When solving mathematical problems, young children will perform better when they can use gestures that match mental representations. However, despite their increasing prevalence in educational settings, few studies have explored this effect in touchscreen-based interactions. Thus, we investigated the impact on young children\'s performance of dragging (where a continuous gesture is performed that is congruent with the change in number) and tapping (involving a discrete gesture that is incongruent) on a touchscreen device when engaged in a continuous number line estimation task. By examining differences in the set size and position of the number line estimation, we were also able to explore the boundary conditions for the superiority effect of congruent gestures. We used a 2 (Gesture Type: drag or tap) × 2 (Set Size: Set 0-10 or Set 0-20) × 2 (Position: left of midpoint or right of midpoint) mixed design. A total of 70 children aged 5 and 6 years (33 girls) were recruited and randomly assigned to either the Drag or Tap group. We found that the congruent gesture (drag) generally facilitated better performance with the touchscreen but with boundary conditions. When completing difficult estimations (right side in the large set size), the Drag group was more accurate, responded to the stimulus faster, and spent more time manipulating than the Tap group. These findings suggest that when children require explicit scaffolding, congruent touchscreen gestures help to release mental resources for strategic adjustments, decrease the difficulty of numerical estimation, and support constructing mental representations.

One of the most prominent social influences on human decision making is conformity, which is even more prominent when the perceptual information is ambiguous. The Bayes optimal solution to this problem entails weighting the relative reliability of cognitive information and perceptual signals in constructing the percept from self-sourced/endogenous and social sources, respectively. The current study investigated whether humans integrate the statistics (i.e., mean and variance) of endogenous perceptual and social information in a Bayes optimal way while estimating numerosities. Our results demonstrated adjustment of initial estimations toward group means only when group estimations were more reliable (or \"certain\"), compared to participants\' endogenous metric uncertainty. Our results support Bayes optimal social conformity while also pointing to an implicit form of metacognition.

Various studies have reported an association between musical expertise and enhanced visuospatial and mathematical abilities. A recent work tested the susceptibility of musicians and nonmusicians to the Solitaire numerosity illusion finding that also perceptual biases underlying numerical estimation are influenced by long-term music training. However, the potential link between musical expertise and different perceptual mechanisms of quantitative estimation may be either limited to the visual modality or universal (i.e., modality independent). We addressed this question by developing an acoustic version of the Solitaire illusion. Professional musicians and nonmusicians listened to audio file recordings of piano and trombone notes and were required to estimate the number of piano notes. The stimuli were arranged to form test trials, with piano and trombone notes arranged in a way to form the Solitaire pattern, and control trials, with randomly located notes to assess their quantitative abilities in the acoustic modality. In the control trials, musicians were more accurate in numerical estimation than nonmusicians. In the presence of illusory patterns, nonmusicians differed from musicians in the esteem of regularly arranged vs. randomly arranged notes. This suggests that the association between long-term musical training and different perceptual mechanisms underlying numerical estimation may not be confined to the visual modality. However, neither musicians nor nonmusicians seemed to be susceptible to the acoustic version of the Solitaire illusion, suggesting that the emergence of this illusion may be stimulus and task-dependent.

We propose a method to achieve better wisdom of crowds by utilizing anchoring effects. In this method, people are first asked to make a comparative judgment such as \"Is the number of new COVID-19 infections one month later more or less than 10 (or 200,000)?\" As in this example, two sufficiently different anchors (e.g., \"10\" or \"200,000\") are set in the comparative judgment. After this comparative judgment, people are asked to make their own estimates. These estimates are then aggregated. We hypothesized that the aggregated estimates using this method would be more accurate than those without anchor presentation. To examine the effectiveness of the proposed method, we conducted three studies: a computer simulation and two behavioral experiments (numerical estimation of perceptual stimuli and estimation of new COVID-19 infections by physicians). Through computer simulations, we could identify situations in which the proposed method is effective. Although the proposed method is not always effective (e.g., when a group can make fairly accurate estimations), on average, the proposed method is more likely to achieve better wisdom of crowds. In particular, when a group cannot make accurate estimations (i.e., shows biases such as overestimation or underestimation), the proposed method can achieve better wisdom of crowds. The results of the behavioral experiments were consistent with the computer simulation findings. The proposed method achieved better wisdom of crowds. We discuss new insights into anchoring effects and methods for inducing diverse opinions from group members.

Recent studies have shown that the ability to process number in the face of conflicting dimensions of magnitude is a crucial aspect of numerosity judgments, relying in part on the inhibition of the non-numerical dimensions. Here we report, for the first time, that these inhibitory control processes are specific to the conflicting dimension of magnitude. Using a non-symbolic numerical comparison task adapted to a conflict adaptation paradigm on a group of 82 adults, we show that congruency effects between numerical and non-numerical information were reduced only when the conflicting dimension was the same in the preceding incongruent trial. Attention to number thus involves inhibitory control processes acting at a specific level of information. These results contribute to better characterize the domain general abilities involved in numerical cognition, and provide evidence for a specific interaction between numerosity perception and inhibitory control.

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57-79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group (N = 49), high-performance group (N = 21), and control group (N = 123). The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance.

The current study examined whether discrete numerical estimation is based on the same cognitive process as estimation of continuous magnitudes such as weight and time. While the verbal estimation of numerical quantities has a contingent unit of measurement (e.g., how many cookies fit in a cookie jar? _X_ cookies), estimation of time and weight does not (e.g., how much time does it take to fill a bath with water? _X_ minutes/hours/seconds). Therefore, estimation of the latter categories has another level of difficulty, requiring extensive involvement of cognitive control. During a functional magnetic resonance imaging (fMRI) scan, 18 students performed estimations with three estimation categories: number, time, and weight. Estimations elicited activity in multiple brain regions, mainly: (1) visual regions including bilateral lingual gyrus), (2) parietal regions including the left angular gyrus and right supramarginal gyrus, and (3) the frontal regions (cingulate gyrus and the inferior frontal cortex). Continuous magnitude estimations (mostly time) produced different frontal activity than discrete numerical estimations did, demonstrating different profiles of brain activations between discrete numerical estimations and estimations of continuous magnitudes. The activity level in the right middle and inferior frontal gyrus correlated with the tendency to give extreme responses, signifying the importance of the right prefrontal lobe in estimations.

The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging. The need for these tools dates back to the market crash of 1987, when investors needed better ways to protect their portfolios through option insurance. These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively. The violation of constant volatility and the log-normality assumption of the Black-Scholes option pricing model led to the discovery of the volatility smile, smirk, or skew in options markets. These stylized facts; that is, the volatility smile and implied volatilities implied by the option prices, are well documented in the option literature for almost all financial markets. These are expected to be true for Bitcoin options as well. The data sets for the study are based on short-dated Bitcoin options (14-day maturity) of two time periods traded on Deribit Bitcoin Futures and Options Exchange, a Netherlands-based cryptocurrency derivative exchange. The estimated results are compared with benchmark Black-Scholes implied volatility values for accuracy and efficiency analysis. This study has two aims: (1) to provide insights into the volatility smile in Bitcoin options and (2) to estimate the implied volatility of Bitcoin options through numerical approximation techniques, specifically the Newton Raphson and Bisection methods. The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data. Moreover, the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options. However, the Newton Raphson forecasting technique converges faster than does the Bisection method.

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Subitizing refers to ability of people to accurately and effortlessly enumerate a small number of items, with a capacity around four elements. Previous research showed that \"canonical\" organizations, such as familiar layouts on a dice, can readily improve subitizing performance of people. However, almost all canonical shapes found in the world are also highly symmetrical; therefore, it is unclear whether previously reported facilitative effect of canonical organization is really due to canonicality, or simply driven by spatial symmetry. Here, we investigated the possible effect of symmetry on subitizing by using symmetrical, yet non-canonical, shape structures. These symmetrical layouts were compared with highly controlled random patterns (Experiment 1), as well as fully random and canonical patterns (Experiment 2). Our results showed that symmetry facilitates subitizing performance, but only at set size of 6, suggesting that the effect is insufficient to improve performance of people in the lower or upper range. This was also true, although weaker, in reaction time (RT), error distance measures, and Weber Fractions. On the other hand, canonical layouts produced faster and more accurate subitizing performances across multiple set sizes. We conclude that, although previous findings mixed symmetry in their canonical shapes, their findings on shape canonicality cannot be explained by symmetry alone. We also propose that our symmetrical and canonical results are best explained by the \"groupitizing\" and pattern recognition accounts, respectively.