Observer Based Multi-Level Fault Reconstruction for Interconnected Systems

The problem of local fault (unknown input) reconstruction for interconnected systems is addressed in this paper. This contribution consists of a geometric method which solves the fault reconstruction (FR) problem via observer based and a differential algebraic concept. The fault diagnosis (FD) problem is tackled using the concept of the differential transcendence degree of a differential field extension and the algebraic observability. The goal is to examine whether the fault occurring in the low-level subsystem can be reconstructed correctly by the output at the high-level subsystem under given initial states. By introducing the fault as an additional state of the low subsystem, an observer based approached is proposed to estimate this new state. Particularly, the output of the lower subsystem is assumed unknown, and is considered as auxiliary outputs. Then, the auxiliary outputs are estimated by a sliding mode observer which is generated by using global outputs and inverse techniques. After this, the estimated auxiliary outputs are employed as virtual sensors of the system to generate a reduced-order observer, which is caplable of estimating the fault variable asymptotically. Thus, the purpose of multi-level fault reconstruction is achieved. Numerical simulations on an intensified heat exchanger are presented to illustrate the effectiveness of the proposed approach.

Analysis of Students' Mathematical Literacy Ability in Algebraic Concepts Based on Trends in International Mathematics and Science Study (TIMSS) Problems

This research uses descriptive qualitative. The aim is to examine students' mathematical literacy abilities and types of errors made by students of MTs Mathla’ul Anwar Kedondong in solving algebraic concept questions that were first accessed from TIMSS. The subjects of this study were all students of class VIII-A MTs Mathla’ul Anwar Kedondong, which had 30 students. The data was collected using tests and interviews. All student answers in the analysis of mathematical errors are based on empathy for students' mathematical literacy abilities, namely aspects of knowledge, application, and communication. Furthermore, for further analysis of the students 'mathematical errors, 2 students with the lowest scores were selected, representing the mathematical errors of all students based on the four aspects of the students' mathematical literacy abilities for interviews. The data analysis technique is by reducing data, presenting data, and drawing conclusions. The results of the analysis of the data collected, the mathematical literacy ability of students included in the medium category with an average score of 62.38 scores on a scale of 100. Based on the analysis of the mathematical literacy scheme capability mentioned, students can request solutions according to their needs and sufficient good at solving problems on the criminal aspect. However, judging from the mathematical mistakes made by students, students who made mistakes did not review the answers in solving the problems. This causes the problem solving is not correct.

ANALYSIS OF STUDENTS MATHEMATICAL CONNECTION ERRORS IN TRIGONOMETRIC IDENTITY PROBLEM SOLVING

The trigonometric identity is essential in learning Mathematics because it requires students to think critically, logically, systematically, and thoroughly. Solving trigonometric identity problems requires students to relate conceptual knowledge or procedural knowledge, which then used in questions. This study involved grade X students of senior high school, which were examined to find out the types of mathematical connections errors and causes of the errors. Before task-based interviews were conducted, 36 students were first given a test. Based on several considerations, seven students ( three males and four females) were selected to undergo a task-based interview. This research employed a qualitative research method with a case study design. The results of the analysis indicate that the errors in connecting to conceptual knowledge are most commonly the mistake of connecting the algebraic concept. On the other hand, 86.11% of students experienced errors in connecting to procedural knowledge. This error happened when the students worked on problems with trigonometric identities, which they had rarely encountered in exercises. Errors in mathematical connections in trigonometric identity are caused by the lack of understanding of the algebraic arithmetic operation, emphasis on the concept, and strategic knowledge. It shows that students need a variety of problems to be able to master various forms of trigonometric identities. This research's result also reinforces the critical role of algebraic concepts as prior knowledge in studying trigonometric identity.

Pengembangan Buku Teks Matematika berbasis Investigasi untuk Meningkatkan Penalaran Aljabar

Aljabar merupakan generalisasi aritmatika sehingga dijadikan sebagai titik awal pembelajaran matematika tingkat lanjut. Meski aljabar sangat penting namun siswa memiliki banyak kendala dalam mempelajari aljabar. penelitian pengembangan ini bertujuan untuk mengembangkan buku teks matematika berbasis investigasi dalam meningkatkan kemampuan penalaran aljabar siswa. Tujuan dari buku teks matematika adalah untuk memfasilitasi siswa menyelidiki masalah dalam materi Persamaan Kuadrat. Penelitian pengembangan menggunakan model Four-D yang dimodifikasi diantaranya Define (identifikasi dan pengembangan struktur model bahan ajar), Desain (desain buku teks untuk mendapatkan prototipe yang sesuai dengan pembelajaran yang akan dilakukan), Develop (Pengembangan draft buku teks berdasarkan ahli diperoleh dari uji coba lapangan). Untuk tahap DIseminate pada tahap ini tidak dilakukan. Pada setiap tahappengembangan berisi kegiatan yang menunjukkan adanya urutan langkah-langkah kegiatan. Khususnya, dalam tahap Develop berisi siklus kegiatan. Prosedur penelitian ini dimulai dengan tahap awal studi mendalam tentang eksplorasi kebutuhan buku teks yang dilakukan sebagai cara mengatasi kesalahpahaman siswa dalam menyelesaikan masalah persamaan kuadrat melalui investigasi untuk meningkatkan penalaran aljabar. Langkah selanjutnya adalah menyusun buku teks matematika berbasis investigasi. Buku teks yang dihasilkan divalidasi oleh para ahli dalam pendidikan matematika dan para ahli dalam buku teks matematika. Buku teks diuji secara terbatas untuk mengetahui kepraktisan buku teks. Alhasil dalam penelitian ini diperoleh buku teks matematika berbasis investigasi yang valid dan praktis. Kata kunci: analisis, struktur kognitif, berpikir matematis, bilangan bulat, ABSTRACT The algebraic concept is a generalization of arithmetic so that it is used as an entry point in further learning mathematics. Thus algebra is important. However students have many obstacles in learning algebra. This is development research to develop mathematics textbooks to improve students' algebraic reasoning abilities. The purpose of mathematics textbooks is to facilitate students investigate problems in the material of Quadratic Equations. This development study used a modified Four-D model include Define (identification and development of instructional material model structures), Design (textbook design to obtain prototypes that are appropriate to the learning to be carried out), Develop (Improved draft textbooks based on expert input and data obtained from field trials). At each stage contains activities that indicate the existence of a sequence of steps of activity. Especifically, in the Develop stage contains the activity cycle. The procedure of this study begin with the early stages an in-depth study of the exploration of the needs of textbooks was carried out as a means of overcoming students' misconceptions in solving investigative quadratic equation problems to improve algebraic reasoning abilities. The next step is to compile a textbook that is presented in print media. The resulting textbooks were validated by experts in mathematics education and experts in mathematics textbook. After that the textbook tested in a limited way through developmental experimental studies to study the practicality of textbooks. As a result this study can obtain a valid and practical textbook has obtained. Keywords: Algebraic Reasoning, Investigation, mathematics Textbooks

Students' gestures in understanding algebraic concepts

[English]: The purpose of this qualitative exploratory study was to analyze students’ gestures in understanding algebraic expression. It involved 59 7th-grade students in Semarang city, Indonesia. Students’ gestures were identified through interviews and observations, then analyzed in three stages: data condensation, data display, and drawing and verifying conclusion. Time triangulation was utilized to assure data validity. The results showed that students employed: (1) direct gestures as a representation of coefficients and variables in the form of hand movements forming the shape of objects that they recognize in the everyday environment, (2) indirect gestures as a representation of coefficients and variables in the form of hand movements as if forming the shape of objects that they recognize in the daily environment then followed by consistent and repetitive hand movements as a representation of the coefficients, (3) direct gesture representing constants in the form of hand movements forming a specific number, and (4) writing gestures and pointing gestures to strengthen the explanation given. The present study concludes that the gestures made by the students in understanding the concepts of algebraic expression consist of representation, pointing, and writing. This study yields an important description of students' gestures and types of gestures about the algebraic concept, which provide a further understanding of the topic. Keywords: Gesture, Conceptual understanding, Algebra [Bahasa]: Penelitian kualitatif ini bertujuan untuk menganalisis gestur siswa dalam memahami bentuk aljabar. Penelitian melibatkan 59 siswa di salah satu SMP di Semarang. Data gestur siswa diidentifikasi melalui observasi dan wawancara kemudian dianalisis melalui tahapan kondensasi data, penyajian data, dan penarikan dan verifikasi simpulan. Verifikasi keabsahan data dilakukan menggunakan teknik triangulasi waktu. Hasil penelitian menunjukkan bahwa siswa menggunakan (1) gestur langsung sebagai perwujudan pemahaman konsep koefisien dan variabel dalam bentuk gerakan tangan yang membentuk objek yang dikenali dalam lingkungan sehari-hari, (2) gestur tidak langsung sebagai representasi koefisien dan variabel dalam bentuk gerakan tangan seolah-olah membentuk objek yang dikenali dalam lingkungan sehari-hari kemudian diikuti oleh gerakan tangan yang konsisten dan berulang sebagai representasi koefisien, (3) gestur langsung yang menjadi representasi konstanta melalui gerakan tangan membentuk angka tertentu, dan (4) gestur menulis dan menunjuk untuk memperkuat penjelasan yang diberikan. Penelitian ini menyimpulkan bahwa gestur yang dibentuk siswa dalam memahami konsep bentuk aljabar terdiri dari gestur representasi (gestur representasi langsung dan tidak langsung), gestur menunjuk, dan gestur menulis. Penelitian ini menghasilkan deskripsi penting tentang gestur dan jenis gestur siswa tentang konsep aljabar yang memberikan pemahaman lebih lanjut tentang topik tersebut. Kata kunci: Gestur, Pemahaman konsep, Aljabar

Ethnomathematics on Dayak Tabun Traditional Tools for School Mathematics Learning

The purpose of this study was to describe ethnomathematics on Dayak Tabun traditional tools in school mathematics learning. This study uses a qualitative approach, with descriptive research methods. Observation techniques are direct observation and communication techniques, namely interviews with Dayak Tabun community leaders, especially makers, users, and traditional stakeholders. the results of the research obtained are: 1) form, learning context in geometric concepts, namely flat and wake up space; 2) aspects of the motive, the learning context in the geometry concepts, like are two-dimensional, lines, and angles, besides that the algebraic concept is a number pattern in the form of a constant sequence; 3) the way of making, the learning context in the algebraic concept of numbers, namely fractions in dividing the material into two parts, calculating operations especially on natural numbers, sequential numbers through measurement of materials; 4) in terms of the use of tools, the context of calculating operating learning is the tool used in the dance, namely the tapping of movements and elevation angles in trigonometric material, namely the use of a Sangkuh Akai tool. Therefore, ethnomathematics on traditional Dayak Tabun ethnic instruments can be used as the context of school mathematics learning.

Elementary School Teachers' Mathematical Connections in Solving Trigonometry Problem

This study aims to reveal mathematical connections of elementary school teachers in solving trigonometric problem. The subjects of this study were 22 elementary school teachers as the prospective participants of Professional Teacher Education and Training (PTET). They came from several districts of South Sulawesi Province. The teachers were given trigonometry problem. Trigonometry problems could encourage teachers to connect geometrical and algebraic concept, graphical representation and algebraic representation, as well as daily life context. The result shows that most of the subject teachers of this study solved the problem according to procedures they know without considering everyday life context. On the other hand, there were some subjects who connected problem with everyday life context using graphical, verbal, or numerical representation. Thus, subjects who were able to connect problem information with appropriate concepts and procedures are categorized as substantive connections. While the subjects who were able to connect problem information with mathematical concepts but less precise in using the procedure are categorized as classification connections.

DIAGONALISASI BENTUK KUADRATIK IRISAN KERUCUT

In general, the conic section equation consists of three parts, namely quadratic, cross-product, and linear terms. A conic sections will be easily determined by its shape if it does not contain cross-product term, otherwise it is difficult to determine. Analytically a cone slice is a quadratic form of equation. A concept in linear algebraic discussion can be applied to facilitate the discovery of a shape of a conic section. The process of diagonalization can transform a quadratic form into another form which does not contain crosslinking tribes, ie by diagonalizing the quadrate portion. Hence this paper presents the application of an algebraic concept to find a form of conic sections.