Magnitude knowledge: the common core of numerical development

Money counts: effects of monetary vs. purely numerical values on the mental representation of quantities

It has been established that humans use different cognitive models to represent and process numerical quantities. In this study, we investigated whether the representation of monetary values fundamentally differs from the representation of numbers. We also examined the influence of both socio/economic factors and mathematical ability on such representation. A group of adults (N = 272) were tested anonymously with a variant of the number-to-position task (Siegler & Opfer, 2003). They were asked to position on a horizontal line quantities expressed either in numerical format (e.g., 50) in the "Number" conditions or as monetary values (e.g., 50€) in the "Money" conditions. The extremes of the line consisted either of specific values (i.e. "2 or 2€" and "503 or 503€") in the "Fixed" conditions or of unspecific concepts of quantity (e.g., "little" and "a lot") in the "Fuzzy" conditions. A linear model, as opposed to a logarithmic one, provided the best fit of group average data in all experimental conditions except for the "Money-Fuzzy" condition. The percentages of absolute error were significantly larger for Money stimuli than Number stimuli in both Fixed and Fuzzy conditions. This is consistent with the law of diminishing marginal utility, which entails that the value of monetary quantities is described by a concave curve rather than a linear relationship. As expected from previous research, participants who were more used to spending large quantities of money were closer to the linear representation model. Participants with higher mathematical abilities represented numerical values more closely to a linear model, but no such effect was found for monetary quantities.

Left digit bias in children's and adults' paper-and-pencil number line estimation

Number line estimation tasks are frequently used to learn about numerical thinking, learning, and development. These tasks are often interpreted as though estimates are determined by overall magnitudes of target numerals, rather than specific instantiating digits. Yet estimates are strongly biased by leftmost digits. For example, numbers like "698" are placed too far to the left of numbers like "701" on a 0-1,000 line. This "left digit effect" or "left digit bias" has been investigated little in children, and only on electronic tasks. Here, we ask whether left digit bias appears in paper-and-pencil estimates, and whether it differs for paper-based versus computer-based tasks. In Study 1, 5- to 8-year-old children completed a 0-100 number line task on paper. In Study 2, 7- to 11-year-olds completed a 0-1,000 paper task. In Study 3, adults completed tasks on paper in both ranges. Large left digit effects were observed for children aged 8 years or older and adults, but we did not find evidence for left digit bias in younger children. Study 4 compared paper and computer tasks for adults and children aged 9-12 years. Strong left digit bias was observed in all conditions, with a larger effect for the paper-based task in children. Large left digit effects in number line estimation emerge regardless of task format, with a developmental trajectory broadly consistent with other studies. For children in the age range that reliably exhibits left digit bias (but not adults), paper-and-pencil number line estimation tasks elicit even greater bias than computer-based tasks.

Mathematical Proficiency in Adolescents with ASD

This study aimed to investigate the mathematical abilities of adolescents with Autism Spectrum Disorder (ASD) compared to typically developing (TD) peers, focusing on procedural thinking, arithmetic comprehension, and algebraic technique. Sixty-seven adolescents (31 with ASD, 36 TD) participated in the study. A comprehensive mathematics skills test, incorporating oral and written components, was individually administered to assess abilities across three main mathematical domains. The study employed a mixed-methods approach, combining quantitative analyses of group differences with qualitative assessments of response patterns. Significant differences were observed between ASD and TD groups across most mathematical measures, with TD adolescents generally outperforming those with ASD. Large effect sizes were noted in procedural thinking and algebraic procedures. However, no significant differences were found in word problem-solving. Within the ASD group, considerable variability was observed, with some individuals demonstrating age-appropriate mathematical abilities while others showed consistently low performance across all domains. The study highlights the complex nature of mathematical abilities in adolescents with ASD, characterized by significant group differences and within-group variability. These findings highlight the importance of individualized approaches in mathematical education for adolescents with ASD and emphasize the need for early identification and targeted interventions to address specific challenges in mathematical learning.

Dragging but not tapping promotes preschoolers' numerical estimating with touchscreens

When solving mathematical problems, young children will perform better when they can use gestures that match mental representations. However, despite their increasing prevalence in educational settings, few studies have explored this effect in touchscreen-based interactions. Thus, we investigated the impact on young children's performance of dragging (where a continuous gesture is performed that is congruent with the change in number) and tapping (involving a discrete gesture that is incongruent) on a touchscreen device when engaged in a continuous number line estimation task. By examining differences in the set size and position of the number line estimation, we were also able to explore the boundary conditions for the superiority effect of congruent gestures. We used a 2 (Gesture Type: drag or tap) × 2 (Set Size: Set 0-10 or Set 0-20) × 2 (Position: left of midpoint or right of midpoint) mixed design. A total of 70 children aged 5 and 6 years (33 girls) were recruited and randomly assigned to either the Drag or Tap group. We found that the congruent gesture (drag) generally facilitated better performance with the touchscreen but with boundary conditions. When completing difficult estimations (right side in the large set size), the Drag group was more accurate, responded to the stimulus faster, and spent more time manipulating than the Tap group. These findings suggest that when children require explicit scaffolding, congruent touchscreen gestures help to release mental resources for strategic adjustments, decrease the difficulty of numerical estimation, and support constructing mental representations.

New measures of number line estimation performance reveal children's ordinal understanding of numbers

Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance.

The relation between number line performance and mathematics outcomes: Two meta-analyses

Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta-analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta-analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced (β = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning. RESEARCH HIGHLIGHTS: Two meta-analyses examined the link between the number line and mathematics performance. The first revealed significant relations across domains (e.g., algebra and computation). The second controlled for general cognitive ability and resulted in reduced but still significant relations. The relation between number line and fractions performance was stronger than relations to other domains.

Development of number line estsimation in Chinese preschoolers: a comparison between numerical and non-numerical symbols

To examine the level of number line estimation (NLE) in Chinese children with respect to representations of both numerical (Arabic numerals) and non-numerical symbols (dots), a total of 192 Chinese preschoolers aged between 4 and 5 years participated in four different NLE tasks. These tasks were paired to evaluate the accuracy and patterns of children's estimations in both numerical and non-numerical symbol contexts. Our findings indicate that, for Chinese preschoolers, relatively precise numerical symbol representations begin to emerge as early as 4 years of age. The accuracy of number line estimates for both 4- and 5-year-old children gradually increases in tasks involving both numerical and non-numerical symbols. Additionally, the development and patterns observed in the number line estimates of 4- and 5-year-old Chinese preschoolers are similar in both numerical symbol and non-numerical symbol tasks. These results indicate that the initiation of relatively precise numerical symbol representation and the turning point in the developmental trajectory, where the relatively precise representation for numerical symbols surpasses that of non-numerical ones, occur earlier in Chinese children than in their Western counterparts.

From whole numbers to fractions to word problems: Hierarchical relations in mathematics knowledge for Chinese Grade 6 students

It is well established in the literature that fraction knowledge is important for learning more advanced mathematics, but the hierarchical relations among whole number arithmetic, fraction knowledge, and mathematics word problem-solving are not well understood. In the current study, Chinese Grade 6 students (N = 1160; 465 girls; Mage = 12.1 years, SD = 0.6) completed whole number arithmetic (addition, subtraction, multiplication, and division), fraction (mapping, equivalence, comparison, and arithmetic), and mathematics word problem-solving assessments. They also completed two control measures: number writing speed and nonverbal intelligence. Structural equation modeling was used to investigate the hierarchical relations among these assessments. Among the four fraction tasks, the correlations were low to moderate, suggesting that each task may tap into a unique aspect of fraction understanding. In the model, whole number arithmetic was directly related to all four fraction tasks, but was only indirectly related to mathematics word problem-solving, through fraction arithmetic. Only fraction arithmetic, the most advanced fraction skill, directly predicted mathematics word problem-solving. These findings are consistent with the view that students need to build these associations into their mathematics hierarchy to advance their mathematical competence.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

The longitudinal contribution of mapping to arithmetic: Do numeral knowledge, inhibition or analogical reasoning matter?

BACKGROUND

Recent studies have revealed the association between mapping and arithmetic (Ferres-Forga et al., J. Numer. Cogn., 8, 2022, 123; LeFevre et al., J. Numer. Cogn., 8, 2022, 1).

AIM

The underlying mechanism remains unclear.

MATERIALS & METHODS

The current study recruited 118 kindergarten children and followed up on them three times at 6-month intervals. They completed measures to assess mapping skills (T1), non-verbal IQ (T1), numeral knowledge (T2), inhibitory control (T2), analogical reasoning (T2) and arithmetic (T3).

RESULTS

The results showed that mapping accounted for significant variance in arithmetic ability over and above age, gender and non-verbal IQ. Furthermore, analogical reasoning played an important role in the relationship between mapping and mathematics ability.

DISCUSSION

The findings suggest the association between mapping and mathematics ability prior to formal schooling.

CONCLUSION

Analogical reasoning, rather than numeral knowledge or inhibitory control, may drive that association in young children.

Cognitive Foundations of Early Mathematics: Investigating the Unique Contributions of Numerical, Executive Function, and Spatial Skills

There is an emerging consensus that numerical, executive function (EF), and spatial skills are foundational to children's mathematical learning and development. Moreover, each skill has been theorized to relate to mathematics for different reasons. Thus, it is possible that each cognitive construct is related to mathematics through distinct pathways. The present study tests this hypothesis. One-hundred and eighty 4- to 9-year-olds (Mage = 6.21) completed a battery of numerical, EF, spatial, and mathematics measures. Factor analyses revealed strong, but separable, relations between children's numerical, EF, and spatial skills. Moreover, the three-factor model (i.e., modelling numerical, EF, and spatial skills as separate latent variables) fit the data better than a general intelligence (g-factor) model. While EF skills were the only unique predictor of number line performance, spatial skills were the only unique predictor of arithmetic (addition) performance. Additionally, spatial skills were related to the use of more advanced addition strategies (e.g., composition/decomposition and retrieval), which in turn were related to children's overall arithmetic performance. That is, children's strategy use fully mediated the relation between spatial skills and arithmetic performance. Taken together, these findings provide new insights into the cognitive foundations of early mathematics, with implications for assessment and instruction moving forward.

Effects of a 3-factor field intervention on numerical and geometric knowledge in preschool children

The main aim of this study was to develop and test the effects of a field math intervention program on both number and geometry knowledge. The intervention was developed based on three basic skills previously associated with mathematical performance: symbolic number knowledge, mapping processes and spatial reasoning. The participants were 117 preschoolers from six schools in Cali and Bogotá. The children were assigned to an intervention group (N = 55) or a control group (N = 62). The intervention lasted 11 weeks with 3 sessions per week where the children participated in different game-based activities. Tests of numerical and geometric knowledge were administered before and after the intervention. The effects of the intervention were tested twice, immediately after the program ended and six months later. The results show that the children in the intervention group improved more than the control group in both number and geometry. The second posttest revealed a significant intervention effect for geometry, but not for numerical knowledge. The implications of these mixed patterns of results are discussed in the paper.

Attending to what's important: what heat maps may reveal about attention, inhibitory control, and fraction arithmetic performance

Math proficiency is an important predictor of educational attainment and life success. However, developing mathematical competency is challenging, and some content (e.g., fractions) can be enigmatic. Numerous factors are suspected to influence math performance, including strategy knowledge, attention, and executive functions. In two online studies, we investigated the relationship between adults' fraction arithmetic performance, confidence judgments, inhibitory control (a component of executive functions), and attention to strategy-relevant fraction components. We explored the utility of heat maps (based on mouse clicks) to measure adults' attention to strategy-relevant fraction arithmetic components (operationalized according to each mathematical operation). In Study 1, attending to strategy-relevant fraction components was correlated with inhibitory control, but this finding did not replicate in Study 2. Across both studies, inhibitory control and attention to strategy-relevant fraction components were correlated with arithmetic accuracy. Intraindividual variability in participants' attention to strategy-relevant fraction components was also found. Our findings suggest that heat map questions may be a viable alternative to assess participants' attention during fraction tasks and that attention to specific fraction-arithmetic problem features is related to problem-solving accuracy.

Middle-schoolers' misconceptions in discretized nonsymbolic proportional reasoning explain fraction biases better than their continuous reasoning: Evidence from correlation and cluster analyses

Early emerging nonsymbolic proportional skills have been posited as a foundational ability for later fraction learning. A positive relation between nonsymbolic and symbolic proportional reasoning has been reported, as well as successful nonsymbolic training and intervention programs enhancing fraction magnitude skills. However, little is known about the mechanisms underlying this relationship. Of particular interest are nonsymbolic representations, which can be in continuous formats that may emphasize proportional relations and in discretized formats that may prompt erroneous whole-number strategies and hamper access to fraction magnitudes. We assessed the proportional comparison skills of 159 middle-school students (mean age = 12.54 years, 43% females, 55% males, 2% other or prefer not to say) across three types of representations: (a) continuous, unsegmented bars, (b) discretized, segmented bars that allowed counting strategies, and (c) symbolic fractions. Using both correlational and cluster approaches, we also examined their relations to symbolic fraction comparison ability. Within each stimulus type, we varied proportional distance, and in the discretized and symbolic stimuli, we also manipulated whole-number congruency. We found that fraction distance across all formats modulated middle-schoolers' performance; however, whole-number information affected discretized and symbolic comparison performance. Further, continuous and discretized nonsymbolic performance was related to fraction comparison ability; however, discretized skills explained variance above and beyond the contributions of continuous skills. Finally, our cluster analyses revealed three nonsymbolic comparison profiles: students who chose the bars with the largest number of segments (whole-number bias), chance-level performers, and high performers. Crucially, students with a whole-number bias profile showed this bias in their fraction skills and failed to show any symbolic distance modulation. Together, our results indicate that the relation between nonsymbolic and symbolic proportional skills may be determined by the (mis)conceptions based on discretized representations, rather than understandings of proportional magnitudes, suggesting that interventions focusing on competence with discretized representations may show dividends for fraction understanding.

A potential dissociation between perception and production version for bounded but not unbounded number line estimation

BACKGROUND

What, exactly, do number line estimation (NLE) tasks measure? Different versions of the task were observed to have different effects on performance.

METHOD

We investigated associations between the production (indicating the location) and perception version (indicating the number) of the bounded and unbounded NLE task and their relationship to arithmetic.

RESULTS

A stronger correlation was observed between the production and perception version of the unbounded than the bounded NLE task, indicating that both versions of the unbounded-but not the bounded-NLE task measure the same construct. Moreover, overall low but significant associations between NLE performance and arithmetic were only observed for the production version of the bounded NLE task.

CONCLUSION

These results substantiate that the production version of bounded NLE seems to rely on proportion judgment strategies, whereas both unbounded versions and the perception version of the bounded NLE task may rely more on magnitude estimation.

Symbolic and non-symbolic predictors of number line task in Italian kindergarteners

The number line estimation task (NLE) is often used as a predictor for broader measures of mathematical achievement. In spite of its popularity, it is still not clear whether the task is based on symbolic or non-symbolic numerical competence. In particular, there is only a very limited amount of studies investigating the relationship between NLE performance and symbolic vs. non-symbolic math skills in children who have not yet begun formal schooling. This study investigates the strength of the association between NLE performance and symbolic and non-symbolic tasks in young kindergarteners. Ninety two 5-year-old children completed the NLE task (range 0-100) and a battery of early numerical competence tests including symbolic-lexical tasks, symbolic semantic tasks, and non-symbolic semantic tasks. The relationship between symbolic and non-symbolic early numerical competence and NLE performance was analyzed using a regression model based on the Bayesian Information Criterion (BIC). Results show that only symbolic semantic tasks are significant predictors of NLE performance. These results suggest that symbolic numerical knowledge is involved in number line processing among young children, whilst non-symbolic knowledge is not. This finding brings new data to the debate on the relationship between non-symbolic numeral knowledge and symbolic number processing and supports the evidence of a primary role of symbolic number processing already in young kindergarteners.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

The structure of the notation system in adults' number line estimation: An eye-tracking study

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults' numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.

Cognitive predictors of children's arithmetic principle understanding

The understanding of arithmetic principles (APs) is an important component of our conceptual mathematical knowledge, but we have limited knowledge about how children acquire APs. The current study examined this issue through a longitudinal study with 273 Chinese fourth graders. These children were assessed on various cognitive capacities (e.g., verbal and visuospatial working memory, processing speed, inhibition skill, numerical magnitude representation) in Grade 4 as well as on their AP understanding using multifaceted assessment and their arithmetic competence in Grade 5. Results from structural equation modeling suggested that verbal memory and inhibition skill directly predicted AP understanding, which in turn predicted arithmetic competence. Visuospatial working memory predicted AP understanding through numerical magnitude representation. The findings allow researchers to better account for the individual differences in AP understanding.

The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data

Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.

Different aspects of fraction understanding are associated selectively with performance on a fraction learning game

Fraction understanding seems a double-edged sword. On the one hand, it is an important predictor of later mathematical achievement. On the other hand, it is also one of the topics students struggle most in secondary math education. To complement traditional instruction, digital learning games were developed and found to successfully foster fraction understanding. However, so far, it is not known to what degree different aspects of fraction understanding (e.g., part-whole relations, fraction magnitude, fraction arithmetic) may be conveyed by such games. The current study evaluated selective associations of in-game performance of the fraction learning game Semideus with specific aspects of fraction understanding assessed using a comprehensive paper-pencil test. In Semideus, seventh graders (M_age=12.3years) from academic track secondary schools in Germany estimated the location of fractions on a number line and compared fractions according to their magnitude. Results replicated previous findings showing that in-game performance was significantly associated with mathematics achievement (i.e., math grades) and basic, whole number arithmetic skills. Moreover, we observed significant associations for aspects of fraction understanding closely matching mechanics of the learning game such as fraction number line estimation, fraction magnitude comparison and fraction ordering. These associations were observed for accuracy on the game's fraction learning tasks (e.g., estimation accuracy on the number line) but also generalized to game-based metrics such as virtual incentives (i.e., stars awarded in the game). This implies that the actual game mechanic may help to convey aspects of fraction understanding, substantiating ideas of intrinsic integration in game design.

Leveraging Math Cognition to Combat Health Innumeracy

Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias-a tendency to misapply knowledge about natural numbers to all numbers-is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people's penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and how measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.

The development of number line accuracy in elementary school children: A cross‐country longitudinal study

BACKGROUND

Number line accuracy (NL accuracy) shows improvement over the course of a school education. However, there are practically no cross-country longitudinal studies of NL accuracy over the whole course of elementary school.

AIMS

This study investigated the developmental trajectories of NL accuracy and its types across the elementary school years in two countries-Russia and Kyrgyzstan.

SAMPLE(S)

The analyses were carried out on the data collected from the sample of 508 schoolchildren at Grades 1, 2, 3 and 4 (aged 6.4-11.9 years) from Russia and Kyrgyzstan, who were surveyed as part of the 'Cross-cultural Longitudinal Analysis of Student Success' project.

METHODS

The participants were administered the 'Number Line' computerized test task and a paper-and-pencil 'Standard Progressive Matrices' test at the end of each academic year.

RESULTS

During the course of the elementary school education, NL accuracy increases nonlinearly in both samples from Grade 1 to Grade 4, with a pronounced increase in the rate of improvement from the first to the second year. Cross-country differences in NL accuracy were observed during each year of schooling as well as in the growth of NL accuracy. The development of NL accuracy is described by a model with two developmental types: (1) 'high start and growth' (93% of the pooled sample) and (2) 'low start and no growth' (7%).

CONCLUSIONS

Both NL accuracy and the rate of its growth during elementary school depend on educational conditions. Cross-country differences in the distribution of schoolchildren by these two developmental types were statistically insignificant.

Children's spatial language skills predict their verbal number skills: A longitudinal study

The process of number symbolization is assumed to be critically influenced by the acquisition of so-called verbal number skills (e.g., verbally reciting the number chain and naming Arabic numerals). For the acquisition of these verbal number skills, verbal and visuospatial skills are discussed as contributing factors. In this context, children’s verbal number skills have been found to be associated with their concurrent spatial language skills such as mastery of verbal descriptions of spatial position (e.g., in front of, behind). In a longitudinal study with three measurement times (T1, T2, T3) at an interval of about 6 months, we evaluated the predictive role of preschool children’s (mean age at T1: 3 years and 10 months) spatial language skills for the acquisition of verbal number skills. Children’s spatial language skills at T2 significantly predicted their verbal number skills at T3, when controlling for influences of important covariates such as vocabulary knowledge. In addition, further analyses replicated previous results indicating that children’s spatial language skills at T2 were associated with their verbal number skills at T2. Exploratory analyses further revealed that children’s verbal number skills at T1 predict their spatial language at T2. Results suggests that better spatial language skills at the age of 4 years facilitate the future acquisition of verbal number skills.

Children's use of egocentric reference frames in spatial language is related to their numerical magnitude understanding

Numerical magnitude information is assumed to be spatially represented in the form of a mental number line defined with respect to a body-centred, egocentric frame of reference. In this context, spatial language skills such as mastery of verbal descriptions of spatial position (e.g., in front of, behind, to the right/left) have been proposed to be relevant for grasping spatial relations between numerical magnitudes on the mental number line. We examined 4- to 5-year-old's spatial language skills in tasks that allow responses in egocentric and allocentric frames of reference, as well as their relative understanding of numerical magnitude (assessed by a number word comparison task). In addition, we evaluated influences of children's absolute understanding of numerical magnitude assessed by their number word comprehension (montring different numbers using their fingers) and of their knowledge on numerical sequences (determining predecessors and successors as well as identifying missing dice patterns of a series). Results indicated that when considering responses that corresponded to the egocentric perspective, children's spatial language was associated significantly with their relative numerical magnitude understanding, even after controlling for covariates, such as children's SES, mental rotation skills, and also absolute magnitude understanding or knowledge on numerical sequences. This suggests that the use of egocentric reference frames in spatial language may facilitate spatial representation of numbers along a mental number line and thus seem important for preschoolers' relative understanding of numerical magnitude.

Confidence in COVID problem solving: What factors predict adults' item-level metacognitive judgments on health-related math problems before and after an educational intervention?

The advent of COVID-19 highlighted widespread misconceptions regarding people's accuracy in interpreting quantitative health information. How do people judge whether they accurately answered health-related math problems? Which individual differences predict these item-by-item metacognitive monitoring judgments? How does a brief intervention targeting math skills-which increased problem-solving accuracy-affect people's monitoring judgments? We investigated these pre-registered questions in a secondary analysis of data from a large Qualtrics panel of adults (N = 1,297). Pretest performance accuracy, math self-efficacy, gender, and math anxiety were associated with pretest item-level monitoring judgments. Participants randomly assigned to the intervention condition, relative to the control condition, made higher monitoring judgments post intervention. That is, these participants believed they were more accurate when answering problems. Regardless of experimental condition, those who actually were correct on health-related math problems made higher monitoring judgments than those who answered incorrectly. Finally, consistent with prior research, math anxiety explained additional variance in monitoring judgments beyond trait anxiety. Together, findings indicated the importance of considering both objective (e.g., problem accuracy) and subjective factors (e.g., math self-efficacy, math anxiety) to better understand adults' metacognitive monitoring.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11409-022-09300-3.

Numeracy skills in young children as predictors of mathematical competence

As mathematical competence is linked to educational success, professional achievement, and even a country's economic growth, researchers have been interested in early predictors for quite some time. Although there have been numerous studies on domain-specific numerical abilities predicting later mathematical competence in preschool children, research in toddlers is scarce, especially regarding additional influential aspects, such as domain-general cognitive abilities and the children's social background. Using a large-scale dataset, the present study examined predictive effects of numeracy skills in 17-month-olds for later mathematical achievement. We found small, positive effects, even when controlling for child-related variables (i.e., age and sex) and the children's social background (i.e., maternal education and household language). Additionally, we compared results with a domain-general categorization task and found no distinct effect on mathematical competence. The present results are discussed with regard to the specificities of the dataset, as well as implications for future studies on predictors of mathematical competence.

Deal Me in: Playing Cards in the Home to Learn Math

Recent meta-analyses have demonstrated a significant association between children’s early math achievement and their experiences with math at home, including their caregivers’ talk about math. However, few studies have investigated the relations between caregiver math talk and children’s learning with experimental designs. Eighty-six children (M = 5.0 years) and their caregivers were randomly assigned to play either a numeracy or a shape card game at home for six weeks. Data were collected on children’s number and shape knowledge and families’ math talk during gameplay. There was substantial participant attrition (42% did not return completed materials), however, both an intent-to-treat analysis of the sample that received study materials and a subgroup analysis of study completers showed that children who played the shape game significantly improved their shape naming and matching skills relative to children who played the number game. Children who played the number game did not significantly improve their numerical skills relative to children who played the shape game. Mathematical talk during gameplay varied between families but was correlated over time within families. Caregivers’ and children’s talk about matching cards by shape or color predicted children’s learning from the shape game. The results suggest that despite receiving uniform instructions and materials, there was significant variability in children’s home math experiences that predicted their learning from the card game.

It matters how you start: Early numeracy mastery predicts high school math course-taking and college attendance

Using data from the Applied Problems subtest of the Woodcock-Johnson Tests of Achievement (Woodcock & Johnson, 1989/1990, Woodcock-Johnson psycho-educational battery-revised. Allen, TX: DLM Teaching Resources) administered to 1,364 children from the National Institute of Child Health and Human Development (NICHD) Study of Early Childcare and Youth Development (SECCYD), this study measures children's mastery of three numeric competencies (counting, concrete representational arithmetic and abstract arithmetic operations) at 54 months of age. We find that, even after controlling for key demographic characteristics, the numeric competency that children master prior to school entry relates to important educational transitions in secondary and post-secondary education. Those children who showed low numeric competency prior to school entry enrolled in lower math track classes in high school and were less likely to enrol in college. Important numeracy competency differences at age 54 months related to socioeconomic inequalities were also found. These findings suggest that important indicators of long-term schooling success (i.e., advanced math courses, college enrollment) are evident prior to schooling based on the levels of numeracy mastery.

Development of the Mental Number Line Representation of Numbers 0–10 and Its Relationship to Mental Arithmetic

The internal representation of numbers on the mental number line (MNL) was demonstrated by performing the computerized version of the number-to-position (CNP) task on a touchscreen while restricting response time. We found that the estimation pattern is best fit by a sigmoid function, further denoted as the “sigmoidal model”. Two developmental leaps occurring during elementary school were recognized: (1) the division of the number line into two segments and (2) consistent use of different anchor points on the number line—the left endpoint in first grade, the right endpoint in second grade, and finally the midpoint in third grade. Additionally, when examining the differences between the breakpoints, we found that first graders demonstrated a breakpoint close to 6, which linearly decreased over the years until stabilizing close to 5. The relation between the ability to place individual numbers on a number line and performance of mental arithmetic showed that the consistent use of anchor points correlated significantly with faster responses in mental arithmetic.

Numerical Training Videos and Early Numerical Achievement: A Study on 3-Year-Old Preschoolers

Early numerical abilities predict later math achievement and could be improved in children by using various training methods. As the literature on the use of training videos to develop numerical abilities is still surprisingly scant, the aim of the present study was to test the efficacy of a numerical training video on the development of counting and number line knowledge in 3-year-old preschoolers. Far transfer effects to cardinality and working memory were also examined. The study involved 86 children randomly assigned to two intervention groups: a numerical training group exposed to videos on counting and number lines; and a control group exposed to videos on colors and animal names in a foreign language. After the video training, there was an improvement in the numerical training group's counting skills, but not in their number line knowledge, and this improvement persisted six months later. The numerical training group also showed a far-transfer enhancement of cardinality six months after the intervention. Based on our results, numerical training videos could be effective in helping to enhance early numeracy skills in very young preschoolers.

Perceptions of ease and difficulty, but not growth mindset, relate to specific math attitudes

BACKGROUND

People report negative attitudes towards fractions and percentages relative to whole numbers (WNs, Sidney, Thompson, Fitzsimmons, & Taber, 2021), and these attitudes may relate to an individual's interpretation of what experiences with these number types signify. Because fractions are challenging, individual differences related to beliefs about challenge, such as endorsement of a growth versus fixed mindset (Dweck, 2006) and interpretations of easy or difficult experiences (Fisher & Oyserman, 2017), could relate to attitudes towards fractions relative to other number types.

AIMS

Two studies tested whether gender, math skills, mindset beliefs, and perceptions of difficulty relate to negative math attitudes towards specific number types.

SAMPLES

Two samples of college students (Study 1: N = 491; Study 2: N = 415), approximately 19 years of age (17% male, 51% first year students) participated.

METHODS

Participants rated attitudes pertaining to WNs, fractions, and percentages, endorsement of a growth mindset, and perceptions of ease and difficulty.

RESULTS

Replicating prior work (Sidney, Thompson, Fitzsimmons, & Taber, 2021), college students endorsed more negative attitudes about fractions than WNs and percentages. Self-reported ACT scores related to all number-type attitudes, endorsement of the belief that 'difficult tasks/goals are important' related to fraction attitudes, and endorsement of the belief that 'easy tasks/goals are possible' related to whole number attitudes. Endorsement of a growth mindset did not relate to specific math attitudes.

CONCLUSIONS

People struggle to integrate their whole number and rational number representations, and one reason people hold negative attitudes about fractions may be that they view them as difficult and even impossible.

The new unbounded number line estimation task: A systematic literature review

In 2011, Cohen and Blanc-Goldhammer introduced a new unbounded version of the number line estimation task which they argued to overcome limitations of its 'traditional' bounded counterpart. In line with this proposition, there is increasing evidence that the unbounded number line estimation task may indeed reflect a purer measure of the underlying representation of number magnitude. However, only few studies used this task version yet. In the present literature review, we aimed at evaluating all studies employing the unbounded task version so far to provide an overview of the current state of research. To identify all relevant articles and to evaluate the validity of the task, we conducted a systematic literature search in different databases following the PRISMA guidelines in May 2021. Methodological differences and commonalities of the 16 studies that met the inclusion criteria are discussed here. Our evaluation indicated considerable differences between studies with respect to the number range covered or methodological features such as display size. Additionally, five studies observed evidence for estimation biases in the unbounded task as well. Nevertheless, this review also substantiated the claim that the unbounded task version might indeed be a more valid and purer measure of the mental representation of number magnitude as the results of 14 studies confirmed this hypothesis.

Bidirectional Mapping Between the Symbolic Number System and the Approximate Number System

Abstract. Previous studies have discussed the symmetry of bidirectional mapping between approximate number system (ANS) and symbolic number system (SNS). However, these studies neglected the essential significance of bidirectional mapping in the development of numerical cognition. That is, with age, the connection strength between the ANS and SNS in ANS-SNS mapping could be higher than that in SNS-ANS mapping. Therefore, this study attempted to explore the symmetry of bidirectional mapping by examining whether the connection between the ANS and SNS is the same. Using two types of dot array materials (extensive and intensive) and sequence priming paradigms, this study found a stable negative priming effect in the ANS-SNS priming task, but no priming effect in the SNS-ANS priming task. In addition, although sensory cues (extensive and intensive) could affect performance in the ANS-SNS mapping task, these cues did not affect performance in the ANS-SNS priming task. In general, this study provides valuable insight into the symmetry of bidirectional mapping.

Preschoolers' mastery of advanced counting: The best predictor of addition skills 2 years later

The current study addressed the following question: Among preschoolers' basic numerical abilities, what are the best predictors for the later addition skills? We measured numerical abilities at preschool age and used dominance analysis to determine the dominant predictor for addition skills 2 years later. We tested seven numerical specific predictors (counting, advanced counting, enumeration, Give-N, collection comparison, number-word comparison, and approximate addition). Both quantitative and qualitative aspects (accuracy, strategy choice, and fluency) of addition skills were measured. The results show that the predictor weights for addition skills were 39% (counting), 37% (advanced counting), and 25% (collection comparison). We concluded that counting ability and especially advanced counting measured in early preschool is the most robust predictor of addition skills 2 years later (even after controlling for global cognitive abilities). This study generalized the previous findings found for Western children to Vietnamese preschoolers (N = 157, M_age = 4.8 years); extended and highlighted the role of advanced counting (count from a number other than 1) to later addition performance, mature strategy, and calculation fluency; and suggested further implications.

The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception

A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.

When one size does not fit all: A latent profile analysis of low-income preschoolers' math skills

On average, preschoolers from lower-income households perform worse on symbolic numerical tasks than preschoolers from middle- and upper-income households. Although many recent studies have developed and tested mathematics interventions for low-income preschoolers, the variability within this population has received less attention. The goal of the current study was to describe the variability in low-income children's math skills using a person-centered analysis. We conducted a latent profile analysis on six measures of preschoolers' (N = 115, mean age = 4.6 years) numerical abilities (nonsymbolic magnitude comparison, verbal counting, object counting, cardinality, numeral identification, and symbolic magnitude comparison). The results showed different patterns of strengths and weaknesses and revealed four profiles of numerical skills: (a) poor math abilities on all numerical measures (n = 13), (b) strong math abilities on all numerical measures (n = 41), (c) moderate abilities on all numerical measures (n = 35), and (d) strong counting and numeral skills but poor magnitude skills (n = 26). Children's age, working memory, and inhibitory control significantly predicted their profile membership. We found evidence of quantitative and qualitative differences between profiles, such that some profiles were higher performing across tasks than others, but the overall patterns of performance varied across the different numerical skills assessed.

Predicting Middle School Profiles of Algebra Performance Using Fraction Knowledge

Fraction knowledge and algebraic skill are closely linked. Algebra is a gatekeeper for advanced courses (Booth & Newton, 2012; Brown & Quinn, 2007). This study uses the person-centered approach of latent profile analysis to examine individual differences in middle schoolers' (N = 350) algebra performance at the end of the year (EOY). The relative importance of a range of fraction skills is considered in predicting the likelihood of displaying a particular profile of EOY algebra knowledge, measured by feature knowledge, equation encoding, equation solving, and word-problem skills. Notably, fraction number line estimation did not predict algebra performance on any of the four measures. Fraction arithmetic at the beginning of the year is most predictive of displaying the highest performing EOY algebra profile.

Reconceptualizing Symbolic Magnitude Estimation Training Using Non-declarative Learning Techniques

It is well-documented that mathematics achievement is an important predictor of many positive life outcomes like college graduation, career opportunities, salary, and even citizenship. As such, it is important for researchers and educators to help students succeed in mathematics. Although there are undoubtedly many factors that contribute to students' success in mathematics, much of the research and intervention development has focused on variations in instructional techniques. Indeed, even a cursory glance at many educational journals and granting agencies reveals that there is a large amount of time, energy, and resources being spent on determining the best way to convey information through direct, declarative instruction. The proposed project is motivated by recent calls to expand the focus of research in mathematics education beyond direct, declarative instruction. The overarching goal of the presented experiment is to evaluate the efficacy of a novel mathematics intervention designed using principles taken from the literature on non-declarative learning. The intervention combines errorless learning and structured cue fading to help second grade students improve their understanding of symbolic magnitude. Results indicate that students who learned about symbolic magnitude using the novel intervention did better than students who were provided with extensive declarative support. These findings offer preliminary evidence in favor of using learning combination of errorless learning and cue fading techniques in the mathematics classroom.

Training basic numerical skills in children with Down syndrome using the computerized game "The Number Race"

Individuals with Down syndrome (DS) present reduced basic numerical skills, which have a negative impact on everyday numeracy and mathematical learning. Here, we evaluated the efficacy of the adaptive (non-commercial) computerized game “The Number Race” in improving basic numerical skills in children with DS. The experimental group (EG; N = 30, M_{age-in-months} 118, range 70–149) completed a training playing with “The Number Race”, whereas children in the control group (CG; N = 31, M_{age-in-months} 138, range 76–207) worked with software aiming at improving their reading skills. The training lasted 10 weeks with two weekly sessions of 20–30 min each. We assessed both groups’ numerical and reading skills before and immediately after the end of the training, as well as at a 3-months follow-up. We found weak evidence for post-training groups differences in terms of overall numeracy score. However, the EG displayed substantial improvements in specific numerical skills and in mental calculation, which were maintained over time, and no improvement in reading. Conversely, the CG showed improvements in their reading skills as well as in number skills but to a lesser extent compared to the EG. Overall, “The Number Race” appears as a suitable tool to improve some aspects of numeracy in DS.

Measurement of Cognition for the National Children's Study

The National Children's Study Cognitive Health Domain Team developed detailed plans for assessing cognition longitudinally from infancy to early adulthood. These plans identify high-priority aspects of cognition that can be measured efficiently and effectively, and we believe they can serve as a model for future large-scale longitudinal research. For infancy and toddlerhood, we proposed several paradigms that collectively allowed us to assess six broad cognitive constructs: (1) executive function skills, (2) episodic memory, (3) language, (4) processing speed, (5) spatial and numerical processing, and (6) social cognition. In some cases, different trial sequences within a paradigm allow for the simultaneous assessment of multiple cognitive skills (e.g., executive function skills and processing speed). We define each construct, summarize its significance for understanding developmental outcomes, discuss the feasibility of its assessment throughout development, and present our plan for measuring specific skills at different ages. Given the need for well-validated, direct behavioral measures of cognition that can be used in large-scale longitudinal studies, especially from birth to age 3 years, we also initiated three projects focused on the development of new measures.

Neurofunctional plasticity in fraction learning: An fMRI training study

BACKGROUND

Fractions are known to be difficult for children and adults. Behavioral studies suggest that magnitude processing of fractions can be improved via number line estimation (NLE) trainings, but little is known about the neural correlates of fraction learning.

METHOD

To examine the neuro-cognitive foundations of fraction learning, behavioral performance and neural correlates were measured before and after a five-day NLE training.

RESULTS

In all evaluation tasks behavioral performance increased after training. We observed a fronto-parietal network associated with number magnitude processing to be recruited in all tasks as indicated by a numerical distance effect. For symbolic fractions, the distance effect on intraparietal activation was only observed after training.

CONCLUSION

The absence of a distance effect of symbolic fractions before the training could indicate an initially less automatic access to their overall magnitude. NLE training facilitates processing of overall fraction magnitude as indicated by the distance effect in neural activation.