# Why should We Change Mathematics Teaching?

This is the question that I proposed to my teacher in one of my course. In the discussion, we discussed about the percentage and its relationship with decimal, fractions, ratio and proportion. I argue that we have all been taught about percents and other topics of mathematics in the traditional way, with an emphasis on learning the correct calculation procedures, and very little or not on understanding. Still we all understand what percentages, fractions and decimals are and how they are related. So why should we change the way these topics are taught? Or in any case, why should we not teach the calculation procedures first and let understanding develop later? These questions are the paraphrased ones done by my teacher.

It is an issue that we should discuss, because we hear this argument quite often. Then, my teacher proposed his own question relating to my own question: what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?

Mathematics as a human activity

I try to give an answer to my own question, why should we change the way these topics are taught? In my perspectives, I think we have to change the way mathematics has been taught because when we teach students in the traditional way of teaching, I can be really sure that they will not make sense of what they have been learned. Moreover, as Freudenthal said in his wonderful book, Revisiting Mathematics Education, mathematics is a human activity and therefore it must be connected to reality, stay close to children and be relevant to society in order to be of human value. So, in this sense, the focal point is not mathematics as a ready-made subject full of procedures, formulas or whatsoever. Instead, the focal point is on the activity. Education should gives students a guided opportunity to reinvent the mathematics.

why should we not teach the calculation procedures first and let understanding develop later?
I think, if we do it in this way, students will not develop their understanding of mathematics. Instead, what they get from this way of teaching is that they know the procedures and formulas but they do not know how to use it in their life. It can be seen from the test such as PISA or TIMSS, that many Indonesian students fail to answer questions that dealing with complex situations and asking for advance mathematical thinking and reasoning. Even, I argue that students will not develop their understanding after they know the procedures and formulas. It is because, they will always looking for the formulas and procedures when they want to solve the problems.
what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?
As Freudenthal said that learning processes, or at least part of them, can be more essential than their products (Cited from Revisiting Mathematics Education book). The focus of the teaching of mathematics is the process of learning it, not the product. In RME, the use of context is very important. In contrast with the traditional way of teaching, RME use the context problem both as a source of learning and to apply mathematical concepts. The students, in RME class, can develop mathematical tools and understanding while working on context problems. In the first time, they develop strategies or models that are closely related to the context. Later on, certain aspects of the context situation can become more general which means that the context can get more or less the character of a model and as such can give support for solving other but related problems (http://www.fisme.science.uu.nl/en/rme/).
One of a good example is the experience of a college professor who experienced the change in her son’s understanding of mathematics after following the RME class in the Netherlands. You can read her story in this link: http://mathematicallysane.com/realistic-math-makes-sense-for-students/

# Constructivist Approaches are Compatible with Human Cognitive Architecture: A Response to Kirschner, Sweller, and Clark (2006)

## Introduction

Learning is much more than memorizing. Learning refers to the acquisition of knowledge through interactions with, and observation of, the physical word and the creatures that inhabit it (Ashman & Conway, 1997). In order to really understand and be able to apply knowledge, students must work to solve problems, to discover things for themselves, and to struggle with ideas. The question of how to help students learn particular knowledge, skills, and concepts that will be useful in their life is at the core of the argument presented by Kirschner, Sweller, and Clark (2006). The authors compare minimally guided instructions with instructional approaches that provide direct instructional guidance of the student learning process. They define minimally guided instruction as ”one in which learners, rather than being presented with essential information, must discover or construct essential information for themselves” and then inversely define direct instruction as “providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture” (p. 1).

In their argument, Kirschner Sweller, and Clark (2006) affirm that minimal guided instruction approaches are less effective and efficient than fully guided instruction approaches because they ignore the structures that constitute human cognitive architecture. On the contrary to this, they put a strong emphasis on direct, strong instructional guidance, as an effective and efficient way to teach students. By referring to several studies concerning the efficacy of direct instruction (e.g., Klahr & Nigam, 2004), they claim that students learn more deeply from strongly guided instruction than from constructivist or discovery approaches. Opposing this claim, Kuhn and Dean (2006) have found that direct instruction does not work so well for robust acquisition or for maintenance knowledge over time.

# 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.

# 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.

# Teori Level Van Hiele dalam Pembelajaran Geometri

Pengalaman dari para guru matematika di sekolah menengah (baik SMP maupun SMA) menunjukkan bahwa banyak siswa mengalami kesulitan dalam belajar geometri, khususnya dalam melakukan pembuktian formal. Apa sebenarnya penyebab dari kesulitan tersebut? Selama periode 1930 sampai 1950, beberapa pendidik matematika dan psikolog dari Soviet mengkaji pembelajaran geometri dan mencoba untuk menjawab pertanyaan tersebut. Wirszup (1976) misalnya, melaporkan bahwa:

This very significant research has influenced the improvement in the teaching of geometry only slightly. The truly radical change and far-reaching innovations in the Soviet geometry curriculum have, in fact, been introduced thanks to Russian research inspired by two Western psychologists and educators.

Orang pertama yang dimaksud adalah Jean Piaget dan kedua adalah P.M. van Hiele, seorang pendidik berkebangsaan Belanda, yang meneliti tentang peranan intuisi dalam belajar geometri menarik perhatian orang-orang Soviet setelah dia mengirim makalah berjudul “La pensee de l’enfant et la geometrie” pada konferensi pendidikan matematika di Sevres, Prancis pada tahun 1957.

Pendekatan yang digunakan dalam mengajarkan geometri biasanya cenderung berbeda dengan materi matematika lain, dimana siswa diperkenalkan tentang belajar dengan menggunakan sistem matematika (melalui penggunaan berbagai macam postulat atau aksioma, teorema, definisi dan mengerjakan dengan pembuktian) dan pada saat yang sama siswa juga belajar tentang materi geometri itu sendiri. Oleh karena itu, meskipun materi geometri tersebut masih sangat mendasar, materi tersebut tetap diajarkan secara abstrak.

# What are Learning trajectories? Sebuah Pengantar tentang Lintasan Belajar

Anak-anak, khususnya yang duduk di kelas rendah (1 – 3) di sekolah dasar, mengikuti suatu pola tingkatan alamiah ketika mereka belajar maupun dalam proses perkembangannya. Sebagai contoh, pada awalnya mereka belajar merangkak, berjalan, lalu berlari, dan melompat dengan kecepatan dan kecekatan yang terus meningkat seiring dengan perkembangan fisiknya. Begitupula ketika mereka belajar. Dalam belajar matematika misalnya, mereka juga mengikuti suatu pola tingkatan alamiah, yakni belajar kemampuan-kemampuan dan ide-ide matematika dengan cara mereka sendiri. Ketika para guru memahami pola tingkatan alamiah tersebut, serta aktivitas-aktivitas yang tersusun didalamnya, maka mereka telah membangun suatu lingkungan belajar matematika yang tepat dan efektif. Pola tingkatan alamiah tersebut merupakan dasar dalam membuat learning trajectories atau lintasan belajar. Lintasan belajar sangat berguna bagi guru, khususnya dalam hal menjawab berbagai pertanyaan seperti:  apa tujuan pembelajaran yang akan kita capai? bagaimana kita memulainya? bagaimana kita mengetahui tentang langkah-langkah yang akan kita lakukan? bagaimana kita bisa mencapai tujuan tersebut? dan seterusnya.

Lintasan belajar mempunyai tiga bagian penting yakni: tujuan pembelajaran matematika yang ingin dicapai, lintasan perkembangan yang akan dikembangkan oleh anak atau siswa dalam mencapai tujuan pembelajaran, dan seperangkat kegiatan pembelajaran ataupun tugas-tugas, yang sesuai dengan tingkatan berpikir yang ada pada lintasan perkembangan yang akan membantu anak tersebut dalam mengembangkan proses berpikirnya bahkan sampai pada proses berpikir tingkat tinggi. Ketiga bagian penting dari lintasan belajar tersebut dibahas secara singkat sebagai berikut.