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    Theoretical models of the role of visualisation in learning formal reasoning

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    Although there is empirical evidence that visualisation tools can help students to learn formal subjects such as logic, and although particular strategies and conceptual difficulties have been identified, it has so far proved difficult to provide a general model of learning in this context that accounts for these findings in a systematic way. In this paper, four attempts at explaining the relative difficulty of formal concepts and the role of visualisation in this learning process are presented. These explanations draw on several existing theories, including Vygotsky's Zone of Proximal Development, Green's Cognitive Dimensions, the Popper-Campbell model of conjectural learning, and cognitive complexity. The paper concludes with a comparison of the utility and applicability of the different models. It is also accompanied by a reflexive commentary[0] (linked to this paper as a hypertext) that examines the ways in which theory has been used within these arguments, and which attempts to relate these uses to the wider context of learning technology research

    Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level

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    Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v: E -> R \ {0}, and let (phi(t))(t >= 0) be defined by phi(t) = integral(0)(t) v(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) hits level y before hitting 0 and prove weak convergence of the conditioned process as y -> infinity. In addition, we show the relationship between the conditioning of the process (phi(t))(t >= 0) with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (phi(t))(t >= 0) with a negative drift to drift to infinity and the conditioning of it to hit large levels before hitting 0
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