Models and Modeling Perspective

The Models and Modeling theoretical perspective [Models and Modeling] (Lesh & Doerr, 2003) is a conceptual framework that theorises that children are developing conceptual schemes or models as they make sense of their world. In this study, the use of the term “mathematics” found in and referred to in Models and Modeling literature includes statistics. Models and Modeling focuses on the way that conceptual tools and knowledge are developed and used by individuals and groups to interpret and inform real-world problem solving in ways that are re-useable in other contexts (Lesh & Doerr, 2003). Models are created, interpreted and re-interpreted for making sense of experiences, and the process of creating and interpreting new

experiences engages representational behaviour (Lesh & Doerr, 2003; Mousilides, Sriraman, & Christou, 2007). Initiating young children into modeling activities that reflect the mathematical systems embedded in their social environment provides access to recognising, interpreting, evaluating and assessing such systems

(Mousilides, et al., 2007). Statistics is one such system where children should have opportunities to create, organise, interpret and analyse using data in contextualised statistical problems. For statistical competency to flourish, children’s conceptual systems need to be challenged, tested, and revised (Lesh & Lehrer, 2003). For researchers, understanding statistical reasoning can be achieved by examining how experiences can be structured to reveal children’s existing conceptual thinking.

Young children develop primitive models to make sense of their everyday experiences, including statistical experiences (Lesh & Doerr, 2003). Pedagogical activities that seek to connect complex concepts to children’s existing statistical experiences through modeling are rarely part of the school curriculum experience (Swan, Turner, & Yoon, 2007). Young children’s school instruction with respect to

40 core mathematical and scientific ideas is either delayed or neglected because they are not ordinarily considered to have the basic skills or concepts deemed necessary for learning disciplinary specific concepts (Lehrer & Schauble, 2006b). Despite available research on young children’s capacities for acquiring, developing and reasoning with a modeling approach to learning (English, 2010, 2011, 2012; English & Watters, 2005; Lehrer & Schauble, 2006b) there were no further studies on using modeling with children at school below the age of 6 years.

Young children are capable of accessing, and should have access to important powerful mathematical ideas in their educational environments (Australian

Association of Mathematics Teachers and Early Childhood Australia, 2006; Clements & Sarama, 2004; Lehrer & Schauble, 2006b; Perry & Dockett, 2008). Young children possess powerful statistical ideas, including problem solving, data and probability, that should inform and support the inclusion of statistical ideas in mathematics research (Hunting, Mousley, & Perry, 2012; Perry & Dockett, 2008). Access to important and powerful mathematical ideas, including statistical ideas, should therefore be reflected in the types of problem solving activities young children engage with in classrooms. Models and Modeling is positioned to

theoretically support researching young children’s statistical learning in a way that is cognisant of existing competency and potential to engage in complex ideas. In addition, Model and Modeling tasks are designed to shift attention away from computation and towards statistics as a thinking process that involves

conceptualisation, description and explanation (English, 2006) and encourages sense- making in problem solving (Doerr & English, 2003; Greer, Verschaffel, &

Mukhopadhyay, 2007). Models and Modeling provide a framework for accessible statistical learning opportunities for young children beginning school, including principles of task design to inform the content and structure of the task context.

2.4.2.1 Task design and contextualised problem solving.

Task design in statistics plays a critical role in the contextualising of the statistical problem as they engage a real-world context. The types of problem solving activities that children encounter at school generally disregard the connection that should be forged between a child’s real-world knowledge and experiences and possible solutions to problems (English, 2003a). Typically, mathematical problem

41 solving activities encountered at school are removed from a real-world context (Lesh & Doerr, 2003), however, statistical problems must by definition, engage real-world data contexts.

Modeling problems support real-world problem connections as, under the principles of their design, they use realistic contexts to authenticate and frame the problem (Mousoulides, Sriraman, & Christou, 2007). Models and Modeling problem solving activities are constructed using principles of instructional design (Lesh & Doerr, 2003; Kaiser & Sriraman, 2006). Modeling tasks are purposefully designed to be relevant to students’ worlds and to motivate and interest them in expressing their understanding and inferences (Doerr & English, 2003). Model-eliciting activities, sometimes referred to as “thought revealing activities” (English & Watters, 2005, p. 60) provide meaningful, engaging real-world problems and design problems that need to be mathematically described and interpreted in order to expose the nature of children’s mathematical thinking and conceptual development (Lesh & Doerr, 2003). The design qualities of modeling activities support researchers and educators to understand and assess mathematical problem solving (Lesh, 2006). Modeling

activities can therefore actively accommodate and connect to the need for real world problems found in the statistical definition determined for this study.

Model and Modeling problem activities are non-routine problem situations that employ instructional principles designed to elicit the development and extension, exploration and refinement of significant mathematical constructs (Lesh & Doerr, 2003). Kennedy (2009) notes that solving non-routine problems moves away from “correct answers” and requires strategic and careful problem solving management that can focus children’s attention on processes, conceptual connections and structure. Approaching problem solving as a process with conceptual connections corresponds to the intellectual method advocated for statistical learning (Moore, 1998) and the multiplicity of possible outcomes from solving statistical problems identified by Gattuso (2008) earlier in this review. The design of a problem solving task therefore forges a conscious link between the theorised learning processes considered in the Models and Modeling perspective and its associated pedagogical activities. Embedding core mathematical constructs that are “mathematically generative” is a central principle (English, 2003; English & Watters, 2005). As a

42 result, the key mathematical ideas are not presented “up front” in modeling

problems, but are “embedded within the problem context and are elicited by children as they work the problems” (English, 2006, p. 96). As well as embedding the

mathematical ideas in the problem, there is an accompanying requirement for children to explain and justify the models to others in the learning environment (English, 2006). This form of modeling activity is conceptually available to young children, and allows them “to access mathematical ideas at varying levels of sophistication” (English & Watters, 2005, p. 93) and in doing so, emphasises a heuristic approach compatible with current constructivist approaches to early childhood education (Perry & Dockett, 2008).

2.4.3 Data Modeling and the Task Context for Statistical Problem Solving