Abstract:

In this paper we describe a study in which we investigated the way in which linguistic and extra-linguistic resources used by students faced with a sequence of activities, are important for the emergence and appropriation of mathematical notions.

In the research, 65 students of about 18 years old participated in a course of University Mathematics for Engineering that took place in 2013 at a private institute in northern Mexico.

During the teaching sequence, addressing qualitative relations between the graph of a function and the graph of its derivative was promoted. These notions are being interpreted by the change of position of an object that moves on a straight line and its velocity, respectively. For this, the SimCalc MathWorlds ® software was used, a dynamic environment in which students can explore mathematics of change and variation in the context of movement. In this software, various representations (graphs, tables, algebraic expressions) are dynamically linked and in its designed infrastructure resides the Fundamental Theorem of Calculus.

The rationale of the study is based on Vygotsky's sociocultural theory, particularly in a mutually formative relation between user and tools, emphasizing the role of the environment in which the tool is being used. This framework allowed us to deepen into the phenomenon of co-action, a type of dynamic and instant feedback that allows describing how the user of a dynamic environment guides the actions on the environment, and in turn is guided by the environment as a fluid activity.

The strategy for data collection during the study relies strongly on the current technological possibilities. Thus, students were asked to work in pairs and make recordings of their laptop screens, showing the interactions that took place in the environment of the software. In turn, they were asked to make video recordings using the webcam from their laptops, integrating the multimodal production of signs generated; particularly, speech and gestures.

Later, audiovisual material was collected and reviewed, finally having 28 videos generated by study participants. We resort to the tradition of the analysis of speech and gestures in interaction, in order to analyze the linguistic and extra linguistic productions employed by students in the appropriation of graphical relationships.

The study findings suggest that the ambivalence of SimCalc, as an object and / or subject, provides a tool that mediates cognitive actions and directs appropriation conversations of mathematical meanings. Moreover, although language was recognized as an essential resource for negotiating meanings, the role of gestures in this process is analyzed. Finally, situations in which students benefit from intellectual tools of their social group are also identified.

Keywords:

Technology, gestures, Calculus, motion context, graphical relations, SimCalc.