Across cultures and development, humans can extract proportional quantities from nonsymbolic, visual displays. Two accounts have been proposed for the contributions of these nonsymbolic skills to fraction understanding. The shared-magnitude account suggests that a cognitive system responsible for these nonsymbolic skills is foundational for symbolic fraction learning, predicting that neural patterns of fraction skills resemble magnitude-driven neural responses to continuous depictions, like unsegmented bars, that emphasize proportional magnitude. However, other depictions of proportions, like discretized, segmented bars, convey both proportional and whole-number information. The shared-interference account proposes that the interference from whole-number information in discretized proportions and symbolic fractions drives the link between these skills and predicts that neural responses to fraction magnitudes would strongly relate to overall activity to discretized depictions, particularly those with misleading whole-number information. We leveraged neural representational similarity (NRS) analyses to test these predictions. Nineteen young adults (mean age = 23.26 years) compared proportions depicted in continuous, discretized, and symbolic formats. Univariate results indicated robust neural distance effects in frontal, parietal, and occipital regions across formats. Region of interest (ROI) analyses of the intraparietal sulcus (IPS) revealed neural rational distance effects for continuous proportions, whereas for discretized and symbolic fractions, only comparisons with whole-number interference exhibited a rational magnitude signal. Critically, NRS demonstrated that the IPS fraction magnitude-related activity showed greater similarity with discretized brain responses than continuous ones, but only in the context of misleading whole-number information. Together, these findings support and refine the two accounts and suggest that symbolic fraction proficiency, as well as adept discretized proportional reasoning, involves accessing proportional magnitude code despite the presence of misleading whole-number information.

Relative to fractions, decimal numbers are thought to be easier for students to learn because they employ the same base-10 system as whole numbers. However, unlike whole numbers, larger decimals can have fewer digits, leading to worse performance when comparing Inconsistent decimal pairs, like 0.8 vs 0.26, than Consistent pairs like 0.86 vs 0.2. Students may be applying the whole number rule: “more digits = larger number” or they could be ignoring the decimal points and comparing 8 vs 26. This study used neuroimaging and our specially designed stimulus set to distinguish between these possibilities. We focused on the intraparietal sulcus (IPS), implicated in numerical magnitude processing, and the anterior cingulate cortex (ACC) and insula, implicated in inhibitory control. We found no neural differences between Consistent and Inconsistent comparisons, suggesting that the number of digits does not drive brain responses in skilled adults (n=21). Instead, for Consistent comparisons, we found that the IPS was sensitive to the actual distance between the decimals, while the ACC showed this pattern for Inconsistent comparisons. Crucially, we also examined the effect of the distance between the decimal pairs when ignoring the decimal point. Here, we found sensitivity to this distance among Inconsistent comparisons in the IPS and insula, suggesting that whole number referents are automatically processed during decimal comparison and require engagement of cognitive control regions to counteract. More broadly, our results underscore the unique challenges of decimal notation, revealing the need for educational practices that emphasize differences to whole numbers rather than highlighting similarities.

Working memory (WM) and socioeconomic status (SES) associations with math achievement are well established, but little work has considered whether WM contributions are the same across SES strata. Here, we focus on the extent to which verbal WM predicts math differently for children in low vs. higher SES homes in the US to disentangle how these relations might play out across development in differently resourced contexts, as well as the magnitude of these relationships across SES. Using data from a nationally representative sample of 13,527 children, we estimated a parallel process model to analyze the relation among the starting points and growth of verbal WM and math skills through elementary school. A multi-group model was used to compare the relative strength of associations between verbal WM and growth in math skills between low-income and not-low-income students. Results indicated that students with higher initial verbal WM levels developed math skills faster than those with lower verbal WM levels, especially in low-income households where verbal WM's effect size was nearly double relative to estimates for children in higher-income homes. Findings highlight the importance of providing families with the necessary resources to help children thrive in adverse contexts.

When learning about decimals, whole number knowledge can be a detriment, leading children to incorrectly report, for example, that 0.26 is larger than 0.8. Two potential sources of whole number interference could lead to such errors. Digit length interference arises from the whole number rule that more digits = larger number, while whole referent magnitude interference arises from ignoring the decimal point and comparing the whole referents (26 > 8). The independent effects of each interference type have been measured in adults, but we still do not know how these effects play out in early decimal learning. Further, inhibitory control has been linked to overcoming whole number interference, but it is unclear which specific type of interference is inhibited. Here, we used carefully designed decimal stimuli to examine these two interference effects in middle school students who are more susceptible to whole number interference than adults. Students in 2 U.S. school districts (grades 6-8, n = 178) completed computerized decimal comparison and inhibitory control tasks. We implemented cluster analysis to account for heterogeneous strategy use. The two most prominent groups, Whole Number Biased and High Performing, differed in their extent of digit length interference, but were equally susceptible to whole referent magnitude interference. Crucially, inhibitory control only related to overcoming digit length interference, not whole referent magnitude interference, in both groups. Taken together, whole referent magnitudes are a pervasive source of interference in decimal comparison, independent of overall task performance or individual differences in inhibitory control.

Fraction thinking poses a challenge for students, and several flawed comparison strategies have been identified: whole-number-strategy (choosing larger numerals), reverse-strategy (choosing smaller numerals), and gap-strategy (choosing the smaller difference between numerator and denominator). The prevalence of these strategies among college students is unknown. Here, we used cluster analysis to identify strategy use among 90 college students. Three cognitive factors were also assessed: general math achievement, inhibitory control, and working memory. The results revealed three clusters: Whole-Number-Strategy (14%), who used whole-number-strategy consistently; Partial-Reverse-Strategy (30%), who used reverse-strategy for more challenging fractions; and Gap-Tendency (56%), who performed well except when gap strategy fails. For cognitive factors, Gap-Tendency performed better than Whole-Number-Strategy, but did not differ from Partial-Reverse-Strategy, on any measure. These findings extended prior research on strategy choices to show that the majority of college students have overcome whole bias, but not yet achieved a complete understanding of fraction magnitude.

Students often rely on flawed strategies to compare fractions, focusing on individual components rather than rational magnitudes. Only a handful of studies have explored whether these strategies result in difficulties in other fraction domains or whether they are the consequence of reduced cognitive capacities or attending to the wrong numerical distances (e.g., numerator and denominator distances). Mexican high school students (N=76, mean age=16.18 years) completed a fraction comparison task with pairs either compatible with whole-number rules (e.g., 18/19 vs. 12/19) or misleading (e.g., 23/49 vs. 23/30). Participants completed conceptual and procedural fraction knowledge tests and three executive function tasks. First, cluster analyses revealed that almost half of the students used flawed componential fraction comparison strategies. Particularly, we found two biased (whole-number bias and reverse bias) groups and a third group with overall high performance. Notably, whole-number biased students had lower math achievement, conceptual and procedural fraction knowledge than reverse biased or high-performance students. Next, we probed differences in rational and componential magnitude processing between these groups. Remarkably, both biased groups showed neither rational nor componential distance effects. In contrast, high-performing students' performance was better explained by robust rational distance effects. Together, these results suggest that while fraction conceptual and procedural knowledge distinguish whole-number bias students from reverse-bias and high-performing students, only rational magnitude processing distinguishes between students with flawed strategies and high-performing students.

Growing evidence highlights the predictive power of cross-notation magnitude comparison (e.g., 2/5 vs. 0.25) for math outcomes, but whether these relations persist into adulthood and the underlying mechanisms remain unknown. Across two studies during the 2021-2022 academic year, we investigated undergraduates' cross-notation and within-notation comparison skills given equivalent fractions, decimals, and percentages (Study 1, N = 220 and Study 2, N = 183). We found participants did not perceive equivalent rational numbers equivalently. Cluster analyses revealed that approximately one-quarter of undergraduates exhibited a bias to select percentages as larger in cross-notation comparisons. Compared with the other cluster of undergraduates who showed little-to-no bias, the percentages-are-larger bias cluster performed worse on fraction number line estimation and fraction arithmetic (exact and approximate), as well as reporting lower Scholastic Aptitude Test/American College Test (SAT/ACT) scores. Hierarchical linear regression analyses demonstrated that cross-notation comparison accuracy accounted for variance in SAT/ACT beyond within-notation accuracy. Mediation analyses were consistent with a potential mechanism: Stronger cross-notation knowledge equips individuals to evaluate the reasonableness of fraction arithmetic solutions. Together, these results suggest the importance of an integrated understanding of rational number notations, which may not be fully assessed by within-notation measures alone. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

Individuals with autism can show intact decoding (i.e., ability to recognize and pronounce written words accurately). However, reading comprehension (i.e., ability to infer meaning from written text) in autistic individuals is often lower than expected based on age or grade level. Having intact decoding skills despite potentially atypical reading comprehension suggests altered reading pathways in autism, particularly when processing semantics (i.e., word meaning). To test for neural differences in word processing between autistic and non-autistic younger adults, we examined behavioral and neural responses to reading aloud words and pronounceable nonsense words (pseudowords). Additionally, we manipulated word imageability, word frequency, and word and pseudoword spelling-sound consistency as probes for different components (i.e., orthography, phonology and semantics) of the reading system. Behaviorally, the autistic group had a greater reduction in reaction time as word imageability increased. Neurally, pseudoword consistency effects, a probe of spelling-sound mappings without semantics, were only observed in the autistic group, where increased consistency was associated with decreased activity in bilateral intraparietal sulcus. Also compared to the non-autistic group, the autistic group showed greater effects of word consistency, where increasing word consistency was associated with increasing activation in the bilateral posterior superior temporal gyrus and ventral occipitotemporal cortex. Finally, the autistic group showed stronger effects of pseudoword consistency than the non-autistic group, that is increasing pseudoword consistency was associated with decreasing activation in the left ventral occipitotemporal cortex. Together, these results point to differences in how neural resources are used for reading, with more bilateral areas recruited during spelling-sound decoding in autistics to achieve comparable performance to non-autistics.

Decimal numbers are generally assumed to be a straightforward extension of the base-ten system for whole numbers given their shared place value structure. However, in decimal notation, unlike whole numbers, the same magnitude can be expressed in multiple ways (e.g., 0.8, 0.80, 0.800, etc.). Here, we used a number line task with carefully selected stimuli to investigate how equivalent decimals (e.g., 0.8 and 0.80 on a 0-1 number line) and proportionally equivalent whole numbers (e.g., 80 on a 0-100 number line) are estimated. We find that young adults (N = 88, Mage = 20.22 Years, SD = 1.65, 57 female) have a linear response pattern for both decimals and whole numbers, but that double-digit decimals (e.g., 0.08, 0.82, 0.80) are systematically underestimated relative to proportionally equivalent whole numbers (e.g., 8, 82, 80). Moreover, decimal string length worsens the underestimation, such that single- digit decimals (e.g., 0.8) are perceived as smaller than their equivalent double-digit decimals (e.g., 0.80). Finally, we find that exposing participants to whole number stimuli before decimal stimuli induces magnitude-based underestimation, that is, greater underestimation for larger decimals. Together, these results suggest a small but persistent underestimation bias for decimals less than one, and further that decimal magnitude estimation is fragile and subject to greater underestimation when exposed to whole numbers.

Mounting evidence points to the predictive power of cross-notation rational number understanding (e.g., 2/5 vs. 0.25) relative to within-notation understanding (e.g., 2/5 vs. 1/4) in predicting math outcomes. Although correlational in nature, these studies suggest that number sense training emphasizing integrating across notations may have more positive outcomes than a within-notation focus. However, this idea has not been empirically tested. Thus, across two studies with undergraduate students (N = 183 and N = 181), we investigated the effects of a number line training program using a cross-notation approach (one that focused on connections among fractions, decimals, and percentages) and a within-notation approach (one that focused on fraction magnitude representation only). Both number line approaches produced positive effects, but those of the cross-notation approach were larger for fraction magnitude estimation and cross-notation comparison accuracy. In a third study (N = 63), we adapted the cross-notation number line training for use in place of typical classroom warm-up activities for middle school students. Similar to the results with undergraduate students, the cross-notation training program yielded positive benefits for middle school students over a typical warm-up activity (fraction arithmetic practice). Together, these results suggest the importance of an integrated approach to teaching rational number notations, an approach that appears to be uncommon in current curricula.

Emerging research suggests that episodic memory challenges are commonly encountered by autistic individuals; however, the specific nature of these memory challenges remains elusive. Here, we address critical gaps in the literature by examining pattern separation memory, the ability to store distinct memories of similar stimuli, and its links to the core autistic trait of repetitive, restricted interests and behaviors. Utilizing a large sample of over 120 autistic children and well-matched non-autistic peers, we found that autistic children showed significantly reduced performance on pattern separation memory. A clustering analysis identified three distinct pattern separation memory profiles in autism, each characterized by reduced or increased generalization abilities. Importantly, pattern separation memory was negatively correlated with the severity of repetitive, restricted interest and behavior symptoms in autism. These findings offer new evidence for challenges in pattern separation memory in autism and emphasize the need to consider these challenges when assessing and supporting autistic individuals in educational and clinical settings. Lay abstract Memory challenges remain understudied in childhood autism. Our study investigates one specific aspect of memory function, known as pattern separation memory, in autistic children. Pattern separation memory refers to the critical ability to store unique memories of similar stimuli; however, its role in childhood autism remains largely uncharted. Our study first uncovered that the pattern separation memory was significantly reduced in autistic children, and then showed that reduced memory performance was linked to their symptoms of repetitive, restricted interest and behavior. We also identified distinct subgroups with profiles of reduced and increased generalization for pattern separation memory. More than 72% of autistic children showed a tendency to reduce memory generalization, focusing heavily on unique details of objects for memorization. This focus made it challenging for them to identify commonalities across similar entities. Interestingly, a smaller proportion of autistic children displayed an opposite pattern of increased generalization, marked by challenges in differentiating between similar yet distinct objects. Our findings advance the understanding of memory function in autism and have practical implications for devising personalized learning strategies that align with the unique memory patterns exhibited by autistic children. This study will be of broad interest to researchers in psychology, psychiatry, and brain development as well as teachers, parents, clinicians, and the wider public.

Fractions, decimals, and percentages are generally assumed to differ in difficulty based on the degree to which their structure aligns with, or differs from, that of whole numbers. Percentages are viewed as most similar to whole numbers with their fixed, unstated denominator of 100. Decimals are often assumed to be easier than fractions because their place-value structure is an extension of the base-ten system for whole numbers, unlike fractions, which have a bipartite structure (i.e., a/b). However, unlike whole numbers, a longer string-length for decimals and fractions does not always signify a larger magnitude. To assess understanding of the four notations, we measured number line estimation of equivalent fractions and decimals with shorter string-lengths (e.g., 8/10 and 0.8) and longer string-lengths (e.g., 80/100 and 0.80), percentages (e.g., 80%), and proportionally equivalent whole numbers on a 0-100 scale (e.g., 80.0). Middle school students (N = 65, 33 female) generally underestimated all numbers (Whole Numbers: 3%, Percentages: 2%, Decimals: 17%, and Fractions: 5% below the actual value). Shorter string-length decimals and fractions were estimated as smaller than equivalent longer string-length ones; and larger magnitude decimals and fractions were underestimated by greater amounts than smaller ones. Overall, percentages were estimated similarly to corresponding whole numbers, fractions had modest string-length effects, and decimals were the most underestimated, especially for single-digit decimals. These results highlight the strengths and weaknesses of children’s understanding of each notation’s magnitudes and challenge the assumption that decimals are always better understood than fractions.

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 middleschoolers' 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, suggest-ing that interventions focusing on competence with discretized representations may show divi-dends for fraction understanding.

The present study tests two predictions stemming from the hypothesis that a source of difficulty with rational numbers is interference from whole number magnitude knowledge. First, inhibitory control should be an independent predictor of fraction understanding, even after controlling for working memory. Second, if the source of interference is whole number knowledge, then it should hinder fraction understanding. These predictions were tested in a racially and socioeconomically diverse sample of US children (N=765; 337 female) in grades 3 (ages 8-9), 5 (ages 10-11), and 7 (ages 12-13) who completed a battery of computerized tests. The fraction comparison task included problems with both shared components (e.g., 3/5 > 2/5) and distinct components (e.g., 2/3 > 5/9), and problems that were congruent (e.g., 5/6 > 3/4) and incongruent (e.g., 3/4 > 5/7) with whole number knowledge. Inhibitory control predicted fraction comparison performance over and above working memory across component and congruency types. Whole number knowledge did not hinder performance and instead positively predicted performance for fractions with shared components. These results highlight a role for inhibitory control in rational number understanding and suggest that its contribution may be distinct from inhibiting whole number magnitude knowledge.

Introduction Executive functions (EFs) are linked to positive outcomes across the lifespan. Yet, methodological challenges have prevented precise understanding of the developmental trajectory of their organization. Methods We introduce novel methods to address challenges for both measuring and modeling EFs using an accelerated longitudinal design with a large, diverse sample of students in middle childhood ( N = 1,286; ages 8 to 14). We used eight adaptive assessments hypothesized to measure three EFs, working memory, context monitoring, and interference resolution. We deployed adaptive assessments to equate EF challenge across ages and a data-driven, network analytic approach to reveal the evolving diversity of EFs while simultaneously accounting for their unity. Results and discussion Using this methodological paradigm shift brought new precision and clarity to the development of these EFs, showing these eight tasks are organized into three stable components by age 10, but refinement of composition of these components continues through at least age 14.

Growing evidence points to the predictive power of cross-notation rational number understanding (e.g., 2/5 vs. 0.25) relative to within-notation understanding (e.g., 2/5 vs. 1/4) in predicting math outcomes. Though correlational in nature, these studies suggest that number sense training emphasizing integrating across notations may have more positive outcomes than a within-notation focus. However, this idea has not been empirically tested. Thus, across two studies with undergraduate students (N=183 and N=181), we investigated the effects of a number line training program using a cross-notation approach (one that focused on connections among fractions, decimals, and percentages) and a within-notation approach (one that focused on fraction magnitude representation only). Both number line approaches produced positive effects, but those of the cross-notation approach were larger for fraction magnitude estimation and cross-notation comparison accuracy. Together, these results suggest the importance of an integrated approach to teaching rational number notations, an approach that appears to be uncommon in current curricula.

Children with autism spectrum disorders (ASDs) often display atypical learning styles; however, little is known regarding learning-related brain plasticity and its relation to clinical phenotypic features. Here, we investigate cognitive learning and neural plasticity using functional brain imaging and a novel numerical problem-solving training protocol. Children with ASD showed comparable learning relative to typically developing children but were less likely to shift from rule-based to memory-based strategy. While learning gains in typically developing children were associated with greater plasticity of neural representations in the medial temporal lobe and intraparietal sulcus, learning in children with ASD was associated with more stable neural representations. Crucially, the relation between learning and plasticity of neural representations was moderated by insistence on sameness, a core phenotypic feature of ASD. Our study uncovers atypical cognitive and neural mechanisms underlying learning in children with ASD, and informs pedagogical strategies for nurturing cognitive abilities in childhood autism.

A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers - the numbers unadorned with decimal points (e.g., 8 < 27) - and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual rational distance, those with larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 < < 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 58 participants, Study 2: n = 78). Across both, we find a main effect of consistency, demonstrating string length effects, and also that whole number distance interferes with processing conflicting decimals, demonstrating semantic interference effects. Evidence for both effects supports the semantic interference account, highlighting that decimal comparison difficulties arise from multiple competing numerical codes. Finally, for accuracy we found no relationship between whole number distance sensitivity and math achievement, indicating that whole number magnitude interference affects participants similarly across the spectrum of math achievement.

Individuals on the autism spectrum often have trouble with social and figurative language. As social language is often figurative, it can be challenging to disentangle the cognitive and neural sources of these difficulties. Neural systems for social cognition and language comprehension overlap in areas involved in retrieving linguistic meaning (semantics), such as the anterior temporal lobe (ATL), ventro-medial prefrontal cortex (vmPFC), posterior cingulate cortex (PCC), and posterior middle temporal gyrus (pMTG). Using adjective-noun phrases, we manipulated social/nonsocial and figurative/literal dimensions, which we expected to activate distinct but overlapping regions. We hypothesized that activation differences in the group with autism (AUT) would be greater for more social and figurative stimuli. During fMRI, participants in the AUT group (N = 19) and those in the non-autistic comparison (NAC) group (N = 22) made familiarity judgments to 192 phrases in a balanced 2 × 2 (social/nonsocial x figurative/literal) design. Social phrases activated the PCC in all participants, but only the NAC group activated the vmPFC. Figurative phrases were rated as more literal by the AUT group, with the figurative-literal phrase contrast showing no activation in the AUT group, but activating the PCC and right pMTG in the NAC group. The one significant group-level neural difference was for the social-figurative condition predicted to be most different between groups: greater activation for the AUT group in the right ATL. Differences in the right ATL and pMTG in the AUT group suggest altered engagement of right homologues of the canonical semantic network being recruited for processing combined social and figurative language.

Fractions have a fundamental importance in the design and construction of mathematical knowledge, being the basis for algebra and other advanced mathematical content (BAILEY et al., 2012; BOOTH; NEWTON, 2012; SIEGLER et al., 2012; TORBEYNS et al., 2015). However, historically this representational form of rational numbers has presented several obstacles in the teaching and learning process, both for students and teachers (PINTO, 2011; SERRAZINA; RODRIGUES, 2018; SIEGLER; THOMPSON; SCHNEIDER, 2011). One of the conceptual bases for the development of fractional thinking is understanding the magnitude of these numbers (SIEGLER et al. 2011, 2013; VAN HOOF et al., 2018), yet how teachers process fractions magnitudes remain unknown. Thus, this study aimed to use the lens of experimental math cognition research to investigate how postgraduate mathematics teachers process the magnitude of fractions. A convergent parallel mixed method was carried out: quantitative data were collected based on research in neuroscience and cognitive psychology, followed by a qualitative task where participants explained their answers on a subset of the comparisons. Teachers’ comparison performance indicated they were not thinking about fractions according to the fractions’ individual components. Instead, they used other strategies to compare symbolic fractions, mostly anchored in the part-whole perspective. The most commonly used strategy was the non-generalizable Gap strategy, where participants choose the fraction with a smaller difference between numerator and denominator rather than fully processing the magnitude of fractions. Thus, we can say that the investigated teachers were not influenced by a componential view, but we cannot affirm a holistic view either, since the most used strategy contains misconceptions and is based on an arithmetic procedure. Interestingly, during the qualitative task, which asked teachers textually to justify the choice of the largest fraction, some users of the Gap strategy recognized that it was not mathematically valid. Therefore, a direct implication of this research is the need for teaching strategies that illustrate failures of the Gap strategy. More broadly, instructional approaches that challenge intuitive thinking, allowing us to understand the magnitude of the fraction (SIEGLER et al., 2011) may prove useful for this population.