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Young children have informal knowledge of fractions before learning about fraction symbols in school. In the current study, we followed 103 children in the Mid-Atlantic United States from the fall to the spring of first grade to characterize the development of individual differences in early informal fraction knowledge, as well as its relation to other mathematical and cognitive skills. Most children in our sample showed some early fraction knowledge at the beginning of first grade, especially with nonsymbolic fractions and halving, and this knowledge improved over the school year without explicit instruction in fractions. However, there were large individual differences in early fraction knowledge at the start of first grade, which explained significant variance in math achievement at the end of first grade, even when controlling for whole number knowledge and a variety of cognitive skills. Start-of-year whole number knowledge, but not spatial scaling or proportional reasoning, also predicted early end-of-year fraction knowledge. These data can inform activities for learning in the early years to foster both early fraction and integer knowledge in parallel, which may better prepare students for later formal instruction in fractions.

Children's beliefs about the contribution of effort and ability to success and failure shape their decisions to persist or give up on challenging tasks, with consequences for their academic success. But how do children learn about the concept of "challenge"? Prior work has shown that parents' verbal responses to success and failure shape children's motivational beliefs. In this study, we explore another type of talk--parent and child talk about difficulty--which could contribute to children's motivational beliefs. We performed secondary analyses of two observational studies of parent-child interactions in the United States (Boston and Philadelphia) from age 3 to fourth grade (Study 1, 51% girls, 65.5% White, at least 43.2% below Federal poverty line) and at first grade (Study 2, 54% girls, 72% White, family income-to-needs ratio M [SD] = 4.41 [2.95]) to identify talk about difficulty, characterize the content of those statements, and assess whether task context, child and parent gender, child age, and other parent motivational talk were associated with the quantity of child and parent difficulty talk. We found that many families did discuss difficulty, with variation among families. Parents and children tended to use general statements to talk about difficulty (e.g., "That was hard!"), and task context affected child and parent difficulty talk. In the NICHD-SECCYD dataset, mothers' highlighting how task features contributed to task difficulty was positively correlated with their process praise, suggesting that this talk could be motivationally relevant.

Relying on shared tasks and stimuli to conduct research can enhance the replicability of findings and allow a community of researchers to collect large data sets across multiple experiments. This approach is particularly relevant for experiments in spatial navigation, which often require the development of unfamiliar large-scale virtual environments to test participants. One challenge with shared platforms is that undetected technical errors, rather than being restricted to individual studies, become pervasive across many studies. Here, we discuss the discovery of a software bug in a virtual environment platform used to investigate individual differences in spatial navigation: Virtual Silcton. The bug, which was difficult to detect for several reasons, resulted in storing the absolute value of a direction in a pointing task rather than the signed direction and rendered the original sign of the direction unrecoverable. To assess the impact of the bug on published findings, we collected a new data set for comparison. Results revealed that although the bug caused suppression in pointing errors and had different effects across people (less accurate navigators had more suppression), the effect of the bug on published data is small, partially explaining the difficulty in detecting the bug. We also used the new data set to develop a tool that allows researchers who have previously used Virtual Silcton to evaluate the impact of the bug on their findings. We summarize the ways that shared open materials, shared data, and collaboration can pave the way for better science to prevent errors in the future.

Despite some gains, women continue to be underrepresented in many science, technology, engineering, and math (STEM) fields. Using a national longitudinal dataset of 690 participants born in 1991, we tested whether spatial skills, measured in middle childhood, would help explain this gender gap. We modeled the relation between 4th-grade spatial skills and STEM majors while simultaneously accounting for competing cognitive and motivational mechanisms. Strong spatial skills in 4th grade directly increased the likelihood of choosing STEM college majors, above and beyond math achievement and motivation, verbal achievement and motivation, and family background. Additionally, 4th-grade spatial skills indirectly predicted STEM major choice via math achievement and motivation in the intervening years. Further, our findings suggest that gender differences in 4th-grade spatial skills contribute to women's underrepresentation in STEM majors.

Research on spatial navigation is essential to understanding how mobile species adapt to their environments. Such research increasingly uses virtual environments (VEs) because, although VE has drawbacks, it allows for standardization of procedures, precision in measuring behaviors, ease in introducing variation, and cross-investigator comparability. Developmental researchers have used a wide range of VE testing methods, including desktop computers, gaming consoles, virtual reality, and phone applications. We survey the paradigms to guide researchers' choices, organizing them by their characteristics using a framework proposed by Girard (2022) in which navigation is reactive or deliberative, and may be tied to sensory input or not. This organization highlights what representations each paradigm indicates. VE tools have enriched our picture of the development of navigation, but much research remains to be done, e.g., determining retest reliability, comparing performance on different paradigms, validating performance against real-world behavior and open sharing. Reliable and valid assessments available on open-science repositories are essential for work on the development of navigation, its neural bases, and its implications for other cognitive domains.

External representations powerfully support and augment complex human behavior. When navigating, people often consult external representations to help them find the way to go, but do maps or verbal instructions improve spatial knowledge or support effective wayfinding? Here, we examine spatial knowledge with and without external representations in two studies where participants learn a complex virtual environment. In the first study, we asked participants to generate their own maps or verbal instructions, partway through learning. We found no evidence of improved spatial knowledge in a pointing task requiring participants to infer the direction between two targets, either on the same route or on different routes, and no differences between groups in accurately recreating a map of the target landmarks. However, as a methodological note, pointing was correlated with the accuracy of the maps that participants drew. In the second study, participants had access to an accurate map or set of verbal instructions that they could study while learning the layout of target landmarks. Again, we found no evidence of differentially improved spatial knowledge in the pointing task, although we did find that the map group could recreate a map of the target landmarks more accurately. However, overall improvement was high. There was evidence that the nature of improvement across all conditions was specific to initial navigation ability levels. Our findings add to a mixed literature on the role of external representations for navigation and suggest that more substantial intervention--more scaffolding, explicit training, enhanced visualization, perhaps with personalized sequencing--may be necessary to improve navigation ability.

Parents provide motivational and cognitive support within the same interaction, yet researchers have investigated these separately. We examined two key aspects of parental support, praise (motivational support) and spatial language (cognitive support), from fathers and mothers during three tasks with their first-grade children (6-7-year-olds; N = 107; 56 girls; 72.0% White, 23.4% Black). Parents' praise and spatial language varied by task but not child sex: Both parents produced more praise in the Etch-a-Sketch and block tasks than the card game and produced more spatial language in the Etch-a-Sketch task than other tasks. We further examined whether praise and spatial language in the two spatial tasks (Etch-a-Sketch and block construction) were related to children's later math and spatial skills. We found neither additive nor multiplicative effects of parents' praise or spatial language. We also did not see additive or multiplicative effects of fathers' and mothers' support. However, fathers' greater spatial language at first grade was negatively associated with boys' (but not girls') math achievement in third grade, with greater father spatial tokens related to their sons' lower math achievement. This suggests that boys may perceive fathers' support more negatively than girls do or that fathers may offer additional support for boys with lower abilities. Taken together, this study emphasizes the importance of considering contexts in examining parental support. The correlational nature of the study warrants future research to establish causal relations and to enhance our understanding of multifaceted parent-child interactions.

Both sketching and self-explanation are widely believed to be effective for problem-solving in science learning. However, it is unclear which aspects of these strategies promote learning and how they might interact. Compared to a read-only baseline, we examined the impact of instructing 11-year-old students to solve science problems to sketch, self-explain, or both. Problems were either high spatial or low spatial. We coded elements and relations in their sketches and think-alouds. Self-explanation led to greater accuracy on problems, but only when used alone. Sketching had no effect, although showing more elements and relations in sketches was associated with higher accuracy and occurred more often for high spatial problems. Students may be more comfortable with self-explanation than with sketching and require encouragement or specific instruction to use sketching routinely for the benefits hypothesized to be associated with sketching to appear.

Understanding and reasoning about phenomena at scales outside human perception (for example, geologic time) is critical across science, technology, engineering, and mathematics. Thus, devising strong methods to support acquisition of reasoning at such scales is an important goal in science, technology, engineering, and mathematics education. In two experiments, we examine the use of analogical principles in learning about geologic time. Across both experiments we find that using a spatial analogy (for example, a time line) to make multiple alignments, and keeping all unrelated components of the analogy held constant (for example, keep the time line the same length), leads to better understanding of the magnitude of geologic time. Effective approaches also include hierarchically and progressively aligning scale information (Experiment 1) and active prediction in making alignments paired with immediate feedback (Experiments 1 and 2).

Popular sources often assume the existence of a male advantage in navigation, but the scientificdata are inconsistent. This meta-analysis evaluates the literature on behavioral sex differences inhuman navigation. We quantify the overall magnitude of sex differences in a variety ofparadigms and populations and examine potential moderators in large-scale navigation skills,using 694 effect sizes from 266 studies and a multilevel linear modeling approach. Overall, wefound that male participants outperform female participants, with a small to medium effect size(d= 0.34 to 0.38). The type of task, the type of dependent variable and the testing environmentsignificantly contribute to variability in effect sizes. Pointing and recall tasks show larger sexdifferences than distance estimation tasks or learning to criterion; among the dependent variables, the deviation scores associated with pointing tasks show larger effect sizes. The largestestimate was d = .55 for tasks than required coordinating indoor and outdoor views.Interestingly, studies with children younger than 13 years showed very small effect sizes (d= .15) as compared to older age groups. We discuss the implications of these findings for thestudy of sex differences and identify avenues for future navigation research.

Research on spatial thinking needs reliable and valid measures of individual differences in skills.Visuospatial Perspective Taking (PT)—the ability to mentally maintain and transform spatialrelationships between objects within an environment—is one kind of spatial skill that isespecially relevant to navigation and building cognitive maps. However, the psychometricproperties of various PT tasks have yet to be examined. The present study examines three mainpsychometric properties of PT tasks: 1) the reliability of two tasks developed for children butadapted in difficulty level for use in adult populations, 2) item difficulty and discriminabilitywithin and between four tasks using item response theory, and 3) relation of scores with generalintelligence, working memory, and mental rotation. Results showed that two of the four PT taskshave promising psychometric properties for measuring a wide range of PT ability based on itemdifficulty, discriminability, and efficiency of a test information function.

Background: One criterion of adaptive learning is appropriate generalization to new instances based on the original learning context and avoiding overgeneralization. Appropriate generalization requires understanding what features of a solution are applicable in a new context and whether the new context requires modifications or a new approach. In a series of three experiments, we investigate whether searching for an algebraic formalism before receiving direct instruction facilitates appropriate generalization. Results: (1) Searching buffers against negative transfer: participants who first searched for an equation were less likely to overgeneralize compared to participants who completed a tell-and-practice activity. (2) Likelihood of creating a correct new adaptation varied by performance on the searching task. (3) Asking people to sketch alleviated some of the negative effects of tell-and-practice, but sketching did not augment the effect of searching. (4) When participants received more elaborate tell-and-practice instruction, the advantages of searching were less notable. Conclusions: Searching for an algebraic formula prior to direct instruction may be a productive way to help learners connect a formula to its referent and avoid overgeneralization. Tell-and-practice instruction that only described the mathematical procedures led to the greatest levels of overgeneralization errors and worst performance. Tell-and-practice instruction that highlighted connections between the mathematical structure of the formula and the visual referent performed at similar or marginally worse levels than the search-first conditions.

Many scientific theories and discoveries involve reasoning about extreme scales, removed from human experience, such as time in geology, size in nanoscience. Thus, understanding scale is central to science, technology, engineering, and mathematics. Unfortunately, novices have trouble understanding and comparing sizes of unfamiliar large and small magnitudes. Relational reasoning is a promising tool to bridge the gap between direct experience and phenomena at extreme scales. However, instruction does not always improve understanding, and analogies can fail to bring about conceptual change, and even mislead students. Here we review how people reason about phenomena across scales, in three sections: (a) we develop a framework for how relational reasoning supports understanding extreme scales; (b) we identify cognitive barriers to aligning human and extreme scales; (c) we outline a theory-based approach to teaching scale information using relational reasoning, present two successful learning activities, and consider the role of a unified scale instruction across STEM education. [At the time of submission to ERIC this article was in press with "Educational Psychology Review."]

How do toddlers learn the names of geometric forms? Previous work suggests that preschoolers have fragmentary knowledge and that defining properties are not understood until well into elementary school. The current study investigated when children first begin to understand shape names and how they apply those labels to unusual instances. We tested 25- and 30-month-old children's (N = 30 each) understanding of names for canonical shapes (commonly encountered instances, e.g., equilateral triangles), noncanonical shapes (more irregular instances, e.g., scalene triangles), and embedded shapes (shapes within a larger picture, e.g., triangular slices of pizza). At 25 months, children knew very few names, including those for canonical shapes. By 30 months, however, children had acquired more shape names and were beginning to apply them to some of the less typical instances of the shapes. Possible mechanisms driving this initial development of shape knowledge and implications of that development for school readiness are explored. [This article was published in the "Journal of Cognition and Development" (EJ1090613).]

Episodic memory relies on discriminating among similar elements of episodes. Mnemonic discrimination is relatively poor at age 4, and then improves markedly. We investigated whether motivation to encode items with fine-grain resolution would change this picture of development, using an engaging computer-administered memory task in which a bird ate items that made her healthier (gain frame), sicker (loss frame), or led to no change (control condition). Using gain-loss framing led to enhanced mnemonic discrimination in 4- and 5-year-olds, but did not affect older children or adults. Despite this differential improvement, age-related differences persisted. An additional finding was that loss-framing led to greater mnemonic discrimination than gain-framing across age groups. Motivation only partially accounts for the improvement in mnemonic discrimination.

Previous research has shown sketching to be useful tostudents solving math problems. The present studyexamines which aspects of middle school students’sketching are related to, or predict, successfullyanswering math problems. The effects of individualdifferences in cognitive factors – working memory,spatial ability, and prior math knowledge – on answeraccuracy are also analyzed. Stepwise regressionanalysis indicates that prior math knowledge and theinclusion of numerical representations of key problemrelationships in sketches positively predict answeraccuracy, whereas including irrelevant relationships ina sketch is associated with lower answer accuracy.Methodological implications for future research arediscussed.

Spatial skills are consistently linked to mathematical reasoning, and are sensitive to intervention. One important spatial skill is spatial scaling. We evaluated whether (1) a playful scaling game might promote learning by providing feedback during play, and (2) spatial scaling is related to number-line estimation (NLE) based on the mutual reliance on relative magnitude reasoning. Forty-eight children, ages 5.5-8.3, completed a playful scaling game and a NLE task. Results show that children improve from the first to second half of the task, especially for more difficult trials and for the lowest performing children. In addition, scaling and NLE relate when controlling for age and vocabulary. Similar improvement on the task and relations to NLE were observed in a conceptual replication (N = 52). These results provide support for further study of improving spatial scaling in children, with the possibility to test whether scaling could support related mathematical skills as well.

Four principles of cognitive science were used to make systematic revisions in middle school science instructional modules from two kinds of curriculum: one popular textbook series and one popular hands-on series (two modules each). Schools were randomly assigned to 1 of the 3 arms (cognitive science modifications with professional development, active control with professional development, or business-as-usual). Two cohorts of students were followed in each arm for each setting. There were significant benefits of the cognitive science intervention, but the nature of effects varied for the two settings and curricula. For the text-based curriculum, positive effects of cognitive science modifications were concentrated in classrooms with lower proportions of underrepresented minority students. For the hands-on curriculum, there were positive effects that were not linked to school composition. Participation in the active control did not significantly improve student learning. Implications for policy and research are discussed.

The study of navigation is informed by ethological data from many species, laboratory investigation at behavioural and neurobiological levels, and computational modelling. However, the data are often species-specific, making it challenging to develop general models of how biology supports behaviour. Wiener et al. outlined a framework for organizing the results across taxa, called the 'navigation toolbox' (Wiener et al. In Animal thinking: contemporary issues in comparative cognition (eds R Menzel, J Fischer), pp. 51–76). This framework proposes that spatial cognition is a hierarchical process in which sensory inputs at the lowest level are successively combined into ever-more complex representations, culminating in a metric or quasi-metric internal model of the world (cognitive map). Some animals, notably humans, also use symbolic representations to produce an external representation, such as a verbal description, signpost or map that allows communication of spatial information or instructions between individuals. Recently, new discoveries have extended our understanding of how spatial representations are constructed, highlighting that the hierarchical relationships are bidirectional, with higher levels feeding back to influence lower levels. In the light of these new developments, we revisit the navigation toolbox, elaborate it and incorporate new findings. The toolbox provides a common framework within which the results from different taxa can be described and compared, yielding a more detailed, mechanistic and generalized understanding of navigation. [ABSTRACT FROM AUTHOR]

In the real word, we perceive our environment as a series of static and dynamic views, with viewpoint transitions providing a natural link from one static view to the next. The current research examined if experiencing such transitions is fundamental to learning the spatial layout of small-scale displays. In Experiment 1, participants viewed a tabletop array from 4 orientations in 1 of 3 conditions. The control condition presented the array sequentially, as a series of static views. In the remaining conditions, participants experienced the transition between viewpoints by rotating the array or moving around it. Both transitions improved spatial performance. Experiment 2 added a passive rotation condition to examine the effect of watching the transition without actively generating it. Spatial performance was equivalent across active and passive rotation conditions, with both outperforming static views. Together, these findings suggest that continuous visual flow is key to small-scale spatial learning.

Being able to estimate quantity is important in everyday life and for success in the STEM disciplines. However, people have difficulty reasoning about magnitudes outside of human perception (e.g., nanoseconds, geologic time). This study examines patterns of estimation errors across temporal and spatial magnitudes at large scales. We evaluated the effectiveness of hierarchical alignment in improving estimations, and transfer across dimensions. The activity was successful in increasing accuracy for temporal and spatial magnitudes, and learning transferred to the estimation of numeric magnitudes associated with events and objects. However, there were also a number of informative differences in performance on temporal, spatial, and numeric magnitude measures, suggesting that participants possess different categorical information for these scales. Educational implications are discussed.

Many scientific theories and discoveries involve reasoning about extreme scales, removed from human experience, such as time in geology and size in nanoscience. Thus, understanding scale is central to science, technology, engineering, and mathematics. Unfortunately, novices have trouble understanding and comparing sizes of unfamiliar large and small magnitudes. Relational reasoning is a promising tool to bridge the gap between direct experience and phenomena at extreme scales. However, instruction does not always improve understanding, and analogies can fail to bring about conceptual change, and even mislead students. Here, we review how people reason about phenomena across scales, in three sections: (a) we develop a framework for how relational reasoning supports understanding extreme scales; (b) we identify cognitive barriers to aligning human and extreme scales; and (c) we outline a theory-based approach to teaching scale information using relational reasoning, present two successful learning activities, and consider the role of a unified scale instruction across STEM education. [For the full text of this report, see ED566931.]

We explored whether existing curricula can be adapted to increase students' skills at comprehending and learning from diagrams using an intervention delivered by regular middle-school teachers in intact classrooms. Ninety-two teachers in three states implemented our modified materials over six curricular units from two publishers: "Holt" (a reading-focused curriculum) and "Full Option Science System" ("FOSS") (an inquiry-focused curriculum). Results were compared between two interventions--one based on selected principles of cognitive science (cognitive-science-based) that included instruction in diagram comprehension and one providing professional development in science content only (content-only)--and a business-as-usual control. We analyze posttest items involving different degrees of reliance on diagrams to show how instruction in diagram comprehension can improve comprehension of diagrams during reasoning. At the classroom level, there were significant advantages of the cognitive-science-based intervention over both content-only and business-as-usual with large effect sizes in all "FOSS" units (d = 0.41-0.52), but only one "Holt" unit (d = 0.11). Analyses by type of diagram suggested these effects were largest for transfer to diagrams from an uninstructed domain. Further analyses of high-stakes state test scores available for the participants implementing the "Holt" units showed improved use of diagrams (d = 0.45-0.66). Results suggest that making sense of visualizations (diagrams) is not effortless and self-evident, but that students who receive supports for comprehending can improve their comprehension, and the learning can transfer to new domains.

This report summarizes the major conclusions from a 2-day meeting of educators and researchers from more than a dozen disciplines that regard spatial thinking as a core activity that warrants consideration for a more prominent role in the education of undergraduate students.To make the case for space in the curriculum, the meeting concluded that basic research must come to a more fundamental understanding of what is meant by spatial thinking, including different varieties of spatial thinking across the college curriculum, and identifying what is common to spatial thinking across disciplines. At the same time there is a need to continue to develop methods of teaching spatial thinking. Additionally, there is a need for research on teaching spatial thinking, including assessments of what is learned from programs that aim to teach spatial thinking. Finally, there is a need to document and demonstrate where and how spatial thinking prepares students for academic success and allows them to better compete in the job market and global economy.

Spatial thinking skills are associated with performance, persistence, and achievement in science, technology, engineering, and mathematics (STEM) school subjects. Because STEM knowledge and skills are integral to developing a well-trained workforce within and beyond STEM, spatial skills have become a major focus of cognitive, developmental, and educational research. However, these efforts are greatly hampered by the current lack of access to reliable, valid, and well-normed spatial tests. Although there are hundreds of spatial tests, they are often hard to access and use, and information about their psychometric properties is frequently lacking. Additional problems include (1) substantial disagreement about what different spatial tests measure—even two tests with similar names may measure very different constructs; (2) the inability to measure some STEM-relevant spatial skills by any existing tests; and (3) many tests only being available for specific age groups. The first part of this report delineates these problems, as documented in a series of structured and open-ended interviews and surveys with colleagues. The second part outlines a roadmap for addressing the problems. We present possibilities for developing shared testing systems that would allow researchers to test many participants through the internet. We discuss technological innovations, such as virtual reality, which could facilitate the testing of navigation and other spatial skills. Developing a bank of testing resources will empower researchers and educators to explore and support spatial thinking in their disciplines, as well as drive the development of a comprehensive and coherent theoretical understanding of spatial thinking. [ABSTRACT FROM AUTHOR]