Visual Insight in Geometry

According to a traditional rationalist proposal, it is possible to attain knowledge of certain necessary truths by means of insight—an epistemic mental act that combines the 'presentational' character of perception with the a priori status usually reserved for discursive reasoning. In this dissertation, I defend the insight proposal in relation to a specific subject matter: elementary Euclidean plane geometry, as set out in Book I of Euclid's Elements. In particular, I argue that visualizations and visual experiences of diagrams allow human subjects to grasp truths of geometry by means of visual insight. In the first two chapters, I provide an initial defense of the geometrical insight proposal, drawing on a novel interpretation of Plato's Meno to motivate the view and to reply to some objections. In the remaining three chapters, I provide an account of the psychological underpinnings of geometrical insight, a task that requires considering the psychology of visual imagery alongside the details of Euclid's geometrical system. One important challenge is to explain how basic features of human visual representations can serve to ground our intuitive grasp of Euclid's postulates and other initial assumptions. A second challenge is to explain how we are able to grasp general theorems by considering diagrams that depict only special cases. I argue that both of these challenges can be met by an account that regards geometrical insight as based in visual experiences involving the combined deployment of two varieties of 'dynamic' visual imagery: one that allows the subject to visually rehearse spatial transformations of a figure's parts, and another that allows the subject to entertain alternative ways of structurally integrating the figure as a whole. It is the interplay between these two forms of dynamic imagery that enables a visual experience of a diagram, suitably animated in visual imagination, to justify belief in the propositions of Euclid’s geometry. The upshot is a novel dynamic imagery account that explains how intuitive knowledge of elementary Euclidean plane geometry can be understood as grounded in visual insight.

THE ROLE OF RULES, EXAMPLES AND INDIVIDUAL DIFFERENCES IN THE ACQUISITION OF DECLARATIVE AND PROCEDURAL SECOND LANGUAGE KNOWLEDGE

The relevance of explicit instruction has been well documented in SLA research. Despite numerous positive findings, however, the issue continues to engage scholars worldwide. One issue that was largely neglected in previous empirical studies - and one that may be crucial for the effectiveness of explicit instruction - is the timing and integration of rules and practice. The present study investigated the extent to which grammar explanation (GE) before practice, grammar explanation during practice, and individual differences impact the acquisition of L2 declarative and procedural knowledge of two grammatical structures in Spanish. In this experiment, 128 English-speaking learners of Spanish were randomly assigned to four experimental treatments and completed comprehension-based task-essential practice for interpreting object-verb (OV) and ser/estar (SER) sentences in Spanish. Results confirmed the predicted importance of timing of GE: participants who received GE during practice were more likely to develop and retain their knowledge successfully. Results further revealed that the various combinations of rules and practice posed differential task demands on the learners and consequently drew on language aptitude and WM to a different extent. Since these correlations between individual differences and learning outcomes were the least observed in the conditions that received GE during practice, we argue that the suitable integration of rules and practice ameliorated task demands, reducing the burden on the learner, and accordingly mitigated the role of participants’ individual differences. Finally, some evidence also showed that the comprehension practice that participants received for the two structures was not sufficient for the formation of solid productive knowledge, but was more effective for the OV than for the SER construction.