Medina Ibarra et al, 2019;Singer and Voica, 2003), or contradictory intuitions when working with infinity (Fuentes and Oktac, 2014;Tirosh, 2002;Wijeratne and Zazkis, 2015).…”

Section: Introductionmentioning

confidence: 99%

Exploring a Mathematics Teacher’s Conceptions of Infinity: The Case of Louise

Diaz-Espinoza,

Juárez-López,

Juárez-Ruiz

2023

indonesian.j.of.mathematics.education

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Several papers studied infinity from the difficulties that students and teachers show in developing the concept. For this study, it was considered the analysis of the equality 0.999 … = 1. Mainly, this research aims to show that a mathematics teacher presents erroneous conceptions just like a student; that is, both students and teachers have the same difficulties in the concept of infinity. To this aim, a semi-structured interview was conducted with an in-service mathematics teacher in Tlaxcala, Mexico. The purpose of this research is to exhibit a high school math teacher’s misconceptions about the concept of infinity. In general, misconceptions found here can be divided into four groups: without a clear picture of the concept of infinity, an infinite periodic decimal number cannot be a representation of a finite number, a decreasing infinite sum cannot lead to a finite number and an infinite process is limited in real life is finite and has ended. The results obtained were compared with those already available in references about the difficulties with students and teachers, finding that the results shown here are like those reported in the literature. This highlights the need to overcome the teacher’s conceptions of infinity in future research.

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On Painter’s Paradox: Contextual and Mathematical Approaches to Infinity

Cited by 12 publications

References 16 publications

Dark Matter on the Closed Region Connected With Black Hole by Wormhole

Dark Matter on the Closed Region Connected With Black Hole by Wormhole

Capturing Conceptual Development through the Embodied-Based Experience in Infinite Sets Comparison

Exploring a Mathematics Teacher’s Conceptions of Infinity: The Case of Louise

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