The Ability to Prove Real Analysis Problems: Investigation with APOS Conception

Abstract

English

Proving is an important ability for pre-service mathematics teachers, but previous studies have rarely examined it in Real Analysis using the APOS framework, which is essential for revealing their cognitive development in constructing mathematical proofs. Real Analysis is one of the courses that develops proving ability, but their level of ability in prove problems based on the APOS conception remains unclear. Therefore, this study aims to analyze pre-service mathematics teachers’ proving ability in real analysis based on the APOS conception framework. This study uses a qualitative approach with 59 university students as participants. Data were collected from a test with four proof problems designed to represent different levels of reasoning in Real Analysis. The data were then analyzed using content and thematic analysis techniques guided by APOS indicators (action, process, object, and schema) to identify students’ cognitive structures in proving. The results indicate that students proved the problem without deep reflection. They did not recognize which method was required. They used some examples to prove a statement. They had not yet encapsulated the process of proof as a conceptual object. Thus, this study concludes that the ability of pre-service mathematics teachers to prove Real Analysis problems using the APOS conception approach was that most of them are still in the action conception, only a few have reached the process conception, and none have fulfilled the object and schema conception. These findings indicate that most students operate at the action level, relying on procedural manipulation without conceptual understanding, while only a few demonstrate process-level thinking by beginning to internalize proof structures. This suggests that students’ difficulties stem from a lack of conceptual understanding and insufficient experience in constructing formal proofs. Therefore, the curriculum should be designed to strengthen prerequisite abilities before studying Real Analysis and to develop a learning model that enables students to prove various Real Analysis problems. This condition reflects the need for instructional strategies that facilitate the transition from action to higher APOS levels.

Indonesian

Pembuktian merupakan kemampuan penting yang harus dimiliki mahasiswa calon guru matematika. Salah satu mata kuliah yang melatih kemampuan pembuktian adalah analisis riil. Permasalahannya adalah bagaimana kemampuan mereka dalam membuktikan masalah analisis riil ketika diinvestigasi dengan konsepsi APOS. Penelitian ini menggunakan pendekatan kualitatif dengan subjek penelitian 59 mahasiswa di sebuah universitas. Data yang diperoleh dari tes dengan empat soal kemudian dianalisis menggunakan teknik analisis isi dan tematik untuk mengidentifikasi pola berpikir dan konsepsi APOS yang muncul dalam proses pembuktian mahasiswa. Hasil penelitian ini menunjukkan bahwa mereka membuktikan masalah tanpa refleksi yang mendalam. Mereka tidak menyadari metode apa yang dibutuhkan untuk membuktikan. Mereka menggunakan beberapa contoh untuk membuktikan suatu pernyataan. Mereka belum mengemas proses pembuktian sebagai objek konseptual. Jadi, penelitian ini menyimpulkan bahwa kemampuan mahasiswa calon guru matematika dalam membuktikan masalah analisis riil menggunakan pendekatan konsepsi APOS sebagian besar masih dalam konsepsi aksi, hanya sedikit yang telah mencapai konsepsi proses, dan tidak ada yang memenuhi konsepsi objek dan skema. Oleh karena itu, kurikulum perlu dirancang untuk memperkuat kemampuan prasyarat sebelum mempelajari analisis riil dan merancang model pembelajaran yang memungkinkan siswa membuktikan berbagai permasalahan analisis riil.

Keywords

• APOS conception
• proving ability
• real analysis problems