RME Lesson Plan

25 Sep

  An example of a lesson material plan based on realistic approach


   Lesson: Introduction to the linear equation

  Time : (2 x 40 minutes)

   Grade: Eight, Junior Secondary School students


Goals: After the students follow the lesson, they are able to:

  • communication about mathematics during the discussion;
  • learn to improve their reasoning skills by presenting the result of their work;
  • learn how to improve their critical attitude by conflict to other students opinion;
  • construct the relevant mathematics concept (in this case: linear equation) from a contextual problem, and
  • produce a solution in their own strategies.


  • Give the students a contextual problem that related to the topic as the starting point.

For example: Anisa bought an apple and two mangoes in a supermarket. She gave Rp. 3.500,- to the seller. Right after that, Ilham paid Rp. 4.000,- for two apples and a mango. So, can you determine 1) how much is the prize for an apple. 2) how much is the prize for a mango?

  • By moving around find out which students or groups have the ‘clever’ or intended strategy. This information is important in discussing session.
  • Stimulate the students to compare their solutions.
  • Ask the student or the group of students to present their answer in front of the class.
  • Guide the students in a class discussion


  • during the instruction, give another problem in the same context ( see figure below).

For example: Salma paid Rp. 5.000,- for two apples and two mangoes when she bought them in a supermarket. In the same supermarket, Ihsan had to pay Rp. 2.500,- to the seller for an apple and a mango. So, can you determine 1) how much is the prize for an apple. 2) how much is the prize for a mango?

  • As a home work, asking the students to write a short essay (on a piece of paper) about their experiences after they have learned the two lessons

RME, History, and Founding Principles

25 Sep

A Young Student In A Mathematics Class

RME, History and founding principles

This text is based on the NORMA-lecture, by Marja van den Heuvel-Panhuizen, held in Kristiansand, Norway on 5-9 June 1998

The development of what is now known as RME started almost thirty years ago. The foundations for it were laid by Freudenthal and his colleagues at the former IOWO, which is the oldest predecessor of the Freudenthal Institute. The actual impulse for the reform movement was the inception, in 1968, of the Wiskobas project, initiated by Wijdeveld and Goffree. The present form of RME is mostly determined by Freudenthal’s (1977) view about mathematics. According to him, mathematics must be connected to reality, stay close to children and be relevant to society, in order to be of human value. Instead of seeing mathematics as subject matter that has to be transmitted, Freudenthal stressed the idea of mathematics as a human activity. Education should give students the “guided” opportunity to “re-invent” mathematics by doing it. This means that in mathematics education, the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization (Freudenthal, 1968).
Later on, Treffers (1978, 1987) formulated the idea of two types of mathematization explicitly in an educational context and distinguished “horizontal” and “vertical” mathematization. In broad terms, these two types can be understood as follows.
In horizontal mathematization, the students come up with mathematical tools which can help to organize and solve a problem located in a real-life situation.
Vertical mathematization is the process of reorganization within the mathematical system itself, like, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries.
In short, one could say — quoting Freudenthal (1991) — “horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols.” Although this distinction seems to be free from ambiguity, it does not mean, as Freudenthal said, that the difference between these two worlds is clear cut. Freudenthal also stressed that these two forms of mathematization are of equal value. Furthermore one must keep in mind that mathematization can occur on different levels of understanding.

Misunderstanding of “realistic”

Despite of this overt statement about horizontal and vertical mathematization, RME became known as “real-world mathematics education.” This was especially the case outside The Netherlands, but the same interpretation can also be found in our own country. It must be admitted, the name “Realistic Mathematics Education” is somewhat confusing in this respect. The reason, however, why the Dutch reform of mathematics education was called “realistic” is not just the connection with the real-world, but is related to the emphasis that RME puts on offering the students problem situations which they can imagine. The Dutch translation of the verb “to imagine” is “zich REALISEren.” It is this emphasis on making something real in your mind, that gave RME its name. For the problems to be presented to the students this means that the context can be a real-world context but this is not always necessary. The fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student’s mind.

The realistic approach versus the mechanistic approach

The use of context problems is very significant in RME. This is in contrast with the traditional, mechanistic approach to mathematics education, which contains mostly bare, “naked” problems. If context problems are used in the mechanistic approach, they are mostly used to conclude the learning process. The context problems function only as a field of application. By solving context problems the students can apply what was learned earlier in the bare situation.
In RME this is different; Context problems function also as a source for the learning process. In other words, in RME, contexts problems and real-life situations are used both to constitute and to apply mathematical concepts.
While working on context problems the students can develop mathematical tools and understanding. First, they develop strategies closely connected 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. Eventually, the models give the students access to more formal mathematical knowledge.
In order to fulfil the bridging function between the informal and the formal level, models have to shift from a “model of” to a “model for.” Talking about this shift is not possible without thinking about our colleague Leen Streefland, who died in April 1998. It was he who in 1985*  detected this crucial mechanism in the growth of understanding. His death means a great loss for the world of mathematics education.
Another notable difference between RME and the traditional approach to mathematics education is the rejection of the mechanistic, procedure-focused way of teaching in which the learning content is split up in meaningless small parts and where the students are offered fixed solving procedures to be trained by exercises, often to be done individually. RME, on the contrary, has a more complex and meaningful conceptualization of learning. The students, instead of being the receivers of ready-made mathematics, are considered as active participants in the teaching-learning process, in which they develop mathematical tools and insights. In this respect RME has a lot in common with socio-constructivist based mathematics education. Another similarity between the two approaches to mathematics education is that crucial for the RME teaching methods is that students are also offered opportunities to share their experiences with others.

In summary, RME can be described by means of the following five characteristics (Treffers, 1987):

  • The use of contexts.
  • The use of models.
  • The use of students’ own productions and constructions.
  • The interactive character of the teaching process.
  • The intertwinement of various learning strands.

RME in Brief

24 Sep

RME in brief
Realistic Mathematics Education, or RME, is the Dutch answer to the world-wide felt need to reform the teaching and learning of mathematics. The roots of the Dutch reform movement go back to the early seventies when the first ideas for RME were conceptualized. It was a reaction to both the American “New Math” movement that was likely to flood our country in those days, and to the then prevailing Dutch approach to mathematics education, which often is labeled as “mechanistic mathematics education.”
Since the early days of RME much development work connected to developmental research has been carried out. If anything is to be learned from the Dutch history of the reform of mathematics education, it is that such a reform takes time. This sounds like a superfluous statement, but it is not. Again and again, too optimistic thoughts are heard about educational innovations. The following statement indicates how we think about this: The development of RME is thirty years old now, and we still consider it as “work under construction.”

That we see it in this way, however, has not only to do with the fact that until now the struggle against the mechanistic approach to mathematics education has not been completely conquered— especially in classroom practice much work still has to be done in this respect. More determining for the continuing development of RME is its own character. It is inherent to RME, with its founding idea of mathematics as a human activity, that it can never be considered a fixed and finished theory of mathematics education.

“Progress” issues to be dealt with
This self-renewing feature of RME explains why it is work in progress. But, there are at least two more aspects. One significant characteristic of RME, is the focus on the growth of the students’ knowledge and understanding of mathematics. RME continually works toward the progress of students. In this process, models which originate from context situations and which function as bridges to higher levels of understanding play a key role. Finally, considering the TIMSS results, it seems that RME really can elicit progress in achievements.


24 Sep

RME (Realistic Mathematics Education) diketahui sebagai pendekatan yang telah berhasil di Belanda. Gagasan pendekatan pembelajaran matematika dengan realistik ini tidak hanya populer di Negeri Belanda saja, banyak negara maju telah menggunakan pendekatan baru yaitu pendekatan realistik. Matematika realistik banyak ditentukan oleh pandangan Freudenthal tentang matematika. Dua pandangan penting beliau adalah ‘mathematics must be connected to reality and mathematics as human activity ’. Pertama, matematika harus dekat terhadap siswa dan harus relevan dengan situasi kehidupan sehari-hari. Kedua, ia menekankan bahwa matematika sebagai aktivitas manusia, sehingga siswa harus di beri kesempatan untuk belajar melakukan aktivitas semua topik dalam matematika.

Realistic Mathematics Education adalah pendekatan pengajaran yang bertitik tolak dari hal-hal yang ‘real‘ bagi siswa, menekankan keterampilan ‘proses of doing mathematics’, berdiskusi dan berkolaborasi, berargumentasi dengan teman sekelas sehingga mereka dapat menemukan sendiri (‘student inventing‘ sebagai kebalikan dari ‘teacher telling’) dan pada akhirnya menggunakan matematika itu untuk menyelesaikan masalah baik secara individu maupun kelompok. Pada pendekatan ini peran guru tak lebih dari seorang fasilitator, moderator atau evaluator sementara siswa berfikir, mengkomunikasikan, melatih nuansa demokrasi dengan menghargai pendapat orang lain.

Prinsip prinsip penemuan kembali dapat diinspirasi oleh prosedur-prosedur pemecahan informal. Proses penemuan kembali menggunakan konsep matematisasi. Dua jenis matematisasi diformulasikan oleh Treffers (Suherman, 2001) yaitu matematisasi horizontal dan vertikal. Dalam matematisasi horizontal siswa menggunakan matematika sehingga dapat membantu mereka mengorganisasikan dan menyelesaikan suatu masalah yang ada pada situasi nyata. Sedangkan pada matematisasi vertikal proses pengorganisasian kembali menggunakan matematika itu sendiri.

Teori Pembelajaran Matematika Realistik terdiri dari lima karakteristik yaitu: (1) penggunaan real konteks sebagai titik tolak belajar matematika; (2) penggunaan model yang menekankan penyelesaian secara informal sebelum menggunakan cara formal atau rumus; (3) mengaitkan sesama topik dalam matematika; (4) penggunaan metode interaktif dalam belajar matematika dan (5) menghargai ragam jawaban dan kontribusi siswa. Dalam kerangka Pembelajaran Matematika Realistik, Freudenthal (Suherman, 2001) menyatakan bahwa “Mathematics is human activity”. Sejumlah pakar RME, diantaranya adalah De Lange (1987) dan (1996), Streefland (1991), Gravemeijer (1994), dan Treffers dan Goffree (1985), merumuskan karakteristik pembelajaran matematika realistic sebagai berikut:- Penggunaan masalah-masalah kontekstual, masalah kontekstual tersebut terutama dimaksudkan sebagai titik tolak dimana matematika yang diinginkan dapat muncul.

  • Penggunaan model atau jembatan dengan instrumen vertikal, diarahkan pada pengembangan strategi, skema dan simbolisasi yang cenderung menolak pentransferan rumus atau matematika formal (standar) secara langsung.
  • Penggunaan kontribusi siswa dalam proses belajar mengajar, hal ini dilakukan dalam rangka mengantar siswa dari metode informal menuju kepada proses matematika yang lebih formal atau standar.
  • Adanya interaktifitas, meliputi negosiasi secara eksplisit, intervensi, kooperasi dan evaluasi.
  • Adanya integrasi antar topik-topik pembelajaran, merupakan model holistik yang menunjukkan bahwa unit-unit belajar tidak akan dicapai jika diajarkan secara terpisah melainkan keterkaitan dan keterintegrasian dalam proses pemecahan masalah.

Selain itu, terdapat juga prinsip-prinsip pembelajaran realistik dalam kurikulum matematika realistik yaitu:

  1. Didominasi oleh masalah-masalah dalam konteks, melayani dua hal yaitu sebagai sumber dan sebagai terapan konsep matematika.
  2. Perhatian diberikan kepada pengembangan model-model, situasi, skema, dan simbol-simbol.
  3. Sumbangan dari para siswa, sehingga siswa dapat membuat pembelajaran menjadi konstruktif dan produktif, siswa memproduksi sendiri dan mengkonstruksi sendiri sehingga dapat membimbing para siswa dari level matematika informal menuju matematika formal.
  4. Interaktif sebagai  karakteristik dari proses pembelajaran matematika
  5. Interwinning (membuat jalinan) antar topik atau antar pokok bahasan.

Menurut Treffers dan Goffree (Alimuddin, 2004) bahwa masalah kontekstual dalam kurikulum realistik, berguna untuk mengisi sejumlah fungsi:

  1. Pembentukan konsep: Dalam fase pertama pembelajaran, para siswa diperkenankan untuk masuk ke dalam matematika secara ilmiah dan termotivasi.
  2. Pembentukan model: Masalah-masalah konstekstual memasuki fondasi siswa untuk belajar operasi, prosedur, notasi, aturan, dan mereka mengerjakan ini dalam kaitannya dengan model-model lain yang kegunaannya sebagai pendorong penting dalam berpikir.
  3. Peerapan : masalah konstektual menggunakan reality sebagai sumber dan domain untuk terapan.
  4. Praktek dan latihan dari kemampuan spesipik dalam situasi terapan.

Matematika dalam pembelajaran matematika realistic merupakan proses yang sangat penting. Berkaitan  dengan hal ini, Freudental (Suherman, 2001) mengemukakan dua alasan yang sangat mendasar. Pertama, matematisasi bukan merupakan aktivitas ahli matematika saja, melainkan juga aktivitas siswa dalam memahami situasi sehari-hari. Kedua, berkaitan erat dengan penemuan kembali (reinvention) ide atau  gagasan dari siswa. Artinya siswa diarahkan selah-olah menemukan kembali suatu konsep dalam matematika pada saat pembelajaran berlangsung.

Freudenthal memandang matematisasi sebagai aktivitas umum yang meliputi matematika murni dan matematika  terapan. Pandangan Freudenthal yang lain cenderung kepada interpretasi masalah dalam kehidupan sehari-hari ke dalam model matematika, De Lange bahkan memberi istilah matematika informal sebagai matematisasi horisontal dan matematika formal sebagai matematisasi vertikal.

Treffers dan Goffree (Ermayana, 2003 : terdapat dua tipe matematisasi yang dikenal dalam Realistic Mathematic Education (RME) yaitu:

1. Matematika horizontal – Proses matematika horizontal pada tahapan menengah persoalan sehari-hari menjadi persoalan matematika sehingga dapat diselesaikan atau situasi nyata diubah ke dalam simbol-simbol dan model-model matematika.

2. Matematika vertikal – Proses matematika pada tahap penggunaan simbol, lambang kaidah-kaidah matematika yang berlaku secara umum.

Langkah-langkah tahap pendekatan Realistic Mathematics Education yaitu :

  • Memberikan masalah dalam kehidupan sehari-hari.
  • Mendorong siswa menyelesaikan masalah tersebut, baik individu maupun kelompok.
  • Memberikan masalah yang lain pada siswa, tetapi dalam konteks yang sama setelah diperoleh beberapa langkah dalam menyelesaikan masalah tersebut.
  • Mempertimbangkan cara dan langkah yang ditentukan dengan memeriksa dan meneliti, kemudian guru membimbing siswa untuk melangkah lebih jauh ke arah proses matematika vertikal.
  • Menugaskan siswa baik individu maupun kelompok untuk menyelesaikan permasalahan lain baik terapan maupun bukan terapan.

Sintaks Implementasi Matematika Realististik

Aktivitas Guru Aktivitas Siswa
  • Guru memberikan siswa masalah kontekstual.
  • Guru merespon secara positif jawaban siswa. Siswa diberikan kesempatan untuk memikirkan strategi siswa yang paling efektif.
  • Guru mengarahkan siswa pada beberapa masalah kontekstual dan selanjutnya meminta siswa mengerjakan masalah dengan menggunakan pengalaman mereka.
  • Guru mengelilingi siswa sambil memberikan bantuan seperlunya
  • Guru mengenalkan istilah konsep.
  • Guru memberikan tugas dirumah yaitu mengerjakan soal atau membuat masalah cerita beserta jawabannya yang sesuai dengan matematika formal.
    • Siswa secara sendiri atau kelompok kecil mengerjakan masalah dengan strategi-strategi informal.
  • Siswa secara sendiri-sendiri atau kelompok menyelesaikan masalah tersebut.
  • Beberapa siswa mengerjakan di papan tulis. Melalui diskusi kelas, jawaban siswa dikonfrontasikan.
  • Siswa merumuskan bentuk matematika formal.
  • Siswa mengerjakan tugas rumah dan menyerahkannya kepada guru.

Sering terlontar bahwa penjelasan yang diberikan oleh seorang guru  hanya dapat dimengerti pada saat penjelasan tersebut diberikan di kelas, tetapi ketika siswa sampai di rumah mereka sudah lupa. Hal ini menunjukkan mereka belum mengerti dengan pengetahuan konseptual. Mereka hanya mengerti pengetahuan prosedural.

Menurut Mitzel (Suherman, 2001) mengatakan bahwa, hasil belajar siswa secara langsung dipengaruhi oleh unjuk kerja guru. Bila siswa dalam belajarnya bermakna atau terjadi kaitan antara informasi baru dengan jaringan representasi maka siswa akan mendapatkan suatu pengertian. Tanpa pengertian, tidak dapat mengaplikasikan prosedur, konsep, ataupun proses. Melalui pengalamannya dalam kehidupan sehari-hari mereka mengembangkan ide-ide yang lebih kompleks. Jadi pembelajaran di sekolah akan lebih bermakna bila guru mengaitkan dengan apa yang telah diketahui anak, sehingga akan mempunyai kontribusi yang sangat tinggi dengan pengertian siswa.


(Source : Fitriani Nur, Mahasiswa PPs UNM Makassar | Prodi Pendidikan Matematika, 2008)