Michal Ayalon

Students' Conceptualisations of Function Revealed through Definitions and Examples.

ABSTRACT This study aims to explore the conceptualisations of function that some students express when they are responding to fictitious students’ statements about functions. We also asked them what is meant by “function” and many voluntarily used examples in their responses. The task was developed in collaboration with teachers from two curriculum systems, England and Israel. It was given to 10 high-achieving English students from each of the years 10–13 and to 10 high-achieving Israeli students from comparable years (total of 80 students). Data analysis included identifying students’ dominant ideas for functions as expressed in their responses, and analysing the types of examples that students used to explain their responses. Differences found between the samples from the countries led to conjectures about the influence of curriculum and teaching, and in particular, about the role of word, in this case “function”, in concept image development. Whereas most students showed that they had a meaning for the word, those students whose relevant experience of earlier concepts had been organised around the word “function” generally showed stronger understanding of function as object.

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:

Teachers Editing Textbooks: Transforming Conventional Connections Among Teachers, Textbook Authors, and Mathematicians

The M-TET (Mathematics Teachers Edit Textbooks) project invites mathematics teachers to collaborate in editing the textbooks they use in their classes as a means of transforming conventional connections among teachers, curriculum developers, and mathematicians into more productive connections. The unique aspects that characterize the work environment offered to teachers include the following: designing a textbook for a broad student population, producing a textbook by making changes to a textbook designed by expert curriculum developers, and consulting with professionals that are not part of the teachers’ usual milieu (textbook authors and mathematicians). This chapter explores the nature of the connections between teachers and textbook authors, and between teachers and mathematicians that participation in the M-TET project made possible, and it discusses what might be gained by offering such a work environment.