This research aimed to describe the levels of feeling of rightness (FOR) of students. This research used a qualitative method with an explorative type. The subjects of this research were 3 students of 5th grade selected from 77 other students in Indonesia. In uncovering FOR subject of this research, instruments were used in the form of problems about the open-ended polygon perimeter and interview guidelines. The data of this research were the subjects' answers to the problems of polygon perimeter and the results of interviews with subjects related to these answers. The data were analyzed using the FOR subject level indicator rubric. There were three levels of FOR which were the findings in this research, namely low, medium, and high. Low FOR level was indicated by the answers crossed out and the objectives or goal changed. Medium FOR level was indicated by crossed out answers, objectives or goals changed, problems text read repeatedly, indecisive statements about the truth of the answers that have been generated, and doubts in determining the steps to be taken. High FOR level was indicated by answers that were not crossed and goals that were not changed.

This study aimed to describe teacher interventions in students’ problem-solving. The subjects were three upper-class students at an elementary school in Indonesia who used system 2 when solving problems. This study used a qualitative case study approach. Data were obtained from students’ written answers and audio-visual recordings of teacher interventions to students. The results showed that the subjects needed teacher interventions to induce their awareness when involving system 2. Each subject needed intervention different stages. Subject 1 required intervention stage 3, subject 2 required intervention stage 2, and subject 3 only required intervention at stage 1. From the research results, it was known that the active moment of system 2 in all three subjects was the same, that is after the core problem was known. The core of the problem was ascertained after a doubtful feeling arose on the truth of the resulting answers. This feeling arose because the teacher intervened in the form of questions conducted dialogically.

Fenomena menunjukan kegagalan mahasiswa dalam mengonstruksi bukti secara sintaksis dikarenakan mahasiswa tidak berpikir analitik. Kemampuan mahasiswa dalam mengonstruksi bukti sangat menentukan keberhasilannya mempelajari matematika tingkat lanjut, sehingga penelitian untuk mengungkap fenomena ini penting dilakukan. Penelitian ini bertujuan mendeskripsikan berpikir analitik mahasiswa dalam mengonstruksi bukti secara sintaksis. Untuk mengetahui berpikir analitik mahasiswa, peneliti memberikan masalah pembuktian kepada sekelompok mahasiswa yang telah menempuh Geometri. Dilihat dari keanalitikan jawaban mahasiswa, peneliti menemukan tiga kategori berpikir, yaitu; (1) analitik, (2) semi analitik, dan (3) pre analitik. Berpikir analitik ditandai oleh adanya konstruksi bukti yang jelas, kejelasan algoritma, keruntutuan penalaran, penggunaan simbol secara tepat, setiap pernyataan dilandasi oleh alasan yang tepat dan langkah yang efektif. Berpikir semi analitik ditandai oleh adanya “elemen pengganggu” pemutus rantai implikasi. Berpikir pre analitik ditandai oleh penggunaan gambar sebagai bukti.Kata Kunci : Berpikir Analitik, Bukti, Sintaksis.

Elementary students� understanding of polygon areas concept is important in succeeding in their academic and daily life because the concept is broadly applied at schools and homes. Hence, a comprehensive understanding of the concept is required. The understanding can be seen from students� analytical thinking in facing a complicated problem. If a student does analytical thinking, the student can create a link between the concepts and predict what will happen. In fact, students frequently use procedural thinking to solve almost any type of problem, including non-routine problems. A study to reveal this phenomenon is thus important to conduct. This study aimed to describe students� analytical thinking in solving the polygon areas problems. To know the students� analytical thinking, the researcher gave problems toa team of mathematics Olympiad of elementary school students. Based on students� analytical answers, the researcher found (1) analytical and (2) semi-analytical thinking. Analytical thinking was characterized by algorithm clarity, chronological reasoning, valid argumentation, and effective steps. Semi-analytical thinking was characterized by a presence of �disturbing elements� which broke the chain of implications. The result of the study can be teachers� consideration in selecting teaching methods tailored to the students� thinking possibilities so that knowledge and learning experiences are well internalized.

Penelitian ini bertujuan mengkaji kasus pemecahan masalah modus tollens yang berkaitan dengan keyakinan. Penelitian ini adalah penelitian kualitatis dengan kenis studi kasus. Kasus yang dikaji dalam penelitian ini adalah kasus kolektif. Subjek penelitian adalah tiga mahasiswa Pendidikan matematika yang telah belajar logika matematika. Instrumen penelitian ini adalah Peneliti, masalah modus tollens, rubrik indikator, lembar validasi, buku catatan dan pedoman wawancara. Hasil penelitian ini adalah proses mental yang terkategori menjadi tiga dalam pemecahan masalah modus tolens. Tiga kategori proses mental itu adalah logic beyond belief, belief surrounds thinking1, dan belief surrounds thinking 2. Karakteristik logic beyond belief adalah menghasilkan jawaban benar dengan alasan benar, keyakinan terkontrol oleh sistem 2. Karakteristik belief surrounds thinking 1 adalah menghasilkan jawaban benar dengan alasan salah, mengubah jawaban berdasarkan keyakinan. Karakteristik belief surrounds thinking 2 adalah menghasilkan jawaban salah dengan alasan salah berdasarkan keyakinan.