|Ph.D Thesis||Department of Education in Science and Technology|
|Supervisor:||Assoc. Prof. Zaslavsky Orit|
|Full Thesis text|
Examples play a central role in learning, teaching, and doing mathematics since they are central to the body of knowledge - mathematics - and to the way humans think. This research aims at exploring and characterizing the process of the generation of mathematical examples by junior-high-school students. This includes characterization of the situations in which students choose to give examples, the actual ways in which they construct examples, and the interplay between the strategies they employ and their understanding of the mathematical issues. This research is part of a wider study that examines the use of examples by teachers and students while teaching and learning mathematics, which was funded by the Israel Science Foundation (ISF).
The study consists of two main parts. The first is naturalistic in nature and focuses on observations of 64 mathematics lessons in junior-high-school classes; the second consists of 12 task-based interviews, designed on the basis of the findings from the first stage. All data were analyzed according to grounded theory techniques in a spiral manner in which new data lead to new categorizations, leading to reanalysis of the previous data.
Conditions for example generation: The findings indicate that the main condition that led students to choose to generate an example is their sense of need. The need can be one of three types: (1) an external need, for instance when the teacher asks for an example; (2) a communicational need whereby the example serves as a means to explain mathematics to others; and (3) an intellectual need whereby the example serves to resolve a mathematical uncertainty or dilemma.
The strategies: To construct examples, students used one of five strategies: trial and error, educated trial, modification, immediate retrieval, and search. The analysis shows that different strategies were used in different contexts (i.e., mathematical topics) and as a response to different needs.
The theoretical significance of this research rests on the explicit identification of conditions that encourage or discourage students to generate examples as well as the strategies they apply. In addition, it characterizes situations in which students generate examples without being prompted to do so. The research findings offer a unifying multidimensional conceptual framework to analyze and discuss example generation processes and thus constitute the theoretical contribution of the study. The research suggests pedagogical implications, as the findings demonstrate ways in which teachers can encourage their students to generate examples and indicate the merits of such approach.