Sex differences in the association of math achievement with visual-spatial and verbal working memory: Does the type of math test matter?

Previous research on sex differences in mathematical achievement shows mixed findings, which have been argued to depend on types of math tests used and the type of solution strategies (i.e., verbal versus visual-spatial) these tests evoke. The current study evaluated sex differences in (a) performance (development) on two types of math tests in primary schools and (b) the predictive value of verbal and visual-spatial working memory on math achievement. Children (N = 3175) from grades 2 through five participated. Visual-spatial and verbal working memory were assessed using online computerized tasks. Math performance was assessed five times during two school years using a speeded arithmetic test (math fluency) and a word problem test (math problem solving). Results from Multilevel Multigroup Latent Growth Modeling, showed that sex differences in level and growth of math performance were mixed and very small. Sex differences in the predictive value of verbal and visual-spatial working memory for math performance suggested that boys seemed to rely more on verbal strategies than girls. Explanations focus on cognitive and emotional factors and how these may interact to possibly amplify sex differences as children grow older.     

On the language specificity of basic number processing: transcoding in a language with inversion and its relation to working memory capacity

Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.     

To carry or not to carry — Is this the question? Disentangling the carry effect in multi-digit addition

Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic.     

Children’s early mental number line: Logarithmic or decomposed linear?

Recently, the nature of children’s mental number line has received much investigation. In the number line task, children are required to mark a presented number on a physical number line with fixed endpoints. Typically, it was observed that the estimations of younger/inexperienced children were accounted for best by a logarithmic function, whereas those of older/more experienced children were reflected best by a linear function. This led to the conclusion that children’s mental number line transforms from logarithmic to linear with age and experience. In this study, we outline an alternative interpretation of children’s performance in a number line task. We suggest that two separate linear representations for one- and two-digit numbers may exist in young children and that initially the integration of these two representations into the place value structure of the Arabic number system is not fully mastered. When testing this assumption in a sample of more than 120 first graders, we observed that the two-linear model consistently provided better fit indexes. We conclude that instead of assuming a transition from logarithmic to linear coding, performance differences could also be accounted for by an improvement in integrating tens and units into the Arabic place value system.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Multi-digit number processing beyond the two-digit number range: a combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.