This study aims to reveal the thinking process of field independent and field dependent student in solving polyhedron problems based on information processing theory. The data collected in this study consist of verbal data derived from the problem-solving process, which reflect students’ thinking processes. This study focusses in-depth exploration of the students’ thinking processes and therefore adopts a qualitative approach with a descriptive exploratory design. This study was conducted on February 10–21, 2025.

This study follows the characteristics of qualitative research as proposed by Creswell (2014), which are: (1) researchers collect data from various sources, such as interviews, observations, and documentation, then review and interpret the data by assigning, meaning, and organizing them into categories or themes across data sources; (2) researchers build patterns, categories or themes inductively, by processing data into more abstract units of information; (3) researchers try to create a complex picture of the problem being studied; and (4) the data obtained in this study are primarily verbal in nature.

The study was conducted with 22 eight-grade students of SMP IT Nurul Hasan, Ternate. The selection of research subjects was based on the consideration that the research subjects had already studied the topic of polyhedron. In addition, participants were classified by administering the Group Embedded Figure Test (GEFT), which had been validated for the purpose of grouping field independent and field dependent cognitive styles.

The criteria used for the classification were as follows: students who correctly answered 10–18 items were categorized as field independent cognitive style, while those who correctly answered 0–9 items were categorized as having field dependent cognitive style ( Oh & Lim, 2005). The results of the classification of students’ cognitive styles are presented in Table 1.

Based on the results of the Group Embedded Figure Test, 11 (50%) students were classified as having a field independent cognitive style and 11 (50%) students were classified as having field dependent cognitive style. Subsequently, one subject from each cognitive style group was selected. The selection process involved consultation with the mathematics teacher who taught the eight-grade students, with consideration given to students’ communication skills, both oral and written. Finally, one student from the field independent category, referred to as C _B and one student from the field dependent category, referred to as the C _L, were selected. The distribution of students according to field-independent and field-dependent cognitive style categories is presented in Table 2.

After the two subjects were selected, a problem-solving test on the topic of polyhedron was administered, followed by interviews aimed at identifying the students’ thinking process based on the components of information processing theory. The interview procedures for the two subjects varied depending on the communication dynamics between the researcher and each subject.

The instruments used in this study were the Group Embedded Figured Test (GEFT), polyhedrons problem-solving tests, and interview guidelines. The GEFT served as an initial mathematics test consisting of 20 multiple-choice questions covering basic mathematical concepts. The polyhedron problem-solving tests was a written assesment consisting of one essay question designed to address the research objective of this study. The test instrument was developed and modified by the researcher through a supervision process and subsequently validated by expert validators in mathematics and mathematics education. The interview guidelines were designed to collect in-depth information regarding students’ thinking processes in solving polyhedron problems. These guidelines were developed by the researcher through a supervision process and were also validated by expert validators in mathematics and mathematics education.

The stages of data analysis in this study consisted of the following steps: (1) collecting interview data, problem-solving test data, and GEFT data; (2) transcribing the data; (3) organizing data into meaningful units which and categorizing them through coding; (4) checking the data validity; (5) analyzing the thinking structures of field-independent and field-dependent subjects; (6) constructing representations of the thinking structures; (7) analyzing the findings obtained from each subject; (8) drawing conclusions based on the research questions. The data analysis stages are illustrated in Figure 1.

Furthermore, to examine the thinking process of field-independent and field-dependent students in solving polyhedron problems, the data were analyzed using the components of information processing theory, namely attention, perception, retrieval, rehearsal, and encoding. In the attention component, students focus on reading the problem carefully and thoroughly to identify, understand and retain relevant information. In the perception component, students interpret the information and formulate a problem-solving plan. In the rehearsal component, the information presented in the question was repeated to strengthen memory retention. In the retrieval component, information stored in long-term memory is recalled into short-term memory to support problem solving. Finally, in the encoding component, students write down each step of problem-solving process and draw conclusions based on the obtained solutions.

The determination of the research subjects began with the administration of the Group Embedded Figured Test (GEFT) to 22 eighth-grade students. The GEFT was used to classify students into field-independent and field-dependent cognitive styles. The test consisted of 20 questions covering basic geometry concepts presented in the form of word problems. These basic geometry questions were included because the students had previously studied geometry in both the elementary and junior high school levels, providing them with relevant prior learning experiences. The results of the GEFT are presented in Table 3.

Students who obtained test scores of 50% or higher were categorized as having a field independent cognitive style, whereas students who obtained test scores of less than 50% were categorized as having a field dependent cognitive style. The results of the GEFT showed that 11 students were classified as field independent and 11 students as field dependent.

After grouping students according to cognitive styles, the researcher selected the research subjects by considering their thinking potential and communication skills. Consequently, two students were selected as the subjects of this study; one student from the field independent cognitive style category, referred to as C _B, and one student from the field dependent cognitive style category, referred to as C _L. Following the selection of the two subjects, a problem-solving test on the polyhedron topics was administered, followed by interviews to identify students’ thinking processes based on the components of information processing theory. The selection of these subjects also received written consent from their parents and the research subjects.

Analysis of the thinking processes to two subjects resulted in two distinct categories: the locality category, demonstrated by the field independent subject, and the globality category demonstrated by the field dependent subject. The locality category emerged in the retrieval component, in which information retrieved from long-term memory was partitioned into separate parts, enabling the identification of relationships among the parts. In contrast, the globality category also emerged in the retrieval component; however, information retrieved from long-term memory was integrated into a complete (comprehensive) whole, making it difficult to identify relationships among individual components. The categories of subjects’ thinking processes are shown in Table 4.

The first finding, namely the locality process, was observed in the field-independent subject (C _B) during the solution of the polyhedron problem. Accordingly, C _B was categorized under the locality thinking process. The second finding, namely the globality process, was observed in the field-dependent subject (C _L) during problem solving and was therefore categorized under the globality thinking process. These categorizations were not coincidental but were derived from triangulated data, including interview data, field notes and the written work produced by each subjects during problem solving. The categories of students’ thinking processes in solving polyhedron problems are illustrated in Figure 2.

The description of the locality thinking process by C _B subject was analyzed based on the components of attention, perception, retrieval, and encoding. When given the following polyhedron problem:

The problem-solving process of C _B subject is illustrated in Figure 3.

After the polyhedron test question was given, at the attention stage, C _B subject observed and read the problem. Attention refers to the process of focusing on the information provided in order to solve the problem and determine what is known and what is being asked. The attention demonstrated by C _B subject is presented in the following interview excerpts.

Researcher: In your opinion, what is the question you are reading about?

C _B Subject: I think, this question is about polyhedron.

Researcher: Okay, then what is known and what is being asked in this question?

C _B Subject: The givens in this problem are that there are five teachers and six times as many students, so there are 30 students. One tent can accommodate a maximum of four people. The tent is shaped like a triangular prism.

Researcher: Then, what is asked in that question?

C _B Subject: The question asks how much minimum fabric must be prepared so that all campers can stay there.

Subject C _B demonstrated an understanding that the given problem involved a polyhedron. Next, C _B directed attention to the information received by reading the question carefully and thoroughly. Through this attention process, C _B was able to state what was known and what was asked in the problem. C _B identified that the number of teachers was five, the number of students was six times the number of teachers and that the tent was shaped like a triangular prism with a maximum capacity of four people. C _B also stated that the problem asked for minimum amount of fabric required to accommodate all campers.

Perception emerged after the attention process. C _B subject perception of the information received serves as an initial plan for determining the formula that can be used. C _B subject’s perception is illustrated in the following excerpts.

Researcher: Before solving the problem, do you have an idea how to solve it? If you do, explain.

C _B Subject: The surface area of the tent is equal to the area of the tent door. In other words, the front of the tent is equal to the back of the tent. Then, add the areas of the left and right tent roofs.

Researcher: Next, how do you solve this problem?

C _B Subject: First, the width must be determined using the Pythagorean theorem, because the width ( l ) of tent is unknown.

Based on these statements, C _B subject planned to calculate the front and back areas of the tent, and then add the area of the left and right sides of the tent roof. Then, he would use the Pythagorean theorem formula to find the value of l (the width of the tent). To do so, C _B used the concepts of the triangle area, rectangle area, and the Pythagorean theorem. This perception was realized in short-term memory, where the perceived information was processed simultaneously.

During the realization, C _B subject performed retrieval by recalling relevant concepts from long-term memory. C _B subject’s explanation is shown in the following excerpts.

Researcher: Then, what is the next step?

C _B Subject: To find the front area of the tent, I calculated 2 meters multiplied by 3 meters multiplied by 2 meters, and divided by 2 meters, and obtained 6 m 2 .

Researcher: Ok, what is the next step after that?

C _B Subject: Next, I find the area of the tent roof using multiplication operations: 2 × ( p × l ).

Researcher: What does 2 × ( p × l ) mean?

C _B Subject: It means two times the tent door section, the length of the tent ( p) is 4 meters, while the width of the tent ( l ) is unknown.

Researcher: How do you determine the value of ( l )?

C _B Subject: By using the Pythagorean theorem formula AC ² = AB ² + BC ²

Researcher: How do you apply the formula?

C _B Subject: I substitute the values 2 2 + 1.5 2 = 4 + 2.25 = 6.2 . The square root of 6.2 is 2.5.

Researcher: Where does the value 1.5 come from?

C _B Subject: It comes from half of the tent base, because the total tent base length is 3 meters.

The statements show that C _B subject correctly performed retrieval. C _B calculated the front area of the tent as 6 m 2 and then determined the roof area using the expression 2 × ( p × l ). To find the unknown width, C _B recalled and applied the Pythagorean theorem, obtaining a width of 2.5 meters. The C _B subject then performed rehearsal by rewriting the expression 2 × ( p × l ). This is shown in the following excerpt.

Researcher: Why, did you rewrite 2 × ( p × l )?

C _B Subject: To find the area of the tent roof

Researcher: What is the next step?

C _B Subject: Since the width of the tent is already known to be 2.5 meters, I substitute it into 2 × (4 m × 2.5 m) and get 20 m ².

Through rehearsal, C _B reinforced the retrieved information and correctly calculated the roof area as 20 m ². Next, C _B performed retrieval again to determine the total surface area of the tent by adding the front area (6 m ²) and the roof area (20 m ²), resulting 26 m ².

At the encoding stage, C _B subject provided a complete solution. C _B concluded that one tent requires 26 m ² of fabric. Since the total number of campers (teachers + students) was 35 and one tent could accommodate a maximum of four people, C _B concluded that 35/4 = 9 tents were required.

The description of the globality thinking process demonstrated by C _L subject was analyzed based on the components of attention, perception, retrieval, and encoding. When C _L was presented with the same polyhedron problem, the problem-solving process is shown in Figure 4.

At the attention stage, subject C _L read and observed the problem. Attention is the process of focusing on the information provided to determine what is known and what is asked. Subject C _L’s attention is reflected in the following interview excepts.

Researcher: What do you understand from this question?

C _L Subject: The question is about polyhedron problem.

Researcher: What is known from the question?

C _L Subject: It is known that there are five teachers, 30 students, and one tent can accomodate a maximum of four people.

Researcher: What is asked in the question?

C _L Subject: What is the minimum amount of fabric needed, if one tent is for a maximum of four people.

Although C _L subject did not write down what was known and what was asked in the problem, the interview revealed that C _L understood the information correctly. Perception emerged after attention. C _L subject perceived that the problem could be solved using the formulas for triangle area, rectangle area, and the Pythagorean theorem. This is shown in the following excerpts.

Researcher: What do you understand from this question? Try to explain.

C _L Subject: To solve this problem, I used the formula for the area of triangle, the area of rectangle and the Pythagorean theorem.

Researcher: How do you solve this problem? Please explain in more detail.

C _L Subject: To solve this problem, I used the formula for the area of triangle L ∆ ABC = ½ × a × t, the formula for the area of rectangle L ⎕ BCEF =  p × l and Pyithagorean theorem c ² =  a ² +  b ²

Researcher: Why should you use the Pythagorean theorem?

C _L Subject: Because the width of the tent is unknown, I search it using the Pythagorean theorem.

This excerpt shows that the C _L subject recalled the relevant formulas as a complete set, without explicity identifying relationships among individual geometric components. When this perception was realized in short-term memory, C _L subject performed retrieval by recalling mathematical concepts from long-term memory. The retrieval process is shown in the following interview excerpt.

Researcher: Why did you write ½ × 3 × 2 and 4 × l?

C _L Subject: Because, the area of triangle ABC is ½ × base ( a) × height ( t). The base is 3 meters and the height is 2 meters, so the result is 3 m ².

Researcher: What’s the next step?

C _L Subject: To find the area of rectangle BCEF, we can multiply the length ( p) and width ( l ). The length is 4 meters but the width is unknown.

Researcher: How do you determine the width ( l ) value?

C _L Subject: I used Pythagorean theorem. The value of a is 2 meters and b is half of the tent surface, which is 1.5 m.

Researcher: Next, how do you get the value 2.5 meters?

C _L Subject: The values are substituted into the formula c ² =  a ² +  b ², so it becomes c ² = 2 ² + 1.5 ² = 6.25 and the square root of 6.25 is 2.5.

Researcher: Does this result provide the area of rectangle BCEF?

C _L Subject: Not yet, it only gives the width of the tent width. Then, 4 meters multiplied by the tent width 2.5 meters equals 10 m ².

Based on this excerpt, C _L subject retrieved mathematical information globally by applying formulas sequentially without decomposing the problem into explicitly related subcomponents. The subject proceeded to calculate the surface area of the tent by summing the areas of two triangular faces and two rectangular faces, resulting in a total area of 26 m ².

At the encoding stage, C _L subject did not provide a complete conclusion regarding the number of tents required based on the total number of campers. This indicates that the information processed in short-term memory was not fully encoded back into long-term memory. Therefore, C _L subject thinking process is categorized as globality, particularly at the retrieval and encoding stages. The similarities and differences between these thinking processes are presented in Table 5.

The difference between the thinking processes of field-independent and field-dependent subjects in solving polyhedron problems lie in three components: (1) attention, where field-independent subject explicitly identified what is known and asked, while field-dependent subject did not; (2) rehearsal, where the field independent subject performed repetition and reinforcements, while the field-dependent did not; (3) encoding, where the field-independent subject provided a complete conclusion, while field-dependent subjects did not.