Silvia Pixner

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

Recently, the nature of childrens 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 childrens mental number line transforms from logarithmic to linear with age and experience. In this study, we outline an alternative interpretation of childrens 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.

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.

Whorf reloaded: Language effects on nonverbal number processing in first grade—A trilingual study

The unit-decade compatibility effect is interpreted to reflect processes of place value integration in two-digit number magnitude comparisons. The current study aimed at elucidating the influence of language properties on the compatibility effect of Arabic two-digit numbers in Austrian, Italian, and Czech first graders. The number word systems of the three countries differ with respect to their correspondence between name and place value systems; the German language is characterized by its inversion of the order of tens and units in number words as compared with digital notations, whereas Italian number words are generally not inverted and there are both forms for Czech number words. Interestingly, the German-speaking children showed the most pronounced compatibility effect with respect to both accuracy and speed. We interpret our results as evidence for a detrimental influence of an intransparent number word system place value processing. The data corroborate a weak Whorfian hypothesis in children, with even nonverbal Arabic number processing seeming to be influenced by linguistic properties in children.

Language Effects on Children’s Nonverbal Number Line Estimations:

The mental number line of children is usually assumed to be language-independent; however, this independency has not yet been studied. In this cross-cultural study, we examined the influence of language properties on a nonverbal version of the number line task in Italian- and German-speaking first graders. The essential difference between the two languages concerns the inversion property of most German multi-digit numbers (e.g., 48 → “eight-and-forty”), whereas in Italian number-words no inversion is found. The analyses revealed two language-specific differences in the number line task: (a) Generally, the estimates of Italian children were more accurate than those of Austrian children, even when controlling for general cognitive abilities. (b) Italian children performed particularly better when inversion errors led to large estimation errors. In conclusion, these findings indicate that the organization of children’s mental number line is indeed influenced by language properties even in nonverbal settings.

Language affects symbolic arithmetic in children: the case of number word inversion.

Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in childrens mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.

Language influences on numerical development—Inversion effects on multi-digit number processing

In early numerical development, children have to become familiar with the Arabic number system and its place-value structure. The present review summarizes and discusses evidence for language influences on the acquisition of the highly transparent structuring principles of digital-Arabic digits by means of its moderation through the transparency of the respective languages number word system. In particular, the so-called inversion property (i.e., 24 named as “four and twenty” instead of “twenty four”) was found to influence number processing in children not only in verbal but also in non-verbal numerical tasks. Additionally, there is first evidence suggesting that inversion-related difficulties may influence numerical processing longitudinally. Generally, language-specific influences in childrens numerical development are most pronounced for multi-digit numbers. Yet, there is currently only one study on three-digit number processing for German-speaking children. A direct comparison of additional new data from Italian-speaking children further corroborates the assumption that language impacts on cognitive (number) processing as inversion-related interference was found most pronounced for German-speaking children. In sum, we conclude that numerical development may not be language-specific but seems to be moderated by language.

Decomposed but Parallel Processing of Two-Digit Numbers in 1st Graders

It has been suggested that decomposed processing of two-digit numbers develops from sequential (left-to-right) to parallel with age (Nuerk et al., 2004). However, task demands may have provoked sequential processing as a specific rather than a universal processing style. In the current study a standard unit-decade compatibility effect observed in two-digit number magnitude comparison indicated that first graders were already able to process the single digit magnitudes of tens and units separately and in parallel. Consequently, previous findings of sequential processing may be specific for stimulus characteristics in which such a processing style is useful. It is concluded that even first graders seem to be able to adapt their individual processing styles depending on stimulus properties. More generally, this suggests that the manner by which children process two-digit numbers is strategically adaptive rather than fixed at a particular developmental stage.

On the development of Arabic three-digit number processing in primary school children

The development of two-digit number processing in children, and in particular the influence of place-value understanding, has recently received increasing research interest. However, place-value influences leading to decomposed processing have not yet been investigated for multi-digit numbers beyond the two-digit number range in children. Therefore, we evaluated the separate influences of hundreds, tens, and units on three-digit number processing by means of the hundred distance effect, the decade-hundred compatibility effect, and the unit-hundred compatibility effect in a longitudinal design from Grade 2 to Grade 4. In a number magnitude comparison task, a strong hundred distance effect indicated that the magnitudes of the hundreds digits were predominantly processed. We also observed indexes of decomposed parallel processing of hundreds and units digits but not of hundreds and tens digits. Regarding the developmental trajectories, the hundred distance effect and the unit-hundred compatibility effect showed a reliable trend to increase with grade level. However, both the significance and the increase with grade level of decomposed parallel processing were observed to be less consistent than expected. The latter is discussed in terms of different processing strategies as well as specificities differentiating between two- and three-digit numbers. Taken together, these are the first data showing decomposed processing of three-digit numbers in children. Yet, it must be noted that the results also indicate that findings from two-digit number processing cannot simply be generalized to the three-digit number range.

Neurocognition and brain structure in pediatric patients with type 1 diabetes

Recent findings suggest that in addition to severe hypoglycemia, chronic hyperglycemia may also hamper the cognitive development of patients with type 1 diabetes. Executive and memory dysfunctions mediated by frontoparietal and temporal brain structures are frequently reported to be associated with type 1 diabetes. However, most studies investigating pediatric patients with diabetes focus on either brain function or brain structure. The current study combines neuropsychological and structural brain imaging methods (i.e., voxel-based-morphometry) to study the neurofunctional integrity of frontoparietal brain areas. We investigated 30 children with type 1 diabetes and 19 healthy controls. Children with diabetes were divided into two groups representing better (HbA1c ≤ 7.9%) and worse (HbA1c ≥ 8.0%) glycemic con- trol. Our findings were threefold: First, results revealed significant group differences with respect to neuropsychological performance (i.e. response accuracies on a marker task tapping frontoparietal brain functions). Second, structural imaging disclosed significant group differences between patients and controls regarding gray matter volume in frontal (anterior cingulate) and occipital (cuneus, bordering precuneus) brain regions and regarding white matter in middle temporal and occipital gyri as well as in the ventromedial temporal lobe (uncus). Third, disease duration, age at diagnosis and white matter volume in a hippocampal region-of-interest (but not HbA1c levels, intelligence, total gray/white matter or other white/gray matter regions-of-interest) explained 56% of neuropsycholo- gical performance variance. Taken together, our findings are among the first to provide evidence of a direct link between brain function and brain structure in pediatric patients with type 1 diabetes.

Contribution of working memory in multiplication fact network in children may shift from verbal to visuo-spatial: a longitudinal investigation.

Number facts are commonly assumed to be verbally stored in an associative multiplication fact retrieval network. Prominent evidence for this assumption comes from so-called operand-related errors (e.g., 4 × 6 = 28). However, little is known about the development of this network in children and its relation to verbal and non-verbal memories. In a longitudinal design, we explored elementary school children from grades 3 and 4 in a multiplication verification task with the operand-related and -unrelated distractors. We examined the contribution of multiplicative fact retrieval by verbal and visuo-spatial short-term and working memory (WM). Children in grade 4 showed smaller reaction times in all conditions. However, there was no significant difference in errors between grades. Contribution of verbal and visuo-spatial WM also changed with grade. Multiplication correlated with verbal WM and performance in grade 3 but with visuo-spatial WM and performance in grade 4. We suggest that the relation to verbal WM in grade 3 indicates primary linguistic learning of and access to multiplication in grade 3 which is probably based on verbal repetition of the multiplication table heavily practiced in grades 2 and 3. However, the relation to visuo-spatial semantic WM in grade 4 suggests that there is a shift from verbal to visual and semantic learning in grade 4. This shifting may be induced because later in elementary school, multiplication problems are rather carried out via more written, i.e., visual tasks, which also involve executive functions. More generally, the current data indicates that mathematical development is not generally characterized by a steady progress in performance; rather verbal and non-verbal memory contributions of performance shift over time, probably due to different learning contents.