Sequencing and patterning

Patterning ability comprises the ability to recognise, extend, create and copy patterns (Waters, 2004:321). Patterning is found within most preprimary school mathematics curricula of the world. It is regarded as a foundational skill for algebra and algebraic thinking (Tipps et al., 2011:137), but until recently, was an infrequently researched topic in the realms of education.

In an attempt to create a link between patterning and algebraic functioning, Lee et al. (2011) investigated the relationship between proficiencies on pattern tasks and algebraic word problems in 9- and 10-year-olds. Their findings suggest a significant correlation between proficiency in number patterning and algebraic performance (Lee et al., 2011:280). They conclude that algebraic reasoning will be difficult if the child has either poor computational facility or poor ability to recognise patterns in information and generalise rules about those patterns (Lee et al., 2011:280).

But patterning incorporates more than just algebraic functioning. In a longitudinal and cross- cultural study on reasoning abilities, English argues that patterning knowledge influences analogical reasoning in young children and that identifying, extending and generalising patterns are important components of inductive reasoning (English, 2004 in Waters, 2004:322).

In spite of the gains expected from patterning, it would appear that teachers have limited understanding of the types, levels or complexities of patterning tasks (Waters, 2004:327). There are far more varieties of patterning than the simple colour patterning one often finds in preprimary school classes. One gets pattern structures like hopscotch patterns (which explore the child’s ability to rotate a unit of repeat) and growing patterns (which increase or decrease systematically) (Papic, 2007:10-12). Warren and Cooper divide early childhood patterning into two broad categories, namely repeated patterns and growing patterns (Warren & Cooper, 2006:11). Patterning not only improves reasoning, but could be used as an intervention strategy in the lives of young children. Mulligan and associates (Mulligan et al., 2006) studied 683 low achieving

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students aged 5-12 by involving them in a project which largely aimed at, among other things, improving children’s ability to identify and apply patterns. The research implied that if low achievers had poor awareness of patterns and structure, their achievement could be improved through explicit teaching in mathematical patterns and structures (Mulligan et al., 2006:377). Participating children showed a marked improvement in school-based and system-wide measures of mathematical achievement, as well as PASA-scores (Pattern and Structures Assessment Scores), particularly in the early grades (Mulligan et al., 2006:377).

Although the abovementioned studies have been conducted in a range of foundational contexts, young children are capable of developing complex patterning concepts prior to formal schooling (Papic & Mulligan, 2007:599). Researchers conclude that patterning promotes other mathematical processes like multiplicative thinking and transformation skills (Papic & Mulligan, 2007:599). Papic (2007:8) describes how patterning is an essential skill in early mathematics, and is vital for the development of spatial awareness, sequencing and ordering, comparison and classification. She goes on to explain how patterning is also integral for the development of counting and arithmetic structure, base ten and multiplicative concepts, units of measure, proportional reasoning and data exploration (Papic, 2007:8). Patterning can also lead to functional thinking and the understanding of variation between data sets (Warren & Cooper, 2006:14).

A noteworthy research project into the effects of patterning and academic achievement was undertaken by Hendricks and her colleagues (2006). After a four-month patterning intervention programme researchers concluded that teaching children to understand the relations involved in patterns may promote abstract thinking and may also be an additional way to support and strengthen the development of age-appropriate mental abilities, boosting overall intellectual growth (Hendricks et al., 2006:88).

3.4.9 Measurement

“Measurement is an important elementary mathematical and scientific competence, but…appears to be poorly learned” (Smith et al., 2011:617).

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Measurement is regarded as a fundamental aspect of a preprimary school mathematics programme as it bridges two important areas, namely geometry and number (National Council of Teachers of Mathematics, 2000 in Cross et al., 2009:79). It is a mathematics topic that is used most directly in students’ daily lives (Reys et al., 2012:403). It is also regarded as beneficial in that it encompasses many other topics in mathematics and can be used as a pedagogical tool to engage students who would otherwise be less motivated (Reys et al., 2012:404).

The process of measurement is a surprisingly complex task due to the principle of compensation, which stipulates that the smaller a measurement unit is, the more of these units are required to measure an attribute (Tipps et al., 2011:478). The difficulty associated with this inverse proportion concept generalises back to the classical conservation problems presented by Piaget (Carpenter & Lewis, 1976:53) (see “Piaget’s theory of cognitive development” par 2.2.2.1.f ). In these activities the distracting cues are not different-shaped containers, but rather different-sized units of measure. It is suggested that just as children fail to recognise the height and width compensation relationship in liquid conservation tasks, so children struggle with the unit size and number of units relationship in a measurement problem.

This Piagetian conservation problem regarding measurement can be overcome if teachers make use of ready-made systems like rulers (Cross et al., 2009:199-200). It would appear, however, that most recent curricula for young children follow a sequence of instruction in which children first compare lengths, measure with non-standard units and then progress to formal units of measurement. This is rooted in the belief that children need to gain experience with non-standard, informal units before progressing to standard units (Reys, 2012:413). However, Boulton-Lewis et al. (1996:329) argue that young children should be introduced to standard units of measurement from the initial stages and not non-standard units. This idea was also confirmed by Clements, who describes several studies that challenge the conventional wisdom regarding the teaching of non- standard before standard units (Clements, 1999).

That being said, Carpenter and Lewis investigated whether young children (Grade 1 and 2) would be able to predict an inverse relationship between unit sizes and number of units (Carpenter & Lewis, 1976:54). They concluded that children do, in fact, have some notion of this relationship at

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a much earlier stage than predicted by previous studies (Carpenter & Lewis, 1976:57). Strangely, they hypothesised that this notion was acquired independently of experience, and ultimately that manipulating different units of measure did not contribute to children understanding unit-size- number-of-units relationships (Carpenter & Lewis, 1976:57). This, however, does not take into account the fact that children may have had measurement experiences outside of realm of the research experiment.

Unlike Carpenter and Lewis, Tipps et al. (2011:474) advocate that children do need a variety of experiences to help them understand the concept of measurement and to become skilled with measurement tools and appropriate units. This idea is supported by Wall and Posamentier (2007:48), who claim that children need direct experiences with comparing objects, counting units and making connections between spatial concepts and numbers in order to establish a foundation in measurement concepts. It would therefore seem that children’s ability to engage in measuring activities may depend on both their level of development and their experiences (Irwin et al., 2004:3).

Estimation is regarded as a useful aspect of a measurement curriculum and general functioning in daily life (Hildreth, 1983:50; Siegler & Booth, 2004:428). Unfortunately, almost no documented research has been conducted in the possible mathematical and cognitive gains of implementing the process of estimation in a measurement context at preprimary school level.