The effect of mathematics anxiety on the processing of numerical magnitude

In an effort to understand the origins of mathematics anxiety, we investigated the processing of symbolic magnitude by high mathematics-anxious (HMA) and low mathematics-anxious (LMA) individuals by examining their performance on two variants of the symbolic numerical comparison task. In two experiments, a numerical distance by mathematics anxiety (MA) interaction was obtained, demonstrating that the effect of numerical distance on response times was larger for HMA than for LMA individuals. These data support the claim that HMA individuals have less precise representations of numerical magnitude than their LMA peers, suggesting that MA is associated with low-level numerical deficits that compromise the development of higher level mathematical skills.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks.     

Numerical and physical magnitudes are mapped into time

In two experiments we investigated mapping of numerical and physical magnitudes with temporal order. Pairs of digits were presented sequentially for a size comparison task. An advantage for numbers presented in ascending order was found when participants were comparing the numbers' physical and numerical magnitudes. The effect was more robust for comparisons of physical size, as it was found using both select larger and select smaller instructions, while for numerical comparisons it was found only for select larger instructions. Varying both the digits' numerical and physical sizes resulted in a size congruity effect, indicating automatic processing of the irrelevant magnitude dimension. Temporal order and the congruency between numerical and physical magnitudes affected comparisons in an additive manner, thus suggesting that they affect different stages of the comparison process.     

Musicians outperform nonmusicians in magnitude estimation: Evidence of a common processing mechanism for time,space and numbers

It has been proposed that time, space, and numbers may be computed by a common magnitude system. Even though several behavioural and neuroanatomical studies have focused on this topic, the debate is still open. To date, nobody has used the individual differences for one of these domains to investigate the existence of a shared cognitive system. Musicians are known to outperform nonmusicians in temporal discrimination tasks. We therefore observed professional musicians and nonmusicians undertaking three different tasks: temporal (participants were required to estimate which of two tones lasted longer), spatial (which line was longer), and numerical discrimination (which group of dots was more numerous). If time, space, and numbers are processed by the same mechanism, it is expected that musicians will have a greater ability, even in nontemporal dimensions. As expected, musicians were more accurate with regard to temporal discrimination. They also gave better performances in both the spatial and the numerical tasks, but only outside the subitizing range. Our data are in accordance with the existence of a common magnitude system. We suggest, however, that this mechanism may not involve the whole numerical range.     

Physical and cognitive effort discounting of hypothetical monetary rewards

Effort discounting refers to the decrease in the subjective value of a reward as the effort required to obtain the reward increases. The main aims of this study were to ascertain whether the amount of the reward affects the steepness of the effort discounting process for hypothetical monetary rewards, to identify whether this steepness depends on the type of effort that is required, and to determine whether the steepness of different types of effort covary at the individual level. Two types of effort were studied under hypothetical choice situations: physical effort and cognitive effort. Both physical and cognitive effort discounting were well described by the hyperbolic model. Large rewards were discounted less steeply than small rewards for both types of effort. This finding agrees with the results of prior studies which have found that larger rewards have greater motivational power. In addition, the steepness of physical effort discounting was positively correlated with the steepness of cognitive effort discounting, which suggests that the effort discounting process is a trait‐like characteristic within an individual.     

Delay discounting of qualitatively different reinforcers in rats

Humans discount larger delayed rewards less steeply than smaller rewards, whereas no such magnitude effect has been observed in rats (and pigeons). It remains possible that rats' discounting is sensitive to differences in the quality of the delayed reinforcer even though it is not sensitive to amount. To evaluate this possibility, Experiment 1 examined discounting of qualitatively different food reinforcers: highly preferred versus nonpreferred food pellets. Similarly, Experiment 2 examined discounting of highly preferred versus nonpreferred liquid reinforcers. In both experiments, an adjusting-amount procedure was used to determine the amount of immediate reinforcer that was judged to be of equal subjective value to the delayed reinforcer. The amount and quality of the delayed reinforcer were varied across conditions. Discounting was well described by a hyperbolic function, but no systematic effects of the quantity or the quality of the delayed reinforcer were observed.     

概率大小、损益结果和认知闭合需要对模糊规避的影响

模糊规避是指在相同奖赏的情况下,决策者会力图规避从主观上判断具有模糊概率的事件而偏好具有相同精确概率的事件。本研究探讨了概率大小、损益结果和认知闭合需要对模糊规避的影响。研究发现,在小概率受益的情况下,个体倾向于模糊寻求;在中概率受益的情况下,个体倾向于模糊规避;在高概率受益的情况下,个体倾向于模糊规避;在小概率损失的情况下,个体倾向于模糊规避;在中概率损失的情况下,个体倾向于模糊规避;在高概率损失的情况下,个体倾向于模糊寻求。但是,研究并未发现认知闭合需要对模糊规避有预测作用。