Abstract

The visualization of figures throughout task-solving has often posed significant challenges to students engaged in geometry learning. While Duval emphasizes the importance of rapidly identifying relevant features in a given figure, there is limited understanding of how students specifically comprehend geometrical figures and how this comprehension can either facilitate or hinder their task-solving abilities. Several researchers (Parzysz, 1988; Duval, 1998, 2014; Fischbein, 1993; Laborde, 1994; Mesquita, 1998) have highlighted different functions of geometric figures. According to Mesquita (1998), figurative representations of geometrical objects help individuals better understand the relationships among these objects. Although figures can mobilize "multiple relationships," they cannot distinguish between what the figures provide (i.e., their assumptions) and what they request. Duval (2014) explains that there are two fundamentally different approaches to interpreting figures: the spontaneous perception of any visual representation and the mathematically constrained recognition of properties based on given attributes. The cognitive gap between these approaches often leads to difficulties and inconsistencies in students' understanding, as noted by Duval (2011).To define the heuristic role of geometrical figures, Duval (1988) introduced the term "cognitive apprehension," which encompasses four types: perceptual, sequential, discursive, and operative apprehension. Studies conducted by Michael-Chrysanthou (2013), Michael-Chrysanthou & Gagatsis (2013, 2015), and Karpuz & Atasoy (2019) in line with Duval (1999) demonstrated that although geometrical figures are expected to aid in solving geometry problems, many students do not leverage this potential benefit effectively. Instead, they experience inhibited operative apprehension and a lack of interplay between perceptual and discursive apprehension during geometrical tasks.

This study aims to investigate the cognitive apprehensions of middle school students when solving geometrical tasks. The research involved a sample of 305 ninth, tenth, and eleventh-grade students from five schools in Roudehen, Iran. The test included four tasks from previous studies (Michael-Chrysanthou, 2013) designed to examine students' geometrical figure apprehensions based on Duval's (1999) cognitive interactions involved in geometrical activity. The students' responses were analyzed both qualitatively and quantitatively, with coding and classification of responses based on prior research (Creswell, 2014).
Findings
Task 1, Perceptual Apprehension: This task involved figures resembling squares. Despite this visual resemblance, students could not definitively classify these figures as squares due to the absence of supporting information in the task prompt. The results revealed that ninth graders displayed a higher recognition rate of seven squares (61.96%) in the figure, whereas only around half of tenth graders (51.11%) identified the same number of squares. Some tenth and eleventh graders (13.33% and 10.31%, respectively) used a measurement-based approach to conclude that there were no squares in the figure, emphasizing the insufficiency of the information provided. This approach was more prevalent among tenth and eleventh graders, suggesting potential differences in geometry education. Further exploration of students' responses to Task 1 revealed diverse reasoning patterns. Some ninth graders employed a painting method to count squares accurately, while others perceived a large four-sided shape as a square and identified additional squares by drawing symmetrical lines.

Task 2, Operative Apprehension: Figures were placed on a checkered surface with mathematical attributes. The results indicated that 43.65% of ninth graders used operative apprehension to answer the task. Surprisingly, 22.50% of tenth graders did not consider operative apprehension sufficient, favoring the use of the concept of area to determine the answer. Some students, inspired by the descriptive role of the figure, chose to rely on area calculations, demonstrating diverse problem-solving strategies. In Task 2, students' answers were shaped by various cognitive processes. They employed natural discursive reasoning, estimation, conjecture, and the concept of area to provide solutions. A significant portion used operative apprehension to express their desire for additional information.

Task 3, Discursive Apprehension: Students were tasked with comparing the lengths of MH and NH. The majority of ninth graders (99.19%) relied on natural discursive reasoning to answer, with a smaller percentage (8.87%) transitioning from perceptual apprehension towards discursive apprehension. Surprisingly, a shift from perceptual apprehension to discursive apprehension was not evident among tenth and eleventh graders. A fraction of tenth graders (11.11%) applied perceptual apprehension due to differences in geometry education methods. Notably, several students followed various paths of theoretical discursive apprehension when approaching Task 3. They employed congruence of triangles, accepted equality of lines or angles without proper reasoning, or resorted to visual perception in cases where the figure did not align with the problem's description. The students' cognitive processes often prioritized the visual aspect over revisiting the verbal description, leading to mathematically incorrect responses.

Task 4, Sequential Apprehension: Students were asked to draw a parallelogram based on a given triangle with equal areas. Results showed that 72.41% of ninth graders employed natural discursive reasoning to solve the task, demonstrating their reliance on heuristically interpreting geometric figures. Tenth and eleventh graders displayed lower usage of this method, with many opting for theoretical discursive apprehension. The study identified two primary paths of theoretical discursive reasoning among students, each leading to the creation of a parallelogram with an equal area to the triangle. Students used various strategies to accomplish this, such as halving the base of the triangle and employing congruence, ultimately arriving at the correct answer. Some students chose alternative paths of theoretical discursive reasoning, demonstrating a nuanced understanding of geometric concepts.
Discussion and Conclusion: The study enhances existing literature by identifying new reasoning patterns derived from additional coding, highlighting how students understand and solve geometrical tasks. The findings align with previous research, particularly that of Michael-Chrysanthou & Gagatsis (2013) and Mesquita (1998), which indicated diverse reasoning strategies among students. The research confirms the significant role of cognitive apprehension in geometry learning and underscores the need for instructional strategies that address the cognitive gaps identified by Duval. Educators often find it hard to resist intervening in student work. However, it’s vital to help students progress from basic to discursive understanding of geometric figures for problem-solving. (Cannizzaro & Menghini, 2006). Gagatsis et al. (2023) found that just solving problems correctly doesn’t equip teachers to foresee students’ difficulties. Incorporating tailored geometry tasks into the curriculum is a deliberate approach that considers the various apprehensions and challenges students encounter with different geometric concepts throughout their secondary school years (Michael–Chrysanthou, et al. 2024). These highlight Duval’s (1999) emphasis on the need to integrate mathematical and cognitive elements in task assignments. Duval (2014) also advocates for more research on helping students understand figures mathematically, beyond just visual understanding, to solve problems independently. Specialized geometry courses can enhance teachers’ understanding of different mental representations of geometric figures, thereby improving students’ reasoning abilities. However, the study had limitations such as potential fatigue from long tests and the use of convenience sampling, which requires careful interpretation of results.