Understanding Percent using the Percentage Bar

Summary of the article:
Van Galen, F. & D. v. Eerde. (2013). Solving Problems with the Percentage Bar. IndoMS. J.M.E, 4(1), 1-8.

In an observation toward 14 Grade VII students of a achool in the Netherlands, surprising result (only 4 students could solve the problem correctly) was performed by the students when dealing with the following problem:

Trying to solve the problem, the students directly did calculations. None of them tried to make visual representation to see the relations between the given numbers. As the consequence, they often went wrong.

Actually, the students have tried several ways to find the answer like working with equal proportion or playing with the numbers in the question.

The former one was supposed to bring them to the expected result. However, they were stuck since they have to work with unfriendly proportion (15 out of 100) which led them to confused. While the latter one was really out of concept. The students might think that the problem must involve simpler calculation. So, they just tried to divide 600 over 15 and conclude that the answer should be 40.

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.

Learning Percentage through Role-Playing ‘Surveyor-Respondent’

One of the very-closed mathematical concepts to us is percentage, of which the idea is represented in such common forms as discount, interest in bank, battery, polling result, and some other social activities. This familiarity would be a great potential to bridge students to learn the concept using PMRI (Pendidikan Matematika Realistik Indonesia) learning approach.

Therefore with, my friend and I have tried to design a set of activities to encourage students’ understanding regarding the concept. The design has been tried out to Grade V-E students of SD Pusri Palembang.

The first activity was conducted to identify how closed the concept of percentage to students is. To begin with, we showed a tagged paper ‘50%’ and asked the students where they use to meet such writing. Almost all the students answered ‘discount’. Several other answers like ‘battery’ and ‘polling result’ was also mentioned.

Representation of percents in real life

Introducing Concept of Angles with RME Based Learning

Summary of:
Bustang, Zulkardi, Darmowijoyo, Dolk, M., van Eerde, D. (2013). Developing a local instruction theory for learning the concept of angle through visual field activities and spatial representations. International Education Studies, 6(8), 58-70. doi: 10.5539/ies.v6n8p58

The notoriety of geometry among Indonesian students as one of the hardest topics is not without proof. Studies found many misconceptions experienced by the students, especially in angle representation which might impact on their difficulties in learning geometry in the higher level.

In order to deal with the problem, Bustang has designed a four-step learning activitiy for teaching and learning the concept of angle. The design has been tried out to the third grade students of a public school in Palembang, Indonesia.

At first, the students were familiarized with the real situations involving vision lines and blind spots through an activity called ‘Now you see it, now you don’t’. Here, the students were provided with three problems related to a cat’s vision over a group of mice by representing the ‘top view’, ‘side view’, and additional case if cat do a ‘movement’.

Learning Decimals

JOURNAL SUMMARY

Pramudiani, P., Zulkardi, Hartono, Y. & Ameron, B.V. (2011). A concrete situation for learning decimals. Indonesian Mathematical Society Journal on Mathematics Education, Vol. 2, No. 2, July, (2011), 215-230.

It was found that decimals is an essential part of mathematics whose concepts need to be learned meaningfully in order to prevent students from misconception. The fact, however, showed that Indonesian textbooks could not provide such condition and neither do the learning and teaching activities which finally led the students to the lack of understanding and misconception toward the concept. RME underlying the design of context (in this case precise measurement) and activities seemed to be an appropriate solution regarding this problem. Therefore, the study was conducted, that is, to study how measurement activities promote students’ notion of decimals.

The research was conducted in three main steps, namely, preliminary design (to produce a conjectured local instruction theory containing learning goals, planned activities, and learning process), teaching experiment (involving 26 students from class 5A SDN 21 Palembang in 6 lesson hours, prior to this, 73 students were involved in the pre-assessment and pilot experiment was given to 7 of them each from high, average, and low level students), and retrospective analysis (analyzing data collected and comparing the hypothetical learning trajectory with the students’ actual learning).

Introduction to RME

Summary of Subchapter 2.2 (Understanding RME) of the following reference:
Zulkardi. (2002). Developing a Learning Environment on Realistic Mathematics Education for Indonesian Student Teachers (Doctoral dissertation). University of Twente, Enschede.

The concept of RME is based on Hans Freudenthal’s views of mathematics as a human activity which implies that the students should be encouraged to find and reinvent the mathematics themselves. In order to reach the target, the learning process should start from the real world problem, or things which is well known by the students. Such strategy is then called ‘didactical phenomenology’.

In addition to that, Van Hiele identified three levels of learning mathematics. It starts when students can play with the pattern which is familiar to them. The next phases are when they could recognize the relationships among the patterns, and elaborate its internal characteristics.

Following the ideas, RME is resulted with five main characteristics (tenets), that is:
1. The use of context
This tenet best matches with ‘conceptual and applied mathematization’ proposed by de Lange (1987). The idea positions ‘real world’ as both the starting and final point of learning cycle. So, the students would find mathematics concept in reality, explore it, identify the related-mathematics concept, generalize, and apply it into the other aspects of life.

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.