The development of polar coordinates dates back to the 17th century (Boyer & Merzbach, 2011; Eves, 1990). Polar coordinates are widely used in various disciplines such as mathematics, physics, engineering, and STEM (Borji & Voskoglou, 2016; Haro & Aguilar, 2025; Paoletti et al., 2013). At the collegiate level, the polar coordinate system is an essential concept for studying advanced mathematics and is involved across a variety of courses, including pre-calculus and multivariable calculus (Borji et al., 2020; Habre, 2017; Paoletti et al., 2013). In the context of calculus-related instruction, students are introduced to the relationship between polar and Cartesian coordinate systems, how to convert between them, and sketch polar curves (Borji et al., 2020).

The polar coordinate system is defined as “a two-dimensional coordinate system where each point on the plane has two indicators: a distance from the pole, r, as the first coordinate and an angle from a reference direction, θ, as the second coordinate” (Borji et al., 2020, p.408). Unlike Cartesian coordinates, where each point has a unique representation, a point in polar coordinates can be represented in infinitely many ways (Borji & Voskoglou, 2016; Haro & Aguilar, 2025). This difference is considered one of the sources of students’ difficulties with polar coordinates (Haro & Aguilar, 2025). When engaging with polar coordinates, students often remain constrained by the habits and modes of thinking developed through their prior experiences with Cartesian coordinates (Moore et al., 2014).

Even though it is critical for students’ mathematics learning, research on the polar coordinate system is notably limited (Moore et al., 2014). Moreover, the extant literature indicates that undergraduate students have difficulties with polar coordinates (Habre, 2017; Haro & Aguilar, 2025; Moore et al., 2014). More attention should be devoted to investigating how students construct their understanding of the polar coordinate system, and the challenges they encounter during this process (Montiel et al., 2008; Moore et al., 2014; Sayre & Wittman, 2007). In addition, although graphing has received considerable attention in mathematics education, comparatively little research has focused on graphing in polar coordinates (Habre, 2017). In this respect, this study aims to investigate the concepts that prospective mathematics teachers consider and the difficulties they encounter when sketching graphs of polar equations. Hence, this study aims to answer the following research questions:

Which concepts do prospective mathematics teachers take into account when sketching graphs of polar equations?

What difficulties do prospective mathematics teachers encounter when sketching graphs of polar equations?

This study was designed as a case study (Merriam, 2009) focusing on prospective mathematics teachers’ approaches and difficulties when sketching graphs of polar equations. Participants were second-year students enrolled in the Elementary Mathematics Education program at a state university. Within the scope of an undergraduate course, prospective mathematics teachers were instructed over a two-week period on fundamental concepts of polar coordinates. The instructional content included plotting points in polar coordinates, determining equivalent polar coordinates, converting between polar and Cartesian coordinate systems, transforming equations between polar and Cartesian forms, and sketching graphs of polar equations. Following the instruction period, Question 1 on sketching a polar graph was posed, and 70 prospective mathematics teachers responded. Question 2, which was similar in nature, was posed at the end of the semester and received 55 responses. Questions 1 and 2 are presented below. Question 1. Sketch the graph of the polar equation r=1+2sin3θ Question 2. Sketch the graph of the polar equation r=1+2cos2θ In the data analysis process, the six-step qualitative data analysis framework proposed by Creswell (2013) was employed to analyze both research questions. Initially, the data were organized and prepared through categorization. Next, the entire data set was examined to obtain a general sense of the responses. In the subsequent step, the data were coded by identifying meaningful units, and these codes were used to develop categories, descriptions, and themes. The results were then presented using tables. Finally, the findings were interpreted by relating them to the existing literature and drawing conclusions.

According to the results, prospective mathematics teachers considered the following key concepts while sketching graphs of polar equations: examining possible symmetries of the graph with respect to the polar axis, the line θ=π/2, and the pole, considering the signs of r and θ values, determining the appropriate interval for θ to ensure the entire graph is captured, deciding how many points should be calculated to adequately represent the shape of the graph, identifying the points, and using them as references in the sketch. Regarding the second research question, it was found that some participants had limited content knowledge in particular concepts, such as trigonometry and functions. In addition, they had difficulties in plotting polar points with negative r and θ values. Some participants did not conceptualize polar graphs as curves, which may be attributed to their prior experience with Cartesian coordinates. Consistent findings were reported in the studies of Borji and Voskoglou (2016) and Haro and Aguilar (2025). Borji and Voskoglou (2016) stated that students’ difficulties in polar coordinate problems are related to limited prior knowledge of trigonometric angles and functions, limited understanding of Cartesian coordinates and polar coordinates and equations, and transitions between them. According to Haro and Aguilar (2025), learning obstacles in the polar coordinates can be grouped into two categories: epistemological and didactical. Epistemological obstacles included reliance on Cartesian knowledge and challenges with negative values, multiple representations of the pole, and intersections of polar graphs, while didactical obstacles involved graphing in polar coordinates, interpreting polar functions, and converting between Cartesian and polar forms.

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