Recruitment of magnitude representations to understand graded words.

Language understanding and mathematics understanding are two fundamental forms of human thinking. Prior research has largely focused on the question of how language shapes mathematical thinking. The current study considers the converse question. Specifically, it investigates whether the magnitude representations that are thought to anchor understanding of number are also recruited to understand the meanings of graded words. These are words that come in scales (e.g., Anger) whose members can be ordered by the degree to which they possess the defining property (e.g., calm, annoyed, angry, furious). Experiment 1 uses the comparison paradigm to find evidence that the distance, ratio, and boundary effects that are taken as evidence of the recruitment of magnitude representations extend from numbers to words. Experiment 2 uses a similarity rating paradigm and multi-dimensional scaling to find converging evidence for these effects in graded word understanding. Experiment 3 evaluates an alternative hypothesis - that these effects for graded words simply reflect the statistical structure of the linguistic environment - by using machine learning models of distributional word semantics: LSA, word2vec, GloVe, counterfitted word vectors, BERT, RoBERTa, and GPT-2. These models fail to show the full pattern of effects observed of humans in Experiment 2, suggesting that more is needed than mere statistics. This research paves the way for further investigations of the role of magnitude representations in sentence and text comprehension, and of the question of whether language understanding and number understanding draw on shared or independent magnitude representations. It also informs the role of machine learning models in cognitive psychology research.

Strategy variability in computational estimation and its association with mathematical achievement.

Computational estimation requires a breadth of strategies and selection of the relevant strategy given a problem's features. We used the new Test of Estimation Strategies (TES), composed of 20 arithmetic problems (e.g., 144 x 0.38), to investigate variability in strategy use in young adults. The TES targets the five estimation strategies that adults use most frequently, which fall into two Classes. The three Class One strategies are general-purpose and taught in schools. Proceed Algorithmically entails applying an algorithm (e.g., shifting a decimal place). Round One and Round Two are defined as rounding one or both operands, respectively. The two Class Two strategies are more advanced, requiring application of conceptual knowledge of mathematics. Known-and-Nice is used when a participant relies on a well-known mathematical fact (e.g., 25 × 4 = 100) to form an estimate. Fractions uses a fraction or percentage in the estimation process (e.g., 943 x 0.48 is about 50% or half of 900). We divided our sample of adult participants into two groups (i.e., high, average) based on their estimation performance on the TES. The high-performance group used a broader range of strategies and more frequently applied the most relevant strategy given a problem's features. Overall estimation accuracy was correlated with mathematical achievement, as were strategy breadth and strategy relevance. However, none of these associations survived first controlling for verbal achievement. Participants' strategy reports suggested that the TES problems were generally successful in eliciting the five target strategies and provided evidence for a new strategy, Partitioning. These findings provide a basis for future instructional studies to improve students' computational estimation.

The psychological reality of the learned "p < .05" boundary.

The .05 boundary within Null Hypothesis Statistical Testing (NHST) "has made a lot of people very angry and been widely regarded as a bad move" (to quote Douglas Adams). Here, we move past meta-scientific arguments and ask an empirical question: What is the psychological standing of the .05 boundary for statistical significance? We find that graduate students in the psychological sciences show a boundary effect when relating p-values across .05. We propose this psychological boundary is learned through statistical training in NHST and reading a scientific literature replete with "statistical significance". Consistent with this proposal, undergraduates do not show the same sensitivity to the .05 boundary. Additionally, the size of a graduate student's boundary effect is not associated with their explicit endorsement of questionable research practices. These findings suggest that training creates distortions in initial processing of p-values, but these might be dampened through scientific processes operating over longer timescales.

Executive function predictors of science achievement in middle-school students.

Cognitive flexibility as measured by the Wisconsin Card Sort Task (WCST) has long been associated with frontal lobe function. More recently, this construct has been associated with executive function (EF), which shares overlapping neural correlates. Here, we investigate the relationship between EF, cognitive flexibility, and science achievement in adolescents. This is important because there are fewer educational neuroscience studies of scientific reasoning than of other academically relevant forms of cognition (i.e., mathematical thinking and language understanding). Eighth grade students at a diverse middle school in the Midwestern US completed classroom-adapted measures of three EFs (shifting, inhibition, and updating) and the WCST. Science achievement was indexed by students' standardized test scores and their end-of-the-year science class grades. Among the EF measures, updating was strongly predictive of science achievement. The association between cognitive flexibility and science achievement was comparatively weaker. These findings illuminate the relationship between EF, cognitive flexibility, and science achievement. A methodological contribution was the development of paper-and-pencil based versions of standard EF and cognitive flexibility measures suitable for classroom administration. We expect these materials to help support future classroom-based studies of EF and cognitive flexibility, and whether training these abilities in adolescent learners improves their science achievement.

The role of executive function abilities in interleaved vs. blocked learning of science concepts.

This study investigated the relative efficacy of interleaved versus blocked instruction and the role of executive function in governing learning from these instructional sequences. Eighth grade students learned about three rock concepts (igneous, sedimentary, metamorphic) and their attributes (origin, texture, composition). Consistent with prior studies and as predicted by current theoretical accounts, students who received interleaved instruction showed better memory (i.e., accuracy on true-false questions) when tested 2 weeks later, whereas those who received blocked instruction showed better memory when tested on the same day as instruction. Also consistent with prior studies and theoretical accounts, the blocked group showed greater transfer when tested after a retention interval, although this advantage was not significant. Critically, and as predicted, the shifting and inhibition executive function abilities were more predictive of learning from interleaved vs. blocked instruction. These findings lay the groundwork for future studies investigating the role of executive function in learning from different forms of instruction.

Competing numerical magnitude codes in decimal comparison: Whole number and rational number distance both impact performance.

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.

A language compatibility effect in fraction processing.

A language compatibility effect occurs when there is a match between what a language provides and what a mathematical task demands. Here, we investigated whether such an effect exists for fraction processing in English, which names the numerator first, versus Korean, which names the denominator first. We developed two new tasks: a fraction span task where participants view and then recall four fractions and a fraction identification task where they view one fraction and then another and judge whether the two fractions are the same or not. We generally found that English speakers were advantaged when the numerator drove task performance and Korean speakers were advantaged when the denominator was critical. These findings, particularly from the fraction identification task, were inconsistent with the attentional focus hypothesis, which proposes that the serialisation bias of a language guides which fraction component is attended to first. Rather, they were better explained by the verbal encoding hypothesis, which states that a necessary condition for observing language compatibility effects may be that the fraction components must be encoded in verbal working memory and rehearsed there.

Graded human sensitivity to geometric and topological concepts.

In a seminal study, Dehaene et al. (2006) found evidence that adults and children are sensitive to geometric and topological (GT) concepts using a novel odd-one-out task. However, performance on this task could reflect more general cognitive abilities than intuitive knowledge of GT concepts. Here, we developed a new 2-alternative forced choice (2-AFC) version of the original task where chance represents a higher bar to clear (50% vs. 16.67%) and where the role of general cognitive abilities is minimized. Replicating the original finding, American adult participants showed above-chance sensitivity to 41 of the 43 GT concepts tested. Moreover, their performance was not strongly driven by two general cognitive abilities, fluid intelligence and mental rotation, nor was it strongly associated with mathematical achievement as measured by ACT/SAT scores. The performance profile across the 43 concepts as measured by the new 2-AFC task was found to be highly correlated with the profiles as measured using the original odd-one-out task, as an analysis of data sets spanning populations and ages revealed. Most significantly, an aggregation of the 43 concepts into seven classes of GT concepts found evidence for graded sensitivity. Some classes, such as Euclidean geometry and Topology, were found to be more domain-specific: they "popped out" for participants and were judged very quickly and highly accurately. Others, notably Symmetry and Geometric transformations, were found to be more domain-general: better predicted by participants' general cognitive abilities and mathematical achievement. These results shed light on the graded nature of GT concepts in humans and challenge computational models that emphasize the role of induction.

Categorical Perception of p-Values.

Traditional statistics instruction emphasizes a .05 significance level for hypothesis tests. Here, we investigate the consequences of this training for researchers' mental representations of probabilities - whether .05 becomes a boundary, that is, a discontinuity of the mental number line, and alters their reasoning about p-values. Graduate students with statistical training (n = 25) viewed pairs of p-values and judged whether they were "similar" or "different." After controlling for several covariates, participants were more likely and faster to judge p-values as "different" when they crossed the .05 boundary (e.g., .046 vs. .052) compared to when they did not (e.g., .026 vs. .032). This result suggests a categorical perception-like effect for the processing of p-values. It may be a consequence of traditional statistical instruction creating a psychologically real divide between so-called statistical "significance" and "nonsignificance." Such a distortion is undesirable given modern approaches to statistical reasoning that de-emphasize dichotomizing the p-value continuum.

Linking inhibitory control to math achievement via comparison of conflicting decimal numbers.

The relationship between executive functions (EF) and academic achievement is well-established, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Starting from the premise that executive functions contribute to general math skills both directly - supporting the execution of problem solving strategies - and indirectly - supporting the acquisition of precursor mathematical content, we hypothesize that the contribution of domain-general EF capacities to precursor skills that support later learning can help explain relations between EF and overall math skills. We test this hypothesis by examining whether the contribution of inhibitory control on general math knowledge can be explained by inhibition's contribution to processing rational number pairs that conflict with individual's prior whole number knowledge. In 97 college students (79 female, age = 20.58 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g., 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g., 0.2 vs. 0.87). Individual differences in the Stroop task predicted performance on inconsistent decimal comparisons, which in turn predicted general math achievement. With respect to relating inhibitory control to math achievement, Stroop performance was an independent predictor of achievement after accounting for age, working memory and cognitive flexibility, but decimal performance mediated this relationship. Finally, we found inconsistent decimals performance mediated the relationship of inhibition with rational number performance, but not other advanced mathematical concepts. These results pinpoint the specific contribution of inhibitory control to rational number understanding, and more broadly are consistent with the hypothesis that acquisition of foundational mathematical content can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares.

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Taking an educational psychology course improves neuroscience literacy but does not reduce belief in neuromyths.

Educators are increasingly interested in applying neuroscience findings to improve educational practice. However, their understanding of the brain often lags behind their enthusiasm for the brain. We propose that educational psychology can serve as a bridge between basic research in neuroscience and psychology on one hand and educational practice on the other. We evaluated whether taking an educational psychology course is associated with increased neuroscience literacy and reduced belief in neuromyths in a sample of South Korean pre-service teachers. The results showed that taking an educational psychology course was associated with the increased neuroscience literacy, but there was no impact on belief in neuromyths. We consider the implications of these and other findings of the study for redesigning educational psychology courses and textbooks for improving neuroscience literacy.

Extending the seductive allure of neuroscience explanations effect to popular articles about educational topics.

BACKGROUND: The seductive allure of neuroscience explanations (SANE) is the finding that people overweight psychological arguments when framed in terms of neuroscience findings. AIM: This study extended this finding to arguments concerning the application of psychological findings to educational topics. SAMPLE: Participants (n = 320) were recruited from the general public, specifically among English-speaking Amazon Mechanical Turk workers residing in the United States. METHODS: We developed eight articles that orthogonally varied two processes (learning vs. development) with two disciplines (cognitive vs. affective psychology). We increased neuroscience framing across four levels: psychological finding alone, with an extraneous neuroscience finding (verbal), with an extraneous neuroscience finding (verbal) and graph, and with an extraneous neuroscience finding (verbal) and brain image. Participants were randomly assigned to one level of neuroscience framing and rated the credibility of each article's argument. RESULTS: Seductive allure of neuroscience explanations effects were not ubiquitous. Extraneous verbal neuroscience framings, either alone or accompanied by graphs, did not influence the credibility of the application of psychological findings to educational topics. However, there was a SANE effect when educational articles were accompanied by both extraneous verbal neuroscience findings and brain images. This effect persisted even after controlling for individual differences in familiarity with education, attitude towards psychology, and knowledge of neuroscience. CONCLUSION: The results suggest that there is a SANE effect for articles about educational topics among the general public when they are accompanied by both extraneous verbal neuroscience findings and brain images.

A retrieval-based approach to eliminating hindsight bias.

Individuals exhibit hindsight bias when they are unable to recall their original responses to novel questions after correct answers are provided to them. Prior studies have eliminated hindsight bias by modifying the conditions under which original judgments or correct answers are encoded. Here, we explored whether hindsight bias can be eliminated by manipulating the conditions that hold at retrieval. Our retrieval-based approach predicts that if the conditions at retrieval enable sufficient discrimination of memory representations of original judgments from memory representations of correct answers, then hindsight bias will be reduced or eliminated. Experiment 1 used the standard memory design to replicate the hindsight bias effect in middle-school students. Experiments 2 and 3 modified the retrieval phase of this design, instructing participants beforehand that they would be recalling both their original judgments and the correct answers. As predicted, this enabled participants to form compound retrieval cues that discriminated original judgment traces from correct answer traces, and eliminated hindsight bias. Experiment 4 found that when participants were not instructed beforehand that they would be making both recalls, they did not form discriminating retrieval cues, and hindsight bias returned. These experiments delineate the retrieval conditions that produce-and fail to produce-hindsight bias.

The principles and practices of educational neuroscience: Comment on Bowers (2016).

In his recent critique of Educational Neuroscience, Bowers argues that neuroscience has no role to play in informing education, which he equates with classroom teaching. Neuroscience, he suggests, adds nothing to what we can learn from psychology. In this commentary, we argue that Bowers' assertions misrepresent the nature and aims of the work in this new field. We suggest that, by contrast, psychological and neural levels of explanation complement rather than compete with each other. Bowers' analysis also fails to include a role for educational expertise-a guiding principle of our new field. On this basis, we conclude that his critique is potentially misleading. We set out the well-documented goals of research in Educational Neuroscience, and show how, in collaboration with educators, significant progress has already been achieved, with the prospect of even greater progress in the future. (PsycINFO Database Record

Comparison of opportunities to respond and generation effect as potential causal mechanisms for incremental rehearsal with multiplication combinations.

Incremental rehearsal (IR) is an intervention with demonstrated effectiveness in increasing retention of information, yet little is known about how specific intervention components contribute to the intervention's effectiveness. The purpose of this study was to further the theoretical understanding of the intervention by comparing the effects of opportunities to respond (OTR) and generation demand on retention of multiplication combinations. Using a between subject 2 × 2 factorial design, 103 4th and 5th grade students were taught seven multiplication combinations using one of four versions of IR that orthogonally varied OTR (high versus low) and generation demands (high versus low). A two-way ANOVA revealed main effects for OTR, generation demands, and an interaction of the two factors. The effect of generation demands was large (d=1.31), whereas the overall effect of OTR was moderate (d=0.66). Critically, the two factors interacted, with the largest learning gains observed when OTR and generation demands were both high. The results of this study suggest that generation demand is an important factor in the effectiveness of rehearsal interventions.

The subjective meaning of cognitive architecture: a Marrian analysis.

Marr famously decomposed cognitive theories into three levels. Newell, Pylyshyn, and Anderson offered parallel decompositions of cognitive architectures, which are psychologically plausible computational formalisms for expressing computational models of cognition. These analyses focused on the objective meaning of each level - how it supports computational models that correspond to cognitive phenomena. This paper develops a complementary analysis of the subjective meaning of each level - how it helps cognitive scientists understand cognition. It then argues against calls to eliminatively reduce higher levels to lower levels, for example, in the name of parsimony. Finally, it argues that the failure to attend to the multiple meanings and levels of cognitive architecture contributes to the current, disunified state of theoretical cognitive science.

Understanding decimal proportions: discrete representations, parallel access, and privileged processing of zero.

Much of the research on mathematical cognition has focused on the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9, with considerably less attention paid to more abstract number classes. The current research investigated how people understand decimal proportions--rational numbers between 0 and 1 expressed in the place-value symbol system. The results demonstrate that proportions are represented as discrete structures and processed in parallel. There was a semantic interference effect: When understanding a proportion expression (e.g., "0.29"), both the correct proportion referent (e.g., 0.29) and the incorrect natural number referent (e.g., 29) corresponding to the visually similar natural number expression (e.g., "29") are accessed in parallel, and when these referents lead to conflicting judgments, performance slows. There was also a syntactic interference effect, generalizing the unit-decade compatibility effect for natural numbers: When comparing two proportions, their tenths and hundredths components are processed in parallel, and when the different components lead to conflicting judgments, performance slows. The results also reveal that zero decimals--proportions ending in zero--serve multiple cognitive functions, including eliminating semantic interference and speeding processing. The current research also extends the distance, semantic congruence, and SNARC effects from natural numbers to decimal proportions. These findings inform how people understand the place-value symbol system, and the mental implementation of mathematical symbol systems more generally.

Infusing Neuroscience into Teacher Professional Development.

Bruer (1997) advocated connecting neuroscience and education indirectly through the intermediate discipline of psychology. We argue for a parallel route: the neurobiology of learning, and in particular the core concept of plasticity, have the potential to directly transform teacher preparation and professional development, and ultimately to affect how students think about their own learning. We present a case study of how the core concepts of neuroscience can be brought to in-service teachers - the BrainU workshops. We then discuss how neuroscience can be meaningfully integrated into pre-service teacher preparation, focusing on institutional and cultural barriers.