Spatial Skills, Structuring, and Strategy Choice

SPATIAL SKILLS AND STRUCTURING

With the development of number sense, children discover easier ways to count and learn how to use them in solving mathematical problems more accurately and more efficiently. They gain deeper understanding not only of magnitudes but also of a variety of their representations and ways of operating them, and in doing so they employ spatial sense. Within the very broad term of spatial sense, there are three components which contribute to mathematical thinking: spatial visualisation, geometry and spatial orientation (Van Nes and De Lange, 2007). Spatial visualisation enables us to mentally move, rotate or transform images of objects or their spatial representations. Geometry provides the knowledge and understanding of shapes and figures as well as geometrical structures and patterns which might be transferred into more general domains. Spatial orientation allows the comparison of figures and shapes, understanding relationships and proportions between objects and relating these experiences to surroundings and positioning within space. These skills are crucial in organising newly acquired knowledge as they provide a platform for structuring new concepts into previously gained frameworks. This ability is not only essential in mathematics but also in other scientific and more general learning. In their learning, children improve the ways that they organise new information and gain the ability to amend their strategies depending on the requirements of a problem and in doing so reach a higher level of understanding. According to Carr and Hettinger (2003), structures and strategies provide the means to organize and process information, they “allow us to create symbolic representations of our experiences and to reorganize and compile information into larger, logical units” (p.34).

In mathematics, the awareness of spatial structures and the ability to use them in a variety of contexts shortens the process by which children determine quantities, and compare and calculate them. Facing a group of objects, children’s initial reaction is to count them unitarily. Although successful with small magnitudes, this strategy becomes time consuming and difficult with larger numbers. Many children then employ a way of organising the objects in a way which enables them to count them reliably, whether this is undertaken physically or mentally. However, some children do continue to count unitarily and not develop that form of spatial awareness. As using structures simplifies the development of more formal mathematical concepts and operations, such inability can hamper mathematical development and is a predictor

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of lower achievement and possible difficulties in proceeding to more sophisticated numerical procedures (Butterworth, 1999; Mulligan and Mitchelmore, 2009).

Spatial structuring is essential to many mathematical activities of a numerical or geometrical nature. Van Nes and Dorman (2011) described mathematical skills which rely on spatial structures:

 composing and decomposing of quantities (understanding that 6 = 3 + 3 but also 4 + 2 or 1 + 5),

 counting and grouping,

 part – whole knowledge in addition, multiplication and division,  comparing a number of objects,

 patterning,

 building a construction of blocks,  ordering, generalising and classifying,

 and more sophisticated mathematical operations like algebra, proving, predicting, mental rotation on manipulation of structures.

The authors also suggest a set of activities which if used at an early age can develop children’s spatial structuring ability and support children who already at preschool might be experiencing learning difficulties in mathematics. Those activities were: recognizing and comparing configurations (for example symmetric like dots on a dice, double-structures like egg cartons or five-structures like sets of fingers), recognizing and comparing structured and unstructured objects like dominoes, dice, building blocks), creating and describing patterns, building and analysing 3D constructions and determining the number of blocks in the construction, determining and comparing of unstructured quantities.

Within mathematical development, spatial structuring together with an early spatial sense contributes to an early number sense. Van Nes and de Lange (2007) proposed that the ability to imagine a spatial structure (related to a specific magnitude) and to mentally manipulate it helps in understanding quantities and the process of counting and makes it less time consuming. That description of visualising a spatial structure and manipulating it is parallel to the definition of spatial-temporal reasoning offered by Shaw (2000) and later demonstrated to develop as a result of participation in music (Rauscher et al., 1993; Rauscher et al., 1995; Rauscher and Zupan, 2000). There are many different ways of developing spatial skills. Any of them can be beneficial and engaging in music making is one possibility. Structuring as a way of organising components is regularly used in both mathematics and music with regularity or patterning being important in both. In mathematics, numerical or spatial patterns provide

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structures for problem solving whilst in music patterns and relationships between elements of a pattern give sounds a structure which enables music perception. The role of patterns, relations and transformations in both mathematics and music is paramount and those who recognise such structures are likely to acquire understanding of representations, operations and concepts. THE USE OF STRATEGIES AND STRATEGY CHOICE IN MATHEMATICS

In solving arithmetic problems children use a variety of procedures and choose them depending on the difficulty of the problem. Carr and Hettinger (2003) defined mathematical strategy as a “method used to solve a mathematics problem” (p.34). In the theoretical framework proposed by Siegler and Booth (2005) they described four characteristics of the use of strategies. They were:

 “variability of strategies and representations” – in solving problems most people use a range of strategies rather than just single one,

 “strategy choice” – decisions about which procedure is to be used from the whole range are not random but are taken after consideration of the problem whilst efficiency and accuracy are the principles of strategy choice,

 “changes in strategy use” – with experience and understanding gained, the way people use strategies changes and that change often determines success in mathematics,

 “individual differences” – as the strategy choice is closely related to other cognitive abilities, there are differences between individuals in adaptability and proficiency in using strategies (p.199).

Children are inconsistent in choosing procedures to solve arithmetic problems and the way they make those decisions is not yet fully understood (Rasmussen et al., 2003). Typically, children in solving simple addition problems count both addends (e.g., 4 + 2) with or without using fingers in the process. In doing so children adopt one of three counting procedures:

 min – in which they start with a larger number and then count on the value of a smaller number,

 max – smaller number is a starting point and the larger number is counted on,  sum – counting both quantities starting from 1.

Preschool children use strategies based on counting or retrieval, and within the last group three procedures have been identified: guessing, retrieval and decomposition (Bisanz et al., 2005). Retrieval and decomposition are closely related. In retrieval children use long-term

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memory representations to solve a problem. Decomposition involves retrieving a partial sum and then manipulating it, for example, 7 + 8 might be solved as 7 + 7 with the answer retrieved from memory and adding 1 to the sum. Bisanz et al. (2005) also described four counting strategies divided into overt, for example counting fingers, finger recognition where a child looks at the fingers but does not count them, counting without an external representation; and covert when children do not show overt signs of counting and probably retrieve solutions from memory. Facing an arithmetic problem children attempt first to use covert procedures and gain the answer from memory, if this is unsuccessful they move to overt strategies.

The main purpose of using different procedures is to increase efficiency and accuracy, but what exactly determines the further choices remains a continuous research subject. There are significant differences between individuals in adapting successful strategy choice. Some children are able to make the right strategy choices which are characterised by a balance between the amount of time needed to solve the problem (efficiency) and the likeliness of obtaining the correct answer (accuracy). On the other hand, some children are consistent in choosing poor strategies or giving any answers that come to mind, even though they are unlikely to be accurate. Children who approach all mathematical problems by using the same strategy, for example memory retrieval or finger counting in all calculations, are likely to fall behind their peers who use alternative strategies. Their mathematical achievement is usually much lower. What is crucial for the study reported in this thesis is the fact that choosing strategies is also associated with spatial skills hence more developed spatial skills influence arithmetical achievement (Geary, 1994).

What are the factors influencing children’s acquisition of strategies and how does the choice of strategies develop? The ability to make good choices appears to be determined by good understanding of basic number and arithmetic concepts and by working memory. According to Siegler and Booth (2005), the choice of strategies is not a set skill but increases with the adoption of new strategies, with using relatively advanced strategies more often and with building proficiency in executing existing strategies. The more advanced use of alternatives in problem solving is not the goal in itself. It also helps develop further mathematical skills, for example, building representations on a number line. Siegler and Booth (2005) proposed that with age, children improve their accuracy of number line estimations through an increase of strategy choices. That has a strong relationship with the development of counting skills and the more thorough understanding of numerical magnitudes.

The use of problem-solving strategies develops through increased understanding of numerical facts and mathematical operations but also through children becoming more familiar with a variety of strategies. Experience of using alternative procedures helps them in gaining

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further expertise in mathematical processing and in turn developing more adaptable use of strategies. This enrichment continues to a point of attainment described by Geary (1994). He suggested that as expertise in mathematics grows, pupils become less reliant on strategy choices because they have gained enough knowledge which can easily be retrieved from memory or they can use procedures learned and practised to the extent of becoming automatic. As soon, as those methods start being more efficient and reliable in terms of obtaining the correct result, strategy choice is used less often, mostly in situations where the other methods fail to solve the problem. Those children who continue to have to choose strategies are likely to struggle with mathematics.