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

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having (...) noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

After writing my response to the commentaries, I sat back and reflected on the fascination and frustration of work on this topic. There is the ancient fascination of trying to understand the nature of the sensory bridge linking us to the external world. Also, discussing the measurability of sensation brings to the surface concepts we use and take for granted when we are working in other areas of psychology; and it holds them before us for critical examination. The frustration lies (...) chiefly in the difficulty of formulating concepts clearly, in a way understood by all. At least when constructing the original Tower of Babel, workers hearing strange words knew that communication was not occurring. The psychophysical Tower of Babel sometimes has workers using the same words, but with different meanings. Much of my response is based on attempts to clarify concepts and clear up confusions. (shrink)

Continuous magnitude estimation and continuous cross-modality matching with line length can efficiently track the momentary loudness of time-varying sounds in behavioural experiments. These methods are known to be prone to systematic biases but may be checked for consistency using their counterpart, magnitude production. Thus, in Experiment 1, we performed such an evaluation for time-varying sounds. Twenty participants produced continuous cross-modality matches to assess the momentary loudness of fourteen songs by continuously adjusting the length of a line. In (...) Experiment 2, the resulting temporal line length profile for each excerpt was played back like a video together with the given song and participants were asked to continuously adjust the volume to match the momentary line length. The recorded temporal line length profile, however, was manipulated for segments with durations between 7 to 12 s by eight factors between 0.5 and 2, corresponding to expected differences in adjusted level of −10, −6, −3, −1, 1, 3, 6, and 10 dB according to Stevens’s power law for loudness. The average adjustments 5 s after the onset of the change were −3.3, −2.4, −1.0, −0.2, 0.2, 1.4, 2.4, and 4.4 dB. Smaller adjustments than predicted by the power law are in line with magnitude-production results by Stevens and co-workers due to “regression effects.” Continuous cross-modality matches of line length turned out to be consistent with current loudness models, and by passing the consistency check with cross-modal productions, demonstrate that the method is suited to track the momentary loudness of time-varying sounds. (shrink)

Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, ). While (...) about half of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words “thousand, million, billion, trillion” constitute a uniformly spaced count list. Participants in this group also tended to be optimistic in evaluations of largely ineffective political strategies, relative to linear number-line placers. The results indicate that the surface structure of number words can heavily influence processes for dealing with numbers in this range, and it can amplify the possibility that analogous surface regularities are partially responsible for parallel phenomena in children. In addition, these results have direct implications for lawmakers and scientists hoping to communicate effectively with the public. (shrink)

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or (...) results from changes in precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

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. 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. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single (...) events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

The reliability of a test score is discussed from the viewpoint of underestimation of and, specifically, deflation in estimates or reliability. Many widely used estimators are known to underestimate reliability. Empirical cases have shown that estimates by widely used estimators such as alpha, theta, omega, and rho may be deflated by up to 0.60 units of reliability or even more, with certain types of datasets. The reason for this radical deflation lies in the item–score correlation embedded in the estimators: because (...) the estimates by Rit are deflated when the number of categories in scales are far from each other, as is always the case with item and score, the estimates of reliability are deflated as well. A short-cut method to reach estimates closer to the true magnitude, new types of estimators, and deflation-corrected estimators of reliability, are studied in the article. The empirical section is a study on the characteristics of combinations of DCERs formed by different bases for estimators, different alternative estimators of correlation as the linking factor between item and the score variable, and different conditions. Based on the simulation, an initial typology of the families of DCERs is presented: some estimators are better with binary items and some with polytomous items; some are better with small sample sizes and some with larger ones. (shrink)

BackgroundThe amount and value of researchers’ peer review work is critical for academia and journal publishing. However, this labor is under-recognized, its magnitude is unknown, and alternative ways of organizing peer review labor are rarely considered.MethodsUsing publicly available data, we provide an estimate of researchers’ time and the salary-based contribution to the journal peer review system.ResultsWe found that the total time reviewers globally worked on peer reviews was over 100 million hours in 2020, equivalent to over 15 thousand years. (...) The estimated monetary value of the time US-based reviewers spent on reviews was over 1.5 billion USD in 2020. For China-based reviewers, the estimate is over 600 million USD, and for UK-based, close to 400 million USD.ConclusionsBy design, our results are very likely to be under-estimates as they reflect only a portion of the total number of journals worldwide. The numbers highlight the enormous amount of work and time that researchers provide to the publication system, and the importance of considering alternative ways of structuring, and paying for, peer review. We foster this process by discussing some alternative models that aim to boost the benefits of peer review, thus improving its cost-benefit ratio. (shrink)

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and (...) response time are expected near the boundaries between categories. In contrast to previous work that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers. (shrink)

The disambiguation of a syntactically ambiguous sentence in favor of a less preferred parse can lead to slower reading at the disambiguation point. This phenomenon, referred to as a garden‐path effect, has motivated models in which readers initially maintain only a subset of the possible parses of the sentence, and subsequently require time‐consuming reanalysis to reconstruct a discarded parse. A more recent proposal argues that the garden‐path effect can be reduced to surprisal arising in a fully parallel parser: words consistent (...) with the initially dispreferred but ultimately correct parse are simply less predictable than those consistent with the incorrect parse. Since predictability has pervasive effects in reading far beyond garden‐path sentences, this account, which dispenses with reanalysis mechanisms, is more parsimonious. Crucially, it predicts a linear effect of surprisal: the garden‐path effect is expected to be proportional to the difference in word surprisal between the ultimately correct and ultimately incorrect interpretations. To test this prediction, we used recurrent neural network language models to estimate word‐by‐word surprisal for three temporarily ambiguous constructions. We then estimated the slowdown attributed to each bit of surprisal from human self‐paced reading times, and used that quantity to predict syntactic disambiguation difficulty. Surprisal successfully predicted the existence of garden‐path effects, but drastically underpredicted their magnitude, and failed to predict their relative severity across constructions. We conclude that a full explanation of syntactic disambiguation difficulty may require recovery mechanisms beyond predictability. (shrink)

ABSTRACTAdults use a variety of strategies to reason about fraction magnitudes, and this variability is adaptive. In two studies, we examined the relationships between mathematics anxiety, working memory, strategy variability and performance on two fraction tasks: fraction magnitude comparison and estimation. Adults with higher mathematics anxiety had lower accuracy on the comparison task and greater percentage absolute error on the estimation task. Unexpectedly, mathematics anxiety was not related to variable strategy use. However, variable strategy use was linked (...) to more accurate magnitude comparisons, especially among adults with lower working memory performance or those who use mathematics less frequently, as well as lower PAE on the estimation task. These findings shed light on the role of strategy variability in fraction problem solving and demonstrate a link between mathematics anxiety and fraction magnitude reasoning, a key predictor of general mathematics achievement. (shrink)

The anchoring heuristic refers to phenomena when an arbitrary number affects subsequent numerical estimations. Oppenheimer, LeBoeuf and Brewer showed that it is not necessary for the anchor to be a numerical value, yet current models describing the anchoring heuristic do not fully account for the mechanism of non-numerical anchoring. However, this effect shows similarity to the basic anchoring effect - obtained without the comparative question and based on the availability of the given number in working memory. In this study, we (...) attempt to verify whether those two effect share the same psychological mechanism. In Experiment 1, we show that non-numerical anchoring based on magnitude priming cannot be obtained when the lines are just observed. The examined mechanism proves to be dependent on the act of drawing, displaying limitations similar to the basic anchoring effect, previously pointed out by Brewer and Chapman. By using the same numerical anchors in different size formats, in Experiment 2 we showed that anchoring based on magnitude priming occurs even when the numerical values do not affect the estimations. The results are discussed in the light of a possible mechanism that underlies the investigated effect. (shrink)

Adjusting for covariates is a well-established method to estimate the total causal effect of an exposure variable on an outcome of interest. Depending on the causal structure of the mechanism under study, there may be different adjustment sets, equally valid from a theoretical perspective, leading to identical causal effects. However, in practice, with finite data, estimators built on different sets may display different precisions. To investigate the extent of this variability, we consider the simplest non-trivial non-linear model of a v-structure (...) on three nodes for binary data. We explicitly compute and compare the variance of the two possible different causal estimators. Further, by going beyond leading-order asymptotics, we show that there are parameter regimes where the set with the asymptotically optimal variance does depend on the edge coefficients, a result that is not captured by the recent leading-order developments for general causal models. As a practical consequence, the adjustment set selection needs to account for the relative magnitude of the relationships between variables with respect to the sample size and cannot rely on purely graphical criteria. (shrink)

According to Kant’s Critique of the Power of Judgement, in the end all estimation of magnitude is sensible, or ‘aesthetic’, and the absolutely great in aesthetic estimation is called ‘the mathematical sublime’. This article identifies the relevant sensible element with an inner sensation of a temporal tension: in aesthetic comprehension, the imagination encounters an inevitable tension between the successive reproduction of a magnitude’s individual parts and the simultaneous unification of these parts. The sensation of this tension (...) varies in degree and facilitates aesthetic estimation in general. But in the special case where it comes to involve a sense of exceeding our imagination’s limit, we judge a magnitude to be mathematically sublime. Pace Kant, I argue that this limit is private and contingent rather than transcendental, such that the judgement of the mathematical sublime is neither universal a priori nor necessary. (shrink)

In the discipline of software development, effort estimation renders a pivotal role. For the successful development of the project, an unambiguous estimation is necessitated. But there is the inadequacy of standard methods for estimating an effort which is applicable to all projects. Hence, to procure the best way of estimating the effort becomes an indispensable need of the project manager. Mathematical models are only mediocre in performing accurate estimation. On that account, we opt for analogy-based effort (...) class='Hi'>estimation by means of some soft computing techniques which rely on historical effort estimation data of the successfully completed projects to estimate the effort. So in a thorough study to improve the accuracy, models are generated for the clusters of the datasets with the confidence that data within the cluster have similar properties. This paper aims mainly on the analysis of some of the techniques to improve the effort prediction accuracy. Here the research starts with analyzing the correlation coefficient of the selected datasets. Then the process moves through the analysis of classification accuracy, clustering accuracy, mean magnitude of relative error and prediction accuracy based on some machine learning methods. Finally, a bio-inspired firefly algorithm with fuzzy analogy is applied on the datasets to produce good estimation accuracy. (shrink)