Domain-specific and Metacognitive knowledge: The Good Approaches to Effective Teaching Thinking and Problem Solving?

Introduction

Educators and researchers for many years have been concerned about how to teach thinking and problem solving effectively. However, a lot of researches indicated that students failed to develop their ability in thinking and problem solving when facing everyday phenomena and problems. It seems that students mainly acquired knowledge that remained inert. Their knowledge cannot be useful tools to deal with everyday problems and only ended as a lesson to be learned in the schools. Domain-specific and metacognitive knowledge, however, are pronounced as two approaches that can make teaching reasoning, thinking and problem solving effectively. The goal in this essay is to discuss about domain-specific and metacognitive knowledge in order to answer the following question: do domain-specific and metacognitive knowledge can make teaching thinking and problem solving effectively to encounter everyday situations and solving problems?

Evidence from the literature

In this essay, I emphasize to focus on the discussion about the literature-based study done by Bransford et al. (1986) presenting about two theoretical perspectives – executive or metacognitive processes and domain-specific knowledge – that can improve the teaching thinking and problem solving. Bransford and his colleagues explain the role of specific knowledge by showing several results of well-known research done by the experts and researchers. Taking as an example the study of deGroot, (Brandsford et al., 1986), comparing chess masters and novice ones, gives a description that conditional knowledge can help individuals to solve their problems effectively. In the deGroot research, the chess masters who already have knowledge base in playing chess, performed better in remembering the position of the game when it was meaningful for them. Furthermore, from the other research, it is also noticed that specific knowledge determines the learners’ strategies and their perspectives in solving meaningful problems. For example, Chi (Brandsford, 1986) found that 10-year-old chess enthusiasts can remember the position of the chess pieces more accurately than the college students do who are not experienced in playing chess.

On the other hand, metacognition also play an important role for improving teaching thinking and problem solving. Consider for example the investigation lead by A. Brown, Campione, and Day (Brandsford, 1986) who studied about the effects of different types of teaching environments in transfer tasks. It is noted that many individuals who were failed in transfer tasks are actually do not know why the strategies that they have learned are useful and when they would be used.

Combining both metacognitive and domain-specific knowledge, indeed, can strengthen the teaching thinking and problem solving. By having the ability to use their strategies and knowing the specific knowledge relating to the problems, the learners can increase their critical thinking and become an effective problem solver.

Conclusion

The two theoretical perspectives – metacognitive and domain-specific knowledge – provide a framework for students to think and solve the problems. Students not only have the ability about the general strategies and skills in problem solving, but also conditional knowledge that available for them to understand how concepts and procedures can be used as tools to solve the problems.

References

Brandsford, J., Sherwood, R., Vye, N., & Rieser, J. (1986). Teaching thinking and problem solving: Research Foundations. American Psychologist, 41, 1078 – 1089.

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My Math Story in the Third Grade

 

Mathematics is always being challenging subject for me, and also for many other students as well. I remember, when I entered primary school as a third grader, my mathematics teacher gave me and also my friends a lot of assignments to be done. What a pain was that most of them were about remembering or memorizing such as multiplication tables or formulas. At that time, mathematics was not favorite subject of mine.

Meanwhile, looking back at the classroom, I remember my math teacher at the third grade. She was actually my home teacher at that class, since in Indonesian school contexts, there only one teacher for all subjects in the grade 1 until 3. She was very kind and nice teacher. But, in the certain moment, when the students could not memorize the multiplication table for example, she would be angry and gave the students a punishment like a lot of home works. Also, since she was a teacher for all subjects in the third grade, she could not arrange and think deeper about the topics in mathematics or even how to teach those topics to the students.

Furthermore, there are also several bad evidences in my classroom at that time. The teacher taught the students in a book-oriented way. We as a student have to buy the book, either from the teacher or the book shops, in order to use it in the classroom. The students who do not have it would find difficult to follow the learning process in the classroom and gave a very bad effect on their score. Apart from it, the teacher tended to teach the students in a so called traditional approach of teaching mathematics. She, as a teacher, was the one who mastered the topics and have to transfer all of the information and knowledge to the students who were considered as a receiver. If I and my friends fail to understand something or in the exams, it would be considered as our faults and we have to take remedial in order to get a better score.

Facing those kinds of teaching and learning process made me confused, boring and sometimes frustrated. I struggled with mathematics, especially when it came to the problems requiring formulas or rules. Instead of being an active learner, I became a very quite person in the classroom. The causes, perhaps, due to the rules of the classroom or because of my lack of understanding about mathematics or both. Listening carefully and quietly to the teacher’s explanation were considered as a rule that every student has to take into account. Otherwise, they would be considered as a naughty student. One of anecdotes that was rising among students about learning mathematics is that, mathematics means “mati” or dead. It derived from Indonesian language – matematika – and students just take “mati” from the middle of that word.

In conclusion, mathematics in the time when I studied it in the third grade was taught in a traditional way of teaching. I was struggling in dealing with mathematics problems and formulas. Also, the teacher only transferred the information and knowledge to the students, without paying attention the development of students.

 

Integrating Technology into Mathematics Education: Theoretical Perspectives (A Summary)

For as long as mathematics education has been considered to be a serious scientific domain, researchers, educators, and teachers have been theorizing about the learning and teaching of mathematics. This has led to an overwhelmingly broad spectrum of theoretical approaches, ranging from the philosophical to the practical, from the global to the local, some focusing on learning in general and others very much based in mathematical knowledge. This body of theoretical knowledge is still growing.

Now that the issue of integrating technological tools into the teaching and learning of mathematics has become urgent, one can wonder what the existing theoretical perspectives have to offer. Can they be applied to this new context? Or do we need specific theories appropriate for the specific situation of using tools for doing – and learning – mathematics? If yes, what is so specific about the integration of technology that justifies the need for such new paradigms? Are these paradigms to be considered as part of the body of knowledge of mathematics education, or do they rather belong to theories about humans’ interactions with technology? What kinds of problems do we want the theoretical frameworks, old or new, to solve for us? It is our conviction that theoretical frameworks are needed in order to guide the design of teaching, to understand learning, and to improve mathematics education.

From the time of the development of the mainframe computer in 1942, the first four-function calculator in 1967, the microcomputer in 1978, and the graphing calculator in 1985 (Kelly 2003), both mathematicians and mathematics educators have been intrigued by the possibilities offered by technology. However, it was not until the late 1960s when, according to Fey (1984), mathematicians and mathematics educators began to feel that computing could have significant effects on the content and emphases of school-level and university-level mathematics.

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Pipet, Kartu Berangka dan Pembelajaran Nilai Tempat

Laporan ini memaparkan tentang proses belajar mengajar Nilai Tempat pada kelas II di SD Negeri 98 Palembang, Sumatera Selatan, Indonesia dengan menggunakan pendekatan Realisitic Mathematics Education (RME). Pembelajaran ini melibatkan 32 orang siswa dalam dua hari.

Pada hari pertama, saya bertindak sebagai guru di kelas. Proses belajar mengajar berlangsung selama 60 menit yang dibantu oleh Ibu Maryani dan Novita Sari untuk mendokumentasikan dan membantu siswa untuk mengikuti pembelajaran. Pada pertemuan ini, kami memperkenalkan konsep Faktor Persekutuan Terbesar (FPB) kepada siswa dengan memberikan masalah realistik.

Pada observasi ini, Ibu Mariani bertindak sebagai guru di kelas. Aktivitas pembelajaran berlangsung selama kurang lebih 60 menit yang dibantu oleh Novita Sari dan saya untuk mendokumentasikan dan membantu siswa untuk mengikuti pembelajaran. Pada pertemuan ini, kami memperkenalkan konsep nilai tempat kepada siswa melalui cerita berjudul “Ibu Ros dan Pempek”.

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Permen, Tepuk Tangan dan Pembelajaran KPK & FPB

Laporan ini memaparkan tentang proses belajar mengajar Kelipatan Persekutuan Terkecil (KPK) dan Faktor Persekutuan Terbesar (FPB) pada kelas V di SD Negeri 98 Palembang, Sumatera Selatan, Indonesia dengan menggunakan pendekatan Realisitic Mathematics Education (RME). Pembelajaran ini melibatkan 33 siswa dalam dua hari.

Pada hari pertama, saya bertindak sebagai guru di kelas. Proses belajar mengajar berlangsung selama 60 menit yang dibantu oleh Ibu Maryani dan Novita Sari untuk mendokumentasikan dan membantu siswa untuk mengikuti pembelajaran. Pada pertemuan ini, kami memperkenalkan konsep Faktor Persekutuan Terbesar (FPB) kepada siswa dengan memberikan masalah realistik.

Pada pertemuan kedua, di hari kedua, Novita Sari bertindak sebagai guru dan dibantu oleh Ibu Maryani dan saya untuk membimbing siswa dan mendokumentasikan proses belajar mengajar. Pada kesempatan tersebut, kami memperkenalkan konsep Kelipatan Persekutuan Terkecil (KPK) dengan menggunakan masalah realistik sebagaimana yang kami lakukan pada pertemuan pertama.

Dengan menggunakan pendekatan RME tersebut, kami memulai pembelajaran dengan sebuah masalah yang berasal dari lingkungan siswa sendiri, sesuai dengan materi pembelajaran, membimbing siswa menemukan sendiri konsep KPK dan FPB, serta memberikan kesempatan kepada siswa untuk saling berinteraksi satu sama lain.

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Menghitung Semua Kemungkinan (Sebuah Strategi Pemecahan Masalah)

Mempertimbangkan semua pilihan dapat menjadi cara yang efektif untuk memecahkan sebuah masalah. Meskipun mungkin ada beberapa hal dimana strategi ini bukanlah merupakan prosedur yang paling canggih, strategi tersebut mungkin adalah strategi yang paling mudah digunakan, karena strategi tersebut tidak terlalu abstrak. Namun, dengan menghitung semua kemungkinan merupakan hal yang krusial dalam menggunakan strategi ini. Jika kita tidak mempunyai langkah-langkah pengaturan untuk menghitung semua kemungkinan, strategi tersebut sering kali tidak berhasil dengan baik. Hal ini terlihat dalam aplikasi matematika dan contoh-contoh dalam kehidupan sehari-hari yang lebih rumit dengan menggunakan strategi ini.

Kita sering kali menggunakan strategi pemecahan masalah ini dalam kehidupan sehari-hari tanpa menyadari bahwa strategi ini sebenarnya telah kita gunakan. Misalnya anda diundang untuk datang pada suatu pertemuan di hotel yang berjarak sekita 150 mil. Cara yang paling banyak digunakan oleh orang untuk memutuskan jalan terbaik untuk pergi ke pertemuan tersebut adalahh dengan mendaftar semua kemungkinan jenis perjalanan (misalnya kereta, pesawat, mobil, bus, helikopter, dll) yang dapat digunakan. Baik tertulis maupun secara mental, dan kemudian memilih metode yang paling efisien dengan mengeliminasi atau memilih secara langsung (disebabkan oleh waktu, biaya, dll).

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Landskape Pembelajaran Matematika

Masih ingat dengan tulisan sebelumnya tentang kelas Madeline? Klo belum, saya menyarankan kepada para pembaca untuk membaca tulisan sebelumnya berjudul “matematika atau matematisasi” agar tidak mengalami kebingungan.

Secara historis, para pembuat kurikulum tidak menggunakan kerangka pengembangan seperti yang digunakan oleh Madeline ketika mereka menyusun sebuah standar kurikulum, seperti halnya mereka tidak melihat matematika sebagai suatu proses matemamatisasi – sebagai aktivitas.  Mereka menggunakan kerangka pembelajaran berdasarkan akumulasi konten mata pelajaran.  Mereka menganalisis struktur matematika dan menggambarkan tujuan-tujuan pembelajaran seperti sebuah garis. Kemampuan-kemampuan dan ide-ide kecil diasumsikan terakumulasi kedalam konsep-konsep (Gagne 1965; Bloom dkk, 1971). Sebagai contoh, ide sederhana tentang pecahan dianggap sesuai bagi siswa jika mereka diajarkan dengan cara menunjukkan bagian yang diarsir dari keseluruhan suatu bentuk atau dengan pola blok – blok. Selanjutnya, di kelas tiga, kesamaan pecahan kemudian diperkenalkan, dan berlanjut sampai pada kelas lima dan enam, operasi pada pecahan. Tahapan perkembangan hanya dipertimbangan dalam hal hubungannya dengan konten: dari konsep-konsep dan kemampuan-kemampuan sederhana sampai pada yang kompleks.

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