Student Thinking Levels in Solving Open-Ended Geometric-Function Problem by Algebraic Representation Approach

Pembuktian dalam matematika adalah suatu aktivitas yang penting, tetapi aktivitas ini tergolong sulit bagi mahasiswa calon guru matematika. Masalah ini salah satunya dipengaruhi oleh kepercayaan-diri. Tujuan penelitian ini adalah untuk menganalisis karakteristik pencapaian kemampuan pembuktian matematis dan kepercayaan-diri mahasiswa melalui metode Moore. Penelitian ini menggunakan metode campuran bertahap yaitu tahap kuantitatif dan tahap kualitatif. Pada tahap kuantitatif disimpulkan bahwa kemampuan pembuktian pada kelas yang menggunakan metode Moore lebih baik daripada kelas yang menggunakan model pembelajaran langsung. Metode Moore dapat mengungkap proses perkembangan capaian pembelajaran mahasiswa dalam pembuktian, sehingga dosen dapat memberikan umpan balik untuk mengembangkannya. Pada tahap kualitatif, dihasilkan karakteristik kemampuan pembuktian beberapa mahasiswa. Karakteristik ini ditinjau berdasarkan respon mahasiswa terhadap masalah pembuktian. Pada pembelajaran dengan metode Moore, mahasiswa tidak diperbolehkan membuka bahan ajar, sehingga dosen harus mengikuti alur berpikir mahasiswa dan mengarahkan proses berpikirnya. Sebagai implikasi, metode Moore baik digunakan dengan catatan mahasiswa harus belajar terlebih dahulu sebelum pembelajaran di kelas.Proving in mathematics is an important activity, but this activity is classified as difficult for prospective mathematics teacher students. This problem is influenced by self-confidence. The purpose of this study was to analyze the characteristics of achievement of students' mathematical proving ability and self-confidence through the Moore method. This study uses a phased mixed method, namely quantitative and qualitative stages. In the quantitative stage, it was produced: "Based on the overall and PAM categories, the ability to prove the class using the Moore method is better than the class that uses the direct learning model". Moore's method can reveal the process of developing student learning outcomes in proof, so that lecturers can provide feedback to develop it. In the qualitative stage, the characteristics of the ability of several students are produced. these characteristics are reviewed based on student responses to the problem of proof. In the Moore method of learning, students are not allowed to open teaching materials, so the lecturer must follow the flow of student thinking and direct the thinking process. As an implication of the results of this study, the Moore method is well used with the notes that students must study before learning in class.

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On mappings with finite length distortion on Riemann surfaces

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2019-33-4 ◽

2019 ◽

Vol 33 ◽

Cited By ~ 1

Author(s):

Serhii Volkov ◽

Vladimir Ryazanov

Keyword(s):

Finite Length ◽

Riemann Surfaces ◽

Boundary Behavior ◽

Main Tool ◽

Natural Generalization ◽

Finite Distortion ◽

Lipschitz Mappings ◽

Geometric Function ◽

Mappings With Finite Distortion ◽

Length Distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

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Focus and contrastive topic in questions and answers, with particular reference to Turkish

Theoretical Linguistics ◽

10.1515/tl-2020-0001 ◽

2020 ◽

Vol 46 (1-2) ◽

pp. 1-71 ◽

Cited By ~ 1

Author(s):

Beste Kamali ◽

Manfred Krifka

Keyword(s):

Algebraic Representation ◽

Similar Role ◽

Discourse Coherence ◽

Dynamic Interpretation ◽

Contrastive Topic ◽

Questions And Answers ◽

Alternative Questions

AbstractMuch recent research has recognized the importance of focus and contrastive topic in assertions for discourse coherence. However, with few exceptions, it has been neglected that focus and contrastive topic also occur in questions, and have a similar role in establishing coherence. We propose a framework of dynamic interpretation based on the notion of Commitment Spaces that show that a uniform interpretation of focus and contrastive topic is possible. The algebraic representation format is rich enough so that a separate introduction of discourse trees is not necessary. The paper discusses these phenomena for Turkish, a language with an explicit focus marker for polar and alternative questions, which distinguishes focus from contrastive topic.

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Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

European Journal of Applied Mathematics ◽

10.1017/s0956792521000127 ◽

2021 ◽

pp. 1-26

Author(s):

ELENA CHERKAEV ◽

MINWOO KIM ◽

MIKYOUNG LIM

Keyword(s):

Composite Materials ◽

Series Expansion ◽

Function Theory ◽

Effective Conductivity ◽

Two Dimensions ◽

Boundary Integral ◽

Geometric Function Theory ◽

Boundary Integral Formulation ◽

Geometric Function ◽

Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

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On the algebraic representation of projectively embeddable affine geometries

Journal of Geometry ◽

10.1007/bf01222863 ◽

1995 ◽

Vol 54 (1-2) ◽

pp. 155-160 ◽

Cited By ~ 1

Author(s):

Stefan E. Schmidt

Keyword(s):

Algebraic Representation ◽

Affine Geometries

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Bianchi type–I Model with Time Varying Λ and G: The Generalized Solution

Open Astronomy ◽

10.1515/astro-2020-0012 ◽

2020 ◽

Vol 29 (1) ◽

pp. 89-93

Author(s):

Alnadhief H. A. Alfedeel

Keyword(s):

Analytical Solution ◽

Bianchi Type ◽

Generalized Solution ◽

Type I ◽

Time Varying ◽

Field Equations ◽

Geometric Function ◽

Varying Λ ◽

The Universe ◽

Average Scale Factor

AbstractIn this paper, we have investigated the homogeneous and anisotropic Bianchi type–I cosmological model with a time-varying Newtonian and cosmological constant. We have analytically solved Einstein’s field equations (EFEs) in the presence of a stiff-perfect fluid. We show that the analytical solution for the average scale factor for the generalized Friedman equation involves the hyper-geometric function. We have studied the physical and kinematical quantities of the model, and it is found that the universe becomes isotropic at late times.

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Implications of Lesson Study for Tertiary-Level EFL Teachers’ Professional Development: A Case Study From Turkey

SAGE Open ◽

10.1177/21582440211023771 ◽

2021 ◽

Vol 11 (2) ◽

pp. 215824402110237

Author(s):

İlknur Bayram ◽

Fatma Bıkmaz

Keyword(s):

Professional Development ◽

Lesson Study ◽

English Language ◽

Research Question ◽

Teaching Style ◽

Language Teachers ◽

Student Thinking ◽

Pedagogical Content ◽

Starting Point

This qualitative case study carried out at a Turkish university with four English language teachers aims to explore what teachers experience in the planning, implementation, analysis, and reporting phases of the lessons study process and what the implications of lesson study for teacher professional development can be. Data in this four-month study were gathered through observations, interviews, whole group discussions, and reflective reports. Findings revealed that lesson study had potential challenges and benefits for the professional development of teachers. The model poses challenges in finding a topic and research question, determining the lesson design and teaching style, making student thinking observable and analyzing qualitative data. On the other hand, it benefited teachers in terms of increasing their pedagogical content knowledge, reflectivity, research skills, collaboration, and collegiality. This study suggests that lesson study might be a good starting point for institutions wishing to adopt a more teacher-led, inquiry-driven and collaborative perspective for professional development.

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