Construction of the mathematical concept is important to support students' success in learning. Factorization is one of the important subjects to be mastered by students in order to solve advanced mathematics problems. Reflective abstraction is a mental mechanism which someone can build all mathematical concepts at all levels. This study used the mental structure of APOS theory (Action, Process, Object, Schema) to describe the mental mechanisms of interiorization, encapsulation, and thematization of students in reconstructing the mathematical concepts. This study aims to reveal and describe the reflective abstraction of students in reconstructing the factorization concept.

 

The type of this research is explorative-descriptive by using qualitative approach. Participants in this study were two volunteer students at 8^th grade in one school in Surabaya consist of one male student and one female student with medium mathematics ability. The data were collected by interview-based on tasks.

 

The results obtained that male student at the action phase connected the knowledge that he got to perform to do the tasks such as using knowledge of multiplication, knowing what was known from the tasks as could distinguish variables, coefficients, and constants. In the phase of interiorization and process, he could explain how he got the answer. In the encapsulation and object phase, he could explain the rules used such as using distributive properties. In the tematization and scheme phase, he could connected the actions, processes, objects, and schemes that have been formed to form a factorization scheme and he could done more complex tasks, while female student only reach to the interiorization and process phase, where at the action phase she was connected the knowledge that he got to perform to do the tasks,  mentioned what was known from the tasks. At the interiorization and process phase, she could explain how she got the answer. It can be concluded that the reflective abstraction of male student is better than female student (medium mathematics ability) in reconstructing the concept of factorization.

 

The instrument in this study consists of two types, namely type 1 and type 2. Where algebra form of type 1 contains one variable and algebra form of type 2 contains two variables. And each of them are second-degree.

 

The reflective abstraction indicators in this study can be seen in Table 1.

Table 1. Reflective Abstraction Indicator

Reflective Abstraction

Phases

Indicators

Action

- Connecting the mathematical knowledge that belongs to the factorization content to find the answers.

- In this phase, students could not finish the task until the end.

Interiorization and Process

- Connecting more mathematical knowledge related to the factorization content to find the answers.

- In this phase, students were able to complete the tasks until the end, but the steps used are still wrong.

Encapsulation and Objects

- Connecting mathematical knowledge with the knowledge of factorization that is wholly owned to get the right results.

- Awaring of the total process by realizing the transformation done and able to explain the rules used in every step of the settlement done.

Thematization and Schema

- Connecting actions, processes, and objects, and leads to formation of a factorization scheme.

- Using schemes that have been formed through actions, processes, objects that in mind to form new objects that can be used in accomplishing more complex tasks.

(Adapted from Cetin & Dubinsky, 2017)