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7 INSPIRING CLASSROOM ACTIVITIES USING REALISTIC MATHEMATICS
(Part 2)
Composed by : Ratih Ayu Apsari
The content is summarized from seven articles of National Conference of Mathematics XIV (2008), Sriwijaya University.
[3] D.N. Handayani: ButterflyWing Context to Support The Earlier Development of Number Sense.
One of mathematical phenomenology that deal with the early number sense, specially in first grade students of primary school is the cardinality. In the formal mathematics, cardinality of a set is a measure of the number of element of the set. The problem is, the beginner students often not realize the aim of counting. For instance, when they aregiven five candies and asked to count how many candies is it, they might start counting as “one, two, three, four, five”. However, even though they stop at “five”, the teacher should investigate it carefully: are they allready understand that 5 means five objects or they think that the fifth object is 5.
To help the students reach the basic understanding in number sense, the butterflywing context had been choosen in this research, because it has pattern which similar in the left and right side. The students were asked to observed the pattern in the butterfly wings and guided to recognize the pattern. At the end of lesson, the students are expected to count “how many” in sets of objects using various structures.
[4] Meiliasari: The Context of Structured Candy on Earlier Addition and Substraction Up to 20.
At the earlier phase of learning, children generally use many kind of counting strategies, such as count all objects one by one and count from larger addend, which usually use their fingers to keep the track. When deal with larger number, some students might realize that the counting method seems to be no longer effective. The children will develop new strategies, such as doubling (i.e., 6+8 = 6+6+2) and splitting (i.e., 6+8 = 6+4+4=10+4). The development of this process may take quite long time.
In order to help te students working in addition in easier and quicker way – and less possibility to do computational error, this study was conducted using the context of candies, which is very familiar in student’s daily live. It is had been tested that the question such as; Dinda as 4 candies and then her mother give another 3 candies. How many candies does Dinda has? Is more acceptable in low class students rather than directly use the abstract symbol as 3+4 = …?
The lesson activity is organize as follow. At the first time, teacher told students to count a number of candies. They can manually count it (unstructured way) or they can arrange it first, suc as by making two rows of candies which has some number for each row, and then count it (structured way). The purpose of this activity is to make students realized that structure helps them to counting faster.
Picture 4: Illustratation of Unstructured and Structured Candies (Picture Modified)
The next activity is purpose to guide the students to recognize the double structure. Before enroll the worksheet phase, the students are invited to sing the “double song” (satu ditambah satu sama dengan dua, dua ditamba dua sama dengan empat, etc.) which is populer in Indonesia. This song is purpose to stimulate the students’ understanding in double sum. In this stage, the worksheet still use the context of candies which are arranged in one or two packs of two rows of five.
Picture 5: Illustration of Question in Worksheet 2
The third activity is illustrated on the following picture.
Picture 6: Illustration of 5+2=7
The task is about a candy box which just fullfil by 7 canies (the white circles is means the place is not covered by candy). At first, the students expected can tell that it all 7 candies on the box. Second, they can explain that since there are 5 candies in first row and 2 candies in second rows, te total is 7. In mathematical sentence, they can write the mathematical sentence 5 + 2 = 7.
To help the students construct their understanding in subtraction, the researcher embeded the concept on the subtraction as the invers of addition, rather than using counting strategies, such as counting up (such as (5 – 3 = 4,5 so 53=2) or counting backward (5 – 3 = 5,4 so 5 – 3 = 2). That’s why, the students will make the correlation between addition and subtraction. For intance, 6 + 4 = 10, so that 10 – 6 = 4 and 10 – 4 = 6.
[5] Neni Mariana : School Building Context to Develop Spatial Ability.
Spatial ability is the skill to mentally manipulate, understand, reorganize, or interpret relationships of shapes and space. This comprehension is related to matematics, specially geometry achievement for the students (Casey et.al., 2008; Melancon, 1994; Tartre, 1990; Tracy, 1987).
The researcher actually designed for nine activities for learning, but for this chance, the disscussion will just about five of it. The activities are set as the following step.
Activity 1 : My School Building
The goal of this activity is the children are able to localize for main buildings of the school on a grid paper. This activity use several cube with same size.
Activity 2 : Where Our Classroom Is
The goal of this is activity is the children are able to localize their alssroom position on a grid paper. Through this activity, children are expected to gradually emerge 2dimension scale model. The previous activity is just use the samesize cube. This condition not match with real situation, because in the school some building have different weight, height, and even the shape. By starting tis activity, a conflict will appear about the shape of the school. Consequently, the pupils will come up with a conclusion that not all buildings can be represented by one cube since some of them higher, wider, etc.
Activity 3 : Removable Buildings
The purpose of this activity is to make sure that children are able to: (a) localize all rooms by making a map of the basements on a grid paper, and (b) connect between 3D constructions and 2D shapes of their basements.
Activity 4 : Make a Route to an Important Place in the School
The goal for this activity is the students are able to spatially explain routes to some important places using directions, such as to the left, straight ahead, to the right, and so forth.
Activity 5 : A Mini Lesson – Where Do You Have to Stand?
The aim for this activity is to help the students to denote onetoone correspondence between a certain point of standing viewer and a certain point of a part of the school buildings.
7 INSPIRING CLASSROOM ACTIVITIES USING REALISTIC MATHEMATICS
(Part 1)
Composed by : Ratih Ayu Apsari
The content is summarized from seven articles of National Conference of Mathematics XIV (2008), Sriwijaya University.
[1] Al Jupri : Supermarket Context in Learning Estimation Problems.
The study was caused by the minimum attention given by the mathematics curriculum in term of the importance of computational estimation skill. However, many experts in mathematics education believe that this comprehension is needed by the children to solve their daily problem experience, since most of it use nonexact calculation (Carpenter et all (1976), Driscoll et all (1981)). There are several kind of question which is appear in daily life, such as are there enough ?, could this be correct ?, and approximately how much is it ?. This kind of question, tend to be easier to solve with estimation strategies rather than the exact one. The researcher use supermarket as the context in teaching estimation for fifth grader students since it is familiar and easy to imagine by the Indonesian pupils. One example of its problem given in the class is:
Picture 1: Illustration of Problem Given on the Research (Picture Modified)
This kind of question can stimulated students to try other way (estimation strategy) to find the answer, since it will quite complicated to do an exact calculation like what they use to. However this context might be helpful for students either teachers, the researcher himself realize that many students still doubting the result they get through estimation strategy since their prior paradigm told them that mathematics should not be answered by approximately answer. That’s why the researcher suggest to do class discussion after doing students’ worksheet.
[2] Ariyadi Wijaya : Indonesian Traditional Games in Learning Measurement of Length.
The researcher use Gundu and Benthik (Javanese traditional games) as his context to encourage the basic concept understanding in measurement for second grade elementary school’s students. This research conducted since the measurement topic usually teach directly in formal level. Even though children engaged in a number of daily activities which use measurement context, it not guaranteed that they can make a relation between what they use to do and what they learn in the class.
The idea behind applied the gundu game is to emerge the concept of indirect comparison. In this game, each player should throw their marbles from starting point to a pole on the ground. The winner determined by compare the distance of the marbles from the pole.
Picture 2: Illustration of Gundu Game
The Benthik game itself is doing on this way: all players in each team have to hit a short stick and then the distance of the fallen stick is measured. The winner determined by the team which has the longer distance of the stick. Students are free to choose measurement tools either the unit which use to compare the length of distance. The conjecture for this activity is the students, while measure the distance of their stick, may use different tool and unit, such as their footsteps. It will lead them to different result since every students have different size of step. From this situation, they will realize that they need to use same measurement to obtain same result for same distance. This basic concept can be used as the first step in introduce the standard measurement for the students.
Picture 3: Illustration of Benthik Game
The report of a small study on percentages

Introduction
This paper is the report of a small study on percentages conducted in a seventh grade of one international secondary school in Rotterdam from September until October 2011. The study was conducted as a part of one course the researcher took, namely Integrative Practical (IP). The course aims at giving experience to master students in doing a design research. The study aims at supporting seventh graders in developing their understanding on percentages. In the light of it, the general research question posed in this study is how can we support pupils in developing their understanding of percentages? This general research question can be elaborated into three specific sub questions: Read more…
Review: Ghosts of Diophantus
Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18, 7590
The main issue of this article is the relationship between the historical evolution of algebraic ideas and their conceptual development that indicates how an historical analysis can guide and inform teaching. It is based on a conjecture that the ontogenetic development of mathematical ideas, development of ideas during the life time of the individual mathematician, might parallel with its phylogenetic evolution, evolution throughout the historical time. Related to this idea, some mathematics education experts such as Poincare, Branford, and Polya suggested that curriculum content in mathematics, in order to make it useful for students, should be presented in precisely the same order in which that content evolved during the history of mathematics. Furthermore, Polya suggested that students should be facilitated to ‘rediscover’ all the ‘great steps’ taken by mathematicians throughout history. This idea is similar with Freudenthal’s suggestion that students should be encouraged to ‘reinvent’ mathematics ideas. Read more…
Pendaftaran Beasiswa S2 International Master Program on Mathematics Education (IMPoME) 2012
Pendaftaran beasiswa IMPoME untuk periode 2012 telah dibuka, dengan syaratsyarat sebagai berikut:
1. Mengisi aplication form dengan lengkap, download di sini: stunedformimpome2012
2. Mengisi CV dengan lengkap, download di sini: cvformneso2012
3. Fotocopy Kartu Tanda Penduduk (KTP)
4. Pas Photo 4 x6 (1 lembar)
5. Ijazah S1
6. Transkrip nilai dengan nilai IPK minimal 3, 00
7. Sertifikat TOEFL dengan score minimal 500
8. SK CTAB (Surat Keputusan Calon Tenaga Akademik Baru) dari Rektor
Persyaratan di atas dibuat dengan ramgkap 3 ( 1 asli, 2 fotokopi) menggunakan kertas A4 di bundel berdasarkan nomer urut di atas dan di jilid menggunakan plastik mika warna putih (bening).
Mohon tidak melampirkan dokumen yang tidak kami cantumkan di atas.
Semua berkas harap dikirimkan ke:
Martha Metrica, S.E
PMRI – PPPPTK IPA Bandung
Jalan Diponegoro No.12
Bandung
Telp/Fax: 0224213950/022 4213949
Paling lambat tanggal 31 Desember 2011, berkas sudah kami terima.
Terima kasih
Realistic Mathematics Education; a solution to prevent the linearity illusion (Extended)
Introduction
Linearity illusion, a tendency to see any relation between two quantities as a proportional or linear one, is a phenomenon in mathematics education, especially among students in lower education level. Students tend to improperly utilize the linear model in solving word problems related to lengths, areas, and volumes of similar planes, figures, and solids (De Bock et al, 2002). The best known example of this misconception (Van Dooren et al, 2004), originated from what is called “synthetic model of linearity”, is that if a geometrical shape is enlarged k times, its area or volume will also become k times larger too.
In some extend, the linearity illusion can be considered as an obstacle in promoting the mathematical literacy since it provides some restriction to the students in solving ‘real’ problems related to mathematics. Referring to de Lange (2006), mathematical literacy is about dealing with ‘real’ problems. That means that these problems are not ‘purely’ mathematical but are placed in some kind of a ‘situation’. In short, the students have to ‘solve’ a real world problem requiring the use of the skills and competencies they have acquired trough schooling and life experiences. A fundamental role in that process is referred as ‘mathematization’. Read more…
Counting on, commonly used to deal with situation related to addition, is a handy strategy to cope with some subtraction problems. When children having constructed this kind of strategy are adding 8 + 4, rather than counting from one, they start with 8 and say “9, 10, 11, 12”. Teacher should arrange learning environment that can support students constructing counting on strategy when working with subtraction problem. It can be done by promoting a right context, a situation intended to develop mathematizing, suggesting counting on strategy such as this drawing problem. Read more…