Using APOS Theory Framework: Why Did Students Unable To Construct a Formal Proof?

Abstract

AbstrakBerpikir matematis sangat dibutuhkan dalam belajar matematika, apalagi di perguruan tinggi. Salah satu aktifitas belajar matematika ialah pembuktian matematis. Penelitian ini mendeskripsikan proses berpikir mahasiswa yang gagal mengonstruksi bukti-matematis formal. Analisis dalam deskripsi tersebut menggunakan Teori APOS dalam menjelaskan struktur mental dan mekanisme mental yang muncul ketika melakukan pembuktian matematis. Penelitian ini menggunakan pendekatan kualitatif. Penelitian ini melibatkan 26 mahasiswa pendidikan matematika di salah satu perguruan tinggi di Banten, Indonesia. Data diperoleh melalui think-aloud ketika menyelesaikan tugas pembuktian dan dilanjutkan dengan wawancara berbasis tugas. Hasil penelitian ini menyatakan bahwa mahasiswa gagal mengonstruksi bukti matematis formal karena proses berpikir mahasiswa tersebut hanya melalui tahapan interiorisasi masalah, interiorisasi objek, dan koordinasi. Dengan demikian, mekanisme mental lainnya, seperti: enkapsulasi, de-enkapsulasi dangeneralisasi belum muncul. Hal tersebut menunjukkan bahwa struktur mental Action telah terbentuk, namun Process dan Object belum muncul. Oleh karena itu perlu dipertimbangkan suatu strategi pembelajaran yang mendukung mekanisme mental yang dibutuhkan dalam mengonstruksi bukti. Kata Kunci: proses berpikir, konstruki bukti, pembuktian, bukti matematis, teori APOS AbstractMathematical thinking is necessary for mathematics learning, especially in undergraduate level. One of the activities in undergraduate mathematics learning is proving. This article describes students' thinking process who unable to construct a mathematical formal proof. The description uses APOS Theory to explore students' mental mechanism and students' mental structure while they do proving. This research is qualitative research that conducted on 26 students majored in mathematics education in public university in Banten province, Indonesia. Data was obtained using think-aloud when solving the proving task and then interview based the task. Results show that the students could not construct a formal proof because the mental mechanism that appeared was interiorization and coordination. Based on the results, some suitable learning activities should be designed to support the construction of these mental mechanisms.

Improving the Reasoning Ability of Elementary School Student through the Indonesian Realistic Mathematics Education

“…It is hard to imagine how the learning process runs and what the learning result is towards the people who do not possess the listening ability and aims to know the beautiful chirp of birds, even more, towards the people who have the listening ability but never hear the birds chirp (Syamsuri, Purwanto, Subanji, & Irawati, 2017). They might give a written answer, but they cannot understand it and it is not the knowledge they have.…”

Exploring the Mental Structure and Mechanism: How the Style of Truth-Seekers in Mathematical Problem-Solving?

“…In general, Mathematics student-teachers do not perform the process of encapsulation to object conception upon solving the problem of an infinite set (Brijlall & Maharaj, 2015). That statement is in line with the notion claiming that students fail to construct formal mathematical evidence because the process of thinking in its mental mechanism has yet to manifest encapsulation, de-encapsulation, and generalization (Syamsuri, et al 2017).…”

“…The findings of this study have contradicted the results of other studies, which suggest that Mathematics students, upon working on the problem of algebra, only prove object which has been formed while the process, action, and scheme are not constructed. What is more, the tendency of their mental mechanisms is only marked by interiorization and co-ordination (Syamsuri, et al 2017). But, in the process of constructing their knowledge, the Mathematics students performing the truth-seeking process also perform the process of understanding problems, exploring, formulating, justifying, and proving possible incorrect information in both questions (Astawa, Budayasa, & Juniati, 2018).…”

“…However, no research delves into the relationship between critical thinking dispositions, especially the truth-seeking of Mathematics students with the ability to construct their knowledge when solving problems based on their mental mechanisms and structures. Some researchers conduct studies focusing on the mental mechanism and structure of students in solving mathematical problems or proving a theorem without activating the disposition of critical thinking, especially the truth-seeking component (Brijlall & Maharaj, 2015;Syamsuri, Purwanto, Subanji, & Irawati, 2017). In general, Mathematics student-teachers do not perform the process of encapsulation to object conception upon solving the problem of an infinite set (Brijlall & Maharaj, 2015).…”

The instruments of higher order thinking skills

“…Several researchers already documented their research about students' mathematics thinking ability [10][11][12][13][14]. Based on their studies, we can conclude that the thinking ability of high-level PGSD students is still low.…”