Academic journal article International Education Studies

# Proportional Reasoning: How Do the 4th Graders Use Their Intuitive Understanding?

Academic journal article International Education Studies

# Proportional Reasoning: How Do the 4th Graders Use Their Intuitive Understanding?

## Article excerpt

Abstract

In Indonesia, the proportion is being taught formally in Grade 5 (10-11 years old). However, the existing learning approach does not support the development of the students' proportional reasoning. The way to teach proportion by giving cross multiplication is not meaningful for the students. They just memorize the procedure without understanding how it works. Within a design research, a learning sequence was developed for Grade 4 students (9-10 years old) in order to develop their proportional reasoning as well as their ability to solve the proportional problem before they learned more formally in Grade 5. The students of Grade 4 might have an intuitive understanding about proportionality and they might be able to deal with the comparison problem. How do they use this intuitive understanding to solve the comparison problem and what kind of difficulties that they faced? These questions were addressed through the analyzing of the students' work on the pretest and the video of students' interview. The result shows that the students' intuitive understanding, in principle, can help them to deal with the comparison problem. The students' experience may lead them to use the concept of proportionality instead of the absolute value in simple comparison problem.

Keywords: proportion, proportional reasoning, intuitive understanding, missing value problem, comparison problem

1. Introduction

Proportion is one of the concepts in mathematics which can be found everywhere (van Galen, Feijs, Figueiredo, Gravemeijer, Herpen, & Keijzer, 2008). Even when we cook, we determine the recipe by using the concept of proportion. The recipe to make four pans of pizza is a double recipe of two pans of pizza. If we need two kilograms of rice to make fried rice for 10 people then we need a kilogram of rice for 5 people. It means that the concept of proportion is needed in our life.

Based on the curriculum in Indonesia (Depdiknas, 2006), the proportion was taught in Grade 5 (10-11 years old). According to Zulkardi (2002), most of textbook that used in Indonesia contain mainly the set of mies and algorithm which is already formal and they lack of application which is needed by the students in order to make the concept be real for them. In line with that, in general the teachers teach the proportion by using the notion of algebra and cross multiplication to solve the proportional problem. The students may be able to use that formal procedure to find the answer. However, it is not a guarantee that they understand the insight of the proportionality. As stated by van Galen and van Eerde (2013), the procedures will quickly become vulnerable tricks if they are insufficiently anchored in understanding. Moreover, Lesh, Post and Behr as cited in CPRE, CAMS & El Paso state the cross multiplication method may not facilitate the development of proportional reasoning; even tend to avoid the proportional reasoning. The students are told to put the numbers on formulas and then compute them. Meanwhile, in Parish (2010), Kilpatrick, Swafford and Findeil; Lamon; Lesh, Post and Behr consider the proportional reasoning as a "capstone" of elementary school mathematics. Therefore, we need a learning design which can support the students to develop the proportional reasoning as well as the ability to solve the proportional problem. Within a design research, a learning sequence was designed and developed based on the principle of Pendidikan Realistik Matematika Indonesia (PMRI)-the Indonesian perspective of Realistic Mathematics Education (RME).

In their study, Boyer and Levine (2012) proposed that the students' formal mathematics understanding about proportion can be enhanced by instruction which builds on their early intuitive understanding of proportional relations. Furthermore, the Math Learning Study Committee as cited in CPRE, CAMS and El Paso believe that teaching the cross multiplication before the students understand the proportional relationship is meaningless. …

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