Advanced Mathematical Thinking

The results show that there is an association between students’ Self-Renewal Capacity (SRC) and Advanced Mathematical Thinking (AMT). These findings are based on statistical test results. These results indicate that if students have high SRC, then they also have high AMT and if students have low SRC, then they also have low AMT. The association between students’ SRC and AMT can be seen from sub-indicator ‘exploration’ on SRC and components of AMT. If person has creative ideas, has an interest in the process of generalization, proof, and representation, and has a high curiosity about new something relatively, then they can develop representation, abstractions, creative thinking, and mathematical proofs ability. Otherwise, if person hasn’t creative ideas, hasn’t an interest in the process of generalization, proof, and representation, and has n’t a high curiosity about new something relatively, then they can not develop representation, abstractions, creative thinking, and mathematical proofs ability well.

Dengan ini saya menyatakan bahwa disertasi dengan judul “ Peningkatan Kemampuan Advanced Mathematical Thinking dan Disposisi Berpikir Kreatif Matematis Mahasiswa melalui Pendekatan M-APOS ” ini adalah benar-benar karya saya sendiri, dan saya tidak melakukan plagiarisme atau pengutipan dengan cara-cara yang tidak sesuai dengan etika yang berlaku dalam tradisi keilmuan. Atas pernyataan ini, saya siap menerima tindakan/sanksi yang dijatuhkan kepada saya apabila kemudian ditemukan adanya pelanggaran atas etika akademik dalam karya saya ini, atau ada klaim terhadap keaslian karya saya ini.

This study examines the enhancement of college students’ advanced mathematical thinking ability and disposition of mathematical creative thinking through M- APOS approach. The aim of this study is to examine comprehensively the contribution of M-APOS approach application toward achievement and enhancement of AMT ability and Disposition of Mathematical Creative Thinking (DMCT) of college students. This study applied quasi-experiment design. The samples of this study were all students in Mathematics Education of Faculty of Teacher Training and Educational Sciences who took Algebra subject. The research instrument used pretest and posttest for AMT ability, attitude scale for DMCT, observation sheet, and interview guidance. The results of this study are: (1) there is no achievement and enhancement difference between AMT ability of the students who got M-APOS approach and the students who got conventional learning; (2) there is no achievement and enhancement difference between AMT ability of the students who got M-APOS approach and the students who got conventional learning in upper and middle of PMA, while in the lower level of PMA, the enhancement of the students ’ AMT ability of M-APOS class is the higher than that of conventional class; (3) there is no interaction between teaching and the level of PMA toward the achievement and enhancement of students’ A MT ability; (4) there is no the achievement and enhancement difference between DMCT of the students who got M-APOS approach and the students who got conventional learning, but the achievement of students’ DMCT who got M -APOS approach is significantly higher than the students who got conventional learning; and (5) there is interaction between teaching and the level of PMA toward the achievement and enhancement of the students’ DMCT.

1. a. Pencapaian dan peningkatan Advanced Mathematical Thinking mahasiswa yang memperoleh pembelajaran Model PACE, baik secara keseluruhan maupun untuk semua level kemampuan awal matematis (tinggi, sedang, dan rendah), lebih baik daripada mahasiswa yang memperoleh pembelajaran konvensional. Pencapaian dan peningkatan Advanced Mathematical Thinking mahasiswa secara keseluruhan dan berkemampuan awal matematis tinggi yang memperoleh pembelajaran Model PACE berada pada kategori sedang, sementara untuk mahasiswa berkemampuan awal matematis sedang dan rendah, pencapaiannya berada pada kategori rendah, tetapi peningkatannya berada pada kategori sedang. Lain halnya dengan mahasiswa yang memperoleh pembelajaran konvensional, pencapaiannya secara keseluruhan dan tiap level kemampuan awal matematis (tinggi, sedang, dan rendah) berada pada kategori rendah, tetapi peningkatannya berada pada kategori sedang.

The main purpose of this research is to analize of achievement and enhancement of the students` Advanced Mathematical Thinking and Self-Renewal Capacity comprehensively as a result of the implementation of PACE (Project, Activity, Cooperative learning, Exercise) model learning and conventional learning. This research used a quasi-experimental with pretest-posttest control group design. The population of this research included all students of regular class in the mathematics education department of one of the private universities in East Jakarta and the sample was group of students joining Mathematical Statistic Subject. This research used various instruments. They were test of mathematical prior knowledge, test of advanced mathematical thinking, self-renewal capacity scale, observation sheet, and interview sheet. For data analysis, this research used parametric and non-parametric statistic. The result of this research are: (1) the achievement and enhancement of the students` Advanced Mathematical Thinking and Self-Renewal Capacity taught by using PACE model learning are better than the achievement and enhancement of those who were taught by using conventional learning; (2) there is no interaction between learning (PACE model learning and conventional learning) and mathematical prior knowledge (high, intermediate, low) towards the achievement and enhancement of the students` Advanced Mathematical Thinking and Self-Renewal Capacity; and (3) there is association between the students’ Advanced Mathematical Thinking and Self- Renewal Capacity.

Mathematical problem solving is the heart of mathematics and visualization is the core of solving mathematics. Visual thinking is the most effective and powerful in mathematics instruction. The way we learn, and then remember, sustaining a strong relationship with the way our senses operate solve math problems. Schools have the freedom to manage and choose approach to learning mathematics. How can character education is taught to students and therefore will have the opportunity to emphasize the special character of the school. While each school has the freedom to choose the kind of values that, for nation building must have some values taught in every school. Those values must be the most important value to support the Indonesian nation-building project. First, multicultural values. Indonesia consists of various ethnic, religious, and cultural. If the nation wants to be stronger in the future, we must accept differences between us. Without accepting the differences, we will easily slip into fights and conflicts. The spirit of multiculturalism and religious diversity would encourage all people to accept others as members of the Indonesian nation. This is the spirit of Bhineka Tunggal Ika: although different, but still one. We sometimes lack of respect that spirit. Some of us want to impose our ideas on others Second, honesty. One important reason why corruption is very difficult to be eradicated in Indonesia is the lack of honesty. Most of us are not honest anymore. The spirit of honesty in the learning of mathematics must be inculcated in school. Students should learn about the value of honesty and implementation. Students must learn to be honest in school, to be honest with yourself, honest with others, be honest with their life and honest about what they do. Students should be honest when completing math problems, tests and national

Gambaran psikologis siswa terhadap pemahaman matematik akan terlihat lemah pada masa awal pembelajaran, soal – soal/ tugas – tugas matematika yang dimunculkan pada masa ini didesain untuk menstimulasi ide – ide, strategi dan representasi (Develop Understanding). Setiap ide, strategi dan representasi ini, kemudian akan dikoreksi untuk melihat ketepatan dan kelengkapannya, serta diperluas dan dikoneksikan dengan beberapa materi terkait, melalui pemberian stimulant atau pengalaman pada siswa sehingga ide – ide, strategi dan representasi yang telah mereka dapatkan sebelumnya menjadi lebih nyata, kuat dan bermakna (Solidify Understanding). Dalam kerangka CMI, ide-ide yang telah menjadi lebih kuat dan tegas disebut konsep; strategi yang kuat menjadi algoritma; dan representasi yang bermakna menjadi alat. Meskipun kemampuan pemahaman telah dikembangkan dan dikuatkan, akan tetapi, tetap masih perlu perbaikan lebih lanjut untuk menjadi mencapai kemahiran (Practice Understanding). Dalam kerangka CMI, ide yang telah disempurnakan akan berkembang menjadi definisi atau properti; algoritma yang tepat berkembang menjadi prosedur; dan alat-alat yang telah disempurnakan berkembang menjadi sebuah model matematik. Definisi dan sifat, prosedur, dan model yang telah diperoleh, kemudian harus disesuaikan dengan hasil – hasil pemikiran yang dilakukan oleh teman – temannya, melalui kegiatan diskusi, sehingga setaip siswa memperoleh keyakinan bahwa proses pemikiran mereka telah tepat. Pada komponen Continuum of Mathematical Understanding ini, proses – proses konseptualisasi matematik, doing mathematics dan representasi matematik berlangsung pada sepanjang garis kontinum, seperti tergambar pada gambar 4 di bawah, dengan demikian, diharapkan tujuan pembelajaran matematika yakni pencapaian pemahaman matematik yang mendalam akan tercapai.

Recognizing the importance of a strategy and learning approach to develop the students thinking skills, it is absolutely necessary to mathematics learning that more actively involve students in the learning process itself. This can be realized through an alternative learning that is designed such that it reflects the involvement of students actively and constructively. Students as learners need to get used to being able to construct their own knowledge and being able to transform into other that more complex situations so that such knowledge will become the property of the learner itself, which is attached forever. The process of constructing knowledge can be done by the learners themselves based on the experience that has been previously owned, or may also be a result of the discovery that involve environmental factors. Based on the views of constructivism, a learning strategy must have characteristics as follows: use more time to develop an understanding that can enhance the ability of learners to use the knowledge, involve students in the learning process so that the abstract concepts presented more concrete, implementation of small group discussions, presentation of the problems that are not routine.

Studi ini dirancang dalam bentuk eksperimen dengan disain kelompok kontrol dan postes saja yang bertujuan menelaah peranan pembelajaran yang mengajarkan berpikir metaforik terhadap kemampuan bertanya matematis guru SMA. Populasi dalam penelitian ini adalah guru SMA mata pelajaran matematika di Provinsi Jawa Barat, sedangkan sampel penelitian ini adalah 124 orang guru SMA mata pelajaran matematika yang ditetapkan secara purposif kemudian ditetapkan secara acak yang termasuk ke dalam kelas eksperimen dan kelas kontrol. Berdasarkan hasil dan pembahasan diperoleh kesimpulan: (1) Kemampuan bertanya matematis guru yang memperoleh pembelajaran Metaphorical Thinking lebih baik daripada yang memperoleh pembelajaran biasa; (2) Faktor pembelajaran dan KAM masing-masing mempengaruhi ketercapaian kemampuan bertanya matematis guru. Selain itu, terdapat efek interaksi antara pembelajaran dan KAM secara bersama-sama dalam mengembangkan kemampuan bertanya matematis guru; (3) Ketercapaian penguasaan kemampuan bertanya matematis guru masih belum tercapai dengan baik pada indikator pengajuan permasalahan berupa pertanyaan non-rutin dan pertanyaan terbuka.

This research was a quasi-experimental research, which aimed to determine: the effect comparison of Rigorous Mathematical Thinking (RMT), Problem Based Learning (PBL), and Direct Learning (DL) to mathematical conceptual understanding and mathematical strategic competence of Junior High School„s students in Ngawi on Academic year 2016/2017. In addition to the factor of learning model, in this research we also examined the factors of sex difference, and the interaction between learning model and sex difference.

Selain itu, rendahnya Advanced Mathematical Thinking mahasiswa juga diungkapkan oleh Herlina (2015) dalam studi pendahuluannya, yaitu mahasiswa mengalami kesulitan dalam memahami konsep dalam bentuk notasi matematika, membuktikan, mengaitkan antar konsep, serta menghasilkan ide-ide kreatif dalam menyelesaikan permasalahan matematika. Berkaitan dengan rendahnya Advanced Mathematical Thinking, Tall (2002) mengatakan bahwa salah satu penyebabnya adalah dosen masih terbiasa mengajar secara prosedural dan akan membenarkan jawaban mahasiswa jika mengikuti prosedur tersebut.