Young Children and Statistical Reasoning

Studies have highlighted the strength and vulnerability of young children’s use of their life experiences in reasoning and reaching decisions. Young children have the capacity to intuit and reason in the absence of systematic instruction but they are also prone to be bound by their beliefs, interpretations and take a subjective approach to problem solving (Nikiforidou & Pange, 2009, 2010). Children’s use of intuition has received some close attention because of the impact it has on reasoning and because of the notion that for young children, reliance on the appearance of things takes precedence over logic and objective fact (Bjorklund, 2011). Intuitive understanding exists in many domains and across disciplinary competencies, and for children, is about using what they already know. Under constructivist approaches to teaching and learning, intuition should be considered as a starting point for further learning (Gelman & Brenneman, 2004). Intuition is also associated with creative thinking, as it is a quality that uses insight “to reach sound conclusions from minimal evidence” (Fisher, 2005, p. 26). The use of intuition is also an important factor in abductive reasoning and the generation of hypotheses in statistical reasoning.

56 Intuitive capacity is therefore an essential element in statistical learning which needs to be preserved and nurtured.

Early capacity for inferential reasoning in statistics is a focus of recent statistics education research. Young children can engage with and intuit about statistical inference in appropriately supportive environments and where there is a range of possible data available to draw from (Paparistodemou & Meletiou- Mavrotheris, 2010). Inferential reasoning includes considering the probabilistic language that young children may use to explain and reason about data (Makar & Rubin, 2009). Children’s probabilistic language is a means of expressing uncertainty about the inferences they are drawing from available evidence. The link between context and the language of explanation is an important indicator of the way that the data may be connected to the creative generation of “a tentative hypothesis” (Makar & Rubin, 2009, p. 87). Makar and Rubin further emphasise the critical nature of context in moving thinking towards generalising or looking “beyond the data”. Inference is the means of moving thinking and reasoning beyond the description of the immediate data to hand to the wider context in which the data have been generated.

2.5.2.1 Knowledge and reasoning.

The relationship between knowledge and reasoning in statistical learning for young children is underplayed. Lohman (2005) laments that despite the relationship between levels of reasoning and prior knowledge “the contributions of knowledge to reasoning are often ignored” (p. 228). The ability to classify both events and objects is present from early infancy and is essential to reasoning (Lehrer & Schauble, 2000). Seeing relationships between events and objects, being aware of and working out what is important is determined by one’s knowledge base. A young child’s ability to perceive similarity and difference provides the foundation for the ability to form judgments (Lipman, 2003). Judgements about relationships, connections and distinctions allow comparisons to be made and form an ensemble that are requisite thinking skills needed for concept formation and all other reasoning (Lipman, 2003). Differences in reasoning in individuals however, may result from limitations in both experience and knowledge, reducing the ability to know what knowledge is

57 Watters, 1998). The specificity, breadth and type of knowledge bases available determine young children’s analogical reasoning abilities, including conceptual knowledge (English, 2004).

Young children’s knowledge bases include informal knowledge drawn from everyday experiences and interaction with the world. Prior to entering school, children undertake significant informal mathematical learning and possess powerful mathematical ideas as they begin formal schooling (Perry & Dockett, 2008).

Informal mathematical knowledge acts as a starting point and underpins the learning of formal mathematical knowledge (Zeiffler, Garfield, delMas, & Reading, 2008). Further, because a core component of statistics is its grounding in data context, everyday knowledge that people possess can interfere with the use of data-based evidence and the types of connections and relationships that are made when working statistically (delMas, 2004; Garfield & Ben-Zvi, 2007). The potential impact of everyday knowledge on statistical reasoning places additional emphasis on a need to understand the prior-to-school knowledge young children bring to their early

statistical experiences. delMas (2004) argues that reasoning from everyday

knowledge can produce errors in thinking and reasoning that are difficult to change and yet the area of research into understanding statistical reasoning is one of the most neglected. Therefore, there is a need for studies that probe for “an understanding of the processes and mental structures that support both erroneous and correct statistical reasoning” (delMas, 2004, p. 92) which this study aimed to explore.

Young children’s beliefs are the source of their theories about how the world is and how it operates. Masnick, Klahr, and Morris (2007) saw the importance of young children’s beliefs in statistical reasoning, arguing that engaging in reasoning is where theory, knowledge and data interact. The theories and knowledge that children hold about data impact on how they reason, including children’s use of statistically specific knowledge and their searching for patterns as they consider data.

Considering young children’s knowledge is also important when distinguishing between mathematics and statistics influences and in defining statistical knowledge. The difference in reasoning processes distinguishes and shapes statistical thinking (Shaughnessy & Pfannkuch, 2002; Burgess, 2009). The continual interaction between statistical knowledge, data context knowledge and knowledge of the data

58 plays a critical role in statistical problem solving (Wild & Pfannkuch, 1999).

Statistical knowledge is intimately tied with notions of statistical literacy, statistical thinking and statistical reasoning, although each of these three complex terms are contested for what defines them and their importance in shaping statistical learning foci (e.g.,; Budgett & Pfannkuch 2010; Chance, 2002; Gal, 2002, 2005; Garfield & Ben-Zvi, 2008; Gould, 2010; Wallman, 1993; Watson, 2009).

Gaining understanding of what knowledge young children bring to statistical problem solving is important. Young children beginning school do not have formal statistical knowledge and their life experience is limited by their age, criteria that can constrain their assessment of data properties, data patterns and forming data

expectations (Masnick, Klahr, & Morris, 2007). Wild and Pfannkuch (1999) propose five types of statistical thinking, with the fifth being the integration of statistical and contextual knowledge, information and concepts. Arguing for this category of thinking, they state:

one has to bring to bear all relevant knowledge, regardless of the source, on the task in hand, and then to make connections between existing context-knowledge and the results of the analysis to arrive at meaning. (p. 228)

Wild and Pfannkuch suggest that statistical knowledge is the relevant knowledge brought to statistical problem solving, that is, knowledge used to handle and make sense of data. Approaching statistical knowledge as knowledge that is relevant to data handling provides an entry point for exploring the knowledge young children bring to statistical problem solving. Defining statistical knowledge for the purposes of this study draws on Wild and Pfannkuch’s work and is stated as knowledge

children bring to judgments they make or actions they take to make statistical sense of data.

Young children’s real world context knowledge and beliefs are a major influence in how they reason with data that can impact on how they resolve data that contradicts it or falls outside their sphere of experience. Young children for example, usually see data as an isolated incident, rather than within the context of a

distribution (Sheaffer, 2002). Masnick, Klahr, and Morris (2007) considered that although children may recognise characteristics of and variations in data, in order to shift deeply held views and knowledge about the world, inconsistencies between pre-

59 existing knowledge and objective evidence may need to be both sizeable and

consistent. Data based explanations draw out both contextual and statistical

knowledge (Gil & Ben-Zvi, 2011), and yet the interplay between context knowledge and handling data is under researched, as is the role of context in how statistics is learned (Langrall, 2010). Metz (1998) comments that knowledge of “the key ideas that children bring to instruction is particularly important in a domain as complex as statistics and probability” (p. 150). Although there has been research on the effect of context on deductive reasoning, research on statistical context and reasoning has been limited (Schwartz & Goldman, 1996). This study aimed to explore the knowledge young children brought to statistical problems solving and the reasons young children revealed as they made statistical sense of data. The study specifically aimed to examine young children’s use of pre-existing or prior-to-school knowledge and knowledge of the data context when making data handling decisions and the inductive reasoning used as data handling decisions were made.

2.5.2.2 Context and reasoning.

Making sense of context has been described as central to statistical literacy (Chick & Pierce, 2012) because the very essence of a statistics problem is the context it is embedded in. Statistical reasoning processes are shaped by contact between the data context and the collected data when finding a solution. The ability to form conceptual and evidenced connections to, and reason creatively from data with context in mind is where hypotheses, predictions, inferences and new knowledge are made (Ben-Zvi, Maker, & Bakker, 2011; Pfannkuch, 2011).

The importance of reasoning with context in statistical problem solving is found in research that first, confirms the close and critical interaction between the context of data and statistical reasoning (e.g., Langrall et al., 2011; Moore, 1990; Watson & Callingham, 2003; Wild & Pfannkuch, 1999) and second, highlights that reasoning abilities, including the ability to make connections, may be impacted by knowledge bases children have to draw on (Diezmann & Watters, 1998; English, 2004). A growing body of research has examined the development of informal statistical inferential reasoning and the role of context in its development (Ben-Zvi et al., 2012; Dierdorp, Bakker, Eijkelhof, & van Maanen, 2011; Gil & Ben-Zvi, 2011; Langrall et al., 2011; Makar, Bakker, & Ben-Zvi, 2011; Makar & Ben-Zvi, 2011;

60 Pfannkuch, 2011). These studies have emphasised how context knowledge impacts on interpreting data and the fundamental role that the relationship between context and statistical knowledge has for statistical reasoning, particularly inferential reasoning.

Data context is of critical importance in the process of inferential reasoning in statistics, where data needs to be moved from being simply read to being used for sense making (Chick & Pierce, 2012). The role of data context in a statistical

investigation however, creates a contextual contradiction, as the context of a problem has the capacity to both motivate and mislead (Ben-Zvi, Makar, & Bakker, 2009). Students can be motivated by the data context to engage in statistical sense making when reasoning inferentially. Children’s informal and personal knowledge of the data context can bring additional information and insight to data that can influence interpretation and explanation of data, justification for the use of data and

conclusions drawn from data (Masnick, Klahr, & Morris, 2007). Conversely, students’ data context knowledge that is potentially inconsistent or insufficient can mislead them as they consider the statistical knowledge they have from the available data. Makar, Bakker, and Ben-Zvi (2011) state that although distinguishing between statistical and context knowledge is not easily done, students must coordinate between context knowledge and statistical knowledge as they look for evidence for their reasoning in moving to a problem solution. Context therefore has the potential to make a statistical problem more accessible and at the same time constrain it (Langrall, 2010).

The tight connection between data context and a statistical problem is the crux of the contextual dilemma in reasoning in statistics. A real world statistical problem being worked on is drawn from, and is situated within a context. That context also brings with it context-specific knowledge. The goal of finding a solution to the real world problem is to use data as evidence to increase context knowledge and understanding. In data analysis, the relationship between data context and data is described by Wild and Pfannkuch (1999) as involving an interplay or shuffling between the data and context spheres, “finding something out” and “ascertain(ing) meaning of what we have seen” (p. 336). Young children’s ability to reason about data can be complex if it is “extended beyond describing and interpreting data

61 towards making informal inferences that go beyond data” (Ben-Zvi, Makar, &

Bakker, 2009, p. 2).

The central role of data context in statistical problem solving raises questions as to the need for exploratory research in children’s statistical reasoning. The form of the data context for the statistical problem, that is, how the data for the problem is contextualised, may impact children’s reasoning as they find a solution to a statistical problem. One pedagogical approach to teaching young children is to use picture story books as a springboard for learning. Story is a primary means of young children organising, making meaning and sharing experiences (Im, Parlakian, & Osborn, 2007). Picture story books serve to provide contextual bridges between children’s experiences and the informal, vicarious experiences found between the pages of the book. The next section considers how young children’s reasoning with context in a statistical problem may be supported by the use of picture story books.

2.6 Engaging Statistical Context Through Task Design