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By means of the present study we wanted to investigate three research hypotheses regarding the relevance of working memory capacity for early arithmetic skills in kindergarten children and first and second grade students. With regard to Hypothesis 1, we did not find any significant differences between a group of children with age- adequate average arithmetic skills and another group of children with age-delayed arithmetic skills when both groups were parallelized by arithmetic skills rather than age. This finding is inconsistent with the findings of McLean & Hitch (1999) who found that these two groups differed in long-term memory activation (central

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Central Executive Phonological Loop Visual-Spatial Sketchpad Pear son Co rr e lation Co e ff ic ie n t

Working memory subsystem

Group 1 Group 2

executive) with a large effect size (d = 1.11). In our study neither the central

executive component of working memory (including listening span, which also taps this central executive function) nor naming speed as a measure of access speed to long-term memory differed significantly by group. Additionally we did not find any mean differences of cognitive variables between the two groups. The main aim of performance-matching is usually to find out whether below average older children differ from average younger children (1) only in terms of age-adequate performance, (2) in terms of a general delay of cognitive measures or (3) additionally to 2 have specific cognitive deficits. In our study we found evidence for the second idea and thus have to reject Hypothesis 1.

To test Hypothesis 2, we conducted hierarchical regression analyses to reveal the specific weight of each working memory component for predicting early

arithmetic skills under control of other variables. The arithmetic skills test, based on the stage model of Fritz, Ricken & Balzer (2009), included verbal and non-verbal problems as well as rather complex and rather simple problems which could be solved in a very automatic way by older children. Accordingly, we expected the test to tap each component of working memory. It is generally accepted that the visuo- spatial sketchpad is especially important for early arithmetic skills, while the

phonological loop is more involved in verbal and word problems (Andersson, 2007; Rasmussen & Bisanz, 2005; Swanson, 2006). Furthermore, it is assumed that the central executive plays a role when solving complex problems the solution of which is not fully automatized (de Smedt et al., 2009; Meyer et al., 2009). In this regard, we expected that all three working memory components would account for unique variance of arithmetic skills.

Analyses revealed that central executive and visuo-spatial sketchpad capacities, but not phonological loop capacities, are predictive for early arithmetic

skills, when controlling for other predictors like age and further cognitive variables. A possible explanation could be that our children were just too young and untrained when dealing with verbal arithmetic problems. Our sample included children from kindergarten to second grade of two different groups. Group 1 children with age- adequate arithmetic skills were, on average, one year younger than Group 2 children with non age-adequate, delayed arithmetic skills. This idea is in line with Rasmussen & Bisanz (2005) who found that there is a shift in the usage of working memory slave systems from kindergarten to primary school, which might be based on training provided at school. It seems that some, especially weak children are rather resistant against using new (verbal) strategies taught at school – which is a novel finding of our study. This finding remains stable in all regression models and under control of several important known predictors of early arithmetic skills. Thus, Hypothesis 2 could be largely confirmed as both central executive as well as visuo-spatial sketchpad capacities predicted arithmetic skills in this early stage of math learning.

Hypothesis 3 stated that children with arithmetic deficits would not only have lower working memory capacities, but that they also would make worse use of it in solving arithmetic math problems. Because we had to reject Hypothesis 1, we have to reformulate Hypothesis 3 to a certain amount: Children with arithmetic deficits make worse use of their working memory capacities compared to younger children who are matched according to their math achievement. To our knowledge there has been no research regarding this question yet, though it would be of importance for the understanding of arithmetic difficulties of young children.

To our understanding, it is a premise of most researchers that the correlation between achievement and working memory capacity is not moderated by

achievement group membership. This premise results from the point of view that correlations display linear relations. With respect to our results, there is reasonable

doubt that this premise is true. There have been similar findings regarding age as a moderator of these correlations by de Smedt et al. (2009) and Meyer et al. (2009). One major difference between these findings and ours is, that they found a

decreasing correlation of one component with math achievement, while the correlation for another component increased with age. In contrast we found a decrease in the visuo-spatial sketchpad correlation, a decrease in the central

executive correlation – which might however result from reduction of range effects – and no change with regard to the phonological loop. Thus, while increasing age leads to different strategies being used (Rasmussen & Bisanz, 2005), which require

different aspects of working memory, math problems lead to a deficit in working memory usage, which possibly relies on low (visuo-spatial) strategy use. We would like to introduce the term usage-deficit hypothesis for this phenomenon.

Kindergarten children and early primary school students who perform at average usually rely heavily on the visuo-spatial sketchpad to solve math problems (Bull et al., 2005; Krajewski et al., 2008; Rasmussen & Bisanz, 2005), which is often attributed to the use of visual strategies like imaging strategies or spatial strategies like finger counting. It could be that these children are unable to connect their

counting abilities with the visualization of the represented sets. Further research with this age group should focus the question whether or not children at risk for math problem struggle at using these strategies, do not use them at all or start to use them later in their life, when they are of little or no help solving the math problems they are facing at that given time. Another explanation offered by the literature is that these children are not able to visually link abstract numbers with concrete sets and understand that both represent the same quantity (Fritz et al., 2009), a problem which is already tackled in current trainings by showing children different ways to connect numbers with sets (i.e. Marko-T, Gerlach, Fritz & Leutner, in press)

Thus on the practical side, our findings suggest that if children with

arithmetic difficulties do have the necessary cognitive capacities but just fail to make proper use of them (as implicated by low correlations), this usage deficit could be remedied by proper strategy training. Marko-T focuses the relation between numbers and sets and helps children to visualize these. Another idea could be to implement a training program on how to use visuo-spatial memory strategies in an arithmetic context, like visualizing an abstract arithmetic problem like 8 divided by 2 into the concrete idea of equally sharing 8 candies with a friend. In a next step it seems very helpful to implement such a training on visualizing abstract arithmetic problems and manipulate these visualizations in ways that helps solving the problem.

So far, our findings on the usage-deficit hypothesis and its implications are limited to preschool and early primary school age as well as the arithmetic domain. To generalize these findings further research should conduct related analyses with children of different age groups and with problems from other math domains like geometry or probability theory. Furthermore our findings are limited as we only compared average achieving children with children of equal arithmetic skill but older age, because we were interested in the question whether or not children of the same achievement level differed in terms of working memory. However, to generalize our findings it seems necessary to find out whether our proposed usage-deficit exists between children of the same age, but with average and low math achievement. Our last impulse for further research is the idea to train young children on how to use, for example, visuo-spatial memory strategies in arithmetic and see if this training has generalized effects on other domains, which might be more effective for school success then a pure math training.