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 :
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 :
- Trying to make proportions equal, for example :
- 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.
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.
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%.
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.
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….
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.
- 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.
- 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.
- 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 :
- Students visualise the side view of the building blocks as the squares, regardless it has an goes-out part or goes-into part
- Students visualise the goes-into and goes-out part in a three-dimensional drawing
- Students visualise the building blocks as three-dimensional drawing
Moreover, researcher categorise students’ interpretation and visualisation on top view into two general categories namely :
- Students interpret the top view part of the cubes that is located on the top layer
- 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.