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Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples. [ABSTRACT FROM AUTHOR]

The emotions that can be considered members of the set of Aesthetic Emotions (AEs) is controversial. The present study investigated the terms used by researchers in peer reviewed studies to exemplify AEs. 100 publications from 2000–2019 exemplifying AE terms were located to produced 111 AEs which were proposed as the basis of an AE lexicon. Awe, (being) moved and wonder were reliable members and without contradiction. One fifth were negatively valenced (e.g., anger, disgust), suggesting that the presence of negative AEs is generally accepted but not reliably. One quarter of the entries were also non-AEs and an additional 20 were exclusively so, producing a total of 131 terms. The lexicon is a concrete, dynamic set of examples against which to investigate extant definitions of AEs and to further develop theory. The robust presence of three terms suggests that calls to abandon the concept of AE may be premature. [ABSTRACT FROM AUTHOR]