Analysis of high school students' mental models of angles

Angles are a foundational concept in mathematics with wide applications in science and engineering, yet students often struggle with understanding the mathematical concepts behind them. Existing research has focused largely on misconceptions, leaving the cognitive structures behind these difficulties underexplored. To this end, we leveraged the “Fidelities Model of Conceptual Development” (short: FG-FF framework) to explore students cognitions of angles. The FG-FF framework, originally developed in physics education, distinguishes two cognitive dimensions in learner's mental models. Firstly, a Gestalt, which refers to its visual structure and appearance, and secondly a Functionality, which encompasses how well a model captures conceptual dynamics. The degrees in which both cognitive dimensions are related to reality—Fidelity of Gestalt (FG) and Functional Fidelity (FF)—enable a distinct typification of learners with respect to their mental models. While promising in science education, this framework has seen limited use in mathematics to date. To explore the explanatory power of this framework with regards to mental models in mathematics education, we conducted a cross-sectional survey with N = 403 high school students, using a newly developed instrument tailored to assess FG and FF in the context of mental models of angles. Confirmatory factor analysis confirmed a strong two-factor model (CFI = 0.96, RMSEA = 0.03, SRMR = 0.05) with solid reliability (ω FF = 0.78, ω FG = 0.66), validating the framework's relevance for mathematics education. The findings further suggest that effective teaching of geometry should intentionally guide students from visually grounded, Gestalt-oriented representations toward more abstract, functionally coherent understandings of mathematical concepts. By focusing on this progression, educators can design learning sequences that better connect perceptual intuition with formal reasoning, fostering deeper conceptual understanding of angles and related geometric ideas.