Teachers need to plan a lesson appropriately based on students’ needs. However, each student has different learning styles, interests, readiness, paces and abilities. It is teacher’s responsibility to know how each student learns and what learning methods appropriate for them.

This issue prompted the researchers to find a learning model that meets the needs of students on the volume of prism topic, and to group the students based on their achievement after implementing problem based learning (PBL) to know a group of students who experience success through the learning model.

The subjects were 66 eighth graders from two classrooms in a junior high school in Banda Aceh, Indonesia. Making use of the discriminant analysis by Fisher, researchers categorized students’ ability into two groups, successful and unsuccessful groups. The finding showed that 50 students were in a successful group and 16 students were considered unsuccessful. That implies PBL met the needs of the 50 students and this finding suggests the implementation of another learning model for the unsuccessful group to find out a learning model best for them in learning solid geometry. Therefore, implementing PBL in teaching geometry is recommended and future research to examine the other learning models is needed.

## Introduction

As teachers, making sure all students learn mathematics well is necessary. To achieve the goal, teachers need to design an instruction that meets students’ needs. However, in a math classroom, students are diverse. Every student has different ways of learning, abilities, paces, readiness and interests. Different students learn well by different learning models.

Students will not learn well if teaching style conflicts students’ learning style [1]. A shift from teacher-centered to student-centered approach is a recognition that learning should be focusing on students’ needs. In student-centered approach, teachers need to learn how to differentiate instruction [1]. This issue prompted the researchers to find a learning model that meet the needs of students on the volume of prism topic, and to categorize students based on their achievement after implementing a learning model to examine the group of students who experience success through the learning model.

One of student-centered learning models that can be used to teach geometry is Problem Based Learning (PBL). PBL appeared as a recognition that students learn by doing and by thinking through problems. Originally, PBL was developed for adults, to train doctors in how to approach and solve medical problems in the real world where in one case it might be difficult to identify symptoms of a patient and in another case a patient might have multiple symptoms.

Through PBL, the medical students researched a situation, developed appropriate questions, and produced their own plan to solve the problem. It was found that PBL developed the students’ abilities to improve their knowledge and to learn how to deal with the new illnesses. They became “self-directed learners” and used available resources to help them to learn [2]. This successful learning then leads PBL adopted to other fields such as mathematics education.

Some researchers have used PBL in teaching mathematics [3], [4]. Schettino used PBL to teach geometry. He defines “PBL as an instructional approach where students’ learning and content materials are constructed through the use, facilitation, and experience of contextual problems … in a discussion-based classroom setting where student voice, experience, and prior knowledge are valued” [3].

He believed that PBL allows students to use skills such as communication, collaboration, and technology literacy required by the current curriculum. Through learning using PBL, he found that students made connections and saw the relationship among the math concepts, had the opportunity to create their own strategies and expressed their ideas. Thus, implementing PBL is one of ways to meet not only the curriculum demand but also the students’ needs.

### Method

This present study used mixed methods, quantitative and qualitative. The quantitative method is used to categorize students’ achievement on the volume of prism topic after they experienced a class session with PBL. The qualitative method is used to describe the learning process using PBL and students’ response to the implementation of the learning model. The study involved 66 students of Grade 8 in a junior high school in Banda Aceh, Indonesia. Each classroom consists of 33 students.

This study was conducted through two phases. In the first phase, students from both classes were given a pre-test about the volume of prism. Then, in the next session, both classes were taught the volume of prism using PBL by the same teacher for one meeting. Two days after the class session, they were given a post-test. Students’ grades on the both tests were collected to see their progress on the topic. Also, data about students’ math scores on the previous grade report were collected.

Based on their achievement on the topic, the students from both classes were categorized into two groups¬, a group who was successful and had not, with minimum completeness criteria 65. To do this process, the researchers used discriminant analysis by fisher method. Discriminant analysis is a statistical technique used to classifying observations, for example, predicting success in academic programs [5]. “Discriminant function analysis produces functions that help define the groups; it distinguishes the first group from others, then the second group from the rest, and so on” [6]. The researchers used SPSS (version 16.0) to get the formula of discriminant analysis. After that, students from each category were interviewed to find out their responses toward the implementation of PBL.

A group of students who had not been successful in a learning using PBL on the volume of prism topic were given another treatment in the second phase. They were taught the same topic using a different learning model, given a test, categorized into two groups using fisher’s discriminant analysis and interviewed to get their responses toward the implementation of the learning model used in the class session.

The group who was not successful on the learning in the second phase would be treated using another method that will lead to the third phase.

The goal is to find a learning model suitable to the students on solid geometry topic. This process will be repeated as in the previous phase. However, this paper only discusses the first phase.

### Result and Discussion

During the first phase, the students were taught using PBL on the topic of prism volume. With PBL, the students were expected to be able to find the volume of prism in real life problems. Previously, the students had learned the volume of cubes and rectangular prisms. Using their prior knowledge, the students figured out the volume of triangular prisms.

To motivate the students, the teacher showed them an application of the volume of prisms in real life. For example, the teacher showed a picture of a shower tub shape that looks like a prism and asked them how to find the volume of the water in the tub. In this way, the students realized that they need to know the volume of the prism to be able to find the volume of the water.

In a similar way, the teacher gave students a real-life problem. However, at this time, the students worked in a group to solve the problem. To launch the problem, the teacher asked the students what they understood from the problem and what they had not. Also, the students were encouraged to pose questions about the problems to help them in solving the problem and write them all. Examples of student questioning are: What is the shape of the swimming pool?

What is the length and the width of the pool? What is the volume of the water? What does it mean by sloping floor in between two parts of the pool? What is the width of each part of the pool? As solving the problem, the students also did some research from the textbook and recalled their previous knowledge. Working in a group allowed them to discuss and learn from others. Interaction among the students help them to communicate their ideas and move them forward.

### The problem assigned to students

While students were working in a group, the teacher assisted them by asking purposeful questions to assess their thinking and to help them think forward. However, the teacher did not tell the students the answers. Instead, the teacher supported students’ productive struggle.

For example, when a group could not get started, the teacher asked them, “What do you understand from the problem? What information do you know from the problem? Could you please rewrite them?” Also, the teacher encouraged the students to look back to the questions they had posed before. This helped them to recall the information needed for solving the problem.

Students recorded their solutions in a poster then presented their strategies in front of the class.

The teacher asked them to compare solutions found by different groups and discuss it in the whole class discussion. This activity allowed students to clarify their thinking and reveal students’ misunderstanding about the volume of prisms. A group of students found the volume of prism by making connection to the volume of rectangular prism. To get the volume of a triangular prism, the students divided the volume of the rectangular prism by two (see Figure 3).

Then, when a student showed a misconception, the teacher brought it to the class for further discussion so that other students will not perform the same misconception in the future. For instance, a student modelled the problem in an incorrect way. She misinterpreted the sentence: “On the surface of the pool, there is sloping floor in between two parts of the pool” and drew a picture as in the Figure 2.

### Student misinterpretation on the problem

Then, the teacher asked a question to the class, “If we model the problem like this (as in Figure 2), how would you use information of the depth 3:4 in the picture? How would you put the numbers in the picture?” Some students reasoned that they could not determine the depth of another part of the pool if the surface is not flat.

Some argued that there is sloping floor in between two parts of the pool; thus, surface of both parts of the pool should be flat. Interestingly, a group of students believed that the volume of water could be varied because the depth of the parts of pool is 3:4. That means that it could be either 3 m and 4 m, 9 m and 12 m, or 30 m and 40 m. Those will affect the volume of water in the pool. Such a discussion promoted students’ reasoning and justification that helped them to understand the concept of the volume of prism.

### Example of student’s solution

After learning the volume of prism, the students were given a post test in the following session. Then, students’ achievements on both pre- and post-tests were examined. The data were normalized using the natural logarithm to minimize the error. By using SPSS software (version 16.0), the fisher’s linear discriminant functions were obtained in the following table.

Table 1. Fisher’s linear discriminant functions

- Classification Function Coefficients
- Group Student
- Successful Students Unsuccessful Students
- Math score 3.463 3.470
- Pre-test .015 -.011
- Post-test .618 .255

**Based on the table above, the discriminant functions were:**

Successful group = 3.463X_1 + 0.015X_2 + 0.618X_3 = 0

Unsuccessful group = 3.470X_1 – 0.011X_2 + 0.255X_3 = 0

where:

- X_1 = Math score on previous grade report
- X_2 = Pre-test on the volume of prism topic before the class session (phase 1)
- X_3 = Post-test on the volume of prism topic after the class session (phase 1)

In SPSS (refer to Table 1), every column to which variables correspond consists of coefficients that form a function for a group. These coefficients are called coefficients of fisher’s linear discriminant functions and could be used directly for group classification. Each coefficient in each formula is used to determine a subject to which group he/she belongs based on the highest score got from the formulas [7].

Referring to the procedure, scores of each student were substituted into the formula to find to which group the student belongs. A student will be in a group that has the highest score. It was found that 50 students belong to a successful group and 16 students were considered unsuccessful. This finding indicates that learning the volume of prism using PBL met the needs of the 50 students. However, the rest of the students needed a different method of learning.

When asked about the implementation of PBL, students from the successful group responded positively. They said the learning method used was different from they usually did and they enjoyed the learning process. A student stated that he liked a whole class discussion because he could share his ideas, which was unusual in a mathematics classroom. Some students from the unsuccessful group did not like discussion. They found it hard to express what they thought and needed more time to process their thinking.

They also were not confident in sharing their thoughts. A student mentioned that she preferred solving problems by herself because she needed to contemplate about the problem. When asked “what would you do if you struggle?”, she answered that she will ask the teacher and go back to think and solve the problems by herself. Other students still struggled to understand the concept of the volume of prism and could not identify a prism in the complex solid geometry. Students in the unsuccessful group were also asked their preference about the learning model. Some of them prefers working in a small group, but not having a whole class discussion. Some of them likes working individually first then having a discussion with partners to ensure their answers.

Those findings suggest that different students have different preferences of learning, paces, and interests. PBL did not meet the needs of 16 students from the unsuccessful group. It recommends that the students be taught using another model of learning, Discovery Learning, for example. Teachers also could use Think Pair Share (TPS) for those who prefer working independently and in pair. The next phase of the implementation of those learning models needs to investigate further. To ensure that the learning fit the students’ needs, teachers should know the students’ background, interests, preferences and paces. In a big classroom, meeting all students’ need would be challenging. However, with a continuous effort, it could be achieved.

### Conclusion

Teaching geometry using PBL is one of the promising ways to enhance students’ learning. PBL enables students to explore the problem by working in a group and students will be center of learning. This activity could actively engage students in problem solving and promote a meaningful discussion among students and teacher.

However, not all students will be convenient in such a model of learning. Some students learn well by contemplating the problem by themselves and cannot get distracted by others. Another prefers learning by discussion. Those differences lead teachers to find the best learning method for students and decide what kind of method teachers should use in their classroom. In a big classroom, it is challenging to figure out each student’ need and interest. One way to deal with that issue is using fisher’s discriminant analysis to classify a group of students to which a learning model fit in.

The discriminant analysis could be a starting tool to investigate what method best meet students’ needs. By this method, teachers will have a picture of how many students succeed by using a certain learning model. After the teachers get to know all students’ needs and interests, teachers could design differentiated instructions that could meet all students’ needs in one lesson plan, without making multiple lesson plans.

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