A framework of growth points of secondary students' developing understanding of function

This research developed a framework describing students' developing understanding of function. The research started with the problem: How might typical learning paths of secondary school students' developing understanding of function be described and assessed? The following principles and research questions guided the development of the framework. Principle 1. The framework should be research-based. Principle 2. The framework should include key aspects of the function concept. Principle 3. The framework should be in a form that would enable teachers to assess and monitor students' developing understanding of this concept. Principle 4. The framework should reflect students' big ideas or growth points which describe students' key cognitive strategies, knowledge and skills in working with function tasks. Principle 5. The framework should reflect typical learning trajectories or a general trend of the growth points in students' developing understanding of function. The following questions guided the development of the framework of growth points: What are the growth points in students' developing understanding of function? What information on students' understanding of function is revealed in the course of developing the framework of growth points that would be potentially useful for teachers? The framework considered four key domains of the function concept: Graphs, Equations, Linking Representations and Equivalent Functions. Students' understanding of function in each of these domains was described in terms of growth points. Growth points are descriptions of students' 'big ideas'. The description of each growth point highlights students' developing conceptual understanding rather than merely procedural understanding of a mathematical concept.;For example, growth points in students' understanding of function under Equations were: 1) interpretations based on individual points; 2) interpretations based on holistic analysis of relationships; 3) interpretations based on local properties; and, 4) manipulations and transformations of functions (in equation form) as objects. The growth points in each domain are more or less ordered according to the likelihood that these 'big ideas' would emerge. To identify and describe these growth points, Year 8, 9 and 10 students in Australia and the Philippines were given tasks involving function that would highlight thinking in terms of the process-object conception and the property-oriented conception of function. Students' performance on these tasks and their strategies served as bases for the identification and description of the growth points. The research approach was interpretive and exploratory during the initial stages of analysis. The research then moved to a quantitative approach to identify typical patterns across the growth points, before returning to an interpretive phase in refining the growth points in the light of these data. The main data were collected from students in the Philippines largely through two written tests. Interviews with a sample of students also provided insights into students' strategies and interpretations of tasks. The research outputs, the research-based framework and the assessment tasks, have the potential to provide teachers with a structure through which they can assess and develop students' growth in the understanding of function, and their own understanding of the function concept.