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Studies which have aimed to assess the relationship between patterning and arithmetic have typically looked at children from the ages of 5-8 and used both alphanumeric (numbers and letters) and non-alphanumeric (e.g. colours, objects or shapes) patterning tasks. The exact relationship between these skills and arithmetic is not clear, although it is logical to assume that alphanumeric tasks are better correlated with arithmetic than nonalphanumeric tasks (Burgoyne, Witteveen, Tolan, Malone, & Hulme, 2017).

The simplest patterning tasks require children to complete sequences which appear in a repeated format with more complex tasks requiring children to complete increasing or rotating items. It is possible that patterning tasks relate to numerical development as many early number abilities involve predictable sequences. For example, counting in 2s involves an alternating pattern of 0s and 2s and the ability to generalise underlying relationships for early arithmetic (e.g. 1+2 is the same as 2+1) is a key concept which may act as a foundation to later arithmetic ability (Burgoyne et al., 2017). However, as research into patterning and arithmetic is in its infancy, the underlying mechanisms are not completely clear.

Correlational evidence

Evidence supporting a concurrent correlation between patterning and arithmetic in children comes from a number of studies. For example, Warren and Miller (2013) examined performance on a simple repeating patterns task involving numbers in 230 children aged 5 years 9 months. Mathematics was measured via a general maths test which examined basic number knowledge (14 questions) and some advanced mathematical knowledge including probabilities and geometry (5 questions). Patterning scores correlated with general maths score (r = .44, p < .05) and explained unique variation in maths score, after controlling for language abilities (patterning unique beta = .24, p < .05; language unique beta = .52, p < .05). Together, language and patterning scores explained 43% of the variation in general math abilities (note that no unique R-squared values were provided). Despite controlling for language, the authors failed to control for other predictors of arithmetic, such as executive function which has been linked to both maths and patterning abilities (e.g. Bull & Lee, 2014; Lee et al., 2012; Miller, Rittle-Johnson, Loehr, & Fyfe, 2016).

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A study by Lee, Ng, Bull, Pe, & Ho (2011) used a number patterning task whilst controlling for executive function in a correlational design involving 151 children (mean age = 10 years 1 month). Number patterns correlated strongly with maths ability, measured via three numerical tasks (rs = .42-.60, p < .001). Structural equation modelling showed that understanding of number patterns, arithmetic and working memory (an executive function measure) explained 63% of the variation in algebra, with number patterns remaining a unique predictor of algebra score concurrently (β = .04, p < .01) and when children were tested again one year later (β = .31, p <.001). For this study algebra, rather than arithmetic, was the main outcome measure (possibly because children were older in this study). Taken together, these studies suggest a role of patterning in maths skills after controlling for predictors of numerical abilities (including language or executive function). However, an important caveat is that the patterning tasks used included numbers, and therefore the relationship between pattering and numerical skills may be simply explained by number knowledge. Schmerold (2015) used an extensive patterning task to assess the relationship with math ability and executive functioning in 74 children in the first grade (mean age = 7 years 2 months). Children completed repeating, increasing and rotating pattern tasks which used numbers, letters, shapes, pictures and object stimuli (a total of 48 patterning questions). As is commonly seen in patterning tasks, missing items were presented at the beginning, middle and end of a sequence and children were asked to identify the correct option from four possible answers. Maths abilities were measured using the Woodcock-Johnson Test of Cognitive Abilities III (W-J Test) (Woodcock, McGrew and Mather, 2001) and included applied problems, quantitative concepts and number series. Executive functioning was measured via working memory, cognitive flexibility and inhibition.

In line with previous evidence, patterning was strongly correlated with math score for the three math measures (r = .52-.54, p < .001). Patterning was also correlated with cognitive flexibility and working memory (r = .41, p < .001 and r = .23, p < .05, respectively) although not inhibition (r = .17, p > .05). Together, the correlated measures were entered into hierarchical regression and explained 40% of the variation in applied problems and quantitative concepts, with patterning explaining 29% of the variation in number series. Patterning remained a unique and significant predictor in

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all three models (β = .18, .08 and .11, respectively). Importantly, the patterning task used in this study used numbers and other stimuli and therefore this relationship cannot be explained purely by number knowledge. However, the results are not fully replicated in a study by Lee et al. (2012) who reports similar correlations between patterning (numbers and shapes test) and arithmetic (number patterns: r = .46, p < .001; shape patterns: r = .25, p < .05) but no unique relationship once executive functioning is controlled.

One reason for this difference may be due to methodology of the patterning tasks for which there were several differences across the two studies. Schmerold (2015) used a patterning task in which children chose one of four options to complete the pattern, as is seen for other pattern measures. Lee et al. (2012) created a combined pattern score based on the child’s ability to choose the correct response and their ability to describe the rules governing the pattern, a process which may involve a number of different factors to choosing a missing piece. Secondly, the stimuli in Lee et al. (2012) were manipulated according to size and relative position of the shape but not in Schmerold (2015). Finally, only repeating shapes and numbers were examined in Lee et al. rather than range stimuli and pattern types used in Schmerold. These different methods may be in part the reason for the difference in results, although more evidence from other studies is required to confirm that patterning plays a unique role in number skills. In summary, there are a number of studies which show that patterning and arithmetic are concurrently correlated and some evidence to support a unique relationship between these variables. What is not clear from these studies is whether pattern abilities predict later arithmetic, and therefore whether these two factors are causally linked.

Longitudinal evidence

Some longitudinal evidence points to a relationship between alphanumeric patterns and later numerical abilities. For example, Pasnak et al. (2016) explored the longitudinal relationship between increasing patterns and later arithmetic. Nighty-six children in Grade 1 (mean age 6 years 6 months) were examined on increasing pattern ability, maths score (W-J III) and reading ability at two time-points (beginning and end of school year). In the patterning task, children were presented with letter and number sequences that increased by one, two, or three numbers or letters. Children

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selected the correct answer from a choice of four responses. The authors used a time- lag correlation design which compares performances across tests at the two time points to indicate a direction of any significant relationship. Number patterning at time 1 was a significant correlate of math score at time 2 (r = .39, p < .01) although the converse was not true (maths at time 2 to number patterns at time 1: r = -.05, p < .05) thus indicating that earlier patterning may lead to later numerical ability (difference in correlation coefficients is significant: z = 3.47, p < .01). These findings mirror those of VanDerHeyden et al. (2011) who showed using a similar patterning task (but repeating rather than increasing) that number patterning correlates longitudinally with addition (r = .42, p < .05). Unfortunately, both studies failed to control for known predictors of arithmetic and therefore the unique relationship between these skills is not known. Moreover, as the relationship between patterning and maths abilities focuses on numerical patterns the relationship reported may be due largely to number knowledge.

One study which assessed nonalphanumeric patterning as a predictor of arithmetic is by Rittle-Johnson, Fyfe, Hofer and Farran (2017). In this large-scale longitudinal study, several predictors of mathematical achievement were examined in 517 low- income children aged 4-11. In the study, children were assessed at four time-points: beginning of pre-school year (mean age = 4 years 5 months) end of pre-school year, end of kindergarten and end of first grade (aged 7) and maths achievement was measured again four years later at time 5 when children were aged 11. Patterning was measured via a repeating nonalphanumeric task whereby children were required to choose the correct coloured cube to complete a repeating sequence presented with coloured cubes (e.g. red-blue-red-blue). Children were assessed on a range of numerical cognitive measures at several time points including general maths achievement, calculation, nonsymbolic comparison, counting, number identification and shape completion. Language measures, reading and general cognitive abilities (e.g. working ability in the classroom) were also taken at the four time-points. Patterning at the end of first grade was a unique and significant predictor of maths achievement five years later (β = .08, SE = .04, p < .05) therefore suggesting an important role for early nonalphanumeric repeating patterning skills in later arithmetic. However, the patterning task was not reliable and despite controlling for

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many known predictors of arithmetic, the authors did not include executive function or intelligence measures, which are strong predictors of both patterning and arithmetic. In a later study, Rittle-Johnson, Zippert, and Boice (2018) used a reliable patterning task (Cronbach’s alpha = .83) and controlled for important predictors of arithmetic including spatial awareness, working memory and general cognitive skills. Seventy- three children were assessed at time 1 (mean age = 4 years 7 months) on repeating number patterns and general maths skills (Research-Based Early Mathematics Assessment; Weiland et al., 2012). Patterning score was significantly correlated with both concurrent mathematical skills, and maths performance measured 6.8 months later (r = .64, r = .65, respectively). Moreover, when other important factors of arithmetic were controlled (e.g. spatial skills, verbal skills and working memory) patterning remained a unique predictor of maths achievement (time 1: β = .30, p = .01; time 2: β = .40, p < .001). Spatial skills correlated significantly to both patterning score (r = .38) and maths scores (time 1: r = .61; time 2: r = .59). What is not known from this study is how different types of patterning task correlate with one another and with arithmetic, as the authors focused upon one type of task (repeating) with one type of stimuli (numbers).