Young children's statistical reasoning : a tale of two contexts

This thesis explored the knowledge and reasoning of young children in solving novel statistical problems, and the influence of problem context and design on their solutions. It found that young children's statistical competencies are underestimated, and that problem design and context facilitated children's application of a wide range of knowledge and reasoning skills, none of which had been taught. A qualitative design-based research method, informed by the Models and Modeling perspective (Lesh & Doerr, 2003) underpinned the study. Data modelling activities incorporating picture story books were used to contextualise the problems. Children applied real-world understanding to problem solving, including attribute identification, categorisation and classification skills. Intuitive and metarepresentational knowledge together with inductive and probabilistic reasoning was used to make sense of data, and beginning awareness of statistical variation and informal inference was visible.

Numeracy and statistical reasoning on entering university

The relationship between mathematics and statistical reasoning frequently receives comment (Vere-Jones 1995, Moore 1997); however most of the research into the area tends to focus on mathematics anxiety. Gnaldi (2003) showed that in a statistics course for psychologists, the statistical understanding of students at the end of the course depended on students’ basic numeracy, rather than the number or level of previous mathematics courses the student had undertaken. As part of a study into the development of statistical thinking at the interface between secondary and tertiary education, students enrolled in an introductory data analysis subject were assessed regarding their statistical reasoning, basic numeracy skills, mathematics background and attitudes towards statistics. This work reports on some key relationships between these factors and in particular the importance of numeracy to statistical reasoning.

Numeracy counts in the statistical reasoning equation

The relationship between mathematics and statistical reasoning frequently receives comment (Vere-Jones 1995, Moore 1997); however most of the research into the area tends to focus on maths anxiety. Gnaldi (Gnaldi 2003) showed that in a statistics course for psychologists, the statistical understanding of students at the end of the course depended on students’ basic numeracy, rather than the number or level of previous mathematics courses the student had undertaken. As part of a study into the development of statistical thinking at the interface between secondary and tertiary education, students enrolled in an introductory data analysis subject were assessed regarding their statistical reasoning ability, basic numeracy skills and attitudes towards statistics. This work reports on the relationships between these factors and in particular the importance of numeracy to statistical reasoning.

When does information about causal structure improve statistical reasoning?

Base rate neglect on the mammography problem can be overcome by explicitly presenting a causal basis for the typically vague false-positive statistic. One account of this causal facilitation effect is that people make probabilistic judgements over intuitive causal models parameterized with the evidence in the problem. Poorly defined or difficult-to-map evidence interferes with this process, leading to errors in statistical reasoning. To assess whether the construction of parameterized causal representations is an intuitive or deliberative process, in Experiment 1 we combined a secondary load paradigm with manipulations of the presence or absence of an alternative cause in typical statistical reasoning problems. We found limited effects of a secondary load, no evidence that information about an alternative cause improves statistical reasoning, but some evidence that it reduces base rate neglect errors. In Experiments 2 and 3 where we did not impose a load, we observed causal facilitation effects. The amount of Bayesian responding in the causal conditions was impervious to the presence of a load (Experiment 1) and to the precise statistical information that was presented (Experiment 3). However, we found less Bayesian responding in the causal condition than previously reported. We conclude with a discussion of the implications of our findings and the suggestion that there may be population effects in the accuracy of statistical reasoning.

Whose statistical reasoning is improved by information about causal structure?

People often struggle when making Bayesian probabilistic estimates on the basis of competing sources of statistical evidence. Recently, Krynski and Tenenbaum (Journal of Experimental Psychology: General, 136, 430–450, 2007) proposed that a causal Bayesian framework accounts for peoples’ errors in Bayesian reasoning and showed that, by clarifying the causal relations among the pieces of evidence, judgments on a classic statistical reasoning problem could be significantly improved. We aimed to understand whose statistical reasoning is facilitated by the causal structure intervention. In Experiment 1, although we observed causal facilitation effects overall, the effect was confined to participants high in numeracy. We did not find an overall facilitation effect in Experiment 2 but did replicate the earlier interaction between numerical ability and the presence or absence of causal content. This effect held when we controlled for general cognitive ability and thinking disposition. Our results suggest that clarifying causal structure facilitates Bayesian judgments, but only for participants with sufficient understanding of basic concepts in probability and statistics.

Introduction to statistical reasoning, a set of notes.

Mimeographed.

Reconceptualizing statistical learning in the early years

This chapter argues for the need to restructure children’s statistical experiences from the beginning years of formal schooling. The ability to understand and apply statistical reasoning is paramount across all walks of life, as seen in the variety of graphs, tables, diagrams, and other data representations requiring interpretation. Young children are immersed in our data-driven society, with early access to computer technology and daily exposure to the mass media. With the rate of data proliferation have come increased calls for advancing children’s statistical reasoning abilities, commencing with the earliest years of schooling (e.g., Langrall et al. 2008; Lehrer and Schauble 2005; Shaughnessy 2010; Whitin and Whitin 2011). Several articles (e.g., Franklin and Garfield 2006; Langrall et al. 2008) and policy documents (e.g., National Council of Teachers ofMathematics 2006) have highlighted the need for a renewed focus on this component of early mathematics learning, with children working mathematically and scientifically in dealing with realworld data. One approach to this component in the beginning school years is through data modelling (English 2010; Lehrer and Romberg 1996; Lehrer and Schauble 2000, 2007)...

Promoting Statistical Literacy Through Data Modelling in the Early School Years

This chapter addresses data modelling as a means of promoting statistical literacy in the early grades. Consideration is first given to the importance of increasing young children’s exposure to statistical reasoning experiences and how data modelling can be a rich means of doing so. Selected components of data modelling are then reviewed, followed by a report on some findings from the third-year of a three-year longitudinal study across grades one through three.

Reasoning paradigms in legal decision support systems

In this paper we discuss the strengths and weaknesses of a range of artificial intelligence approaches used in legal domains. Symbolic reasoning systems which rely on deductive, inductive and analogical reasoning are described and reviewed. The role of statistical reasoning in law is examined, and the use of neural networks analysed. There is discussion of architectures for, and examples of, systems which combine a number of these reasoning strategies. We conclude that to build intelligent legal decision support systems requires a range of reasoning strategies.

Young children's early modelling with data

An educational priority of many nations is to enhance mathematical learning in early childhood. One area in need of special attention is that of statistics. This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling activities. Such modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (i.e., identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Results are reported from the first year of a three-year longitudinal study in which three classes of first-grade children and their teachers engaged in activities that required the creation of data models. The theme of “Looking after our Environment,” a component of the children’s science curriculum at the time, provided the context for the activities. Findings focus on how the children dealt with given complex attributes and how they generated their own attributes in classifying broad data sets, and the nature of the models the children created in organising, structuring, and representing their data.

Data modelling in the beginning school years

This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling. Some of the core components of data modelling are addressed. A selection of results from the first data modelling activity implemented during the second year (2010; second grade) of a current longitudinal study are reported. Data modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Reported here are children's abilities to identify diverse and complex attributes, sort and classify data in different ways, and create and interpret models to represent their data.

Data modeling in elementary and middle school classes : a shared experience

This paper argues for a renewed focus on statistical reasoning in the elementary school years, with opportunities for children to engage in data modeling. Data modeling involves investigations of meaningful phenomena, deciding what is worthy of attention, and then progressing to organizing, structuring, visualizing, and representing data. Reported here are some findings from a two-part activity (Baxter Brown’s Picnic and Planning a Picnic) implemented at the end of the second year of a current three-year longitudinal study (grade levels 1-3). Planning a Picnic was also implemented in a grade 7 class to provide an opportunity for the different age groups to share their products. Addressed here are the grade 2 children’s predictions for missing data in Baxter Brown’s Picnic, the questions posed and representations created by both grade levels in Planning a Picnic, and the metarepresentational competence displayed in the grade levels’ sharing of their products for Planning a Picnic.

Complex modelling in the primary/middle school years

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by continuing education as usual (Katehi, Pearson, & Feder, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualisation. These technologies have led to significant changes in the forms of mathematical and scientific thinking that are required beyond the classroom. Modelling, in its various forms, can develop and broaden children’s mathematical and scientific thinking beyond the standard curriculum. This paper first considers future competencies in the mathematical sciences within an increasingly complex world. Next, consideration is given to interdisciplinary problem solving and models and modelling. Examples of complex, interdisciplinary modelling activities across grades are presented, with data modelling in 1st grade, model-eliciting in 4th grade, and engineering-based modelling in 7th-9th grades.

Data modelling with first-grade students

This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling. Results are reported from the first year of a 3-year longitudinal study in which three classes of first-grade children (6-year-olds) and their teachers engaged in data modelling activities. The theme of Looking after our Environment, part of the children’s science curriculum, provided the task context. The goals for the two activities addressed here included engaging children in core components of data modelling, namely, selecting attributes, structuring and representing data, identifying variation in data, and making predictions from given data. Results include the various ways in which children represented and re represented collected data, including attribute selection, and the metarepresentational competence they displayed in doing so. The “data lenses” through which the children dealt with informal inference (variation and prediction) are also reported.