The development of a semantic model for the interpretation of mathematics including the use of technology

The semantic model developed in this research was in response to the difficulty a group of mathematics learners had with conventional mathematical language and their interpretation of mathematical constructs. In order to develop the model ideas from linguistics, psycholinguistics, cognitive psychology, formal languages and natural language processing were investigated. This investigation led to the identification of four main processes: the parsing process, syntactic processing, semantic processing and conceptual processing. The model showed the complex interdependency between these four processes and provided a theoretical framework in which the behaviour of the mathematics learner could be analysed. The model was then extended to include the use of technological artefacts into the learning process. To facilitate this aspect of the research, the theory of instrumentation was incorporated into the semantic model. The conclusion of this research was that although the cognitive processes were interdependent, they could develop at different rates until mastery of a topic was achieved. It also found that the introduction of a technological artefact into the learning environment introduced another layer of complexity, both in terms of the learning process and the underlying relationship between the four cognitive processes.

Measurements of generalisation based on information geometry

Neural networks are statistical models and learning rules are estimators. In this paper a theory for measuring generalisation is developed by combining Bayesian decision theory with information geometry. The performance of an estimator is measured by the information divergence between the true distribution and the estimate, averaged over the Bayesian posterior. This unifies the majority of error measures currently in use. The optimal estimators also reveal some intricate interrelationships among information geometry, Banach spaces and sufficient statistics.

Are poor mathematics skills associated with visual deficits in temporal processing?

Developmental learning disabilities such as dyslexia and dyscalculia have a high rate of co-occurrence in pediatric populations, suggesting that they share underlying cognitive and neurophysiological mechanisms. Dyslexia and other developmental disorders with a strong heritable component have been associated with reduced sensitivity to coherent motion stimuli, an index of visual temporal processing on a millisecond time-scale. Here we examined whether deficits in sensitivity to visual motion are evident in children who have poor mathematics skills relative to other children of the same age. We obtained psychophysical thresholds for visual coherent motion and a control task from two groups of children who differed in their performance on a test of mathematics achievement. Children with math skills in the lowest 10% in their cohort were less sensitive than age-matched controls to coherent motion, but they had statistically equivalent thresholds to controls on a coherent form control measure. Children with mathematics difficulties therefore tend to present a similar pattern of visual processing deficit to those that have been reported previously in other developmental disorders. We speculate that reduced sensitivity to temporally defined stimuli such as coherent motion represents a common processing deficit apparent across a range of commonly co-occurring developmental disorders.

Modelling the underlying principles of human aesthetic preference in evolutionary art

Our understanding of creativity is limited, yet there is substantial research trying to mimic human creativity in artificial systems and in particular to produce systems that automatically evolve art appreciated by humans. We propose here to study human visual preference through observation of nearly 500 user sessions with a simple evolutionary art system. The progress of a set of aesthetic measures throughout each interactive user session is monitored and subsequently mimicked by automatic evolution in an attempt to produce an image to the liking of the human user.