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Graph-matching situations: some insights from a cross year survey in the UK

There are multiple branching curriculum decisions to be made about how functions develop for learners through school. Our main study, from which the data for this paper is taken, contributes by exploring the development of understanding of functions; it compares UK and Israeli students, who learn formally about functions at different ages. This report focuses on covariation – a central aspect – in the ‘space’ of graph-matching situations. Research acknowledges the importance of activities that develop a graphical view of functions as describing variations and for describing situations in problem solving (e.g., Eisenberg, 1991; Swan, 1980). Research also reports on difficulties: the most frequently cited is interpreting a graph as a literal picture of a situation (e.g. Leinhardt, Zaslavsky, & Stein, 1990). Difficulties with compound variables representing rate and decreasing functions are also reported (ibid.). The aim of this report is to identify some implications of students’ choices in graph-matching situations throughout school in the UK. We report findings from four graph-matching tasks derived from Swan (1980), and data from 120 UK students, 20 from each of years 7 to 11: 10 from a high-achieving class (A), 10 from a middle-achieving class (B) in each school, and 10 from the first and second years of post-16 mathematics study. Students were asked to match four situations to graphs, write their chosen variables on the axes, and provide explanations for their choices. All situations focused on identifying the variables, forming the relation between them (in particular, capturing their covariation), and noticing contextual features. Variables of different kinds were used, which, as indicated above, may be linked to differences in the quality of responses. Due to lack of space we give just one example: “After the concert there was a stunned silence. Then one person in the audience began to clap. Gradually, those around her joined in and soon everyone was applauding and cheering” (unidimensional variables; increasing function). The full task, including the variety of graphs, appears in Ayalon, Lerman, and Watson (2013). An iterative and comparative process of analysing 480 students’ responses led to three codes: (1) No choice, often accompanied by “I don’t know”; (2) Lack of full analysis; (3) Full analysis. Further analysis of code 2 led to four sub-categories of difficulties: