Start from “Common Sense”, End with “Understanding”

Summary of the article:
Gravemeijer, K. (2011). How concrete is concrete. Indonesian Mathematics Society Journal on Mathematics Education, 2(1), 1-8.

This might be the answer of the previous problem given in “When nine doughnuts price as much as the other ten”. The real problem is that students sometimes think that mathematics that they learn in school has no relation with what they see in everyday life.


Dealing with this problem, teachers try to provide students with manipulative which perhaps help them connect the math to the real-world context. Occasionally, those manipulative seemed very realistic for teachers, unfortunately not for students. Teachers are usually trapped with their own thinking and forget that their students are in not in their level.

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Does “context” only match with “Lower-level Math”?


Well, I am not going to ask you to solve nor to discuss the solution of the problem beside.

The question was taken from the ‘Examen Vwo‘ of Wiskunde B, a yearly math event in the Netherlands.

What I am going to show you from the picture is related to the title of this post, “Does context only match with lower level math?” which is rather be a common question, at least, by several Indonesian teachers. Furthermore, some of them used it as an excuse for not applying RME, CTL, or involve any context in their teaching of higher level math. As the consequence, most of our students feel less motivated in learning math since they did not find it useful for them.

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Design of PMRI-based Mathematical Problems (English Version)



In order to make 6 portions of ‘es pisang ijo sauce’, the following ingredients are required:
– 1 litre of coconut milk
– 60 gram of tepung beras
– 100 gram of sugar
– 1/2 tea spoon of salt
– 1 piece of palm leave

Andi wants to make the sauce, and he has 10 litre of coconut milk, 100 gram of tepung beras, 1 kg of sugar, a lot of salt, 10 pieces of palm leave. How many portion of ‘es pisang ijo sauce’ he could make?

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Let’s Play Math (Colourful Necklace for Learning Number Pattern/ A Paper Based Applet)

This paper-based applet adopts the Java version entitled ‘Beads on Chain’. To access this, you may visit:

Number pattern, or commonly known as ‘bilangan loncat’ by the lower grade students in Indonesian primary school, is first taught in the third grade.

The ‘beads on chain’ applet is actually a very good manipulative to stimulate students’ understanding on this concept. However, internet connection which does not support most Indonesian elementary schools has been an undeniable challenge. Therefore, preserving the paper-based version could be an alternative.

I have tried out the ‘Colourful Necklace’ to learn number pattern with a number of Grade III-D students of SD Pusri Palembang.

At first, I invited 2 students to play during the breaktime. Hearing that I would play with them, some other students came closer and got involved. Foto0355 It shows how this tool first impressed and be positively responded by the students.

To begin with, I told them the way to make a colourful necklace. Here, I emphasized that a similar pattern is important to make the necklace more beautiful. So, we first placed two yellow beads, then one brown, then yellow, and then blue again. Next, I tested them for the following sequence to ensure that they understood the problem. Finally, we got the first two sequences of alternating beads colour.

I, then, told them that I would love to use 32 beads for my necklace and I was worried about the colour of the last bead will be. The students were mostly confused and tried to gamble the answers. After a little repeating with emphasis to the questions, several students tried to think.

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Addition and Subtraction up to 100 Using Empty Number Lines

Reflection on Paper:
Design Research on Addition and Subtraction Up to 100
Using Mental Arithmetic Strategies on an Empty Number Line
At the 2nd Grade of SDN Percontohan Komplek IKIP Jakarta
(Puspita Sari, Dede de Haan, Zulkardi, 2008)

Innumeracy problems experienced by many students nowadays might be caused by inappropriate approach in teaching algorithm in primary schools which seems to be premature and less contextual according to some experts. In order to deal with the problems, realistic approach involving mental arithmetic strategies is suggested in advance, that is, emphasising more on number values rather than number digits.

In case of teaching addition and subtraction to 2nd grade students, an empty number line – number line with no numbers on it – seems to fit the need since it could encourage students’ informal counting strategy to develop. The use of context as demanded in RME is highly required to stimulate a meaningful learning toward the students. For the research, the context applied was celebrating the 63rd Indonesian Independence Day, due to the current d-situation.

The Use of Empty Number Lines in Learning Implementation
1. Empty number line as a model of
In this part, the number line was introduced using a string of beads which was coloured alternately every ten beads, the length of which would be measured by students using a paper strip. This would smoothly redirect the students to think of empty number lines.number lines as model of

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