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This paper discusses the importance of considering bilingual learners' non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper focuses on the word-use, gestures and dragging actions in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners' competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic, touchscreen environments, particularly for bilingual learners. [ABSTRACT FROM AUTHOR]

Student autonomy has been an important object of study for mathematics educators for many years. Over time, framings of autonomy have moved from a focus on the individual to considerations of how an individual's autonomy is entangled in classroom-level interactions. What has been less closely studied is how classroom interactions provide uneven access to autonomy for individuals. This study uses a communicational perspective to clarify Piaget's intellectual autonomy and examine students' mathematical interactions. The findings describe barriers and supports to autonomous activity for three students. Students were prevented from engaging in autonomous activity when they were seen as less capable than others, when they felt the need to manage the activities of their peers, or when they focused on being seen as knowledgeable. In contrast, students acted with autonomy when they took up the teacher's request for explanations, noticed a contrast between their answer and the right answer, and worked on making connections across different representations. [ABSTRACT FROM AUTHOR]

This paper investigates how three widely used calculus textbooks in the U.S. realize the derivative as a point-specific object and as a function using Sfard's communicational approach. For this purpose, the study analyzed word-use and visual mediators for the limit process through which the derivative at a point was objectified, and word-use and visual mediators for the derivative process through which the derivative as a function was objectified. The analysis highlighted inconsistency in realizations of the limit process and object, and derivative process and object found in some instances, and implicit discussion on relations among different visual mediators. Specifically, the results of this study led to four observations about issues suggested by one or more of the textbooks: a) visual mediators of the initial and final objects, and the visual mediators of these processes were often inconsistent; b) transformations of visual mediators from initial object, through process, to final object were often implicit both in one visual mediator and across multiple visual mediators; c) visual mediators for the limit processes were sometimes disconnected from the initial and final objects, or the connections were not explicit; and d) realizations of both the derivative at a point and the derivative of a function were mediated with nearly identical symbols suggesting a possible difficulty with understanding the difference between them. [ABSTRACT FROM AUTHOR]

This special issue comprises five studies which vary in their focus and mathematical content, yet they all share an underlying communicational theoretical framework-commognition. Within this framework, learning mathematics is defined as a change in one's mathematical discourse, that is, in the form of communication known as mathematical. Teaching can be defined as the communicational activity the motive of which is to bring the learners' discourse closer to a canonic discourse. Reading the five research articles interconnected via their shared theoretical infrastructure has the potential to further develop insights about various aspects of teaching and learning mathematics. [ABSTRACT FROM AUTHOR]