Universal Framework for Quantum Error-Correcting Codes

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.

The Analysis of Students’ Critical Thinking Ability with Visualizer-Verbalizer Cognitive style in Mathematics

This study aimed to describe the process of critical thinking of students in solving mathematical problems in algebraic viewed from the differences in cognitive styles-Verbalizer Visualizer. The description of the thinking process was based on Polya’s problem-solving indicator namely understanding the problems, developing plans, implementing problem-solving activities, and to re-examining the results obtained. The research instrument includes the researcher himself, tests of cognitive style, problem-solving task, and the interview guideline. The data analysis was performed by means of data reduction, data presentation, interpretation of data, and drawing conclusions. The study concluded that the visualizer subject disclosed the acknowledged information and asked to use the images and algebraic notation, used a variety of strategies to solve problems, performed calculations by taking into account the things we need and recheck the answer in detail. Verbalizer Subject, however, was able to disclose the acknowledged information and questioned using mathematics formula without using pictures and notation. In terms of making plans for the completion, the subject merely used one step-finishing process. In determining the final outcome, verbalizer subject rechecked every step of the finishing by recalling and paying attention to the proper answers

Logic in the 19th century

The nineteenth century was one of the most active periods for logic in Western philosophy. It is regarded foremost as being the first time logic became ‘symbolic’ and ‘mathematical’. There was tremendous diversity and conflict in logic in this period – indeed there were substantial debates over whether there even is a subject called ‘formal logic’ and over what the most basic logical forms were. There was an explicit discussion in the early part of the century of whether logic should be extensional or intensional, opinion eventually settling on a purely extensional conception. By the end of the century, many of these debates had been resolved or had withered away, and the most widespread conception and practice of logic coalesced in the early twentieth century into a view we now identify with the works of Frege, Russell and Whitehead. The nineteenth century brought, for the first time, from Boole and Frege, distinct proposals on how to symbolize logic that were both extensively developed and had widespread influence. Boole is correctly regarded as the father of modern symbolic logic. Frege shared two concerns of many nineteenth-century mathematicians – avoiding incorrect derivations and providing rigorous and clear foundations for the infinitesimal and derivative calculus – and therefore sought to develop a very clear notion of mathematical ‘proof’. His notation has not been taken up but his influence, especially on the move towards symbolization, has been considerable. De Morgan was the first logician extensively and symbolically to discuss the logic of relations. However, a systematic algebraic notation for relations was provided only later by Peirce, and developed also by Schröder.

PENERAPAN PENDEKATAN KETERAMPILAN PROSES DALAM PEMBELAJARAN MATEMATIKA TERHADAP KEMAMPUAN PEMECAHAN MASALAH PADA PESERTA DIDIK KELAS VIII SMP DDI SIBATUA PANGKAJENE

AbstrakTujuan dari penelitian ini adalah untuk mendeskripsikan pemahaman siswa tentang notasi aljabar melalui kemampuan. Pemahaman dalam penelitian ini berarti kemampuan menerjemahkan (translasi), kemampuan menafsirkan (interpretasi), dan kemampuan untuk melakukan ekstrapolasi (ekstrapolasi). Penelitian ini merupakan penelitian kualitatif. Subyek penelitian ini adalah siswa kelas 5 di sekolah dasar, terdiri dari 2 siswa dengan kemampuan verbal tinggi yang disebut subjek verbal tinggi (SVT), 1 siswa dengan subjek verbal menengah (SVS), dan 1 siswa dengan subjek verbal rendah (SVR ). Fokus dari penelitian ini adalah untuk mendeskripsikan pemahaman siswa tentang notasi aljabar berdasarkan kemampuan verbal mereka dalam terjemahan, interpretasi, dan ekstrapolasi. Data dikumpulkan dengan melakukan wawancara berbasis uji coba dengan subyek penelitian. Data penelitian divalidasi dengan metode triangulasi, yaitu wawancara dengan menggunakan pertanyaan utama seperti pertanyaan yang tercantum dalam TPNA. Data penelitian dianalisis dengan (a) mempelajari data subjek dan membandingkan data yang diwawancarai untuk mendapatkan data yang valid, (b) melakukan reduksi data dengan membuat abstraksi, (c) mengklasifikasi dan mengidentifikasi data untuk menarik kesimpulan. Hasil penelitian menunjukkan bahwa (1) pemahaman notasi aljabar dengan subjek verbal tinggi (SVT): (a) pemahaman simbol yang termasuk dalam pemahaman translate (translasi), (b) memahami konsep simbol tapi Blum mampu membedakan antara simbol dan variabel, (c) mampu menafsirkan makna notasi aljabar (simbol dan tanda) dengan baik; (2) pemahaman notasi aljabar medium verbal subject (SVS): (a) pemahaman simbol yang termasuk dalam pemahaman translate (translasi), (b) mengenali simbol namun belum dapat membedakan simbol dan variabelnya, c) mampu menafsirkan makna notasi aljabar (simbol dan tanda) dengan baik; (3) pemahaman notasi aljabar dengan subjek verbal rendah (SVR): (a) pemahaman simbol yang termasuk dalam pemahaman translasi (translasi), (b) mengenali simbol namun belum dapat membedakan simbol dan variabelnya, c) belum bisa menafsirkan makna notasi aljabar (simbol, variabel, dan tanda) dengan baik. AbstractThe purpose of the study is to describe the students’ understanding on algebraic notation through abilities. The understanding in this study means the ability to translate (translation), the ability to interpret (interpretation), and the ability to extrapolate (extrapolation). This study is a qualitative research. The subjects of this study are 4 fifth grade students in elementary school, consisted of 2 students with high verbal ability called as high verbal subject (SVT), 1 student with medium verbal subject (SVS), and 1 student with low verbal subject (SVR). The focus of this research is to describe the students’ understanding on algebraic notation based on their verbal abilities in translation, interpretation, and extrapolation. The data were collected by conducting test-based interview to the research subjects. The research data were validated by conducting triangulation method, namely interview using major questions as the questions stated in TPNA. Research data were analyzed by (a) studying the subject data and comparing interviewed data to obtain the valid one, (b) conducting data reduction by making the abstraction, (c) classifying and identifying the data to draw the conclusion. The results of the study reveals that (1) the understanding of algebraic notation of high verbal subject (SVT): (a) understanding of the symbols included in the comprehension translate (translation), (b) understand the concept of the symbol but Blum was able to distinguish between symbols and variables, ( c) able to interpret the meaning of algebraic notation (symbols and signs) well; (2) the understanding of algebraic notation of medium verbal subject (SVS): (a) understanding of the symbols included in the comprehension translate (translation), (b) recognize symbols but have not been able to distinguish the symbols and variables, (c) is able to interpret the meaning of the notation algebra (symbols and signs) well; (3) the understanding of algebraic notation of low verbal subject (SVR): (a) understanding of the symbols included in the comprehension translate (translation), (b) recognize symbols but have not been able to distinguish the symbols and variables, (c) has not been able to interpret the meaning of algebraic notation (symbol, variable, and marks) well.

PROFIL PEMAHAMAN KONSEP NILAI TEMPAT DITINJAU DARI KEMAMPUAN AWAL MATEMATIKA PADA SISWA KELAS III SDN 133 TAKALALA SOPPENG

Tujuan dari penelitian ini adalah untuk menggambarkan pemahaman siswa dari notasi aljabar pada melalui kemampuan. Pemahaman dalam penelitian ini berarti kemampuan untuk menerjemahkan (terjemahan), kemampuan untuk menafsirkan (interpretasi), dan kemampuan untuk ekstrapolasi. Penelitian ini merupakan penelitian kualitatif. Subyek penelitian ini adalah empat siswa kelas lima di SD, terdiri dari dua siswa dengan kemampuan verbal yang tinggi disebut sebagai subyek tinggi verbal (SVT), 1 siswa dengan mata pelajaran lisan menengah (SVS), dan satu siswa dengan mata pelajaran lisan rendah ( SVR). Fokus penelitian ini adalah untuk mendeskripsikan pemahaman siswa tentang notasi aljabar berdasarkan kemampuan verbal mereka dalam terjemahan, interpretasi, dan ekstrapolasi. Data dikumpulkan dengan melakukan wawancara berdasarkan tes-dengan subyek penelitian. The dara penelitian divalidasi dengan melakukan metode triangulasi item, yaitu menggunakan pertanyaan wawancara utama sebagai pertanyaan dinyatakan dalam tPNA. Data penelitian dianalisis oleh (a) mempelajari subyek data dan diwawancarai Membandingkan data untuk Mendapatkan satu valid, (b) melakukan reduksi data dengan membuat abstraksi, (c) mengklasifikasikan dan mengidentifikasi data adalah untuk menarik kesimpulan. Hasil penelitian menunjukkan bahwa (1) pemahaman notasi aljabar dari pelajaran lisan tinggi (SVT): (a) pemahaman tentang simbol termasuk dalam pemahaman yang menerjemahkan (terjemahan), (b) memahami konsep simbol tapi Blum adalah mampu pemadam antara simbol dan variabel, (c) mampu menafsirkan makna aljabar notasi (simbol dan tanda-tanda) baik; (2) pemahaman notasi algberaic pelajaran lisan menengah (SVS): (a) pemahaman tentang simbol termasuk dalam pemahaman yang menerjemahkan (terjemahan), (b) Mengakui simbol tapi belum Mampu memadamkan simbol dan variabel, (c) adalah Mampu menginterpretasikan arti dari aljabar notasi (simbol dan tanda-tanda) baik; (3) pemahaman notasi aljabar dari pelajaran lisan rendah (SVR): (a) pemahaman tentang simbol termasuk dalam pemahaman yang menerjemahkan (terjemahan), (b) Mengakui simbol tapi belum Mampu memadamkan simbol dan variabel, (c) belum mampu menafsirkan makna aljabar notasi (simbol, variabel, dan tanda) dengan baik.The purpose of the study is to describe the students’ understanding on algebraic notation through abilities. The understanding in this study means the ability to translate (translation), the ability to interpret (interpretation), and the ability to extrapolate (extrapolation). This study is a qualitative research. The subjects of this study are 4 fifth grade students in elementary school, consisted of 2 students with high verbal ability called as high verbal subject (SVT), 1 student with medium verbal subject (SVS), and 1 student with low verbal subject (SVR). The focus of this research is to describe the students’ understanding on algebraic notation based on their verbal abilities in translation, interpretation, and extrapolation. The data were collected by conducting test-based interview to the research subjects. The research dara were validated by conducting triangulation method, namely interview using major questions as the questions stated in TPNA. Research data were analyzed by (a) studying the subject data and comparing interviewed data to obtain the valid one, (b) conducting data reduction by making the abstraction, (c) classifying and identifying the data to draw the conclusion. The results of the study reveals that (1) the understanding of algebraic notation of high verbal subject (SVT): (a) understanding of the symbols included in the comprehension translate (translation), (b) understand the concept of the symbol but Blum was able to distinguish between symbols and variables, ( c) able to interpret the meaning of algebraic notation (symbols and signs) well; (2) the understanding of algebraic notation of medium verbal subject (SVS): (a) understanding of the symbols included in the comprehension translate (translation), (b) recognize symbols but have not been able to distinguish the symbols and variables, (c) is able to interpret the meaning of the notation algebra (symbols and signs) well; (3) the understanding of algebraic notation of low verbal subject (SVR): (a) understanding of the symbols included in the comprehension translate (translation), (b) recognize symbols but have not been able to distinguish the symbols and variables, (c) has not been able to interpret the meaning of algebraic notation (symbol, variable, and marks) well.

An Emoji Is Worth a Thousand Variables

Using icon-based equations can provide multiple entry points into algebraic thinking and formal algebraic notation.