Mathematical Abstraction in learning and teaching mathematics

Mathematical abstraction is an important process in teaching and learning mathematics so pre-service mathematics teachers need to understand and experience this process. One of the theoretical-methodological frameworks for studying this process is Abstraction in Context (AiC). Based on this framework, abstraction process comprises of observable epistemic actions, Recognition, Building-With, Construction, and Consolidation called as RBC + C model. This study investigates and analyzes how pre-service mathematics teachers constructed and consolidated concept of Parallel Coordinates in a group discussion. It uses AiC framework for analyzing mathematical abstraction of a group of pre-service teachers consisted of four students in learning Parallel Coordinates concepts. The data were collected through video recording, students' worksheet, test, and field notes. The result shows that the students' prior knowledge related to concept of the Cartesian coordinate has significant role in the process of constructing Parallel Coordinates concept as a new knowledge. The consolidation process is influenced by the social interaction between group members. The abstraction process taken place in this group were dominated by empirical abstraction that emphasizes on the aspect of identifying characteristic of manipulated or imagined object during the process of recognizing and building-with.

The notion of 'concrete,' from concrete manipulatives to pedagogical sequences such as 'concrete to abstract,' is embedded in educational theories, research, and practice, especially in mathematics education. In this article, the author considers research on the use of manipulatives and offers a critique of common perspectives on the notions of concrete manipulatives and concrete ideas. He offers a reformulation of the definition of 'concrete' as used in psychology and education and provides illustrations of how, accepting that reformulation, computer manipulatives may be pedagogically efficacious.

Penelitian ini dilatarbelakangi oleh siswa SLTA yang masih merasa kesulitan dalam mempelajari geometri, dan lebih khususnya lagi materi dimensi tiga. Siswa mengalami kesulitan dalam melakukan proses abstraksi, padahal abstraksi merupakan hal yang penting dalam pembelajaran matematika. Dalam mempelajari geometri dibutuhkan suatu alat peraga, seiring dengan perkembangan teknologi dan komunikasi dalam pendidikan yang berdampak pula pada alat peraga untuk geometri. Salah satu alat peraga tersebut berbentuk virtual manipulatives atau alat peraga yang dapat dimanipulasi, yaitu program GeoGebra. Salah satu fungsi GeoGebra adalah program untuk membantu siswa mempelajari geometri. Penelitian ini bertujuan untuk melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dimensi tiga dengan bantuan program GeoGebra serta melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dengan metode konvensional. Penelitian ini merupakan penelitian kualitatif dengan subjek penelitian kelas sepuluh MAN Cilegon. Teknik pengumpulan data dalam penelitian ini menggunakan observasi, wawancara, dan tes. Hasil penelitian di analisis dengan teknik analytic induction dan constant comparison. Penelitian ini menghasilkan kesimpulan bahwa proses abstraksi siswa yang mempelajari materi dimensi tiga dengan bantuan GeoGebra adalah proses abstraksi empiris yang terjadi selama proses pembentukan konsep. Sedangkan proses abstraksi siswa yang mempelajari materi dimensi tiga dengan metode konvensional adalah proses abstraksi teoritis yang berlangsung pada proses asimilasi, akomodasi dan ekualibrasi dalam satu rangkaian pemrosesan informasi.

Kata Kunci : Abstraksi, Virtual Manipulatives, GeoGebra.

from my book "selected stories in mathematics and physics"amazon

  Peacock    described the artificial numbers of his time in logical symbols instead of their intuitive interpretation with pictures of the real world. Considered from this point of view ,algebra is severed from its tie to arithmetic , and an algebra becomes a purely formal deductive study. This new Peacock’s algebra is the “symbolical algebra”  the first step to “abstract algebra”. His fundamental logical principle was the “principle of permanence of equivalent forms”.  Yet I give the basic concepts of the  new algebra of Hamilton  as an example of this liberation. .....

O que é a Matemática? Esta é a inquirição que pretendemos abordar. A Matemática poderia ser conceituada como técnica, arte; filosofia ou ciência. Complementando com a descrição sucinta das diversas áreas da Matemática pura, da aplicada, e finalizamos com a educação Matemática.

Salah satu permasalahan yang dihadapi siswa adalah lemahnya kemampuan siswa dalam operasi perkalian. Siswa kesulitan dalam operasi perkalian dengan menggunakan cara bersusun. Sejalan dengan itu, tujuan penelitian ini adalah untuk mengatasi kesulitan siswa dalam operasi perkalian dengan menerapkan metode perkalian latis. Jenis penelitian ini adalah penelitian deskriptif kualitatif yang dilakukan di kelas VII Sekolah Menengah Pertama dengan melibatkan enam siswa yang mengalami kesulitan dalam operasi perkalian. Penelitian ini dilaksanakan dalam 3 tahap yaitu: 1) perencanaan; 2) pelaksanaan yang terdiri dari 3 bagian: tes diagnostik, tes treatment, dan wawancara; 3) evaluasi. Hasil penelitian menunjukkan bahwa mereka sangat tertarik dan senang dengan metode ini. Mereka mampu menyelesaikan operasi perkalian dengan mudah, walaupun masih ada siswa yang kurang teliti. Namun, hal itu tidak membuat mereka merasa jenuh bahkan menjadi lebih semangat untuk memperbaiki kesalahannya. Siswa lebih memilih metode ini dalam menghitung perkalian dari pada metode yang dikenal sebelumnya. Kata kunci: kesulitan siswa, operasi perkalian, metode latis.

Marcus du Sautoy's book "Finding Moonshine" is a rich exploration of the topic of groups, referred to throughout the book as "symmetries". Writing for a general audience, du Sautoy proceeds, chapter by chapter, to record what he does, month by month, in a year of his life as a professional research mathematician. In doing so, he reflects on earlier moments in his life that led to him becoming a mathematician, and on events and people who contributed to the mathematical development of the theory of groups. Most simply, groups can be understood through the example of the mathematical properties of symmetry found in mirror-reflections of 2-dimensional objects, and other symmetries observed in rotating some 2-dimensional objects, such as squares, rectangles, triangles, and pentagons. The "Moonshine" de Sautoy searches for and finds is the mysterious connection between the "Monster", the biggest possible finite group that first exists in a mathematical universe of 196,883 dimensions, and the Fourier expansion of the related Modular function (whose first coefficient is 196,884). This is a brilliant, lucid story!

Abstraction is a fundamental process in learning mathematics. So far, there are still few researchers and professionals discussed about how this process occurs in mathematics classroom, especially in geometry. In Indonesia, 50% of the content of Junior High School Mathematics Curriculum consists of geometry. In line with the development of information technology, dynamic geometry software emerges; one of them is Geometers' Sketchpad (GSP). This software was created to assist students in learning geometry by visualizing and manipulating geometry objects. Unfortunately, this software is still rarely used in Indonesia. The concern of this research is to investigate how the abstraction process in learning geometry using GSP takes place. The objective of this research is to figure out students' abstraction process in learning geometry concepts using GSP and students' abstraction process during solving geometry problems in a classroom. This study is a qualitative research and it was conducted in a public junior high school in Indonesia. The subjects of this research are seventh grade students. The data were collected through observation, test, and interviews and the data were analysed using analytical induction and constant comparative techniques. The results of this research are that both types of the students' abstraction process during construction of geometry concepts and solving geometry problems falls into the category of empirical abstraction. In addition, students' ability in the aspect of representation of mathematical ideas into symbols or mathematics language as a part of students' abstraction is the most dominant aspect that appears when students solved geometry problems.

Uno de los problemas que se presenta en la enseñanza del Cálculo consiste en que prevalece el trabajo mecánico en las aulas, dejando a un lado la construcción de los conceptos, así como la evasión de con-sideraciones visuales al abordarlos. Este trabajo se enfoca en propo-ner elementos que permitan al alumno visualizar situaciones en las que está inmerso el concepto de derivada, como es el caso de la razón de cambio instantánea. Para ello se recurre a la tecnología disponible para los jóvenes, como los dispositivos móviles y aplicaciones de cómputo de uso específico. También se empleó realidad aumenta-da debido a que esta tecnología ha sido reconocida como apoyo en ciertos aspectos educativos tales como: la atención y relación viso-espacial. Para evaluar el sistema se consideró una muestra de 2 grupos de 30 estudiantes cada uno, un grupo fue el de control (GC) y el otro el de estudio (GE), ambos fueron evaluados después de haber trabaja-do una secuencia de enseñanza, en donde se utilizó la misma prueba y diferentes herramientas. Los resultados mostraron que el grupo de estudio identificó el concepto de derivada resolviendo de forma correcta un porcentaje mayor de reactivos en el cuestionario final en relación al grupo de control. El incremento en promedio del GE fue de 2.6 a 7.4 y del GC fue de 2.08 a 4, por lo que se concluye que actividades apoyadas con dispositivos móviles contribuyen a mejorar la visualización y la construcción de los conceptos.

abstract
One of the problems presented in the teaching of calculus is that mechanical work prevails in the classrooms, reducing the time for the construction of concepts, as well as preventing visual consider-ations when they are approached. This paper focuses on proposing elements that allow students to visualize situations that incorporate the concept of the derivative, such as the case of the reason for instantaneous change. To this end, we used the technology available to young people, such as mobile devices and specific computer  applications. We also used Augmented Reality since this technology has been considered a successful support in certain educational aspects, such as attention and visual-spatial relationships. To evaluate the system a sample of two groups of 30 students each was used; one was a control group (CG) and the other a research group (RG). Both were evaluated after having worked through a sequence of teaching in which the same test was used with different tools. The results showed that the research group identified the concept of the derivative and correctly solved a higher percentage of problems in the final questionnaire compared with the control group. The aver-age increase of the RG was from 2.6 to 7.4, and the CG was from 2.08 to 4, which led to the conclusion that activities supported by mobile devices contribute to the improvement of visualization and the con-struction of concepts.

Circle-square crop configurations in the Texas Panhandle (images courtesy of Google Earth). Scholars during the Medieval Period believed that there was something "divine" or "perfect" about the circle. The farmers of today in the Texas Panhandle believe in the practicality and efficiency of the circle via the utilization of center pivot irrigation of their crops.

In the light of the ASEAN Economic Integration, the educational system is now geared towards K-12 landscape. This study presents analyses on the students’ motivation and intention towards learning Mathematics, Mathematics performance and students’ preferred track in the senior high during the first batch of the K-12 implementation. A survey was conducted investigate students’ motivation and intention towards learning mathematics, since motivation is perceived as a pre-requisite of and an indispensable component for student engagement in learning. Saeed and Zyngier also cited that, student engagement in learning is not only an end in itself but it is also a means to the end of students achieving sound academic outcomes. A correlation approach was adapted to show relationship among students’ motivation, students’ intention towards learning Mathematics, Mathematics performance, and preferred track in the senior high school. This was conducted at the Basic Education Department - North C...

Because gestures can ground abstract ideas in the physical world and make connections between ideas, they play a potentially important role in the classroom. This study examined whether teachers can intentionally alter their gesture production. Six teachers taught a brief mathematics lesson three times, once without any special instructions, once attempting to use gestures to link ideas, and once attempting to inhibit all gestures. All 6 teachers were able both to increase and inhibit their gesture production when asked to do so. These results suggest that teachers can alter their gesture production. This is promising evidence that interventions that require teachers to alter their gestures are feasible.