Introduction to RME

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Apakah ini Sekedar Ketidaktelitian Siswa?

Materi pengukuran waktu merupakan materi yang sangat penting. Hal ini karena dengan mempelajari materi ini siswa diharapkan memiliki bekal yang cukup untuk menyelesaikan masalah kehidupan sehari-hari mereka yang berkaitan dengan waktu. Sayangnya, pembelajaran matematika, khususnya tentang pengukuran waktu, masih jarang yang mampu mengkaitkannya dengan dunia siswa yang sesungguhnya. Sejalan dengan apa yang disampaikan oleh Gravemeijer dalam reflection paper saya sebelumnya, pembelajaran ini masih sering disampaikan dalam angka-angka dan rumus-rumus kurang bermakna yang sering menggiring siswa pada dunia yang berbeda dengannya.

https://umipuji.wordpress.com/2013/11/28/common-sense-for-concreting-abstract-mathematics/

Ketika saya dan rekan saya memberikan sebuah soal pengukuran waktu kepada siswa kelas 5, kami mendapati kasus yang hampir sama. Dari kelima kelompok siswa, tiga kelompok belum mampu menggunakan pengalaman sehari-harinya untuk menyelesaikan soal yang pada dasarnya merupakan soal yang biasa mereka temukan dalam kehidupan sehari-hari. Ketika ketiga kelompok ini sampai pada penentuan “Pukul berapakah yang menunjukkan 115 menit sebelum pukul 14.00?” Ketiga kelompok mendapatkan jawaban yang sama, yaitu 12.45. Mengapa demikian? Mari perhatikan uraian jawaban mereka berikut ini.

  • 115 menit = 1 jam 55 menit
  • Siswa menulisnya menjadi 1.55
  • Siswa mengurangkan 1.55 dari 14.00 seperti pada gambar berikut.

time ok 2

Seperti ditunjukkan pada gambar, hasilnya terlihat bahwa 14.00 – 1.55 = 12.45. Pernahkan teman-teman menemukan jawaban siswa yang demikian?

Awalnya saya pikir ini hanya masalah ketidaktelitian. Tetapi setelah saya cermati, nampaknya hal ini tidak demikian. Perhatikan uraian saya berikutnya.

Ketika kami meminta siswa untuk mengecek dengan cara menghitung dari depan (apakah 115 menit setelah pukul 12.45) menunjukkan pukul 14.00? Kali ini siswa mampu mengkaitkan pengalaman sehari-harinya untuk menghitung 115 menit setelah pukul 12.45. Mereka dapat menyadari bahwa 115 menit setelah pukul 12.45 itu lebih dari 14.00, yang berarti jawaban mereka kurang tepat.

Mereka pun mengecek perhitungan mereka (dengan cara susun seperti pada gambar di atas) berkali-kali dan tidak pula menemukan titik kesalahannya. Mereka yakin kalau perhitungan mereka benar. Namun, mengapa ketika di cek jawabannya salah????

Ketika kami menanyakan dalam satu jam ada berapa menit, mereka dengan cepat, tepat, dan kompak menjawab 60 menit. Tetapi mengapa mereka belum mampu mengkaitkan hal ini untuk menyelesaikan persoalan di atas? Mereka menyamakan perhitungan 1.400 – 155 dengan  perhitungan pukul 14.00 dikurangi 1 jam 55 menit. Bukankah ini berarti bahwa siswa belum mampu memaknai angka-angka yang mereka tulis???

Saya pun teringat dengan materi operasi hitung pada pengukuran waktu yang diajarkan mulai kelas empat. Siswa diajarkan tentang tambah kurang pada satuan waktu. Di beberapa buku paket diajarkan cara penyelesaian sebagai berikut.

Contoh soal :

7 jam 20 menit – 2 jam 45 menit =…………..

Caranya :

time ok 1

Dari situ, kita bisa melihat bahwa apa yang diajarkan dalam buku paket adalah tentang lama waktu. Sedangkan soal yang kami berikan berkaitan dengan notasi jam (14.00) dan lama waktu (1 jam 55 menit). Namun, sepertinya siswa menggunakan ingatannya yang parsial tentang operasi tambah kurang pada lama waktu diatas untuk menyelesaikan persoalan yang kami berikan. Selain penggunaan notasi yang kurang tepat (1 jam 55 menit ditulis 1.55), siswa belum mampu mengkaitkan 1 jam yang terdiri atas 60 menit untuk menyelesaikan persoalan di atas. Bukankah ini tidak hanya masalah ketelitian? Bukankah kesalahan seperti ini bisa dihindari jika pembelajaran tidak hanya mengingat tetapi penuh makna?

Menjadi guru memang tidak mudah. Ketika saya mengajar di sebuah SD, saya pun mengalami hal demikian. Inginnya mengajar begini dan begitu. Tetapi, tidak jarang idealita kita terbentur dengan target-target materi yang begitu banyak atau pun waktu yang serasa tidak cukup untuk mempersiapkan bahan ajar dengan maksimal. Namun, marilah kita belajar untuk pantang patah arang. Belajar bersama untuk menjadi guru yang lebih baik. Belajar bersama untuk mendidik anak-anak Indonesia menjadi anak yang lebih cerdas.

Carpediem. Majulah Pendidikan Indonesia !!!

Common Sense Concrete for Concreting Abstract Mathematics

This paper is the reflection of the article of Koeno Gravemeijer that titles How Concrete is Concrete?  Koeno Gravemeijer is a realistic mathematics education expert from Freudenthal Institute, Utrecht University, Nederland.

Mathematics is abstract, so most of the students think that mathematics is not easy. For making the learning of mathematics easier, the teachers introduce tactile or visual models. They think it is able to make the abstract mathematics concrete. However, Gravemeijer will show that our common way of making things concrete for the students does not work because we help students to make a connection with what we know, not what they know.

The first example that will be shown by Gravemeijer is about how the student of grade 6 solved the comparing fraction problem. The context is about a bakery that would cut banquet bars (a sort of large cookies) to order. They were given paper strips of a given length to enact the cutting process. Then they were asked to cut the banquet bars into eight equal pieces, or six, or ten, and so forth. After they asked to use similar strips to compare 1/3 and 2/6, they solved this problem by comparing the lengths of pieces produced in the two different divisions-either dividing by three or dividing by six. However, students concluded that 1/3 was not equal to 2/6.

Gravemeijer analized the answer of the students then he found that the way students cut the strips was not very precise. It made the answer of the student is like that. It means that the tactile representations do not support an insightful solution. Furthermore, in this case, we may denote the two ways in which ‘concrete’ can be understood as either “material concrete” or as “common sense concrete”. The students’ point of view is sometimes different with teacher’s point of view. In this case, the teacher see the relation between 1/3 and 2/6 in the paper cutting, but the students just see the blocks with various sizes.

So, how is the better one? We prefer to help students to reason how 1/6 relates to 1/3. First, a bar is divided into three parts and each piece into halves next, to get six equal pieces. Then, they will understand that 1/6 is a half of 1/3 and they able to concluse that 2/6 equal to 1/3.

equality

The second example is about addition tasks for the student of grade 1. She is Auburn. When Auburn counting on 16 + 9, she got 25. She also know that if she had 16 cookies and another 9 added, she would have 25 cookies. But, when she filled the worksheet, she wrote 15. When she was interviewed why the answers are different, she said that she didn’t know. Lets look at a part of that worksheet below.

concrete

If we analize the worksheet, we able to know why Auburn’s answer is 15. She just imitated the example from the worksheet. She thought that the mathematics of the worksheets seem different with the world of everyday-life experience. She will not be inclined to use everyday-life knowledge to make sense of ‘school math’ problems. Gravemeijer argue that the knowledge gap between teachers and students is too big to make this work. And manipulatives cannot bridge this gap.

So, what is the better? We could use arithmetic rack to fasilitate the students for communicating and scaffolding.

artmtk rck

In conclution, we may try to follow Freudenthal’s adagio that ‘mathematics should start and stay within common sense’ by trying to foster the growth of what is common sense for the students.

Percentage Bar As The Powerful Model for Learning Percentage

This article is the reflection paper of the article of Frans van Galen and Dolly van Eerde that titles Solving Problems with The Percentage Bar. It was publised in IndoMS JME, Volume 4 Number 1, 1 January 2013, page 1-8.  Frans van Galen and Dolly van Eerde are realistic mathematics education experts from Freudenthal Institute, Utrecht University, Nederland.

Percentage is one of the matematics material that is taught from grade 4 until 6 in primary school. We can imagine, how often this material is taught. However, when a group of IMPOME students carried out a study in grade 7 on students’ understanding of percentages, they were surprised because there are only four students from 14 students who they tested were able to give a correct answer on the following problem :

bike problem

Although the students had a certain area to be a scrap paper, they just used it to wrote calculation. None of them used it to draw model to clarify the relation between the given numbers. It was obvious that most of the students didn’t know a systematics procedure for working with percentages.

A week later, they interviewed the students about the way they solved some of these problems. They find that there are four students’ ways to solve the problem.

  • Converting the percentage into a decimal number and multiplied, for example : 0,15 x 600
  • Multiply with fractions, for example :

soal 1

  • Trying to make proportions equal, for example :

soal 2

  • Guided by the numbers in the problem, for example : 600 is divided by 15

Eventhought two first solutions are true, but they were not consistent and made calculation error. While, the way solution number 3 made the students were confused.

In a teaching experiment, the students were taught the use of the percentage bar.The researchers has 2 approaches.

  • Calculating via 10%. The problem is like that “if the price of the handphone is € 300 and you just pay € 240, how much the percentage of the discount?” The answer of the student is shown below.

PBar

First, student calculated 50% and 25% of € 300, but that didn’t lead to a discount of € 60. As a next step, she started with 10% and found € 60 by doubling.

  • Calculating via 1%. The problem is about the downloading the computer file. “How many Mb if someone download 11% of 600Mb?” The student answer is shown below.

PBar 2

Student’s strategy is 50% equals 300, 25% equals 150 and 10% equals 60. 11% as 10% + 1%.

Although teaching experiment by percentage bar was carried out just one lesson, the result confirm that the percentage bar is a powerful model that deserves a central place in the teaching of the percentage. It helps the students to make a representation for themselves of the relations between what is given and what is asked, offers scrap paper for the intermediate steps in the calculation process, and offers a natural entry to calculating via 1%.

Penyelesaian Soal Persentase oleh Siswa Kelas 5 SD IBA Palembang

Menurut para pakar matematika realistik dari Belanda yang disampaikan oleh Ibu Ratu Ilma, materi persentase merupakan materi tersulit bagi anak. Hal ini sesuai dengan pengalaman saya ketika mengajar di SDIT maupun ketika mempraktekkan pendekatan realistik di SD IBA. Bisa jadi siswa dapat menghitung dengan cepat berbagai macam konversi pecahan dari pecahan biasa atau desimal menjadi persen, tetapi belum tentu siswa memahami betul makna dalam setiap materi persentase itu sendiri. Sehingga ketika siswa dihadapkan dengan soal yang berbeda variasinya, siswa belum bisa memiliki inisiatif untuk menyelesaikannya dengan caranya sendiri. Kebanyakan dari mereka baru menggunakan formula atau rumus yang telah diberikan oleh guru. Berikut saya kutipkan presentasi pengalaman saya ketika menginterview siswa yang sedang menyelesaikan salah satu soal dalam persentase. Karena sebelumnya siswa baru satu kali diajarkan tentang materi persentase dengan pendekatan realistik maka siswa cenderung masih menggunakan formula seperti yang biasa ditemukan di buku paket.

http://www.scribd.com/doc/180102029/Percentage-Problem-Solving-ppsx

Pada tulisan berikutnya, insyaAllah saya akan berikan contoh penyelesaian soal pecahan dimana siswa tidak menggunakan rumus yang tersedia, tetapi dengan caranya sendiri menyelesaikan soal pecahan tersebut.

Terima kasih, semoga bermanfaat….

Building Blocks to Develop Students’ Spatial Visualisation Ability

The title of this research is On Developing Students’ Spatial Visualisation Ability. The research has been done by Dwi Afrini Risma, graduate student of IMPOME, in 2013. Together with Ratu Ilma Indra Putri and Yusuf Hartono as her consultants, researcher published it in International Education Studies, volume 6, nomor 9.

The aims of this research is to support the development of students’ spatial visualisation ability and to study about how students visualise and interprete the building blocks. It is very important because spatial visualisation ability that is taught in geometry topic is one of important abilities is needed in students’ life.  Unfortunatally, many students cannot visualise three-dimensional objects in a two-dimensional perspective.

This research involved 39 students of 3rd grade of Elementary Students of 117 Palembang and a 3rd grade classroom teacher. The learning activities was done in 70 minutes-long meeting. Because of some technical problems, researcher splited the activity became two meetings. During learning in this activity, the students were grouped into some small groups of 3 or 4 students. In the building block activity, the students had to solve three problems. First two problems were solved in the first meeting and the last problem was solved in the second meeting.

Meeting 1

  • Introducing the context
  • Solved the first problem.

Students were asked to construct a building blocks from a small wooden cubes as shown in figure 1 and then draw its side views and top view.

block 1

  • Solved the second problem

The students hd to construct their own building blocks that consist of 5 wooden cubes and then draw its side views and top vies.

Meeting 2

  • Classroom discussion
  • Solved the last problem

Individually, students were asked to construct their own building blocks from four wooden cubes and draw its side view and top view.

As the result, researcher can see that the student interprete and visualise views of the building blocks in a various way. In discussion, researcher has seen how the teacher supported the student by giving some important question and giving important instruction in observing. Furthermore, the teacher also gave space for the students to think and visualise the left view of the building blocks by playing dummies.

In conclution, there is no students represent the building blocks as a three-dimensional drawing after experiencing the building block activity, indicates that the students develop their spatial visualisation ability. Based on analysis of students’ working, researcher categorise the way students interpret and visualize the side view of building blocks into three general ways :

  1. Students visualise the side view of the building blocks as the squares, regardless it has an goes-out part or goes-into part
  2. Students visualise the goes-into and goes-out part in a three-dimensional drawing
  3. Students visualise the building blocks as three-dimensional drawing

Moreover, researcher categorise students’ interpretation and visualisation on top view into two general categories namely :

  1. Students interpret the top view part of the cubes that is located on the top layer
  2. Students interpret the top view as how the cubes arrays are visible if we see it fro the top

Beside that, the role of the teacher by giving some important questions and instructions is to facilitate the development of students thinking and to gave space on students’ creation.

The Example of Statistic, Geometry, and Number Problems with The Context of Purworejo

Meaningful learning is an effective learning to achieve learning objectives. One of ways of learning in order to be meaningful, according to one of characteristics of  RME (Realistic Mathematics Education) is the use of context related to the life and culture around the students. For students who were in the Purworejo, can use the potential and cultural of Purworejo as the context for learning.

Purworejo is one of regencies in Central Java, which is located on the south coast of Java. This area has a lot of potentials, ranging from natural resources such as beaches and waterfall, dances like Dolalak Dance , Bedug Kyai Bagelen or Pendawa as the eldest and largest bedug, as well as a wide variety of culinary appetizing. By making potential and the culture as the context of learning, it is not only make learning more meaningful, but also be able to continue and  maintain the local heritage.

The examples :

STATISTIC PROBLEM

context 1In 2009, students of representatives from 16 districts in Purworejo danced Dolalak Dance in Purworejo Square to celebrate  The National Education Day. It was noted  that each district sent an average of 1,000 students. If  Kutoarjo, one of  district in Purworejo, did not send a representative students, the average number of representatives per district just 965 people. How many students of representative of Kutoarjo?

Answer : 1,525 students

Explanation :

The number of Dolalak dancer from 16 districs = 16 x 1.000 = 16,000

The number of Dolalak dancer without representative from Kutoarjo = 15 x 965 = 14.475

The number of Dolalak dancer from Kutoarjo  =  16,000 – 14.475 = 1,525

                                                                          

GEOMETRY PROBLEM

On 3 May, 1936, membrane of Bedug (it is like drum) Kyai Bagelen  which originally was made ​​from bison leather was replaced  with the leather of Bengal and Pemacek buffalos from Winong, one of the village in Purworejo. As the largest “bedug” in the world, it has diameter that reaches 194 cm for the front, while diameter of the back reaches 180 cm. What is the minimum area of  the leather Bengal and Pemacek buffalo  for each membrane if it takes an extra 15 cm to install membrane on wooden stump?

context 2

Answer : front : 39.424 cm2, behind  : 34.650 cm2

Explanation :

Area of front membrane =  r2 =  ( + 15)2 =  x 1122 = 39.424 cm2 = 3,9424 m2

Area of behind membrane =  r2 =  ( + 15)2 =  x 1052 = 34.650 cm2 = 3,465 m2

 

NUMBER PROBLEM

maknnAt the “Halal Bi Halal” agenda that is held by “Karang Taruna” of Wareng, every 3 guests  share a plate of “tahu pong”. Every 4 guests share a plate of  “geblek”, and every 6 guests share a plate of “clorot”. If  all of the number of plates is 63, how many guests in attendance?

Note :

Halal Bi Halal is one kind of celebrations for Moslem after Idul Fitri

Idul Fitri is one kind of feasts for Moslem

Karang Taruna is one kind of youth organization

Wareng is one kind of village in Purworejo

Tahu pong, geblek, and clorot is some kinds of special culinary from Purworejo

Answer : 84 people

Explanation :

The smallest common multiple of 3, 4, and  6 is 12

Number of Guests

Number of Plates for Tahu Pong

Number of Plates for Geblek

Number of Plates for  Clorot

Total number of Plates

12

4

3

2

9 (less)

60

20

15

10

45 (less)

84

28

21

14

63 (correct)

Do you want to know more about…

1. Dolalak Dance, click ……

http://uptpdankpurworejo.wordpress.com/2013/03/14/sejarah-tari-dolalak-purworejo/

2. Eldest and Largest Bedug in the world, click…….

http://psbgdirgantara.blogspot.com/2009/10/sejarah-bedug-agung-pendowo-purworejo.html

3. Culinary of Purworejo, click…..

http://purworejokedu.blogspot.com/2010/01/makanan-khas-purworejo-dan-kuliner.html