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The purpose of this study was to examine shifts in young children's learning progression levels while they interacted with virtual manipulative mathematics apps on touch-screen devices. A total of 100 children participated in six mathematics learning sequences while using 18 virtual manipulative mathematics touch-screen apps during clinical interviews. Researchers developed a micro-scoring tool to analyze video data from two camera sources (i.e., GoPro camera, wall-mounted camera). Our results showed that it is possible to document evidence of shifts in children's learning progressions while they are interacting with mathematics apps on touch-screen devices. Our results also indicated patterns in the children's interactions that were related to the shifts in their learning progression levels. These results suggest that an open-ended number of tasks with a variety of representations and tasks at varying levels of difficulty led to children refining their understanding and shaping their concept image of mathematical ideas resulting in incremental shifts in learning. The results of this study have important implications about how mathematical tasks in touch-screen apps may prompt children's incremental learning progression shifts to occur, and thereby promote opportunities for learning. We propose that design features in mathematics apps can be created to support and encourage these learning shifts.

In Humboldt County, California, nest exclosures were used between 2001 and 2006 to reduce the predation of eggs aiding in the recovery of a threatened population segment of western snowy plovers (Charadrius nivosus). Due to a sudden increase in adult mortality in 2006, field biologists abandoned the use of nest exclosures. This paper describes a discrete‐time stochastic model designed to compare two different management strategies (with and without exclosures) to predict the change in the plover population. The model uses beta distributions to model demographic parameters, and whenever possible, these distributions were fit to survey data of these populations. The model shows nest exclosures to be effective in increasing chick fledging rates. However, the model also shows that an increase in adult mortality potentially caused by the nest exclosures would counteract this increase in fledging rates. The model predicts that there will be a net negative effect on the population if these exclosures reduce adult survival to 90% of its unexclosed rate. The model also demonstrates that the population is dependent on immigration.