Differential Development of Children’s Understanding of the Cardinality of Small Numbers and Zero

Counting and the understanding of cardinality are important steps in children’s numerical development. Recent studies have indicated that language and visuospatial abilities play an important role in the development of children’s cardinal knowledge of small numbers. However, predictors for the knowledge about zero were usually not considered in these studies. Therefore, the present study investigated whether the acquisition of cardinality knowledge on small numbers and the concept of zero share cross-domain and domain-specific numerical predictors. Particular interest was paid to the question whether visuospatial abilities – in addition to language abilities – were associated with children’s understanding of small numbers and zero. Accordingly, we assessed kindergarteners aged 4 to 5 years in terms of their understanding of small numbers and zero as well as their visuospatial, general language, counting, Arabic number identification abilities, and their finger number knowledge. We observed significant zero-order correlations of vocabulary, number identification, finger knowledge, and counting abilities with children’s knowledge about zero as well as understanding of the cardinality of small numbers. Subsequent regression analyses substantiated the influences of counting abilities on knowledge about zero and the influences of both counting abilities and finger knowledge on children’s understanding of the cardinality of small numbers. No significant influences of cross-domain predictors were observed. In sum, these results indicate that domain-specific numerical precursor skills seem to be more important for children’s development of an understanding of the cardinality of small numbers as well as of the concept of zero than the more proximal cross-domain abilities such as language and visuospatial abilities

More than simple facts: : cross-linguistic differences in place-value processing in arithmetic fact retrieval

Linguistic specificities such as the inversion property of number words (e.g., in German 43 is spoken dreiundvierzig, literally three and forty) moderate Arabic number processing. So far, cross-linguistic studies have mostly focused on inversion-related effects on simple (number comparison) and calculation-based (multi-digit addition) magnitude processing of numerical information. Despite the assumption that multiplication facts are represented in verbal format, not much attention has been paid to inversion-related influences on multiplication fact retrieval. Accordingly, the current study evaluated inversion-related effects on the processing of place-value information in multiplication. In a verification paradigm, the decade consistency effect (i.e., more errors when the decade of a solution probe shares the decade digit with the correct solution) was larger for English- than German-speaking participants for table-related probes. Processing of decade digits might be prioritised in English-speaking participants because the decade-digit is named first in English number words whereas in German number words the unit-digit is named first. Our results indicate that i) the influence of specificities of a verbal number word formation on place-value processing generalise to arithmetic fact retrieval and ii) inversion of number words might even be advantageous in specific cases

Effects of place-value and magnitude processing on word problem solving

Word problems are widely used in math instruction to convey arithmetic operations within the context of everyday situations. Although they require solving an arithmetic problem, there is only little research regarding influences of basic numerical processes on the difficulty of arithmetic word problems. Therefore, the present study set out to investigate influences of place-value and magnitude processing as reflected by the carry/borrowing and the problem size effect. Accordingly, we assessed 128 children between 8 and 11 years of age on two-digit addition and subtraction problems presented in both word problem and digital-Arabic formats. Our results indicated that word problems were more difficult to solve. Closer inspection of basic numerical processes indicated that presentation of arithmetic problems in word problem format significantly increased the effects of place-value and magnitude processing, driving a higher level of difficulty. This finding seems to suggest quantitative as well as qualitative differences in arithmetic operations and their underlying basic numerical processes when embedded in word problems

Motivational and math anxiety perspective for mathematical learning and learning difficulties

Studies on the predictors of mathematical learning and learning difficulties have been receiving growing attention. It is well established in the literature that IQ is a powerful predictor of several psychosocial outcomes, including school achievement. Other well-consolidated literature shows that cognitive–motivational effects exist but are rather small. There seems to be a big interaction between interest and specific knowledge in a domain (expertise). Variance is confounded. In this chapter, motivation and math anxiety are described within the opportunity–propensity (O-P) model including antecedent, opportunity, and propensity indicators to predict mathematical accuracy and fluency. Analyses on secondary data using this O-P model revealed that motivation as a propensity factor accounted for 10% of unique variance in addition to other predictors. Motivation and engagement were stronger predictors from kindergarten till second grade than from third till sixth grade. In addition, math anxiety might be considered a moderator of the relationship between mathematical achievement and motivation. Aside from acute effects, math anxiety might cause long-term negative impairments in mathematical performance that are kept up by a negative approach to mathematics and resulting avoidance behavior

Association between language and early numerical development – the case of quantifiers

Language and numerical development are associated in early childhood. In the present study, we focused on the association of the development of children’s cardinality understanding of small numbers and their knowledge on unspecific quantifiers (i.e., many, a few). In a longitudinal study, we pursued the research question whether children’s knowledge on quantifiers facilitated the acquisition of specific cardinality understanding. Therefore, we followed a total of 76 (34 boys and 42 girls) monolingual German-speaking children aged between 3;6 and 4;6 years at the first measurement time point for two more measurement time points 6 months and one year later. Children’s cardinality understanding of small numbers and their quantifier knowledge were assessed using a give-N task. Results indicated a full mediation effect of children’s quantifier knowledge at time point 2 on the development of their cardinality understanding from time point 2 to time point 3. This indicates a significant contribution of children’s knowledge on unspecific quantifiers to the development of their specific cardinality understanding

Mental simulation and its influence on finger-based numerical representations

BackgroundThere is evidence indicating beneficial effects of mental simulation on athletic and musical performance. We evaluated whether such beneficial effects of mental simulation generalize to the cognitive domain in terms of embodied (finger-based) numerical representations.MethodsWe assessed 70 preschoolers (36 girls, mean age 5;9) on tasks assessing different basic numerical skills (e.g., counting, cardinality understanding, number composition, etc.) as well as different aspects of finger-based numerical representations. A subgroub completed a mental simulation phase prior to testing finger-based representations.ResultsChildren who completed the mental simulation phase, performed better on the tasks assessing finger-based representations compared to, children who did not complete the simulation phase. This held even when controlling for performance in basic numerical skills.ConclusionThis study provides evidence that beneficial effects of mental simulation generalize to embodied (finger-based) numerical representations. Mental simulation may be useful to integrate in the instruction of basic numerical skills.</div

A longitudinal study on basic numerical skills in early numerical development

Basic numerical skills underlie children's numerical development. In this follow-up study, we investigated the associations between the development of counting, cardinality understanding, and awareness of structures in quantities in children at the age of 4;7 and 5;3 years. We assumed that the ability to consider structures in quantities should allow children to capture the cardinality of non-symbolic quantities more efficiently and thus moderate the development of counting and cardinality understanding. We assessed 57 (32 girls) children in a pretest at about four and a half years of age and again in a posttest six months later. Results showed that structured quantities were captured more efficiently than unstructured quantities which might have been counted. Moreover, results substantiated the mediating role of considering structures in quantities on the development of children's counting and cardinality understanding. As such, this follow-up study indicated beneficial contributions of awareness of structures on the development of counting and cardinality understanding. This may suggest that introducing structured quantities early on in elementary math education might be sensible