Fraction knowledge is foundational for later mathematics achievement, yet its developmental organization remains debated and is constrained by the limited availability of grade-appropriate measures with established psychometric properties. We developed and evaluated the Fraction Knowledge Assessment (FKA), a set of grade-specific instruments for 2nd, 3rd, 5th, and 6th grades, and examined changes in fraction knowledge across two adjacent-grade transitions. Using an accelerated longitudinal design, we followed 342 children in two cohorts (2nd-3rd grade; 5th-6th grade). Participants completed standardized mathematics and cognitive measures at baseline and grade-appropriate FKAs across two consecutive years. Confirmatory factor analysis and Item Response Theory were used to examine dimensional structure, item functioning, and to link FKA versions across grades prior to developmental analyses. Results provided no strong evidence for a robust separation between conceptual and procedural fraction knowledge, with two-factor solutions showing comparable fit to one-factor models and highly correlated factors. Item-level analyses revealed developmental shifts: fraction magnitude items differentiated ability more strongly in earlier grades, whereas fraction arithmetic items were more discriminative in later grades. Using equated ability scores, significant gains were observed in both cohorts, with larger improvements from 2nd to 3rd grade than from 5th to 6th grade. Baseline mathematics achievement and working memory were longitudinally associated with later fraction knowledge. These findings support an integrated account of fraction knowledge development and illustrate how psychometrically linked, grade-appropriate assessments can be used to characterize developmental change.

A substantial body of research has demonstrated that human and nonhuman animals have perceptually-based abilities to process magnitudes of nonsymbolic ratios (e.g., ratios composed by juxtaposing two line segments). Lewis et al. (2015) extended Dehaene and Cohen’s (2007) neuronal recycling hypothesis to include neurocognitive architectures for nonsymbolic ratio processing, proposing that these systems might support symbolic fractions acquisition. We tested two key propositions: (1) children should show neural sensitivity to nonsymbolic fractions before receiving formal fractions instruction, and (2) they should leverage this foundation by recruiting neural architectures for nonsymbolic fractions processing for symbolic fractions. We compared nonsymbolic and symbolic fractions processing among 2nd-graders (n=28, ages 7.5 - 8.8), who had not yet received formal symbolic fractions instruction, and 5th-graders (n=33, ages 10.3 - 11.9), who had. During fMRI scanning, children performed ratio comparison tasks, determining which of two nonsymbolic or symbolic fractions was larger. Both cohorts showed behavioral and neural evidence of processing nonsymbolic and symbolic fractions magnitudes, with performance modulated by numerical distance between stimuli. Consistent with predictions, 2nd-graders recruited a right parietal-frontal network for nonsymbolic fractions but not for symbolic fractions, whereas 5th-graders recruited a bilateral parietal-frontal network for both, overlapping with but extending beyond that of 2nd-graders. Neural representational similarity analysis revealed significant similarities between nonsymbolic and symbolic fraction processing in the intraparietal sulcus, a key region for numerical processing. These results present the first developmental neuroimaging evidence that neural substrates for nonsymbolic ratios exist before formal learning, which may be recycled to process symbolic fractions.

Learning fractions is essential for academic and daily life success. A critical first step in acquiring fractions is learning to transcode them (e.g., writing ½ when hearing “one half”). However, little is known about how students master fraction transcoding. We addressed this gap by assessing fraction writing in two groups of Brazilian students with limited education: adults in the first year of an adult education program (AEP-1) and 2nd graders. Both groups made frequent transcoding errors. Errors were classified into three categories, Syntactic: correct numerator/denominator values with an incorrect notation (12th for “one half”); Lexical: incorrect numerals with the correct notation (⅓ for “one half”); Combined: incorrect numerals and notation (15th for “one-half”). AEP-1 students’ performance was strongly bimodal: those with weak fraction writing skills made predominantly syntactic errors, whereas those with strong fraction writing skills made mostly lexical errors. Second graders did not transcode any fractions correctly making exclusively syntactic or combined errors. Approximately half the AEP-1 students with the lowest levels of schooling (< 3 years) succeeded in writing fractions, suggesting an important role of informal experiences for this group.

Learning companion robots for young children are increasingly adopted in informal learning environments. Although parents play a pivotal role in their children's learning, very little is known about how parents prefer to incorporate robots into their children's learning activities. We developed prototype capabilities for a learning companion robot to deliver educational prompts and responses to parent-child pairs during reading sessions and conducted in-home user studies involving 10 families with children aged 3-5. Our data indicates that parents want to work with robots as collaborators to augment parental activities to foster children's learning, introducing the notion of parent-robot collaboration. Our findings offer an empirical understanding of the needs and challenges of parent-child interaction in informal learning scenarios and design opportunities for integrating a companion robot into these interactions. We offer insights into how robots might be designed to facilitate parent-robot collaboration, including parenting policies, collaboration patterns, and interaction paradigms.

A substantial body of research has found that human and nonhuman animals are capable of processing the magnitudes of nonsymbolic ratios. Lewis, Mathews and Hubbard (2015) hypothesized that this ability may depend on a neurocognitive architecture called the ratio processing system (RPS). They further hypothesized 1) that the RPS might serve as a neurocognitive startup tool—an evolutionarily conserved cognitive architecture—and 2) that it can be recycled to support the acquisition of symbolic fractions knowledge. We tested these two predictions of the RPS account by comparing neural signatures of the RPS in 2nd-graders, who have not yet received formal symbolic fraction instruction, and 5th-graders, who have. During fMRI scanning, children performed ratio comparison tasks in which they determined which of two ratios or symbolic fractions was larger. Both cohorts showed behavioral and neural evidence of processing symbolic and nonsymbolic fractions magnitudes, with performance modulated by the numerical distance between stimuli. Consistent with our predictions, 2nd-grade children reliably recruited a right parietal-frontal network for nonsymbolic ratio comparisons but not symbolic fractions, and 5th-grade children recruited a bilateral parietal-frontal network for both nonsymbolic and symbolic fractions that overlapped with, but extended beyond, that found for 2nd-graders. These results present the first neuroimaging evidence that neural substrates for nonsymbolic ratios exist prior to formal learning and that this nonsymbolic foundation may be recycled to process symbolic fractions. These findings open the door for pedagogical strategies that focus on supporting this recycling process to improve students’ understanding of symbolic fractions.

Learning fractions is essential for academic and daily life success. A critical first step in acquiring fractions is learning to transcode them (e.g., writing ½ when hearing “one half”). However, little is known about how students master fraction transcoding. We addressed this gap by assessing fraction writing in two groups of Brazilian students with limited education: adults in the first year of an adult education program (AEP-1) and 2nd graders. Both groups made frequent transcoding errors. Errors were classified into three categories, Syntactic: correct numerator/denominator values with an incorrect notation (12th for “one half”); Lexical: incorrect numerals with the correct notation (⅓ for “one half”); Combined: incorrect numerals and notation (15th for “one-half”). AEP-1 students’ performance was strongly bimodal: those with weak fraction writing skills made predominantly syntactic errors, whereas those with strong fraction writing skills made mostly lexical errors. Second graders did not transcode any fractions correctly making exclusively syntactic or combined errors. Approximately half the AEP-1 students with the lowest levels of schooling (< 3 years) succeeded in writing fractions, suggesting an important role of informal experiences. In the early stages of learning fractions transcoding, students know how to write whole numbers, but do not yet know fractions syntax, leading to the preponderance of syntactic errors. As students learn fractions syntax, they make infrequent, predominantly lexical, errors. We conclude by proposing a model of fraction writing development that incorporates formal and informal learning experiences, executive functions, and phonemic awareness to explain the mastery of this important skill.

Parent-child interaction is critical for child development, yet parents may need guidance in some aspects of their engagement with their children. Current research on educational math robots focuses on child-robot interactions but falls short of including the parents and integrating the critical role they play in children's learning. We explore how educational robots can be designed to facilitate parent-child conversations, focusing on math talk, a predictor of later math ability in children. We prototyped capabilities for a social robot to support math talk via reading and play activities and conducted an exploratory Wizard-of-Oz in-home study for parent-child interactions facilitated by a robot. Our findings yield insights into how parents were inspired by the robot's prompts, their desired interaction styles and methods for the robot, and how they wanted to include the robot in the activities, leading to guidelines for the design of parent-child-robot interaction in educational contexts.

Although important for the acquisition of later math skills, fractions are notoriously difficult. Previous studies have shown that higher math anxiety (MA) is associated with lower performance in symbolic fraction tasks in adults and have suggested that MA may negatively impact the acquisition of fractions in children. However, the effects of MA on fraction skills in school-aged children remain underexplored. We, therefore, investigated the impact of MA on the performance of younger (second and third graders) and older (fifth and sixth graders) children in math fluency (MF), written calculation, fraction knowledge (FK), and symbolic fraction and nonsymbolic ratio processing. On the basis of our prior work suggesting a perceptual foundation for fraction processing, we predicted that symbolic, but not nonsymbolic, math skills (especially fractions) would be impaired by MA. As predicted, higher MA was associated with lower performance in general mathematics achievement and symbolic fraction tasks, but nonsymbolic ratio processing was not affected by MA in either age group. Furthermore, working memory capacity partially mediated the effects of MA on general mathematics achievement, FK, and symbolic fraction processing. These results suggest that understanding the bidirectional interactions between MA and fractions may be important for helping children acquire these critical skills.

Children’s early numerical knowledge establishes a foundation for later development of mathematics achievement and playing linear number board games is effective in improving basic numerical abilities. Besides the visuo-spatial cues provided by traditional number board games, learning companion robots can integrate multi-sensory information and offer social cues that can support children’s learning experiences. We explored how young children experience sensory feedback (audio and visual) and social expressions from a robot when playing a linear number board game, “RoboMath.” We present the interaction design of the game and our investigation of children’s (n = 19, aged 4) and parents’ experiences under three conditions: (1) visual-only, (2) audio-visual, and (3) audio-visual-social robot interaction. We report our qualitative analysis, including the themes observed from interviews with families on their perceptions of the game and the interaction with the robot, their child’s experiences, and their design recommendations.

Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition - and that privileging the ANS may be a mistake.

In grapheme-color synesthesia, synesthetes reliably and automatically experience specific colors when viewing black-and-white graphemes. In this chapter, we review the impact of synesthesia on memory. Over 50 years of research has demonstrated that synesthetes have better long-term memory than nonsynesthetes. However, little is known about how synesthesia impacts earlier stages of memory. We explored whether the synesthetic advantage in long-term memory arises from advantages at earlier memory stages by testing the impact of grapheme-color synesthesia on three different stages of memory. We find a consistent synesthetic memory advantage and that this difference is greatest when people are asked to remember larger numbers of items. We argue that these advantages in early memory processing stages may be important for understanding the mechanistic underpinnings of long-term memory enhancement in synesthesia. Finally, we frame these findings in a novel recoding theory of synesthetic memory, which extends the classic dual-coding theory of memory.

A fundamental question about fractions is whether they are grounded in an abstract non-symbolic magnitude code, similar to those postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving non-symbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second and fifth grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, non-symbolic ratios and mixed symbolic/non-symbolic pairs. This paradigm allowed us to test three key questions: 1) whether children show an analog magnitude code for rational numbers, 2) whether that code is compatible with mental representations of symbolic fractions, and 3) how formal education with fractions affects the symbolic-non-symbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth grade children’s response times and error rates showed classic distance and notation effects. Non-symbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions, but a smaller advantage for non-symbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance, and fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a non-symbolic ratio magnitude code, and that symbolic fractions can be translated into that magnitude code.

The attentional spatial-numerical association of response codes (Att-SNARC) effect (Fischer, Castel, Dodd, & Pratt, 2003)-the finding that participants are quicker to detect left-side targets when the targets are preceded by small numbers and quicker to detect right-side targets when they are preceded by large numbers-has been used as evidence for embodied number representations and to support strong claims about the link between number and space (e.g., a mental number line). We attempted to replicate Experiment 2 of Fischer et al. by collecting data from 1,105 participants at 17 labs. Across all 1,105 participants and four interstimulus-interval conditions, the proportion of times the effect we observed was positive (i.e., directionally consistent with the original effect) was .50. Further, the effects we observed both within and across labs were minuscule and incompatible with those observed by Fischer et al. Given this, we conclude that we failed to replicate the effect reported by Fischer et al. In addition, our analysis of several participant-level moderators (finger-counting habits, reading and writing direction, handedness, and mathematics fluency and mathematics anxiety) revealed no substantial moderating effects. Our results indicate that the Att-SNARC effect cannot be used as evidence to support strong claims about the link between number and space.

Research on how humans understand the relative magnitude of symbolic fractions presents a unique case of the symbol-grounding problem with numbers. Specifically, how do people access a holistic sense of rational number magnitude from decimal fractions (e.g. 0.125) and common fractions (e.g. 1/8)? Researchers have previously suggested that people cannot directly access magnitude information from common fraction notation, but instead must use a form of calculation to access this meaning. Questions remain regarding the nature of calculation and whether a division-like conversion to decimals is a necessary process that permits access to fraction magnitudes. To test whether calculation is necessary to access fractions magnitudes, we carried out a series of six parallel experiments in which we examined how adults access the magnitude of rational numbers (decimals and common fractions) under varying task demands. We asked adult participants to indicate which of two fractions was larger in three different conditions: decimal-decimal, fraction-fraction, and mixed decimal-fraction pairs. Across experiments, we manipulated two aspects of the task demands. 1) Response windows were limited to 1, 2 or 5 s, and 2) participants either did or did not have to identify when the two stimuli were the same magnitude (catch trials). Participants were able to successfully complete the task even at a response window of 1 s and showed evidence of holistic magnitude processing. These results indicate that calculation strategies with fractions are not necessary for accessing a sense of a fractions meaning but are strategic routes to magnitude that participants may use when granted sufficient time. We suggest that rapid magnitude processing with fractions and decimals may occur by mapping symbolic components onto common amodal mental representations of rational numbers.

Fractions are a critical building block for the development of human mathematical cognition, but the origins of this concept are not well-understood. Recent work has found that a whole number sense is present in deep convolutional neural networks (DCNNs) pre-trained for object recognition and uses them as a model for investigating human numerical cognition. Do DCNNs also have a fraction sense? If so, is it dependent or independent of whole number processing? We investigated the neural sensitivity of a pretrained DCNN to both whole numbers and fractions. We replicated and extended previous research that the sense of whole number emerges in a different DCNN architecture. Further, we showed that DCNN is also sensitive to fraction value, i.e., the ratio of numerosities. Testing this model, our results suggest that the fraction sense relies on the whole number sense.

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 201 second grade and 150 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall math knowledge and skill for second graders, but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children’s fractions skills.

In number-form (NF) synesthesia-a condition in which people report vivid, automatic and consistent mental layouts for numerical sequences-numbers and space are closely linked. These explicit associations are similar to the implicit associations demonstrated by the Spatial-Numerical Association of Response Codes (SNARC) effect. Thus, NF synesthesia offers a unique opportunity to investigate spatial-numerical associations. We tested implicit and explicit representations in NF synesthetes using a multiple case-study design. Over two sessions, synesthetes participated in a semi-structured interview focusing on the nature of their associations, as well as SNARC and number line estimation tasks. Contrary to our hypotheses, only one synesthete demonstrated SNARC effects congruent with her reported form, whereas two others exhibited SNARC effects that were the opposite of their explicit NFs. While this inconsistency between implicit and explicit representations may indicate separate underlying cognitive mechanisms, factors such as task-specific constraints and strategic variability must also be considered.

Synesthesia is a benign neurodevelopmental condition in which stimulation of one sensory modality evokes experiences in a second, unstimulated modality (Simner and Hubbard, 2013). In grapheme-color synesthesia (GCS), which is experienced by 1-2% of adults, synesthetes reliably and involuntarily experience specific colors when viewing blackand-white graphemes. Previous case-studies have identified synesthetes with spectacular memory (Luria, 1968; Smilek, Dixon, Cudahy, & Merikle, 2001) and group studies have found advantages for synesthetes compared to nonsynesthetes in long-term memory (Rothen, Meier, & Ward, 2012). Here, we tested whether similar advantages are also present in earlier stages of memory. We tested visual iconic memory-the sensory store for vision-which has a very large capacity, but decays approximately 1000 ms after stimulus offset (Chow, 1985; Sergent et al., 2013; Sperling, 1960). We tested 20 synesthetes and 20 nonsynesthetes in a direct replication of the Sperling (1960) Partial Report Paradigm using letters (Experiment 1) and non-alphanumeric symbols (Experiment 2) as stimuli. Overall, synesthetes had a greater iconic memory capacity than nonsynesthetes when presented with synesthesia-inducing letter stimuli. This advantage was reduced when they were presented with non-synesthesia inducing symbol stimuli. Critically this advantage was most prominent when memory was stressed by asking participants to remember large arrays. Our results demonstrate that synesthetic memory advantages extend to the earliest stages of memory, and suggest that advantages in later stages of memory may arise from these earlier advantages.