Awareness of number in children with severe and profound learning difficulties: three exploratory case studies

This paper reports on exploratory work investigating how children with severe and profound learning difficulties register an awareness of small quantities and how they might use this information to inform their understanding. It draws on studies of typically developing children and investigates their application to pupils whose response to conventional mathematical tasks are often limited because they lack relevance and interest. The responses of the three pupils to individualized learning contexts mirror the progression suggested in the literature, namely from awareness of number to simple actions using number cues to problem-solving behaviour

Digitisation, representation, and formalisation - Digital libraries of mathematics

One of the main tasks of the mathematical knowledge management community must surely be to enhance access to mathematics on digital systems. In this paper we present a spectrum of approaches to solving the various problems inherent in this task, arguing that a variety of approaches is both necessary and useful. The main ideas presented are about the differences between digitised mathematics, digitally represented mathematics and formalised mathematics. Each has its part to play in managing mathematical information in a connected world. Digitised material is that which is embodied in a computer file, accessible and displayable locally or globally. Represented material is digital material in which there is some structure (usually syntactic in nature) which maps to the mathematics contained in the digitised information. Formalised material is that in which both the syntax and semantics of the represented material, is automatically accessible. Given the range of mathematical information to which access is desired, and the limited resources available for managing that information, we must ensure that these resources are applied to digitise, form representations of or formalise, existing and new mathematical information in such a way as to extract the most benefit from the least expenditure of resources. We also analyse some of the various social and legal issues which surround the practical tasks.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Changing state disruption of lip-reading by irrelevant sound in perceptual and memory tasks

Three experiments investigated irrelevant sound interference of lip-read lists. In Experiment 1, an acoustically changing sequence of nine irrelevant utterances was more disruptive to spoken immediate identification of lists of nine lip-read digits than nine repetitions of the same utterances (the changing-state effect; Jones, Madden, & Miles, 1992). Experiment 2 replicated this finding when lip-read items were sampled with replacement from the nine digits to form the lip-read lists. In Experiment 3, when the irrelevant sound was confined to the retention interval of a delayed recall task, a changing-state pattern of disruption also occurred. Results confirm a changing-state effect in memory for lip-read items but also point to the possibility that, for lip-reading, changing-state effects may occur at an earlier, perceptual stage.

The influence of affective statements on performance on implicit and explicit memory tasks

Consumers are often exposed to brand names presented concurrently with information such as brand claims and warnings, prior to making product choices. As implicit memory has been implicated in the consumer choice process, two experiments were conducted to assess the influence of additional information, namely simple positive and negative statements, on implicit and explicit memory tasks. In Experiment 1, the mere presence of additional statements at study significantly reduced performance on a two-alternatives forced recognition task. However, the same manipulation had no effect on implicit preference judgement task performance. The valence of the accompanying information had no effect on either task. In Experiment 2, using modified implicit and explicit choice tasks, performance on the former was unaffected by statement valence, whereas the latter task was. Hence, positive priming was obtained for brand names previously shown with negative information. The results of the experiments are discussed in relation to both implicit memory and consumer choice.

Negative correlations between control performance and verbalizable knowledge: indicators for implicit learning in process control tasks?

Negative correlations between task performance in dynamic control tasks and verbalizable knowledge, as assessed by a post-task questionnaire, have been interpreted as dissociations that indicate two antagonistic modes of learning, one being “explicit”, the other “implicit”. This paper views the control tasks as finite-state automata and offers an alternative interpretation of these negative correlations. It is argued that “good controllers” observe fewer different state transitions and, consequently, can answer fewer post-task questions about system transitions than can “bad controllers”. Two experiments demonstrate the validity of the argument by showing the predicted negative relationship between control performance and the number of explored state transitions, and the predicted positive relationship between the number of explored state transitions and questionnaire scores. However, the experiments also elucidate important boundary conditions for the critical effects. We discuss the implications of these findings, and of other problems arising from the process control paradigm, for conclusions about implicit versus explicit learning processes.