There is more to examples than meets the eye: Thinking with and through mathematical examples in different settings

The paper provides an overview of the broad field of research on the roles and use of examples in mathematical learning and thinking, specifically in concept learning and proving, and offers a way to organize and categorize selected research in this field. The main contribution of the paper is the distinction between three settings of example-use: spontaneous example-use, evoked example-use, and responsive example-use to a provided example. In the first two – the main source of examples comes from the student/learner, while in the third setting the main source is the teacher/researcher. The illustrations that accompany the description of each setting by research-based cases convey the affordances and challenges of each one. While these three settings are applicable to various kinds of learning, I highlight their affordances mainly for concept formation, and point to papers in this issue that provide a rich collection of cases that convey the place of these settings for conjecturing and proving.