Numerous researchers have pointed out that angles, angle measurements, and angle rotation concepts are central to the development of geometric knowledge, but it remains a difficult concept for students and teachers to grasp (Browning, Garza-Kling, & Sundling, 2008; Clements & Battista, 1989, 1990; Keiser, 2000, 2004; Mitchelmore & White, 1998; Moore, 2013; Yigit, 2014). They claimed that students’ difficulties in learning of the concept of angles are based on learning of the multiple definition of an angle, describing angles, measuring the size of angles, and conceiving different types of angles such as 0-line angles (an angle whose degree is 0 and 360 degrees), 1- line angles (an angle whose degree is 180 degrees), and 2-lines angles (an angle where both rays of the angle are visible). While there are studies that shed light on students’ difficulties with the concept of angles, there is a lack of research that illuminates how students develop their schemas of relationships between angles and side lengths in a right triangle (RASR). Therefore, this study was designed to answer the research question, “What relationships pre-service secondary mathematics teachers (PSMTs) construct between angles and side lengths in a right triangle from the Action-Process-Object-Schema (APOS) theoretical perspective?” (Asiala, Brown, DeVries, Dubinsky, Mathews, & Thomas, 1996). There is need to expand the research in mathematics education concerning PSMTs’ schemas of RASR since there is no specific research concerning students’ learning, or how students’ knowledge of RASR is related to learning of more advanced topics such as trigonometric ratios.  Study of the proposed research question expanded the limited literature on the learning of RASR as well as the concept of angles.

The APOS learning theory was used as a theoretical lens to determine PSMTs’ schema of RASR. Dubinsky and his colleagues (Arnon et al., 2014; Asiala et al., 1996; Dubinsky, 1991; Dubinsky & McDonald, 2001) extended Piaget’s theory of reflective abstraction, and applied it to advanced mathematical thinking to develop APOS learning theory. Their main goal in developing APOS theory was to create a model to investigate and analyze the level of students’ mental constructions of a mathematical concept (Asiala et al., 1996). Specifically, a model is a description of how a schema for a specific mathematical concept develops and how the mental constructions of actions, processes, and objects can be used to construct the schema, and it is a useful guide for researchers to follow when investigating the levels of students’ learning of a concept (Asiala et al., 1996). Specifically, students use their existing knowledge of a physical or mental object to attempt to learn a new action. In order to learn a new concept, students carry out transformations by reacting to external cues that give exact details of which steps to take to perform an operation. Then, an action might be interiorized into a process when an action is repeated, reflected upon, and/or combined with other actions. At the process level, students perform the same sort of transformations that they did at the action level, but the process level is not triggered by an external stimuli; the process level is an internal construction. Once students are able to reflect upon actions in a way that allows them to think about the process as an entity, they realize that transformations can be acted upon, and they are able to construct such transformations. In this case, the process is encapsulated into a cognitive object (Asiala et al., 1996). Students then organize the actions, processes, and objects, as well as prior schemas, into a new schema that accurately accommodates the new knowledge discovered from the mathematical problem. 

Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. New York, NY: Springer. Asiala, M., Brown, A., Devries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In J. Kaput, A. Schoenfeld, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education (Vol 2, pp. 1-32). American Mathematical Society, Providence, RI. Browning, C. A., Garza-Kling, G., & Sundling E. H. (2008). What’s your angle on angles? Teaching Children Mathematics, 14(5), 283-287. Clements, D. H., & Battista, M. T. (1989). Learning of geometrical concepts in a Logo environment. Journal for Research in Mathematics Education, 20(5), 450-467. Clements, D. H., & Battista, M. T. (1990). The effects of Logo on children’s conceptualizations of angle and polygons. Journal for Research in Mathematics Education, 21(5), 356-371. Clements, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 547-589). Hillsdale, NJ: Erlbaum. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-126). Boston, MA: Kluwer Academic Publishers. Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Boston, MA: Kluwer Academic Publishers. Keiser, J. M. (2000). The role of definition. Mathematics Teaching in the Middle School, 5(8), 506–11. Keiser, J. M. (2004). Struggles with developing the concept of angle: Comparing sixth-grade students’ discourse to the history of the angle concept. Mathematical Thinking and Learning, 6(3), 285-306. Mitchelmore, M., & White, P. (1998). Development of angle concepts: A framework for research. Mathematics Education Research Journal, 10(3), 4-27. Moore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics Education, 83(2), 225-245. Piaget, J. (1975). The child's conception of the world. Totowa, NJ: Littlefield, Adams. Yigit, M. (2014). An examination of Pre-service secondary mathematics teachers’ conceptions of angles. The Mathematics Enthusiast, 11(3), 707-736.