![geogebra classic drag axes geogebra classic drag axes](https://www.geogebra.org/resource/X6TePu6B/3kjpl26dJ1Scfsq3/material-X6TePu6B-thumb.png)
In order to establish a link between two core notions, an emphasis on reflections according to the axes, projections onto axes and composition of the reflections as geometric linear transformations (GLT) in ℝ 2, could be a (twofold) heuristic tool for establishing the link between GLT and functions. However, as has been shown in a number of research papers (Bagley, Rasmussen, & Zandieh 2015 Zandieh, Ellis, & Rasmussen 2012, 2017), it is not an easy or trivial task for students to construct a mathematical link between a linear (matrix) transformation and a function, even if the students are aware of the classic Dirichlet-Bourbaki notion of function that is often formulated as f : A → B for two non-empty sets A and B. For instance, the notions of function and linear transformation are strongly connected in this catalogue. Linear algebra is an extensive catalogue, which includes different mathematical objects and representations, where it is not easy for students to build interconnections among them. Finally, a shared environment with action, production and communication conveyed student reasoning and they managed to reinvent a number of geometric linear transformations. The students mostly gestured when they faced a new reflection situation and when describing associated geometric actions.
![geogebra classic drag axes geogebra classic drag axes](https://www.geogebra.org/resource/atfumbnu/VJSMWHF5fgah1m5p/material-atfumbnu-thumb.png)
According to the findings, the artefact use, verbal and written mathematical expressions all interlaced with the emergence of gestures. The data was analysed according to the multimodal paradigm focusing on all semiotic resources, such as gestures and artefact use, in addition to written signs. Data was collected using a video camera observing the students’ working environment, screen recorder software, student production and field notes.
![geogebra classic drag axes geogebra classic drag axes](https://www.geogebra.org/resource/mq77gwax/QeB8aA7bTCP3kEOL/material-mq77gwax-thumb.png)
Task-based interviews were conducted with a pair of linear algebra students, by way of a computer and a teacher. Following the design heuristics of Realistic Mathematics Education, we design a task (in ℝ 2) referring to specific tools and functions of GeoGebra. This paper reports the multimodal resources that attached to students’ reasoning in the reinvention of specific geometric linear transformations (like reflections according to the axes, projections onto axes and composition of reflections) in a dynamic geometry environment.