Browsing by Browse by FOR 2008 "010404 Probability Theory"

In this reflective paper, I explore the thinking of a group of pre-service teachers as they reason about experimental probability and theoretical probability. I am particularly interested in investigating whether pre-service teachers could construct a bidirectional link between the experimental probability and theoretical probability, similar to the tentative model I introduce elsewhere (2008) for coordinating the two perspectives on distribution. Overall, this research study contributes to understanding how pre-service students can build connections to help teachers conceptualize and support students to embrace elements that act as connections between the two approaches to probability.

This paper presents a case study of a group of students (age 14-15) as they use a computer-based domain of stochastic abstraction to begin to view spread or noise as dispersion from the signal. The results show that carefully designed computer tools, in which probability distribution is used as a generator of data, can facilitate the discrimination of signal and noise. This computational affordance of distribution is seen as related to classical statistical methods that aim to separate main effect from random error. In this study, we have seen how signal and noise can be recognised by students as an aspect of distribution. Students' discussion of computer-generated data and their sketches of the distribution express the idea that more variation is centred close to the signal, and less variation is located further away from it.

In this paper I present the background and intentions that have shaped the design of a microworld to study students' understanding of probability distributions. The outcome of this paper is the microworld design itself.

To eradicate or effectively contain a biological invasion, all or most reproductive individuals of the invasion must be found and destroyed. To help find individual invading organisms, predictions of probable locations can be made with statistical models. We estimated spread dynamics based on time-series data and then used model-derived predictions of probable locations of individuals. We considered one of the largest datasets available for an eradication program - the campaign to eradicate the red imported fire ant ('Solenopsis invicta') from around Brisbane, Australia. After estimating within-site growth (local growth) and inter-site dispersal (saltatory spread) of fire ant nests, we modeled probabilities of fire ant presence for > 600 000 1-ha sites, including uncertainties about fire ant population and spatial dynamics. Such a high level of spatial detail is required to assist surveillance efforts, but is difficult to incorporate into common modeling methods because of high computational costs. More than twice as many fire ant nests would have been found in 2008 using predictions made with our method rather than those made with the method currently used in the study region. Our method is suited to considering invasions in which a large area is occupied by the invader at low density. Improved predictions of such invasions can dramatically reduce the area that needs to be searched to find the majority of individuals, assisting containment efforts and potentially making eradication a realistic goal for many invasions previously thought to be ineradicable.

My premise, in line with a constructivist approach and Pratt's (1998) research, is that thinking about distribution must develop from causal meanings already established. The results of the third iteration of a design research study indicate support for my conjecture that it is possible to design an environment in which students' well established causal meanings can be exploited to coordinate the emergent data-centric and modelling perspectives on distribution (Prodromou & Pratt, 2006). In this study, I report on the fourth iteration that investigates how and whether students bridge the two perspectives on distribution.

Our primary goal is to design a microworld which aspires to research thinking-in-change about distribution. Our premise, in line with a constructivist approach and our prior research, is that thinking about distribution must develop from causal meanings already established. This study reports on a design research study of how students appear to exploit their appreciation of causal control to construct new situated meanings for the distribution of throws and success rates. We provided on-screen control mechanisms for average and spread that could be deterministic or subject to stochastic error. The students used these controls to recognise the limitations of causality in the short term but its power in making sense of the emergence of distributional patterns. We suggest that the concept of distribution lies in co-ordinating emergent data-centric and modelling perspectives for distribution and that causality may play a central role in supporting that co-ordination process.

'Aim' At first detection, little information is typically known about an invader's characteristics, true arrival date or spatial extent. Yet, before management options such as control or eradication can be considered, we need to know where a nuisance species has already spread. This is particularly difficult because of stochastic processes. Here, we develop an approach that requires little a 'priori' information, yet accurately delimits the range of a biological invader. 'Location' We used a simulated landscape, subjected to stochasticity inherent in establishment and spread, to test novel theory for delimiting locally spreading populations. 'Methods' We distinguish three stages to identify the boundary of an invasion, which we term Approach, Decline, Delimit (ADD). Our ADD algorithm uses general characteristics of the invasion pattern, obtained during a search for occupied sites, in combination with sampling and probability theory to delimit the invasion. We compare ADD against four naïve delimitation strategies, for long and normal dispersal kernels. 'Results' Our results illustrate the potential difficulty in delimiting invasions. Naïve strategies, such as stopping when the invader is absent, typically failed to properly delimit the invasion. In contrast, ADD operated relatively efficiently, and was robust to habitat heterogeneity and knowledge of the true epicentre, but was sensitive to the sparseness of the invasion. For long-distance dispersal kernels, ADD had 80% accurate delimitations when 'c'. 5% or more of the cells were occupied within the invasion boundary; for normal dispersal kernels, ADD had 95% accurate delimitations when 'c'. 2.5% or more of the cells were occupied. 'Main conclusions' There is virtually no existing theory for delimiting invasions. ADD is efficient and accurate, even with unknown time of invasion, unknown dispersal kernels, stochastic establishment dynamics and spatial heterogeneity, except for very low invasion densities.

This paper presents a case study of two students aged 14-15, as they attempt to make sense of distribution, adopting a range of causal meanings for the variation observed in the animated computer display and in the graphs generated by the simulation. The students' activity is analysed through dimensions of complex causality. The results indicate support for our conjecture that carefully designed computer simulations can offer new ways for harnessing causality to facilitate students' meaning-making for variation in distributions of data. In order to bridge the deterministic and the stochastic, the students transfer agency to specially designed active representations of distributional parameters, such as average and speed.