Mathematics has an abstract object of study. For example, its abstractness can be found in the concept of triangles and linear equations. The concepts in mathematics are constructed from previous concepts (Soedjadi, 2000). Shadiq (2009) argues that the students should construct new concepts independently in learning mathematics which is built on the existing concepts in their scheme. The formation of a new concept is called abstraction (Nurhasanah, Kusumah, & Sabandar, 2013). Ferrari (2003) asserts that abstraction is a fundamental process in mathematics. The abstraction in the process of learning mathematics is inevitable since it plays paramount roles in the formation of mathematical concepts. Thus, the activities involving the process of abstraction is needed as it helps the students construct new concepts appropriately through the concepts they already have in order to make a connection.

Hershkowitz, Schwarz, and Dreyfus (2001) define abstraction as an activity in which students vertically reorganize mathematical structures previously constructed into new mathematical structures. They argue that actions in the process of abstraction tend to be mental actions that cannot be directly observed. In this case, they provided a methodological framework for analyzing students’ abstraction through three epistemic actions. Epistemic actions are mental actions that can be observed through verbal actions or physical actions taken by students. The identification of epistemic actions relevant to the abstraction process is Recognizing, Building-with, and Construction or known as the RBC Model.

In RBC model, recognizing occurs when students realize that prior knowledge is attached to a given mathematical situation (Tsamir & Dreyfus, 2002; Hershkowitz et al., 2007; Dreyfus, Hershkowitz, & Schwarz, 2015; Celebioglu &Yazgan, 2015; Memnun et al., 2017). With regard to the action, recognizing involves trying to compare the results of the previous action and stating that the structure is similar or suitable. Building-with is the activity of combining existing knowledge to meet objectives, for example, to solve problems or prove the truth of a statement (Tsamir & Dreyfus, 2002; Hershkowitz et al., 2007; Dreyfus et al., 2015; Celebioglu &Yazgan, 2015; Memnun et al., 2017). Thus, the same task can be developed by one student, but constructed by another student, depending on the student's personal history (Tsamir & Dreyfus, 2002). Construction is the main step of abstraction (Tsamir & Dreyfus, 2002; Hershkowitz et al., 2007; Dreyfus et al., 2015). This action consists of gathering all the knowledge artifacts to produce a new knowledge structure. In a building-with structure, the goal is achieved by using the knowledge that has been recognized before, while in the construction structure, construction and restructuring of knowledge is the goal of the activity (Tsamir & Dreyfus, 2002).

At the secondary school level, geometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, fields, and space (NCTM, 2000). In the Indonesian school context, geometry is problematic mathematics topics since a large number of students had difficulties in connecting and identifying mathematical ideas to find solutions to problems of the triangles (Agnesya, 2015). Mutia (2017) found that many students have difficulties in understanding the concept of solid geometry plane surfaces. This shows that students' understanding of abstract ideas or concepts in mathematics, especially geometry is still under-achieved.

The way students construct abstract concepts varies. Susanto (2008) explains that the learning process is influenced by individual characteristics, for example, cognitive style. Mayer and Massa (2003) assert that cognitive style delineates how students process and represent information. Prior studies (e.g., Tinajero & Paramo, 1998; Alamolhodaei, 2001; Pitta-Pantazi & Christou, 2009; Chrysostomou et al, 2013; Nurafni, Miatun, & Khusna, 2018) have revealed that cognitive styles contribute to students’ performance in mathematics. Some authors (Witkin et al, 1977; Marashi & Moghadam, 2014; Onwumere & Reid, 2014) categorize cognitive style into field-dependent (FD) and field-independent (FI). These cognitive styles reflect the presence or absence of the nature of dependence on the environment so that the way students process information allows the emergence of differences or obstacles in the process of abstraction when learning geometry activities both in the stages that are able to be reached and the way students express ideas.

Many studies have revealed abstractions using RBC models with different subjects (Dreyfus, 2007; Memnun et al., 2017; Hassan & Mitchelmore, 2006; Celebioglu & Yazgan, 2015; Katranci & Altun, 2013) but researchers have not found any research that revealed students’ abstraction in constructing new knowledge based on cognitive style of students. Considering the importance of abstraction in learning mathematics (e.g., Hershkowitz et al., 2002; Ferrari, 2003), students' difficulties in geometry (Gal & Linchevski, 2010; Mutia, 2017), and the roles played by cognitive style in learning mathematics (Mayer & Massa, 2003; Chrysostomou et al., 2013), the present study aims to reveals the students' abstraction in learning common tangent lines of two circles based on their cognitive style. Besides enriching our knowledge on students' abstraction, this research contributes to mathematics teachers' understanding of how students, based on their cognitive styles, utilize prior knowledge to construct new knowledge.

The present study employs a qualitative approach (Moleong, 2007). Eight of 32 seventh-grade students were selected as the subjects based on three criteria. First, students had FI and FD cognitive styles as a result of the questionnaire. Second, students showed abstraction activities during group work. Third, students answered all the questions given in the task completely. Every four subjects are respectively, FD and FI students.

Data in this study was collected through task-based interview, test, and questionnaire. The questionnaire, Group Embedded Figure Test (GEFT), adopted from Witkin et al. (1977), was used to collect data on students' cognitive styles. It is a standardized test with a reliability index of 0.820 (Witkin, in Bostic, 1988). This instrument has been widely used by the researchers (e.g., Yuen, 2015) for analyzing cognitive style. The abstraction test, which consists of two questions (Table 1), was used as a basis for the interview to examine students' abstraction. The first question aims to help students distinguish the type and the properties of common tangent lines of two circles. Meanwhile, the second question supports the students to determine the length and the difference of the type of common tangent lines of two circles. The subjects were grouped in four; each group consisted of two subjects with similar cognitive styles (Table 3). The purpose of grouping was to see the abstraction that arises when interacting with their partners. The abstraction that arises during group activities as the main consideration in choosing a subject.

The interview was conducted individually based on the results of the abstraction test. During the interview, the researchers played a role as participant observers. The students' process when taking the test and interview was recorded with a video camera to prevent forgetting the situations in which the interviews were being conducted and allow the researchers to review it as often as wanted. The video recording was verbatim-transcribed. The transcripts were analyzed and grouped based on three abstraction activities in the RBC model. The indicators of abstraction were made by referring to abstraction activities according to Hong and Kim (2015) and Memnun et al. (2017) as presented in Table 2. The interviews were conducted twice to verify that the data obtained was credible. The external validity was provided with reporting in detail what has been done so the readers can trace and guide other researchers in determining their data when carrying out similar studies (Sugiyono, 2015).

FI students’ works are presented in Figure 3 and Figure 4. In Figure 3, group 3 made four illustrations showing the possible common tangent lines of two circles based on the relative position of the two circles, the properties of a circle, and the number of tangents they had in each illustration. They also considered the concept of equality of ratio numbers in the second problem. In Figure 4, group 4 made five illustrations exhibiting the possible tangents of the two circles by considering the relative position of the two circles, the properties of a circle, and the number of tangents they had on each illustration. These show the differences in FI students' answers.

In recognizing action, SI-1, SI-2, SI-3, and SI-4 tend to do the same thing. They recognized prior knowledge related to new situations in the given task, such as the concept of a tangent to a circle, the relative position of two circles, and properties of a circle to construct the definition of common tangent lines of two circles. They also recognized prior knowledge about parallel and perpendicular line to construct properties of common tangent lines of two circle. Based on the interviews, the four subjects could easily recognize the definition and properties of common tangent lines of two circles obtained in the previous activity and applied it to determine the length of common tangent lines of two circle so as to form new knowledge as a whole. However, SI-1 and SI-2 needed some guidance to realize that previous knowledge they mentioned, namely Pythagorean theorem needed to be considered in constructing length of common tangent lines of two circles. SI-1 also did not fully recognize that Pythagorean theorem is relevant to the task given although he could mention it when given direct questions by the researcher in [P_I,198] as shown in the following dialogue.

Field-dependent and field-independent students tend to do the same abstraction process on building-with activities but differ on recognizing and construction activities in learning common tangent lines of two circles. Students with field-dependent cognitive style tend to be affected by their learning environment, for example, the context in which students learn. They need more help in organizing information, for instance, recognizing their prior knowledge which is relevant to the task. Meanwhile, field-independent students tend to be able to organize their own information. There is a case where FI students need help in completing the tasks, that is when the problem being faced has a higher level of difficulty that requires prior analysis to be able to recognize relevant prior knowledge. This finding implies that mathematics teachers need to understand each students' cognitive style for consideration in designing effective learning activities. For instance, giving tasks that allow students to form their own mathematical concepts and bringing up discussion activities between groups based on cognitive styles. In addition, the teachers can accustom students to work by recognizing their prior knowledge relevant to the problem given and connecting them to construct new knowledge.