Abstract
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. Following the design heuristics of Realistic Mathematics Education, we design a task (in ℝ2) referring to specific tools and functions of GeoGebra. Task-based interviews were conducted with a pair of linear algebra students, by way of a computer and a teacher. Data was collected using a video camera observing the students’ working environment, screen recorder software, student production and field notes. 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. According to the findings, the artefact use, verbal and written mathematical expressions all interlaced with the emergence of gestures. The students mostly gestured when they faced a new reflection situation and when describing associated geometric actions. Finally, a shared environment with action, production and communication conveyed student reasoning and they managed to reinvent a number of geometric linear transformations.
Highlights
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
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 R2, could be a heuristic tool for establishing the link between GLT and functions
To sum up, adopting both Realistic Mathematics Education (RME) and APC space perspectives, in this paper, we investigate two interrelated research questions: (1) Which multimodal resources emerge while students solve a GLT task in a dynamic geometry environment (DGE)? (2) Which reasoning steps do students follow while reinventing GLT?
Summary
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. 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. 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 R2, could be a (twofold) heuristic tool for establishing the link between GLT and functions. A composition of reflections (as well as compositions of geometric transformations) on the Cartesian plane could enable students to construct a link between the composition of functions and the composition of matrix transformations, and could provide a better understanding of the link between a matrix transformation and a function
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Science and Mathematics Education
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
