Profil Proses Berpikir Mahasiswa dalam Pembuktian Matematis pada Aljabar Abstrak
Abstract
abstrak— Proses berpikir merupakan bagian penting yang dibutuhkan dalam pembuktian matematis. Aljabar abstrak merupakan mata kuliah analisis yang membutuhkan kompetensi pembuktian. Tujuan penelitian ini untuk mendeskripsikan proses berpikir yang terjadi ketika melakukan pembuktikan pada mata kuliah aljabar abstrak. Pada kajian ini pembuktian matematis dibedakan atas pembuktian pemenuhan syarat-syarat pada definisi suatu konsep dan pembuktian pada teorema. Metode penelitian menggunakan pendekatan deskriptif kuantitatif-kualitatif berbasis tugas, disertai instrumen tes dan wawancara. Sampel penelitian yaitu dua kelas mahasiswa yang mengikuti mata kuliah aljabar abstrak sebanyak 25 mahasiswa. Sedangkan 4 subyek penelitian dipilih berdasarkan 2 jenis pembuktian. Hasil penelitian secara kuantitatif menunjukkan proses berpikir mahasiswa yang menggunakan tahapan analogiàabstraksi sebanyak 83 %, konstruksiàanalogi sebanyak 11 %, abstraksiàkonstruksi 6%, sedangkan untuk sampai tahapan konstruksi formal masih sebesar 0%. Secara kualitatif kecenderungan subyek dalam melakukan pembuktian terkait suatu himpunan dengan operasi biner tertentu pemenuhan syarat-syarat grup/subgrup sudah menggunakan tahapan berpikir abstraksiàkonstruksi, sedangkan pembuktian bentuk teorema masih menggunakan tahapan berpikir analogiàabstraksi. Hal ini menunjukkan mahasiswa masih adanya ketergantungan pada contoh-contoh pembuktian teorema yang terdapat pada buku.
Kata kunci—Proses berpikir, Pembuktian matematis, Aljabar abstrak
Abstract— The thinking process is an important part required in mathematical proof. Abstract algebra is an analysis course that requires evidential competence. The aim of this research is to describe the thinking process that occurs when carrying out proofs in abstract algebra courses. In this study, mathematical proof is differentiated into proof of the fulfillment of the conditions in the definition of a concept and proof of theorems. The research method uses a task-based quantitative-qualitative descriptive approach, accompanied by test and interview instruments. The research sample was two classes of students taking abstract algebra courses totaling 56 students. Meanwhile, 4 research subjects were selected based on 2 types of evidence. Quantitative research results show that students' thinking processes use the analogyàabstraction stage as much as 83%, constructionàanalogy as much as 11%, abstractionàconstruction 6%, while up to the formal construction stage it is still 0%. Qualitatively, the subject's tendency to carry out proofs related to a set with certain binary operations to fulfill group/subgroup requirements already uses the abstractionàconstruction thinking stage, while proving the theorem form still uses the analogyàabstraction thinking stage. This shows that students are still dependent on examples of theorem proofs found in books.
Keywords— Thinking process, mathematical proof, abstract algebra
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