scholarly journals Stepping Back: Reflections on a Pedagogical Demonstration of Reflective Abstraction

2015 ◽  
Vol 58 (4-5) ◽  
pp. 245-252 ◽  
Author(s):  
Jedediah W.P. Allen ◽  
Mark H. Bickhard
Keyword(s):  
Reflective Abstraction

scholarly journals Mathematical Abstraction of Year 9 Students Using Realistic Mathematics Education Based on the Van Hiele Levels of Geometry

2019 ◽  
Vol 6 (1) ◽  
pp. 1-12
Author(s):  
Heris Hendriana ◽  
Nelly Fitriani
Keyword(s):  
High School Students ◽  
Mathematics Education ◽  
Traditional Learning ◽  
Realistic Mathematics Education ◽  
Reflective Abstraction ◽  
Van Hiele ◽  
Level Of Abstraction ◽  
Realistic Mathematics ◽  
Van Hiele Levels ◽  
Mathematical Abstraction

Previous research regarding abstraction has not discussed abstraction qualitatively based on van Hiele levels. Thus, it is necessary to study abstraction analysis based on van Hiele levels through Realistic Mathematics Education (RME) approach. The purpose of this research was to analyze mathematical abstraction based on van Hiele levels of geometry (VHLG) through RME and traditional learning approach reviewed from the levels of prior knowledge. This research employed a descriptive qualitative method involving Year 9 junior high school students as the subjects. The instruments were a mathematical abstraction test, van Hiele geometry test, and interview guidelines. The results of the high- and medium-ability students in the classroom using RME approach showed that VHLG was at the Deduction level and the abstraction ability was dominated by Empirical and Reflective Abstraction, whereas the low-ability students are at the level of Abstraction, they had imperfect Empirical and Reflective Abstraction. As for the high-ability students in the traditional learning classroom, the VHLG was at the level of Abstraction; their Reflective Abstraction was at the Representation level. While concerning the low- and medium-ability students, the VHLG was at the Analysis level; they mastered the Reflective Abstraction at the level of Recognition. This study indicates that the RME approach can trigger the development of mathematical abstraction, and accelerate the van Hiele levels of geometry progress.

scholarly journals IDENTIFIKASI KEMAMPUAN PENALARAN SISWA KELAS XI DI MAN 4 BANTUL PADA SUHU DAN KALOR

EDUSAINS ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 30-37
Author(s):  
Nur Arviyanto Himawan ◽  
Jumadi Jumadi ◽  
Edy Purwanto
Keyword(s):  
High School ◽  
21St Century ◽  
Inductive Hypothesis ◽  
Senior High School ◽  
Reflective Abstraction ◽  
Reasoning Skills ◽  
Descriptive Method ◽  
Reasoning Skill

AbstractReasoning skill is essential for students to have to face the challenges of the 21st century. This study aims to determine the percentage of students' reasoning skills based on deductive hypothesis, inductive hypothesis, and reflective abstraction. This research used a descriptive method involving 26 students of Islamic Senior High School 4 Bantul. The data from tests were analyzed quantitatively, while the data from interviews were analyzed qualitatively as supporting data. The results showed that the indicators of deductive hypothesis: Aspects of explaining a fact, there are 62% of students explained the facts correctly but were incomplete and 38% of students explained the facts incorrectly; Aspects of making conclusions deductively, there are 77% of students made correct conclusions but were incomplete and 23% of students made incorrect conclusions. Based on inductive hypothesis indicators: Aspects of making conclusions inductively, all students can make conclusions, but incorrect in associating equations; Aspects of giving reasons, all students gave incorrect reasons. Based on indicators of reflective abstraction: Aspects of developing concepts, there are 50% of students who can connect concepts in a case correctly and 50% of students incorrectly connect concepts in a case. This showed that students' reasoning skill is still low. AbstrakKemampuan penalaran penting dimiliki oleh siswa untuk menghadapi tantangan abad 21. Penelitian ini bertujuan untuk mengetahui persentase kemampuan penalaran siswa berdasarkan hipotesis deduktif, hipotesis induktif dan abstraksi reflektif. Penelitian ini menggunakan metode deskriptif yang melibatkan 26 siswa MAN 4 Bantul. Data dari tes dianalisis secara kuantitatif, sedangkan data dari wawancara dianalisis secara kualitatif sebagai data pendukung. Hasil penelitian menunjukkan bahwa pada indikator hipotesis deduktif: Aspek menjelaskan fakta, terdapat 62% siswa yang menjelaskan fakta secara tepat namun kurang lengkap dan 38% siswa kurang tepat dalam menjelaskan fakta; Aspek membuat kesimpulan secara deduktif, terdapat 77% siswa membuat kesimpulan dengan tepat namun kurang lengkap dan 23% siswa membuat kesimpulan yang kurang tepat. Berdasarkan indikator hipotesis induktif: Aspek membuat kesimpulan secara induktif, semua siswa dapat membuat kesimpulan, namun tidak tepat dalam mengaitkan persamaan; Aspek memberi alasan, semua siswa memberikan alasan yang kurang tepat. Berdasarkan indikator abstraksi reflektif: Aspek pengembangan konsep, sebanyak 50% siswa mampu menghubungkan konsep dalam suatu kasus secara tepat dan 50% siswa kurang tepat dalam menghubungkan konsep pada suatu kasus. Hal ini menunjukkan kemampuan penalaran siswa masih rendah. 

scholarly journals An approach to the design of mathematical task sequences: Conceptual learning as abstraction

2016 ◽  
Vol 10 (4) ◽  
pp. 270-279
Author(s):  
Martin A. Simon
Keyword(s):  
Conceptual Learning ◽  
Teaching Experiment ◽  
Mathematical Task ◽  
Mathematical Concepts ◽  
Student Activities ◽  
Reflective Abstraction ◽  
Division Of Fractions ◽  
Task Sequences

This paper describes an emerging approach to the design of task sequences and the theory that undergirds it. The approach aims at promoting particular mathematical concepts, understood as the result of reflective abstraction. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from approaches in which learning to solve the problem posed is the intended learning. The paper illustrates the approach through data from a teaching experiment on division of fractions. Una aproximación al diseño de secuencias de tareas matemáticas: aprendizaje conceptual como abstracción Este artículo describe una aproximación emergente al diseño de secuencias de tareas y la teoría que la sustenta. La aproximación pretende promover conceptos matemáticos concretos como resultado de una abstracción reflexiva. Es central en esta aproximación la identificación de actividades disponibles para los estudiantes con las que puedan abstraer las ideas pretendidas. La aproximación difiere de aquellas en las que el aprendizaje para resolver problemas es el aprendizaje que se pretende. El artículo ilustra la aproximación a través de datos de un experimento de enseñanza sobre la división de fracciones.Handle: http://hdl.handle.net/10481/41628WOS-ESCI

An Attentional Model for the Conceptual Construction of Units and Number

1981 ◽  
Vol 12 (2) ◽  
pp. 83-94
Author(s):  
Ernst Von Glasersfeld
Keyword(s):  
Central Nervous System ◽  
Nervous System ◽  
Steady State ◽  
Theoretical Model ◽  
Attentional Model ◽  
Reflective Abstraction ◽  
Conceptual Structures ◽  
Novel Approach ◽  
Conceptual Construction ◽  
The Central Nervous System

A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget's postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.

scholarly journals Students’ thinking path in mathematics problem-solving referring to the construction of reflective abstraction

2018 ◽  
Vol 11 (2) ◽  
pp. 155-166
Author(s):  
Patma Sopamena ◽  
Toto Nusantara ◽  
Eddy Bambang Irawan ◽  
. Sisworo
Keyword(s):  
Undergraduate Students ◽  
Limit Problem ◽  
Student Thinking ◽  
Closed Path ◽  
Mathematical Concepts ◽  
Reflective Abstraction ◽  
Mathematics Problem Solving ◽  
Mathematical Problems ◽  
Qualitative Descriptive ◽  
Field Notes

[English]: This research aims to reveal the path of student thinking in solving mathematical problems referring to the construction of reflective abstraction. Reflective abstraction is the process of thinking in constructing logical structures (logico-mathematical structures) by individuals through interiorization, coordination, encapsulation, and generalization. It is an explorative research with the qualitative descriptive approach which involve fourteen undergraduate students enrolled in Calculus course. Data was analyzed through (1) transcribing verbal data (results of think aloud, interviews, observations, field notes, and results of construction of student mathematical concepts), (2) conducting data reduction (coding, drawing thinking structures), (3) analyzing thought processes, and (4) drawing conclusions. We found that the thinking process of students in solving mathematical problems based on the construction of reflective abstraction can occur through the path of interiorization - coordination - encapsulation - generalization then to coordination - encapsulation - generalization. Thus, student’s thinking path in solving mathematical problems is categorized as a simple closed path. Keywords: Thinking path, Limit problem, Reflective abstraction, Simple closed path [Bahasa]: Penelitian ini bertujuan untuk mendeskripsikan terjadinya jalur berpikir mahasiswa dalam menyelesaikan masalah matematika berdasarkan konstruksi abstraksi reflektif. Abstraksi reflektif adalah proses berpikir dalam membangun struktur logis oleh individu melalui interiorisasi, koordinasi, enkapsulasi, dan generalisasi. Penelitian ini tergolong penelitian eksploratif dengan pendekatan deskriptif kualitatif melibatkan empat belas mahasiswa yang mengikuti matakuliah Kalkulus. Proses analisis data dalam penelitian ini dilakukan melalui langkah-langkah: (1) mentranskrip data verbal (hasil thinkalouds, wawancara, pengamatan, catatan lapangan, dan hasil konstruksi konsep matematika mahasiswa), (2) melakukan reduksi data (membuat coding, menggambar struktur berpikir), (3) analisis proses berpikir, dan (4) penarikan kesimpulan. Hasil penelitian menunjukkan bahwa proses berpikir mahasiswa dalam menyelesaikan masalah matematika berdasarkan konstruksi abstraksi reflektif dapat terjadi melalui jalur interiorisasi – koordinasi – enkapsulasi – generalisasi kemudian ke koordinasi – enkapsulasi – generalisasi. Dengan demikian, jalur berpikir mahasiswa dalam menyelesaikan masalah matematika dikategorikan sebagai jalur berpikir tipe lintasan tertutup sederhana. Kata kunci: Jalur berpikir, Masalah limit, Abstraksi reflektif, Jalur tertutup sederhana NB: PDF version of this article will be available in maximum two weeks after this publication

scholarly journals ABSTRAKSI REFLEKTIF SISWA SEKOLAH MENENGAH PERTAMA PADA MATERI SEGIEMPAT DAN SEGITIGA

2021 ◽  
Vol 3 (2) ◽  
pp. 107-121
Author(s):  
Ahmad Choirul Anam
Keyword(s):  
Female Students ◽  
Mathematical Ability ◽  
Male Students ◽  
Research Subject ◽  
Apos Theory ◽  
Student Work ◽  
Ability Test ◽  
Reflective Abstraction ◽  
Math Ability ◽  
New Information

The purpose of this study is to determine a person's ability to perform mental activities mechanically in associating new information received with previously owned information (reconstruction) which is a reflective abstraction. The process of reconstructing the concepts of quadrilaterals and triangles by students uses the mental mechanism approach of APOS theory (interiorization, encapsulation, de-encapsulation, coordination). The process can be known by observing students and analyzing the results of student work, starting with determining 1 class to be given a mathematical ability test. 29 students were present when the math ability test was given. The results of the mathematical ability test were analyzed and then found 2 students of different sexes with moderate ability. The first student is male and the second student is female, hereinafter referred to as the research subject, then the research subject is given the task of quadrilateral and triangle. The results showed that female students had done all stages of reflective abstraction in APOS theory. So, that these students were at the level of relational understanding. Meanwhile, male students only did several stages in the reflective abstraction stage on the APOS theory so that these students were included in the level of instrumental understanding.

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