Approaches, difficulties, and types of `graph as picture' misconceptions in using function to describe movement - a case study of stone throwing

The paper presents the study focused on the use of function graphs to represent familiar movements from practice. It examines how participants solve a simple task that requires identifying the correct graph describing changes in speed over time during vertical projection. From a mathematical perspective, the task requires only the ability to recognise the correct monotonicity of a function. A research method employed in the study was eye-tracking, combined with other approaches, such as interviews and questionnaires. The task was completed by 345 participants with varying levels of mathematical expertise. It was challenging, with an overall success rate of 0.43. The paper presents and categorises various strategies used to solve the task, also examining participants' visual attention. It analyses the difficulties and misconceptions that emerged during problem-solving. Two types of categorisations are introduced: nine mathematical strategies (with 21 subcategories) and eight distinct types of `graph as picture' misconceptions revealed during the task. Initial pedagogical implications for further research and teaching practice are formulated, emphasising the need to address the identified difficulties and misconceptions in schools to prepare students to use function graphs in simple contexts.