Spontaneous and directed attention to number and proportion.

Although difficulties processing both symbolic and nonsymbolic proportion compared with absolute number are well established, the mechanisms involved remain unclear. We investigate four potential explanations to account for better number processing in adulthood: (a) number is more salient than proportion, (b) number is encoded more automatically than proportion, (c) proportion is more effortfully processed than number, and (d) number competes with proportion during decision making. Across three experiments, we used a delayed match-to-sample paradigm in which adults were asked which of two alternatives matched a sample set of red and blue dots. We systematically manipulated which dimension of the sample participants matched (number of red dots, total number of dots, proportion of red dots), the presence/absence of the competing quantity in the choice alternatives, and when they were told which quantitative dimension to encode (before vs. after the sample presentation, or not at all). Overall, data reveal that proportion was less salient than the numerical subset. Additionally, the number of items within the subset, but not the total number of items in the superset, interfered with proportion-based responding. Last, even in the absence of response competition and costly task demands, proportion matching took longer than number matching, highlighting basic processing differences. Together, results reveal pervasive difficulties in representing proportion compared with number, even when task demands are unambiguous. However, this varied depending on the numerical set involved and across encoding, processing, and decision processes. We discuss the implications of these findings for theories of ratio processing and of quantity more generally. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Children's understanding of most is dependent on context.

Children struggle with the quantifier "most". Often, this difficulty is attributed to an inability to interpret most proportionally, with children instead relying on absolute quantity comparisons. However, recent research in proportional reasoning more generally has provided new insight into children's apparent difficulties, revealing that their overreliance on absolute amount is unique to contexts in which the absolute amount can be counted and interferes with proportional information. Across two experiments, we test whether 4- to 6-year-old children's interpretation of most is similarly dependent on the discreteness of the stimuli when comparing two different quantities (e.g., who ate most of their chocolate?) and when verifying whether a single amount can be described with the term most (e.g., is most of the butterfly colored in?). We find that children's interpretation of most does depend on the stimulus format. When choosing between absolutely more vs. proportionally more as depicting most, children showed stronger absolute-based errors with discrete stimuli than continuous stimuli, and by 6-years-old were able to reason proportionally with continuous stimuli, despite still demonstrating strong absolute interference with discrete stimuli. In contrast, children's yes/no judgements of single amounts, where conflicting absolute information is not a factor, showed a weaker understanding of most for continuous stimuli than for discrete stimuli. Together, these results suggest that children's difficulty with most is more nuanced than previously understood: it depends on the format and availability of proportional vs. absolute amounts and develops substantially from 4- to 6-years-old.

Connecting symbolic fractions to their underlying proportions using iterative partitioning.

Fractions are a challenging mathematics topic for many elementary and middle school students, and even for adults. However, a growing body of developmental research suggests that young children can reason about visually presented proportions, well before fraction instruction, providing insight into how fractions might be introduced to improve learning. We designed a card game to teach first and second grade children (N = 195, including a racially and economically diverse sample from the United States) about fractions in one of three ways. In the Actively Divided condition we iteratively divided an area model into equal-sized units, in the Predivided condition we used an area model with the end-state of the Actively Divided condition, and in the Nondivided condition we used a continuous representation of the fraction magnitude that was not divided into unit-sized parts. Children in the actively divided condition demonstrated larger improvements matching symbolic fractions and visual fractions (i.e., pie charts) than children in the other two conditions. Posthoc analyses of children's gameplay revealed that the actively divided condition may have provided a more optimal level of difficulty for young children than the predivided condition, which was particularly difficult, and the nondivided condition, which was trivially easy. These differences in gameplay performance provide insights into possible mechanisms for our results. We discuss open research questions highlighted by this work and implications of these findings for both the development of proportional reasoning and fraction learning. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Children's gesture use provides insight into proportional reasoning strategies.

Children struggle with proportional reasoning when discrete countable information is available because they over-rely on this numerical information even when it leads to errors. In the current study, we investigated whether different types of gesture can exacerbate or mitigate these errors. Children aged 5-7 years (N = 135) were introduced to equivalent proportions using discrete gestures that highlighted separate parts, continuous gestures that highlighted continuous amounts, or no gesture. After training, children completed a proportional reasoning match-to-sample task where whole number information was occasionally pitted against proportional information. After the task, we measured children's own gesture use. Overall, we did not find condition differences in proportional reasoning; however, children who observed continuous gestures produced more continuous gestures than those who observed discrete gestures (and vice versa for discrete gestures). Moreover, producing fewer discrete gestures and more continuous gestures was associated with lower numerical interference on the match-to-sample task. Lastly, to further investigate individual differences, we found that children's inhibitory control and formal math knowledge were correlated with proportional reasoning in general but not with numerical interference in particular. Taken together, these findings highlight that children's own gestures may be a powerful window into the information they attend to during proportional reasoning.

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.

Giving a larger amount or a larger proportion: Stimulus format impacts children's social evaluations.

Young children show remarkably sophisticated abilities to evaluate others. Yet their abilities to engage in proportional moral evaluation undergoes protracted development. Namely, young children evaluate someone who shares absolutely more as being "nicer" than someone who shares proportionally more (e.g., sharing 3-out-of-6 is nicer than sharing 2-out-of-3, because 3 > 2, even though 3/6 < 2/3), whereas adults think the opposite. We investigate the hypothesis that this prior work underestimates children's proportional social reasoning by relying on discrete and spatially separated quantities (e.g., individual stickers), which can hinder proportional reasoning even outside social contexts. In three experiments we examine whether 4- and 5-year-old children's social evaluations are impacted by the discreteness and spatial separation of the resource and compare their behavior to adults (18 to 63 years; across all samples: 38% girls/women, 62% boys/men; no other demographic data was collected). We find that children are sensitive to these features: when the resource was divided into discrete units (Experiment 1) or spatially separated (Experiment 2) children were more likely to use absolute amount, as opposed to proportion, relative to when the resources were not divided and remained spatially connected. However, adults were highly sensitive to proportion regardless of the display's perceptual features (Experiment 3), and children's use of proportion remained below adult-levels. These results suggest that perceptual features influence children's use of absolute versus proportional information in their social evaluations, which has theoretical and methodological implications for understanding children's conceptions of fairness. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Preschoolers' number knowledge relates to spontaneous focusing on number for small, but not large, sets.

Much research has examined the reciprocal relations between a child's spontaneous focus on number (SFON) in the preschool years and later mathematical achievement. However, this literature relies on several different tasks to assess SFON with distinct task demands, making it unclear to what extent these tasks measure the same underlying construct. Moreover, prior studies have investigated SFON in the context of small sets exclusively, but no work has explored whether children demonstrate SFON for large sets and how this relates to children's math ability. In the current study, preschoolers were presented four distinct SFON tasks assessing their spontaneous attention to number for small (Experiment 1) and large (Experiment 2) sets of numbers. Results revealed performance across the four distinct SFON tasks was unrelated. Moreover, preschooler's SFON for small sets (1-4 items) was significantly stronger than that for large sets (10-40 items), and analyses revealed that number knowledge was only associated with SFON for small sets and not large. Together, findings suggest that SFON may not be a set-size-independent construct and instead may hinge upon a child's number knowledge, at least in the preschool years. The role of number language and how it relates to children's SFON are discussed. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Talking about proportion: Fraction labels impact numerical interference in non-symbolic proportional reasoning.

Across two experiments, we investigated how verbal labels impact the way young children attend to proportional information, well before the introduction of formal fraction education. Five- to seven-year-old children were introduced to equivalent non-symbolic proportions labeled in one of three ways: (a) a single, categorical label for multiple fractions (both 3/4 and 6/8 referred to as "blick"), (b) labels that focused on the numerator [e.g., 3/4 labeled as "three blicks" (Experiment 1) or "three-fourths" (Experiment 2)], or (c) labels that had a complete part-whole structure ("three-out-of-four"). Children then completed measures of non-symbolic proportional reasoning that pitted whole-number information against proportional information for novel proportions. Across both experiments, children who heard the categorical labels were more likely to match non-symbolic displays based on proportion than children in any of the other conditions, who demonstrated higher levels of numerical interference. These findings suggest that fraction labels have the potential to shape children's attention to proportional information even in the context of non-symbolic part-whole displays and for children who are not familiar with formal fraction symbols. We discuss these findings in terms of children's developing understanding of proportional reasoning and its implications for fraction education.

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.

Attending to relations: Proportional reasoning in 3- to 6-year-old children.

When proportional information is pit against whole number numerical information, children often attend to the whole number information at the expense of proportional information (e.g., indicating 4/9 is greater than 3/5 because 4 > 3). In the current study, we presented younger (3- to 4-year-olds) and older (5- to 6-year-olds) children a task in which the proportional information was presented either continuously (units cannot be counted) or discretely (countable units; numerical information available). In the discrete conditions, older children showed numerical interference-responding based on the number of pieces instead of the proportion of pieces. However, older children easily overcame this poor strategy selection on discrete trials if they first had some experience with continuous, proportional strategies, suggesting this prevalent reliance on numerical information may be malleable. Younger children, on the other hand, showed difficulty with the proportion task, but showed evidence of proportional reasoning in a simplified estimation-style task, suggesting that younger children may still be developing their proportional and numerical skills in task-dependent ways. Lastly, across both age groups, performance on the proportional reasoning task in continuous contexts, but not discrete contexts, was related to more general analogical reasoning skills. Findings suggest that children's proportional reasoning abilities are actively developing between the ages of 3 and 6 and may depend on domain general reasoning skills. We discuss the implications for this work for both cognitive development and education. (PsycINFO Database Record