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This special issue comprises seven studies and a commentary piece, which relate to various aspects of rituals and the "ritual-exploration" dyad in learning, teaching and learning-to-teach mathematics. The theme of the special issue arose from Sfard and Lavie's (Cognition and Instruction, 23(2), 237–309, 2005) "ritual" vs. "explorative" dyad, on which a Working Session was held during the PME conference in 2016. Three central themes are addressed in the papers of this special issue: (1) the logic of ritual, and why rituals are so persistent in mathematics classrooms; (2) the co-existence of rituals and explorations, including the question of whether rituals and explorations are a binary or a continuum; and (3) alternative theoretical conceptualisations of the ritual-exploration dyad, focusing in particular on what can be learned about rituals from social and critical theory, as well as socio-linguistics. In this introduction, we first give an overview of each of the papers and its unique contribution. We then synthesise the answers given by the papers to the three themes described above. [ABSTRACT FROM AUTHOR]

The reported research attempts to trace possible reasons for third grade learners' limited progress in numeracy in a low socioeconomic status (SES) South African context. This is done through two lenses, both stemming from Sfard's commognitive (The term 'commognition' has been offered by Sfard as an amalgam of 'cognition' and 'communication,' thus expressing the unity of these concepts. Since its original appearance, some authors (including Sfard herself) have preferred using the word 'communicational' to describe Sfard's framework. We chose to stick with 'commognitive' because we believe it clearly points to the specific theoretical stance presented in Sfard , whereas 'communicational' might point to many other theories or frameworks that have something to do with human communication.) framework. One lens aims to analyze two learners' (Mina and Ronaldo (all names are pseudonyms)) mathematical and identity discourse both in one-on-one interviews and in a small group 'math club' lesson led by the second author. The other examines the mathematical milieu in which these learners have participated through the analysis of a school mathematics lesson which exemplifies prevalent instructional practices in this milieu. Relying on the distinction between ritual and explorative participation, we show that while Mina was acting in an extremely ritualized manner, Ronaldo was more explorative in his actions. However, the milieu, as seen in the school lesson, encouraged almost exclusively ritual participation. Thus, while Mina was identified as a good student, Ronaldo was identified as an outcast or 'troublemaker.' We conclude by drawing implications to the tenacious nature of rituals in the mathematics classroom and the effects that these rituals may have on students' identities. [ABSTRACT FROM AUTHOR]