Children See Fractions as Parts, Adults See Them as Wholes, Study Finds

Why do so many children stumble when they first meet fractions and ratios? A new study from researchers at Beijing Normal University, Peking University, and China Women’s University suggests that the answer may lie in how the developing mind represents a ratio in the first place. According to the research, published in Current Psychology, young pupils and college students appear to process non-symbolic ratios, such as the relative amounts of dots in two arrays, in fundamentally different ways. Second graders tend to zero in on the individual components of a ratio, the equivalent of numerators and denominators, while adults are far more likely to perceive the ratio as a single, integrated magnitude. The finding offers a rare direct window into the transition from a componential to a holistic representation of proportional quantities, a transition that educators have long inferred from classroom errors but rarely observed under controlled experimental conditions.

The research team, led by Qinyi Lin of Beijing Normal University’s Institute of Developmental Psychology, recruited forty-eight second-grade pupils and thirty-five college students to complete non-symbolic ratio comparison tasks. In these tasks, participants saw pairs of visual displays, each expressing a ratio, and had to judge which of the two ratios was larger. Crucially, no numbers or fraction symbols were presented; the ratios were conveyed purely through visual quantities. This design matters because symbolic fractions carry their own baggage, including years of formal instruction and well-documented misconceptions. By stripping the stimuli down to bare perceptual quantities, the researchers could ask whether the differences between children and adults reflect deep representational habits rather than merely a failure to manipulate fraction notation.

The study’s central analytical tool was the congruity effect, a signature pattern that reveals which information a person is actually using. On congruent trials, the comparison of individual components, such as the numerators, points to the same answer as the comparison of the overall ratio values. On incongruent trials, the two cues conflict: one display might have the larger numerator while the other expresses the larger overall proportion. If a participant performs noticeably worse on incongruent trials, it indicates that they are leaning on the component values rather than computing the whole ratio. If performance is similar across trial types, it suggests the person is representing the ratio holistically, as a unified magnitude that resists being decomposed into misleading parts.

The results were striking in their asymmetry. Across the experiments, a congruity effect emerged consistently, but its strength depended dramatically on age and on the type of stimulus. For the second-grade pupils, the effect was always significant: their accuracy dropped sharply whenever the component values and the overall ratios told opposing stories. This pattern, the authors argue, indicates a componential representation, in which pupils rely heavily on the individual counts, the perceptual analogues of numerators and denominators, and struggle to integrate them into a single proportional value. The adults told a different story. The congruity effect sometimes diminished or disappeared for them, suggesting that many had shifted toward a holistic pattern in which the overall ratio value, not its parts, drives the comparison.

Experiment 1 used the classic paradigm of the field: arrays of dots, where the ratio between two sets of dots had to be compared with the ratio between two other sets. Experiment 2 extended the logic to new stimulus formats, using line segments and irregular geometric areas to express ratios. This was not a cosmetic change. Much of the existing literature on proportional reasoning relies on discrete, countable stimuli like dots, which arguably invite component-based strategies because the individual elements are so salient. By showing that the age-related differences generalize to continuous quantities, including irregular areas that cannot be counted at a glance, the researchers strengthened the case that the componential-to-holistic shift is a genuine feature of cognitive development rather than an artifact of one particular stimulus type.

At the same time, the three stimulus types produced measurably different performance patterns, and this variation proved to be one of the most educationally provocative results of the study. The differences among the dot, line segment, and irregular area tasks suggest that ratio representation is not entirely format-independent. Some formats appear to scaffold holistic processing more effectively than others, and the authors propose that irregular geometric areas, which force the observer to apprehend a proportion as an undivided spatial relationship, may hold particular promise for classroom materials. If children’s proportional understanding can be scaffolded by the right perceptual format, then curriculum designers may have a powerful, low-cost lever: not more drill with fraction symbols, but a more thoughtful sequence of visual experiences that trains the mind to see the whole before the parts.

The findings resonate with a long tradition in numerical cognition research. Infants as young as six months can discriminate ratios in both numerosity and continuous extent, and neuroimaging work has identified tuning to non-symbolic proportions in the human frontoparietal cortex. Yet a parallel literature documents the natural number bias, the pervasive tendency, even among educated adults, to misapply whole-number intuitions to fractions and proportions. Studies of fraction comparison have repeatedly shown that both children and adults can adopt either component-based or holistic strategies depending on the task, the stimuli, and their expertise. The new study adds a developmental dimension to this picture by comparing learners before and after formal instruction on ratios: the pupils had not yet received systematic teaching on the concept, while the college students had long since mastered it.

That instructional contrast is what makes the study’s framing, from parts to whole, more than a catchy phrase. The authors interpret the disparity between the two groups as evidence that formal instruction on ratios may be accompanied by, and perhaps partly responsible for, a restructuring of how proportional magnitudes are mentally represented. A novice who has only encountered numbers as counts naturally treats a ratio as two counts, and the congruity effect is the behavioral fingerprint of that habit. An expert, by contrast, has compressed the two counts into a single analog magnitude, much as skilled readers perceive whole words rather than individual letters. The transition is substantial, the study concludes, and it is not automatic: without instruction that explicitly promotes relational thinking, children may remain stuck in componential mode, misjudging ratios precisely because the parts are more cognitively accessible than the whole.

For educators and parents, the practical implications are immediate. The persistent errors that children make with fractions, such as claiming that one half plus one third equals two fifths, may be symptoms of a componential representation rather than carelessness or lack of ability. Interventions that encourage children to attend to relations between quantities, rather than to the quantities themselves, have already shown promise in prior work, including studies linking spontaneous focusing on quantitative relations to later rational-number knowledge. The present study suggests that such interventions could be sharpened by incorporating continuous, and especially irregular, area-based ratio displays into early mathematics materials, before symbolic fraction instruction begins. Training the perceptual system to extract overall proportions may lay the representational foundation on which symbolic fraction knowledge can then be built.

The study also leaves open questions that future research will need to address. The samples were modest in size and drawn from a single cultural and educational context, and the cross-sectional design captures age differences rather than tracking the same children over time. Longitudinal work could establish whether the componential-to-holistic shift truly follows instruction, precedes it, or unfolds interactively with it. Still, the core message is clear and compelling: the gap between how children and adults represent ratios is not a gap in effort or intelligence but a gap in mental structure. Understanding that structure, and designing learning experiences that guide children from parts to wholes, may prove to be one of the most effective ways to ease one of mathematics education’s most stubborn stumbling blocks.

Subject of Research: Age-related differences in non-symbolic ratio processing and the developmental shift from componential to holistic proportional representation

Article Title: From parts to whole? age-related differences in non-symbolic ratio processing

Article References: Lin, Q., Chen, C., Zhang, H., Wang, W., & Deng, Z. (2026). From parts to whole? age-related differences in non-symbolic ratio processing. Current Psychology, 45(19), Article 1555. https://doi.org/10.1007/s12144-026-10120-w

Image Credits: AI Generated

DOI: 10.1007/s12144-026-10120-w

Keywords: non-symbolic ratio processing, proportional reasoning, congruity effect, componential representation, holistic representation, numerical cognition, fraction learning, cognitive development, mathematics education, dot arrays, irregular areas, natural number bias