Sex differences in number line estimation: The role of numerical estimation

Sex differences in mathematical performance have frequently been examined over the last decades indicating an advantage for males especially when numerical problems cannot be solved by (classroom‐)learnt strategies and/or estimation. Even in basic numerical tasks such as number line estimation, males were found to outperform females – with sex differences argued to emerge from different solution strategies applied by males and females. We evaluated the latter using two versions of the number line estimation task: a bounded and an unbounded task version. Assuming that women tend more strongly to apply known procedures, we expected them to be at a particular disadvantage in the unbounded number line estimation task, which is less prone to be solved by specific strategies such as proportion judgement but requires numerical estimation. Results confirmed more pronounced sex differences for unbounded number line estimation with males performing significantly more accurately in this task version. This further adds to recent evidence suggesting that estimation performance in the bounded task version may reflect solution strategies rather than numerical estimation. Additionally, it indicates that sex differences regarding the spatial representation of number magnitude may not be universal, but associated with spatial–numerical estimations in particular.     

The influence of number line estimation precision and numeracy on risky financial decision making

This study examined whether different aspects of mathematical proficiency influence one's ability to make adaptive financial decisions. “Numeracy” refers to the ability to process numerical and probabilistic information and is commonly reported as an important factor which contributes to financial decision‐making ability. The precision of mental number representation (MNR), measured with the number line estimation (NLE) task has been reported to be another critical factor. This study aimed to examine the contribution of these mathematical proficiencies while controlling for the influence of fluid intelligence, math anxiety and personality factors. In our decision‐making task, participants chose between two options offering probabilistic monetary gain or loss. Sensitivity to expected value was measured as an index for the ability to discriminate between optimal versus suboptimal options. Partial correlation and hierarchical regression analyses revealed that NLE precision better explained EV sensitivity compared to numeracy, after controlling for all covariates. These results suggest that individuals with more precise MNR are capable of making more rational financial decisions. We also propose that the measurement of “numeracy,” which is commonly used interchangeably with general mathematical proficiency, should include more diverse aspects of mathematical cognition including basic understanding of number magnitude.     

Digit identity influences numerical estimation in children and adults

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7–11 and adults completed 0–1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g., 698 and 701) and “Fifties” boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non‐speeded NLE task. Under both speeded and non‐speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.     

Cross‐Situational Learning: An Experimental Study of Word‐Learning Mechanisms

Cross‐situational learning is a mechanism for learning the meaning of words across multiple exposures, despite exposure‐by‐exposure uncertainty as to the word's true meaning. We present experimental evidence showing that humans learn words effectively using cross‐situational learning, even at high levels of referential uncertainty. Both overall success rates and the time taken to learn words are affected by the degree of referential uncertainty, with greater referential uncertainty leading to less reliable, slower learning. Words are also learned less successfully and more slowly if they are presented interleaved with occurrences of other words, although this effect is relatively weak. We present additional analyses of participants’ trial‐by‐trial behavior showing that participants make use of various cross‐situational learning strategies, depending on the difficulty of the word‐learning task. When referential uncertainty is low, participants generally apply a rigorous eliminative approach to cross‐situational learning. When referential uncertainty is high, or exposures to different words are interleaved, participants apply a frequentist approximation to this eliminative approach. We further suggest that these two ways of exploiting cross‐situational information reside on a continuum of learning strategies, underpinned by a single simple associative learning mechanism.     

A validation of eye movements as a measure of elementary school children's developing number sense

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded during task solution reflect children's use of the number line. By means of a cross-sectional design with 66 children from Grades 1, 2, and 3, we show that eye-tracking data (a) reflect grade-related increase in estimation competence, (b) are correlated with the accuracy of manual answers, (c) relate, in Grade 2, to children's addition competence, (d) are systematically distributed over the number line, and (e) replicate previous findings concerning children's use of counting strategies and orientation-point strategies. These findings demonstrate the validity and utility of eye-tracking data for investigating children's developing number sense and estimation competence.     

Selection of Decision Strategies After Conscious and Unconscious Thought

It is commonly assumed that the use of simple, non‐compensatory strategies is especially pronounced in memory‐based decisions, where information costs are high. At the same time, there is evidence that in memory‐based decisions, a compensatory processing of attributes is facilitated when the processing occurs unconsciously rather than consciously. We applied a strategy classification approach—developed in research on non‐compensatory heuristics—to test two key tenets of unconscious thought theory: the capacity principle and the weighting principle. Participants memorized attribute information about cars and were subsequently either directed to or diverted from thinking consciously about their preferences between the cars (conscious versus unconscious thought). Then, participants indicated in pair‐wise choices which car they would prefer and were classified (based on their choices) as using either one of two compensatory strategies (equal weight or weighted additive) or a non‐compensatory strategy (lexicographic heuristic). In line with the capacity principle, the number of participants best described by a compensatory strategy (the equal‐weight strategy) tended to be higher after unconscious thought than after conscious thought, whereas the number of participants best described by the lexicographic heuristic tended to be lower. Inconsistent with the weighting principle, participants in the unconscious thought condition were better described by the equal‐weight strategy than by the weighted‐additive strategy. In Experiment 2, in which participants were not instructed to form an impression while learning the attribute information, the use of the equal‐weight strategy was not more prevalent after unconscious thought. Copyright © 2013 John Wiley & Sons, Ltd.     

Verbalization of Decision Strategies in Multiple‐Cue Probabilistic Inference

In multiple‐cue probabilistic inference, people choose between alternatives based on several cues, each of which is differentially associated with an alternative's overall value. Various strategies have been proposed for probabilistic inference (e.g., weighted additive, tally, and take‐the‐best). These strategies differ in how many cue values they require to enact and in how they weight each cue. Do decision makers actually use any of these strategies? Ways to investigate this question include analyzing people's choices and the cues that they reveal. However, different strategies often predict the same decisions, and search behavior says nothing about whether or how people use the information that they acquire. In this research, we attempt to elucidate which strategies participants use in a multiple‐cue probabilistic inference task by examining verbal protocols, a high‐density source of process data. The promise of verbal data is in their utility for testing detailed information processing models. To that end, we apply protocol analysis in conjunction with computational simulations. We find converging evidence across outcome measures, search measures, and verbal reports that most participants use simplifying heuristics, namely take‐the‐best. Copyright © 2015 John Wiley & Sons, Ltd.     

儿童数字估计中的心理长度

从儿童对数字属性的认识角度探讨一年级儿童在不同数字范围中采用不同表征形式的根本原因。两个实验均采用数字线估计任务,实验一测量一年级儿童在15cm数字线长度下0～100与0～1000两种范围上的数字估计;实验二测量一年级儿童对0～1000范围10cm和20cm数字线长度的估计。结果显示,无论在不同的数字范围还是在不同的数字线长度下,儿童对低端数字的估计均存在心理长度,即儿童倾向于将低端数字与固定的线段长度对应起来,且这种对应关系不随数字范围与数字线长度的变化而变化。心理长度的存在是儿童在不同数字范围和不同数字线长度中采用不同数字表征形式的根本原因,也是儿童对数概念的认识发展到等距水平时出现的一种独特特点。     

Portfolio Selection from Multiple Benchmarks: A Goal Programming Approach to an Actual Case

This paper deals with benchmark‐based portfolio choice for buy‐and‐hold strategies of investing. Multiple benchmarks for returns are considered, which is more realistic than taking a unique benchmark – a unique aspiration difficult to select in practice among the various aspirations for returns that the investor has in mind. Portfolio selection with multiple benchmarks leads to a multi‐objective problem, which is addressed by mean value – stochastic goal programming. In particular, two benchmarks are considered, which involves two goals. Weights for goals depend on investor's preferences and Arrow's absolute risk aversion coefficients. An efficient frontier of portfolios is obtained. Advantages of this stochastic method in our context are as follows: (i) Mean value‐stochastic goal programming relies on classical utility theory under uncertainty and Arrow's absolute risk aversion, which ensures soundness and strictness, and (ii) the numerical model is easily solved by using available software such as mean–variance software. Numerical results are tabulated and discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Task Importance Effects on Prospective Memory Strategy Use

Mnemonic strategies generally improve prospective memory (PM) performance. However, little is known about why people use such strategies. In the Motivational‐Cognitive Prospective Memory model, task importance is thought to influence performance via multiple mechanisms, including increased strategy use. Our main purpose was to test this mechanism: higher PM task importance was hypothesized to cause greater strategy use. We also tested whether importance would specifically increase the use of more effective (external) strategies. Participants reported their strategy use for two hypothetical PM tasks. As predicted, they listed more strategies for more important tasks. This result demonstrates one mechanism for motivational effects in early phases of the task. We found weaker support for the prediction that participants would selectively increase their use of better strategies for more important tasks. This finding supports a relatively pessimistic view of meta‐memory in PM, at least when it comes to modulating one's use of dependable strategies. Copyright © 2013 John Wiley & Sons, Ltd.     

The Adaptive Use of Recognition in Group Decision Making

Applying the framework of ecological rationality, the authors studied the adaptivity of group decision making. In detail, they investigated whether groups apply decision strategies conditional on their composition in terms of task‐relevant features. The authors focused on the recognition heuristic, so the task‐relevant features were the validity of the group members' recognition and knowledge, which influenced the potential performance of group strategies. Forty‐three three‐member groups performed an inference task in which they had to infer which of two German companies had the higher market capitalization. Results based on the choice data support the hypothesis that groups adaptively apply the strategy that leads to the highest theoretically achievable performance. Time constraints had no effect on strategy use but did have an effect on the proportions of different types of arguments. Possible mechanisms underlying the adaptive use of recognition in group decision making are discussed.     

Association between basic numerical abilities and mathematics achievement

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum‐based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non‐symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.     

Holistic Information Integration in Child Decision Making

Weighted‐additive (WADD) strategies require decision makers to integrate multiple values weighted by their relevance. From what age can children make choices in line with such a WADD‐strategy? We compare multi‐attribute decisions of children (6–7; 8–10; 11–12‐year‐olds) with adults in an open information‐board environment without pre‐decisional information search. In two experiments, we classify decision strategies based on individual choice patterns and find that in all age groups the majority of participants are users of a WADD‐strategy. Simple decision heuristics such as lexicographic strategies were applied rarely by children and not at all by adults. In two additional follow‐up studies, we further investigate the underlying process of WADD‐application by analysing decision latencies in combination with a retrospective think‐aloud study. Results suggest that children did not apply WADD‐strategies in a deliberate fashion in our experiments. Overall, our findings demonstrate that the ability to make good and quick decisions by holistically integrating information is already present in young children. Copyright © 2017 John Wiley & Sons, Ltd.     

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72–67 as “how much do I have to add up to 67 to get 72?”), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.     

Are People Successful at Learning Sequences of Actions on a Perceptual Matching Task?

We report the results of an experiment in which human subjects were trained to perform a perceptual matching task. Subjects were asked to manipulate comparison objects until they matched target objects using the fewest manipulations possible. An unusual feature of the experimental task is that efficient performance requires an understanding of the hidden or latent causal structure governing the relationships between actions and perceptual outcomes. We use two benchmarks to evaluate the quality of subjects' learning. One benchmark is based on optimal performance as calculated by a dynamic programming procedure. The other is based on an adaptive computational agent that uses a reinforcement-learning method known as Q-learning to learn to perform the task. Our analyses suggest that subjects were successful learners. In particular, they learned to perform the perceptual matching task in a near-optimal manner (i.e., using a small number of manipulations) at the end of training. Subjects were able to achieve near-optimal performance because they learned, at least partially, the causal structure underlying the task. In addition, subjects' performances were broadly consistent with those of model-based reinforcement-learning agents that built and used internal models of how their actions influenced the external environment. We hypothesize that people will achieve near-optimal performances on tasks requiring sequences of action-especially sensorimotor tasks with underlying latent causal structures-when they can detect the effects of their actions on the environment, and when they can represent and reason about these effects using an internal mental model.     

Place‐value understanding in number line estimation predicts future arithmetic performance

In multi‐digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place‐value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place‐value understanding in first grade. We hypothesized that estimating the position of two‐digit numbers requires place‐value understanding. Therefore, we fitted a linear function to children's estimates of two‐digit numbers and considered the resulting slope as a measure of children's place‐value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place‐value understanding. Additionally, the slope for two‐digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first‐graders' place‐value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place‐value understanding) may drive the association between number line estimation and arithmetic performance.     

Managers' variance investigation decisions: an experimental examination of probabilistic and outcome ambiguity

Information ambiguity is prevalent in organizations and may influence management decisions. This study examines, given imprecise probabilities or outcomes, how managers decide which department's performance to investigate further when they are provided with performance benchmarks expressed in numerical intervals. Seventy‐nine MBA students participated in two experiments involving investigation decisions. We presented participants with interval benchmarks of a firm's expenses. Being below or above the benchmark should have been seen as equally negative. We found that, when facing outcome ambiguity, our participants consistently preferred to investigate further those departments whose performance was described as having an ambiguous outcome (when the outcome's range was centered either below or above the interval benchmark). However, when facing probabilistic ambiguity, there were two predominant choice patterns: consistently choosing to investigate the department whose performance is described with an ambiguous probability, or consistently choosing to investigate the department with unambiguous performance. To gain further insight, we conducted a follow‐up study collecting written protocols of participants' reasons for making choices involving ambiguous performance information. The results show that our participants displayed similar decision‐making processes when facing outcome ambiguity and probabilistic ambiguity. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparison of human performance in figural and navigational versions of the traveling salesman problem

Performance on a typical pen-and-paper (figural) version of the Traveling Salesman Problem was compared to performance on a room-sized navigational version of the same task. Nine configurations were designed to examine the use of the nearest-neighbor (NN), cluster approach, and convex-hull strategies. Performance decreased with an increasing number of nodes internal to the hull, and improved when the NN strategy produced the optimal path. There was no overall difference in performance between figural and navigational task modalities. However, there was an interaction between modality and configuration, with evidence that participants relied more heavily on the NN strategy in the figural condition. Our results suggest that participants employed similar, but not identical, strategies when solving figural and navigational versions of the problem. Surprisingly, there was no evidence that participants favored global strategies in the figural version and local strategies in the navigational version.     

Bimanual circling in deafferented patients: Evidence for a role of visual forward models

The present study investigated the role of ideation and visual feedback, and their interaction in movement control in the absence of somatosensory feedback, with the hypothesis that visual imagery and internal visual models may play a crucial role in performance even without feedback. Two chronically deafferented participants, GL and IW, circled bimanually two occluded cranks first without vision and then with hand‐congruent and hand‐incongruent visual feedback provided by visible flags. Without vision, GL was unable to circle the cranks. In contrast, IW performed spontaneously a symmetric pattern. Again without feedback, IW performed an instructed symmetric crank pattern well, but was unable to perform anti‐phase cranking. With hand‐congruent visual feedback, GL and IW were able to perform both symmetric and anti‐phase movements, with symmetry being more accurate. Visual feedback during preceding trials made possible trials without visual feedback in GL and improved anti‐phase trials in IW. Frequency‐transformed incongruent visual feedback resulted in poor performance in part due to unsuitable hand‐related strategies. However, IW improved in the latter task after detailed explanations of the condition. In conclusion, we suggest that both participants use visual imagery and visual forward models to control their hand movements. Visual updating of the forward model also improves performance with no vision. In addition, IW seemed to have been able to move from a focus on hand position to one on the transformed visual feedback to improve movement control in the incongruent feedback/movement condition.