Week 2 - Reading Summary

 Fernandez, S. & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind students' interactions with Symmetry

This article describes a quantitative study that explores the experiences of two blind students through multiple experiences with symmetry and geometric translation.

The authors review and critique different theories and conceptions of the body and cognition, including rationalists (e.g., Descartes) and empiricists (e.g., Locke and Diderot) and explain their adoption of the understandings of Merleau-Ponty’s phenonmenology and its alignment to current work in embodied learning. The study is based in the perception that experiences are stored as “multimodal representations” that are “reactivated to simulate states of perception, action and introspection associated with an object, even when it is not present” (p. 40). The authors recognize that neuroscience can be used to support findings of phenomenological studies.

Additionally, the authors draw upon Piaget and Garcia’s stages of development of geometry in learners: intrafigural, interfigural, and transfigural. In the intrafigural stage, learners focus on the properties within a given figure or structure, the internal features. When working in an interfigural stage, learners draw upon references outside of the figure or structure, such as an axis of reflection, and compare separate parts of the figure to this reference point. The final stage, transfigural, sees figures or structures that result from transformations as “objects in their own right” (p. 41). These stages develop linearly and recursively. In this study, the authors reference intrafigural and interfigural stages in relation to the participants.

The participants were two blind students, Edson who had visual memories having lost his sight at the age of 15, and Lucas who lost his sight at the age of two and claimed no visual memories. Neither student had been taught about symmetry or geometric reflections. The method used is based on Vygotsky’s double stimulation where subjects were given tasks beyond their current level of knowledge but also provided assistance from the researcher to develop strategies for the tasks. The tasks in the study were designed to develop a sense of symmetry through folding cardboard geometric figures and reflecting figures across an axis of symmetry using a geoboard. Data was in the form of transcribed interviews with the participants as they worked, including the assistance the researcher provided.

Results of the study were discussed in relation to Piaget and Garcia’s stages of geometric development and to embodied cognition. The authors found that in the task of folding cardboard, the participants worked in the intrafigural stage, attending to how points on the figure related to each other, not to a point of reference outside of the figure. In this task, particpants also drew upon experiences; Edson made reference to folding clothes and the authors were clear that both participants used multimodal images developed within the task as they moved to a new iteration. They also note that while the participants developed a symbolic representation, it did not replace the physical memories of experiences. In the geoboard activities, the authors claim that the participants changed their multimodal resources and were able to work in the interfigural stage. Edson had visual memories of mirrors that aided his work with the assistance of the researcher in pointing out this set of resources he had. Lucas, with no visual memories to draw upon, drew upon guidance from the researcher to complete initial tasks which he then was able to use, in conjunction with his experiences on previous tasks, to complete further tasks. The researchers used their findings to question the linearity of Piaget and Garcia’s stages, suggesting that what stage learners use depends on the resources available. The article concludes with stating that this research helps confirm the link between the body and mathematical abstraction.