The development of the mental representations of the magnitude of fractions

Cross-notation knowledge of fractions and decimals

Understanding fractions and decimals requires not only understanding each notation separately, or within-notation knowledge, but also understanding relations between notations, or cross-notation knowledge. Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. This hypothesis was tested by reanalyzing three published datasets involving fourth- to eighth-grade children from the United States and Finland. All datasets included measures of rational number arithmetic, within-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. fractions and decimals vs. decimals), and cross-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. decimals). Consistent with the hypothesis, cross-notation magnitude knowledge predicted fraction and decimal arithmetic when controlling for within-notation magnitude knowledge. Furthermore, relations between within-notation magnitude knowledge and arithmetic were not notation specific; fraction magnitude knowledge did not predict fraction arithmetic more than decimal arithmetic, and decimal magnitude knowledge did not predict decimal arithmetic more than fraction arithmetic. Implications of the findings for assessing rational number knowledge and learning and teaching about rational numbers are discussed.

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Children's understanding of fraction and decimal symbols and the notation-specific relation to pre-algebra ability

Fraction and decimal concepts are notoriously difficult for children to learn yet are a major component of elementary and middle school math curriculum and an important prerequisite for higher order mathematics (i.e., algebra). Thus, recently there has been a push to understand how children think about rational number magnitudes in order to understand how to promote rational number understanding. However, prior work investigating these questions has focused almost exclusively on fraction notation, overlooking the open questions of how children integrate rational number magnitudes presented in distinct notations (i.e., fractions, decimals, and whole numbers) and whether understanding of these distinct notations may independently contribute to pre-algebra ability. In the current study, we investigated rational number magnitude and arithmetic performance in both fraction and decimal notation in fourth- to seventh-grade children. We then explored how these measures of rational number ability predicted pre-algebra ability. Results reveal that children do represent the magnitudes of fractions and decimals as falling within a single numerical continuum and that, despite greater experience with fraction notation, children are more accurate when processing decimal notation than when processing fraction notation. Regression analyses revealed that both magnitude and arithmetic performance predicted pre-algebra ability, but magnitude understanding may be particularly unique and depend on notation. The educational implications of differences between children in the current study and previous work with adults are discussed.

Developmental changes in the whole number bias

Many students' knowledge of fractions is adversely affected by whole number bias, the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction's magnitude (ratio of numerator to denominator). Although whole number bias appears early in the fraction learning process and under speeded conditions persists into adulthood, even among mathematicians, little is known about its development. Performance with equivalent fractions indicated that between fourth and eighth grade, whole number bias decreased, and reliance on fraction magnitudes increased. These trends were present on both fraction magnitude comparison and number line estimation. However, analyses of individual children's performance indicated that a substantial minority of fourth graders did not show whole number bias and that a substantial minority of eighth graders did show it. Implications of the findings for development of understanding of fraction equivalence and for theories of numerical development are discussed.

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction-decimal mixed pairs and decimal-fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.

The componential processing of fractions in adults and children: effects of stimuli variability and contextual interference

Recent studies have indicated that people have a strong tendency to compare fractions based on constituent numerators or denominators. This is called componential processing. This study explored whether componential processing was preferred in tasks involving high stimuli variability and high contextual interference, when fractions could be compared based either on the holistic values of fractions or on their denominators. Here, stimuli variability referred to the fact that fractions were not monotonous but diversiform. Contextual interference referred to the fact that the processing of fractions was interfered by other stimuli. To our ends, three tasks were used. In Task 1, participants compared a standard fraction 1/5 to unit fractions. This task was used as a low stimuli variability and low contextual interference task. In Task 2 stimuli variability was increased by mixing unit and non-unit fractions. In Task 3, high contextual interference was created by incorporating decimals into fractions. The RT results showed that the processing patterns of fractions were very similar for adults and children. In task 1 and task 3, only componential processing was utilzied. In contrast, both holistic processing and componential processing were utilized in task 2. These results suggest that, if individuals are presented with the opportunity to perform componential processing, both adults and children will tend to do so, even if they are faced with high variability of fractions or high contextual interference.