# The Fifth Instructional Design of PMRI (Indonesian Realistic Mathematics Education): “How many more cake boxes are needed to fulfill the cardboard box?”

(A Description of  Learning of Volume of Cuboid and Cube at Grade 5C SD N 1 Palembang )

by:

Fanni Fatoni
Wisnu Siwi Satiti

bangwachid@gmail.com, fan_math05@yahoo.co.id, siwi.wisnu@gmail.com

A.  Introduction

One of the important topics  taught in the learning of three dimensional space in elementary schools is volume. Activities related to the volume of  three dimensional space  has frequently been performed by students in daily life, such as filling the tub with water until full, observing the sand-laden truck, filling cardboard boxes with smaller cake boxes, up to more complex things like counting the lack of packaging the goods packages that needs to be added to the container so that the car is full. The experiences of students who are associated with these activities can be the starting point for them to learn the concept of volume of three dimensional space.

At primary level, the concept of understanding the volume needs to be implanted to students first before they do the investigation into the discovery of how to find the volume of three dimensional space. This concept can be embedded by providing the experience of how to compare two objects by asking which one is bigger. Learning can begin with a discussion as to the question: which one is usually need more water until full, bath tub or tub lavatory. Through interactive discussion, students will put forward the idea of an answer in accordance with the experience they have gained, such as by considering the shape and the size of each type of tubs or estimate how many times they move water from the well to those both types of tub. This kind of thing is in line with Panhuizen (2005: 54) revealing that  in learning of  volume, students need to be given the experience of comparing the content of objects that are useful for reaching an understanding of the concept of volume.

After students understand the concept of volume of three dimensional space, learning can proceed with investigating how to find the volume of three dimensional. Media such as cube units can be used as the smallest unit forming a cuboid or cube. By identifying the relationship of the length, width, height cuboid/cube with the number of cube units needed to fulfill a model of  cuboid/cube, it is expected that students can find out by themselves the formula of volume of cuboid and cube.

In PMRI (Indonesian Realistic Mathematics Education), the problems in daily life that has been presented above can be the context as starting point to teach  students about the concept of volume of three dimensional space until to find the formula the volume itself. The role of such matters as the context in this regard as expressed Treffers and Goffree (in Wijaya, 2011: 33) that the context plays a role as a tool to shape the concept (concept forming), which in this case is the concept of understanding the volume, and context as a tool for developing a model (model forming), which in this case is student’s strategies to reinvent the formula or how to find the volume of three dimensional space. Through this context, it is expected that the learning of volume will be more useful so that students could understand more and interested to study mathematics.

Realizing the importance of learning volume of three dimensional space using PMRI approach, we need to make a detail instructional design contained in lesson plan and then it is implemented in the classroom learning. Therefore, we (Kohar, Fanni, and Wisnu Siwi) together with partners in the teacher’s class of 5C of SD Negeri 1 Palembang (Mrs. Mega) designed a learning using PMRI approach for material of volume of three dimensional space, which in this case, we focused to the volume of a cube and cuboid. The details of how the design development process (preliminary design), the implementation of the learning (teaching experiment), and the analysis of its learning (retrospective analysis) is described in instructional design in the next section.

B.  Objectives of Design and Observation

Objectives of this instructional design and observation are described as follows.

1. Describe the learning volume of cuboid and cube using PMRI approach at grade 5C of SD Negeri 1 Palembang.
2. Provide experience for the observers to design and implement the instructional design of volume of cuboid and cube using PMRI approach.

C.  Instructional Design

The material taught in this learning process was the volume of a cube and cuboid. This learning activity was implemented in grade 5C of  SDN 1 Palembang. The stages that are conducted in this activity were the preliminary design (analysis of curriculum, learning objectives and determining indicators), followed by teaching experiment (application/instructional design) and performing retrospective analysis (reflection of learning that has already been done) that is described as follows.

1)   Preliminary Design

At this stage, we analyzed curriculum aimed to make learning designed in accordance with the mathematics curriculum applied to grade 5 of elementary  school as a subject in this learning activities. The analysis covers the determination material taught, the purpose of learning, and learning indicators.

Standards competence and basic competence which become a reference in the preparation of lesson plans respectively are to calculate the volume of a cube and cuboid and use it in problem solving and to calculate the volume of a cube and cuboid. From the basic competences, we formulated learning objectives and learning indicators. Because the focus of the lesson are to discuss the concept of the volume of three dimensional space and how to find the volume of a cube and a cuboid, then the expected learning goals is to use cube units and raw rice, students can compare content between the volume of cubes and cuboid, and students may also find the volume of a cube and cuboid by using the given cube units, with indicators of learning are to compare the content between the cubes or cuboid model and to find the formula of volume of cube and cuboid by using the given units.

After determining the goals and indicators of learning, the next activity was designing the content of learning activities that will be carried out. We chose the context of comparing the contents of the water filled bathtub and tub lavatory as a starting point (use of context)  to find the concept of volume, and the problem of shortage of cake box needed so that cardboard boxes full of cake boxes as a context to discover how to find the volume of cuboid and cube.

According to our records at the time of carrying out the previous learning at grade 5C, students were already beginning to look actively in learning activities, either at the time of questioning with the teachers, as well as at the time of the discussion in the group. This shows that the interactivity of students (student’s interactivity) as one of the characteristics of PMRI started to appear. Therefore, in this instructional design of volume of cuboid and cube, the interactivity will again appear in learning activities, i.e. at the time of the initial activities when the teacher doing questioning and asking matter as stimulus of learning, main activities when students work in groups and the presentation of the results of the working groups, and the final activities when teachers with students conclude the lesson. This series of activities aimed students to construct their own understanding of their impact on a given problem to move to the process of further mathematizing (student’s contribution).

Other characteristics, linkages between knowledge/concept with other materials (intertwining strands of learning), is expected to appear at the time when students investigate how to find the volume of a cube and cuboid. The knowledge could be a multiplication skill and square/rectangular concept and how to find it. In relation to the process of mathematizing, built by students through the use of models (use of models) for learning, we conveyed a learning iceberg of volume of cuboid and cube as shown in Figure 1.

1. Instructional Design of PMRI: Volume of Cuboid and Cube (English Version)  (download)

3. Learning Instrument: Lesson Plan and Student’s Activity Sheet (Indonesian version)