The main goal of the study reported in our paper is to characterize teachers' choice of examples in and for the mathematics classroom. Our data is based on 54 lesson observations of five different teachers. Altogether 15 groups of students were observed, three seventh grade, six eighth grade, and six ninth grade classes. The classes varied according to their level-seven classes of top level students and six classes of mixed-average and low level students. In addition, pre and post lesson interviews with the teachers were conducted, and their lesson plans were examined. Data analysis was done in an iterative way, and the categories we explored emerged accordingly. We distinguish between pre-planned and spontaneous examples, and examine their manifestations, as well as the different kinds of underlying considerations teachers employ in making their choices, and the kinds of knowledge they need to draw on. We conclude with a dynamic framework accounting for teachers' choices and generation of examples in the course of teaching mathematics.

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This chapter examines the activities of example-generation and exampleverification from both the teaching and learning perspectives. We closely examine how engaging learners in generating and verifying examples of a particular mathematical concept as a group activity serves both as an indicator of learners' understandings and a catalyst for enhancing their understanding and expanding their example space that is associated with the particular concept. We present two cases that illustrate how the mathematics instruction may look when classroom activities and discussions build on example-generation and example-verification -the first case focuses on the concept of an irrational number and the second on the notion of a periodic function. The learners in these cases are in-service secondary mathematics teachers (MTLs), and the teacher is a mathematics teacher educator (MTE). We show how this kind of learning environment lends itself naturally to genuine opportunities for learners to engage in meaningful mathematics, to share and challenge their thinking, and to sense the need for unpacking mathematical subtleties regarding definitions and ideas. For practicing and prospective mathe matics teachers, engaging in such activity and experiencing the potential learning opportunity that it offers is also likely to convince them to implement this approach in their classrooms.

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In the originally published version of the chapter, Figure 4 in Chapter "What Can You Infer from This Example? Applications of Online, Rich-Media Tasks for Enhancing Pre-service Teachers' Knowledge of the Roles of Examples in Proving" was portrait oriented and downsized, hence the text was virtually unreadable. The erratum of the book has been updated with the change.

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