Dissociation between magnitude comparison and relation identification across different formats for rational numbers

Cross-notation knowledge of rational numbers predicts fraction arithmetic

BACKGROUND

Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered.

AIMS

To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally.

SAMPLE

Three hundred and fifty-four Chinese children (Mage = 112.1 months).

METHODS

During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division.

RESULTS AND CONCLUSIONS

Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.

EXPRESS: People's Preferences for Different Types of Rational Numbers in Linguistic Contexts

Rational numbers, like fractions, decimals, and percentages, differ in the concepts they prefer to express and the entities they prefer to describe as previously reported in display-rational number notation matching tasks and in math word problem compiling contexts. On the one hand, fractions and percentages are preferentially used to express a relation between two magnitudes, while decimals are preferentially used to represent a magnitude. On the other hand, fractions and decimals tend to be used to describe discrete and continuous entities, respectively. However, it remains unclear whether these reported distinctions can extend to more general linguistic contexts. It also remains unclear which factor, the concept to be expressed (magnitudes vs. relations between magnitudes) or the entity to be described (countable vs. continuous), is more predictive of people's preferences for rational number notations. To explore these issues, two corpus studies and a number notation preference experiment were administered. The news and conversation corpus studies detected the general pattern of conceptual distinctions across rational number notations as observed in previous studies; the number notation preference experiment found that the concept to be expressed was more predictive of people's preferences for number notations than the entity to be described. These findings indicate that people's biased uses of rational numbers are constrained by multiple factors, especially by the type of concepts to be expressed, and more importantly, these biases are not specific to mathematical settings but are generalizable to broader linguistic contexts.