This paper is the result of the first phase of the research about the development of students' mathematical understanding and representation ability through Joyful Problem-based Learning (JPBL) at Public Junior High School in North Sumatera, Indonesia. The population is all of the students of public junior high school (PJHS) in North Sumatera. Samples choose based on stratified random sampling. The samples are the students of PJHS 27 Medan, PJHS 1 Percut Sei Tuan, PJHS 1 Tebing Tinggi, and PJHS 2 Pematangsiantar. The techniques used for collecting data is observation, interview, and essay test. The research findings: (1) Based on interview and observation found that conventional approach still uses in all of the class of PJHS, the students engagement in learning activity is very low, and most of the students do not attain minimal mastery achievement. (2) Based on essay test found that performance of the students in mathematical understanding and representation test is categories small.

The purpose of this research was to develop student-centered learning model aiming to improve high order mathematical thinking ability of junior high school students of based on curriculum 2013 in North Sumatera, Indonesia. The special purpose of this research was to analyze and to formulate the purpose of mathematics lesson in high order mathematical thinking ability and also to develop and to try-out the learning model developed. The subject of the research was 7th graders from state and private Junior High School in Medan and Deli Serdang, which were taken proportional randomly. It was elected SMP N 36, SMPN 2 Tembung, and private SMP Bandung Bandar Khalifah. This developmental Research that orientated on developing product was done in three steps. From the first step, it was revealed that either lesson preparation (students' book, teachers' guided, and lesson material) or the instrument was judged valid and need only a little revision. Analysis on data revealed that the students' high order thinking ability especially in mathematical problem solving, mathematical understanding, and mathematical communication enhanced significantly. By analysis, the reliability of instruments on mathematical understanding, mathematical problem solving, and mathematical communication ability was categorized good.

This paper reports some results of a research attempted to cultivate students' mathematical reasoning ability (MRA) by means of implementing problem-based learning (PBL) model. The population was upper secondary students of public schools in Bandung. Cluster sampling technique applied to take sample of amount 158 students from middle-level schools. Data were teacher's assessment on students' performance in last two months, prior mathematical knowledge (PMK) and MRA test, observations, and interview. Two-factors ANOVA then used to analyze the data. Findings showed that overall students in PBL-classroom achieved better MRA than their counterparts in conventional one though the category in both classrooms was low. Interaction between PBL and PMK factor towards MRA did not emerge. Most part of the participants viewed PBL neither positive nor negative. On the contrary, they assessed that the tasks given made and demanded them thinking and learning more actively than ever. The findings showed adopting PBL in schools to facilitate students getting their higher order thinking skills is a reasonable choice, especially schools having middle or high ability students.

<div class="WordSection1"><p align="center"><strong>ABSTRA</strong><strong>K</strong></p><p><strong> </strong></p><p>Tujuan penelitian kuasi eksperimen ini untuk mengetahui: (1) perbedaan peningkatan kemampuan berpikir kritis matematis siswa yang diberi pembelajaran berbasis masalah dengan siswa yang diberi pembelajaran langsung; (2) interaksi pembelajaran  dengan kemampuan awal matematika terhadap peningkatan kemampuan berpikir kritis matematis siswa; (3) perbedaan peningkatan kemandirian belajar siswa yang diberi pembelajaran berbasis masalah dengan siswa yang diberi pembelajaran langsung; (4) interaksi pembelajaran dengan kemampuan awal matematika terhadap peningkatan kemandirian belajar siswa. Populasinya adalah seluruh siswa SMP Swasta Ar-rahman Percut dan sampelnya adalah siswa kelas VIII  sebanyak 60 orang. Instrumen yang digunakan terdiri dari tes kemampuan berpikir kritis matematis dan skala kemandirian belajar. Data dianalisis dengan uji ANAVA dua jalur. Dari hasil analisis diperoleh: (1) peningkatan kemampuan berpikir kritis matematis siswa yang diberi pembelajaran berbasis masalah lebih tinggi daripada yang diberi pembelajaran langsung; (2) tidak terdapat interaksi antara  pembelajaran dengan kemampuan awal matematika terhadap peningkatan kemampuan berpikir kritis matematis; (3) peningkatan kemandirian belajar siswa yang diberi pembelajaran berbasis masalah lebih tinggi daripada yang diberi pembelajaran langsung (4) tidak terdapat interaksi antara pembelajaran dengan kemampuan awal matematika terhadap peningkatan kemandirian belajar.</p><p><strong>Kata Kunci</strong>:   Berpikir Kritis Matematis, Kemandirian Belajar</p><p> </p><p align="center"><strong>ABSTRACT</strong><strong></strong></p><p>The purposes of this research were to know: (1) The difference of improvement in mathematical critical thinking ability of students that given problem-based learning  with students that given direct learning; (2) The interaction between the model of learning with students’ mathematical previous knowledge toward the improvement in mathematical critical thinking ability; (3) The difference of improvement in self-regulated learning of students that given problem-based learning with students that given  direct learning; (4) The interaction between the model of learning with students’ mathematical previous knowledge toward the improvement in self-regulated learning. The population of this research is all students of SMP Swasta Ar-rahman Percut and the sample of this research is grade eight with taken sample two classes (experiment and control) with sample 60 students. The instrument of this research were: test of mathematical critical thinking and scale of self-regulated learning. Based of the results analysis, it showed that: (1) Improvement of the students’ ability in mathematical critical thinking that given problem-based learning was higher than the students’ ability that given direct learning; (2) There did not exist interaction between model of learning and students’ mathematical previous knowledge toward the improvement ability mathematical critical thinking; (3) Improvment  of the students’ self-regulated learning that given problem-based learning was higher than the students’ ability that given direct learning; (4) There did not exist interaction between model of learning and students’ mathematical previous knowledge toward the improvement self-regulated learning.</p></div><p><strong>Keywords</strong>:    Critical Thinking Mathematics, Self-Regulated Learning.</p>

This research aims to develop a student-centered learning model based on local culture and instrument of mathematical higher order thinking of junior high school students in the frame of the 2013-Curriculum in North Sumatra, Indonesia. The subjects of the research are seventh graders which are taken proportionally random consisted of three public junior high schools with 86 students and two private schools with 40 students. As a Developmental Research, the work is done within three stages. The results obtained in the second stage, both learning tools (books of students, teachers, and students' activities sheet) and research instruments are valid with minor revisions, and the results of the trial showed that the reliability scores of the tests comprising of Comparison (MAT-1), Social Arithmetic (MAT-2), Triangle (MAT-3), Quadrilateral (MAT-4), and Transformation (MAT-5) respectively 0. 835, 0.588, 0.438, 0.833, and 0.908. The findings showed that the student-centered learning based on local culture model and the instrument for higher order mathematical thinking ability are valid and effective to use in teaching mathematics for junior high school.

Solving problem either within mathematics or beyond is one of the ultimate goal students learn mathematics. It is since mathematics takes role tool as well as vehicle to develop problem solving ability. One of the supporting components to problem solving is mathematical representation ability (MRA). Nowadays, many teachers and researchers find out secondary school students are so poor in MRA. Considering the problem, this research developed instruction materials based on Joyful Problem-based learning (JPBL) to endorse eighth graders of public secondary school students grasp their MRA. The study took place in the Province of North Sumatera, Indonesia. A number of 88 students engaged as subjects. The study administered MRA test before and after implementing the instruction materials. Data is analyzed descriptively and by t-test. The result showed that the learning materials developed is effective in improving the students' MRA. It increased significantly, though categorized mediocre. The study contributed to the existing instruction materials in pursuing MRA.

In conventional learning, teachers frequently face difficulties to deliver their materials due to limited time and practical materials in teaching network computer. Conventional teaching process, especially practical materials, has still not yet been optimal. Thus, computer and multimedia based learning is required to help the students. Besides it can reduce costs in practical materials procurement, the students can absorb the given knowledge well without thinking of the costs to buy the practical materials. Computer Assisted Instruction Method can present the learning using various media either by picture or video that can assist the effective learning process and simplify the students to manage the learning speed since it is combined with the multimedia. By doing this, the students can practice the lesson materials, study whenever and wherever they want. Compuer learning application prioritize user interface, user friendly, which can make the students be diligent and passionate in learning.

The paper reports part of a study aimed at developing teaching materials in inculcating upper secondary students' mathematical reasoning skills (MRS). To develop the materials, the researcher implemented the Four-D Model. The study took subjects from five public schools in Province of North Sumatera, Indonesia. The researcher designed and developed students' work sheet (SWS) and instrument to measure MRS. Along the teaching ran, which applied problem-based learning model, the researcher observed teachers' and students' activities while nurturing and applying MRS in the frame of solving mathematical problems. Of the four indicators laid to measure the MRS, students lack most in use of pattern relationship to analyse situation, to make analogy, or to generalize. The ways support student's progress in achieving MRS are if (i) the problem faced is much mimicked the task solved in the classroom, (ii) more various problems given to solve under guidance, and (iii) intensive scaffolding is given.