What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

We present a study in which mathematicians and undergraduate students were asked to explain in writing what mathematicians mean by proof. The 175 responses were evaluated using comparative judgement: mathematicians compared pairs of responses and their judgements were used to construct a scaled rank order. We provide evidence establishing the reliability, divergent validity and content validity of this approach to investigating individuals’ written conceptions of mathematical proof. In doing so, we compare the quality of student and mathematician responses and identify which features the judges collectively valued. Substantively, our findings reveal that despite the variety of views in the literature, mathematicians broadly agree on what people should say when asked what mathematicians mean by proof. Methodologically, we provide evidence that comparative judgement could have an important role to play in investigating conceptions of mathematical ideas, and conjecture that similar methods could be productive in evaluating individuals’ more general (mathematical) beliefs

Coastal proximity, health and well-being: results from a longitudinal panel survey.

This is the author's pre print version of an article subsequently published in Health Place. The definitive published version is available from the DOI provided in this record. Copyright © 2013 Elsevier Ltd. All rights reserved.Analysis of English census data revealed a positive association between self-reported health and living near the coast. However that analysis was based on cross-sectional data and was unable to control for potential selection effects (e.g. generally healthier, personality types moving to coastal locations). In the current study we have used English panel data to explore the relationship between the proximity to the coast and indicators of generic and mental health for the same individuals over time. This allowed us to control for both time-invariant factors such as personality and compare the strength of any relationship to that of other relationships (e.g. employment vs. unemployment). In support of cross-sectional analysis, individuals reported significantly better general health and mental health when living nearer the coast, controlling for both individual (e.g. employment status) and area (e.g. green space) level factors. No coastal effect on life satisfaction was found. Although individual level coastal proximity effects for general health and mental health were small, their cumulative impact at the community level may be meaningful for policy makers.European Regional Development FundEuropean Social Fun

Coastal proximity and physical activity: Is the coast an under-appreciated public health resource?

This is the author's pre print version of an article subsequently published in Preventive Medicine. The definitive published version is available from the DOI provided in this record. Copyright © 2014 Published by Elsevier Inc.BACKGROUND: Recent findings suggest that individuals living near the coast are healthier than those living inland. Here we investigated whether this may be related to higher levels of physical activity among coastal dwellers in England, arising in part as a result of more visits to outdoor coastal settings. METHOD: Participants (n=183,755) were drawn from Natural England's Monitor of Engagement with the Natural Environment Survey (2009-2012). Analyses were based on self-reported physical activity for leisure and transport. RESULTS: A small, but significant coastal proximity gradient was seen for the likelihood of achieving recommended guidelines of physical activity a week after adjusting for relevant area and individual level controls. This effect was statistically mediated by the likelihood of having visited the coast in the last seven days. Stratification by region, however, suggested that while the main effect was relatively strong for west coast regions, it was not significant for those in the east. CONCLUSIONS: In general, our findings replicate and extend work from Australia and New Zealand. Further work is needed to explain the marked regional differences in the relationship between coastal proximity and physical activity in England to better understand the coast's potential role as a public health resource.Economic and Social Research CouncilNational Institute for Health Research Health Protection Research Unit (NIHR HPRU

A Proper Motion Survey for White Dwarfs with the Wide Field Planetary Camera 2

We have performed a search for halo white dwarfs as high proper motion objects in a second epoch WFPC2 image of the Groth-Westphal strip. We identify 24 high proper motion objects with mu > 0.014 ''/yr. Five of these high proper motion objects are identified as strong white dwarf candidates on the basis of their position in a reduced proper motion diagram. We create a model of the Milky Way thin disk, thick disk and stellar halo and find that this sample of white dwarfs is clearly an excess above the < 2 detections expected from these known stellar populations. The origin of the excess signal is less clear. Possibly, the excess cannot be explained without invoking a fourth galactic component: a white dwarf dark halo. We present a statistical separation of our sample into the four components and estimate the corresponding local white dwarf densities using only the directly observable variables, V, V-I, and mu. For all Galactic models explored, our sample separates into about 3 disk white dwarfs and 2 halo white dwarfs. However, the further subdivision into the thin and thick disk and the stellar and dark halo, and the subsequent calculation of the local densities are sensitive to the input parameters of our model for each Galactic component. Using the lowest mean mass model for the dark halo we find a 7% white dwarf halo and six times the canonical value for the thin disk white dwarf density (at marginal statistical significance), but possible systematic errors due to uncertainty in the model parameters likely dominate these statistical error bars. The white dwarf halo can be reduced to around 1.5% of the halo dark matter by changing the initial mass function slightly. The local thin disk white dwarf density in our solution can be made consistent with the canonical value by assuming a larger thin disk scaleheight of 500 pc.Comment: revised version, accepted by ApJ, results unchanged, discussion expande

Optical Gravitational Lensing Experiment. OGLE-1999-BUL-19: The First Multi-Peak Parallax Event

We describe a highly unusual microlensing event, OGLE-1999-BUL-19, which exhibits multiple peaks in its light curve. The Einstein radius crossing time for this event is approximately one year, which is unusually long. We show that the motion of the Earth induces these multiple peaks in the light curve, since the relative transverse velocity of the lens projected into the observer plane is very small (v = 12.5 km/s). This is the lowest velocity so far published and we believe that this is the first multiple-peak parallax event ever observed. We also believe that this event may be exhibiting slight binary-source signatures in addition to these parallax-induced multiple peaks. With spectroscopic observations it is possible to test this `parallax plus binary-source' hypothesis and (if this hypothesis turns out to be correct) to simultaneously fit both models and obtain a measurement of the lens mass. Furthermore, spectroscopic observations could also supply information regarding the lens properties, possibly providing another avenue for determining the lens mass. We found that most of the I-band blending is probably caused by light from the lens or a binary companion to the source. However, in the V-band, there appears to be a second blended source 0.35" away from the lensed source. HST observations will be very useful for understanding the nature of the blends. We also suggest that a radial velocity survey of all parallax events will be very useful for further constraining the lensing kinematics and understanding the origins of these events and the excess of long events toward the bulge.Comment: 36 pages, 7 figures. Accepted for publication in MNRA

Should Native Pastures Be Fertilised?

Considerable research has been conducted in recent years demonstrating the ongoing benefits of phosphorus (P) fertiliser in achieving and maintaining higher levels of production from native pastures (Graham 2006; Garden et al. 2003; Lodge et al. 2003; Michalk et al. 2003). However Garden et al. (2003), Hill et al. (2004) and Dorrough et al. (2008) all question the sustainability of this practice by highlighting changes in species composition of these pastures with regard to increasing the introduced annual grass and broadleaf weed components at the expense of the native grasses and forbs but in most cases, these trends have been based on short term replicated data or based on qualitative survey data incapable of establishing causal relationships. In the Monaro region of NSW, 70% of pastures are based on native species and are the cornerstone of livestock production. The Monaro Research, Development and Demonstration of Sustainable Grassland Management Project (MGP) was designed to determine if productivity could be increased without compromising the composition and bio-diversity values of these pastures

Associations between pro-environmental behaviour and neighbourhood nature, nature visit frequency and nature appreciation: Evidence from a nationally representative survey in England

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