This study aims to describe the ability of students' mathematical connections in solving problems in the circle equation material through transposition studies. The method in this research is a descriptive qualitative approach, the techniques used in the study are tests, interviews, and documentation to explore data in research. Test and interview techniques were carried out in order to find out more about the ability of connections and generate knowledge about the transposition in making mathematical problem solving designs. The test is given to 3 students who have high ability, medium ability and low ability. Analysis of the data in this study is through the provision of problem solving tests and interviews using time triangulation, which provides tests and interviews for each student with a different time. The results in this study indicate that highly capable subjects have excellent mathematical connections and didactic knowledge is well utilized and able to provide alternative answers that differ from researchers so that a didactic transposition is formed, that is knowledge to be taught with knowledge, subjects of mathematical ability are able understand the problem well but are not careful enough in solving problem so that previous knowledge is underutilized, the subject of low mathematical ability is not able to understand the problem well and is unable to solve problems in the form of algebra so that the initial knowledge possessed is not well connected.

The study aims to describe students’ creative thinking skills in circle material based on students’ learning obstacle and trajectory. This research applied descriptive qualitative. The instruments used were circular questions and interviews. Data analysis carried out along with the data taken by analyzing students’ creative thinking skills based on students’ learning obstacle and trajectory. The research subjects were VI semester students who were able to handle analytical geometry. The results of the study shown that, most students experience learning obstacle (1). Students have difficulties in finding concepts from the meaning of circles, (2). Students have difficulties working on material circle questions and without re-examining the answers (3). Students found out difficulties in a concept from the general formula of the equation of the circle and make the graph, (4). Learning materials presented in textbooks do not build students’ creative thinking skills, so they do not support meaningful learning. Learning trajectory based on analysis of learning obstacle experienced by students and analysis of causes of difficulties can occur. Preparation of hypothetical learning trajectory through learning situations designed is expected to create students’ creative thinking, indeed, the student reconstruction process is created independent students form the concept of the formula set.

Masalah yang melatarbelakangi penelitian ini adalah rendahnya kemampuan bernalar kritis matematika pada siswa sekolah dasar dan kurangnya penerapan metode pembelajaran yang efektif untuk meningkatkan kemampuan siswa. Oleh karena itu, perlu dilakukan peningkatan kemampuan bernalar kritis matematika melalui penerapan model Problem Based Learning berbasis aplikasi Geogebra. Metode penelitian yang digunakan adalah penelitian tindakan kelas yang dilaksanakan dalam tiga siklus. Subjek penelitian adalah siswa kelas IV SDN Nggembe Tahun Pelajaran 2022/2023 yang berjumlah 25 siswa. Instrumen pengumpulan data yang digunakan adalah tes kemampuan bernalar kritis matematika dan lembar observasi. Data dianalisis menggunakan teknik analisis deskriptif kuantitatif dan kualitatif. Hasil penelitian menunjukkan bahwa penerapan model Problem Based Learning berbasis aplikasi Geogebra dapat meningkatkan kemampuan bernalar kritis matematika siswa SD. Peningkatan tersebut terlihat dari peningkatan skor rata-rata tes kemampuan bernalar kritis matematika siswa dari siklus I (63,6), siklus II (73,6), hingga siklus III (82,4). Selain itu, terdapat peningkatan pada kualitas pembelajaran yang ditunjukkan dari hasil observasi pada setiap siklusnya. Penerapan model Problem Based Learning berbasis aplikasi Geogebra juga mampu memotivasi siswa dalam pembelajaran matematika. Dari hasil penelitian tersebut dapat disimpulkan bahwa penerapan model Problem Based Learning berbasis aplikasi Geogebra dapat meningkatkan kemampuan bernalar kritis matematika pada siswa SD. Model pembelajaran ini dapat dijadikan alternatif pembelajaran matematika yang menarik dan efektif dalam meningkatkan kemampuan bernalar kritis matematika pada siswa SD.

This study aims to formulate an alternative learning design related to the concept of circle equations and tangents to analytic geometry learning through student learning obstacles. This is descriptive qualitative didactic design research (DDR) by identifying the learning obstacle of students obtained equations of circular equations and tangents of circles, then compiling pedagogical didactic anticipations through identified learning obstacles. This research subject is the sixth-semester students of analytic geometry courses. Data collection was carried out using documentation, interview, test and questionnaire techniques. The results showed an alternative didactic design that could be used in the learning equation of circles and tangents of circles and were able to facilitate students as a source of support for learning activities.