Judgments of discrete and continuous quantity: an illusory Stroop effect

Evidence from human cognitive neuroscience, animal neurophysiology, and behavioral research demonstrates that human adults, infants, and children share a common nonverbal quantity processing system with nonhuman animals. This system appears to represent both discrete and continuous quantity, but the proper characterization of the relationship between judgments of discrete and continuous quantity remains controversial. Some researchers have suggested that both continuous and discrete quantity may be automatically extracted from a scene and represented internally, and that competition between these representations leads to Stroop interference. Here, four experiments provide evidence for a different explanation of adults’ performance on the types of tasks that have been said to demonstrate Stroop interference between representations of discrete and continuous quantity. Our well-established tendency to underestimate individual two-dimensional areas can provide an alternative explanation (introduced here as the “illusory-Stroop” hypothesis). Though these experiments were constructed like Stroop tasks, and they produce patterns of performance that initially appear consistent with Stroop interference, Stroop interference effects are not involved. Implications for models of the construction of cumulative area representations and for theories of discrete and continuous quantity processing in large sets are discussed.     

Improving arithmetic performance with number sense training: An investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.     

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.     

Brief non-symbolic,approximate number practice enhances subsequent exact symbolic arithmetic in children

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children’s performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics, as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children’s performance of symbolic mathematics.     

Judgement of discrete and continuous quantity in adults: Number counts!

Three experiments involving a Stroop-like paradigm were conducted. In Experiment 1, adults received a number comparison task in which large sets of dots, orthogonally varying along a discrete dimension (number of dots) and a continuous dimension (cumulative area), were presented. Incongruent trials were processed more slowly and with less accuracy than congruent trials, suggesting that continuous dimensions such as cumulative area are automatically processed and integrated during a discrete quantity judgement task. Experiment 2, in which adults were asked to perform area comparison on the same stimuli, revealed the reciprocal interference from number on the continuous quantity judgements. Experiment 3, in which participants received both the number and area comparison tasks, confirmed the results of Experiments 1 and 2. Contrasting with earlier statements, the results support the view that number acts as a more salient cue than continuous dimensions in adults. Furthermore, the individual predisposition to automatically access approximate number representations was found to correlate significantly with adults' exact arithmetical skills.     

Predicting sights from sounds: 6-month-olds' intermodal numerical abilities

Although the psychophysics of infants’ nonsymbolic number representations have been well studied, less is known about other characteristics of the approximate number system (ANS) in young children. Here three experiments explored the extent to which the ANS yields abstract representations by testing infants’ ability to transfer approximate number representations across sensory modalities. These experiments showed that 6-month-olds matched the approximate number of sounds they heard to the approximate number of sights they saw, looking longer at visual arrays that numerically mismatched a previously heard auditory sequence. This looking preference was observed when sights and sounds mismatched by 1:3 and 1:2 ratios but not by a 2:3 ratio. These findings suggest that infants can compare numerical information obtained in different modalities using representations stored in memory. Furthermore, the acuity of 6-month-olds’ comparisons of intermodal numerical sequences appears to parallel that of their comparisons of unimodal sequences.     

Children’s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children’s predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.     

Components representation of negative numbers: Evidence from auditory stimuli detection and number classification tasks

Past research suggested that negative numbers could be represented in terms of their components in the visual modality. The present study examined the processing of negative numbers in the auditory modality and whether it is affected by context. Experiment 1 employed a stimuli detection task where only negative numbers were presented binaurally. Experiment 2 employed the same task, but both positive and negative numbers were mixed as cues. A reverse attentional spatial–numerical association of response codes (SNARC) effect for negative numbers was obtained in these two experiments. Experiment 3 employed a number classification task where only negative numbers were presented binaurally. Experiment 4 employed the same task, but both positive and negative numbers were mixed. A reverse SNARC effect for negative numbers was obtained in these two experiments. These findings suggest that negative numbers in the auditory modality are generated from the set of positive numbers, thus supporting a components representation.     

The predictive value of numerical magnitude comparison for individual differences in mathematics achievement

Although it has been proposed that the ability to compare numerical magnitudes is related to mathematics achievement, it is not clear whether this ability predicts individual differences in later mathematics achievement. The current study addressed this question in typically developing children by means of a longitudinal design that examined the relationship between a number comparison task assessed at the start of formal schooling (mean age = 6 years 4 months) and a general mathematics achievement test administered 1 year later. Our findings provide longitudinal evidence that the size of the individual’s distance effect, calculated on the basis of reaction times, was predictively related to mathematics achievement. Regression analyses showed that this association was independent of age, intellectual ability, and speed of number identification.     

Both non-symbolic and symbolic quantity processing are important for arithmetical computation but not for mathematical reasoning

This study investigated whether numerical processing was important for two types of mathematical competence: arithmetical computation and mathematical reasoning. Thousand eight hundred and fifty-seven Chinese primary school children in third through sixth grades took eight computerised tasks: numerical processing (numerosity comparison, digit comparison), arithmetical computation, number series completion, non-verbal matrix reasoning, mental rotation, choice reaction time, and word rhyming. Hierarchical regressions showed that both non-symbolic numerical processing (numerosity comparison) and symbolic numerical processing (digit comparison) were independent predictors of arithmetical computation but neither was a predictor of mathematical reasoning (assessed by number series completion). These findings suggest that the cognitive basis of mathematical performance varies depending on the type of mathematical competence measured.     

A comparison of attentional and control shift models of the performance of concurrent tasks

A concurrent verbal task was superimposed upon the performance of a practiced bimanual motor skill by right-handed Ss. Addition of the verbal task did not increase the total number of errors; however, a significant interaction between hands and conditions was observed. The right hand made significantly more errors under the verbal condition, while the left hand made non-significantly fewer errors under that condition. These findings were interpreted as supporting an attentional model rather than a model which proposes that addition of the verbal task causes control of the right hand to shift to the non-verbal right hemisphere.     

Children’s mapping between symbolic and nonsymbolic representations of number

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children’s mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children’s mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children’s mathematical development.     

On the timing basis of bimanual coordination in discrete and continuous tasks.

Motor events are behaviorally meaningful, discrete entities (e.g., key strokes) that are generated at some specific portion of an effector's movement trajectory. Bimanual coordination may be conceptualized with reference to such discrete motor events or with reference to continuous movement trajectories. Studies inspired by the former approach suggest that hand coordination is primarily achieved by assigning a coherent timing goal structure to the motor events produced by each hand. Studies conducted with the latter approach have shown that between-hand interdependence may also arise from the cross-coupling of the command signals that generate each hand's motion. Little is known, however, about the relationships between timing-level coordination and trajectory-level coordination of the hands. Some aspects of these relationships are analyzed using data from experiments that involved bimanual finger tapping and circle drawing at identical and different frequencies.     

How numbers mean: Comparing random walk models of numerical cognition varying both encoding processes and underlying quantity representations

How do people derive meaning from numbers? Here, we instantiate the primary theories of numerical representation in computational models and compare simulated performance to human data. Specifically, we fit simulated data to the distributions for correct and incorrect responses, as well as the pattern of errors made, in a traditional “relative quantity” task. The results reveal that no current theory of numerical representation can adequately account for the data without additional assumptions. However, when we introduce repeated, error-prone sampling of the stimulus (e.g., Cohen, 2009) superior fits are achieved when the underlying representation of integers reflects linear spacing with constant variance. These results provide new insights into (i) the detailed nature of mental numerical representation, and, (ii) general perceptual processes implemented by the human visual system.     

Participant recruitment methods and statistical reasoning performance

Optimal Bayesian reasoning performance has reportedly been elusive, and a variety of explanations have been suggested for this situation. In a series of experiments, it is demonstrated that these difficulties with replication can be accounted for by differences in participant-sampling methodologies. Specifically, the best performances are obtained with students from top-tier, national universities who were paid for their participation. Performance drops significantly as these conditions are altered regarding inducements (e.g., using unpaid participants) or participant source (e.g., using participants from a second-tier, regional university). Honours-programme undergraduates do better than regular undergraduates within the same university, paid participation creates superior performance, and top-tier university students do better than students from lower ranked universities. Pictorial representations (supplementing problem text) usually have a slight facilitative effect across these participant manipulations. These results indicate that studies should take account of these methodological details and focus more on relative levels of performance rather than absolute performance.     

Limiting attentional resources influences performance and output monitoring of an event-based prospective memory task

One of the most frequently used markers in research on numerical cognition is the distance effect. Recently, we have suggested that a distance effect can have different origins depending on the experimental task. By dissociating the comparison distance effect from the priming distance effect we revealed the need to study the origin of this effect before drawing any conclusions from it (van Opstal, Gevers, de Moor, &; Verguts, 2008). Because a distance effect in a same–different task is also commonly used to study number representations (e.g., Dehaene &; Akhavein, 1995), the present study aimed at uncovering the origin of the effect in this task. Computational and empirical results indicate clearly that the distance effect in the same–different task originates from number representations rather than a decision process.     

Reaching towards an end: Numerical end and distance effects in motor movements

Numerical comparisons are affected by the distance between the numbers and by the presence of an end stimulus. In line with embodied cognition approaches, past studies found evidence for the distance effect in continuous motor movements. The present study is the first to provide evidence for the end effect (i.e., faster comparisons for pairs that include an end stimulus of a set) using continuous motor movements. Two digits were presented horizontally on a screen and participants reached towards the larger one using a computer mouse cursor. Response trajectories were straighter (1) when the number pair included the end stimulus of 1, and (2) when the numerical distance between the numbers was large. Importantly, the end effect appeared earlier in the motor trajectory than the distance effect. The implications of this pattern for the cognitive processes underlying the end and the distance effects are discussed.     

Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.     

Event-mapping tasks: investigating the effects of prior information and event complexity on performance

Infants younger than 11.5 months typically fail in event-mapping tasks with complex event sequences, yet succeed when the event sequences are made very simple and brief. The present research explored whether younger infants might succeed at mapping complex event sequences if infants were given information to help them organize and structure the event. Three experiments were conducted with 7.5-month-olds. In all of the experiments, the infants were shown a two-phase test event. In the first phase, infants saw a box–ball occlusion sequence in which the objects emerged at least once to each side of the screen, reversing direction each time to return behind the screen. In the second phase, infants saw a one-ball display. Prior to the test trials, infants were shown an “outline” of the test event that contained the basic components of the event. The experiments varied in (a) the kind of information included in the event outline and (b) the complexity of the box–ball test sequence (i.e., the number of object reversals). The results revealed that the 7.5-month-olds benefitted from viewing an event outline, although the performance of the males was more robust than the females. These results add to a growing body of research indicating that young infants can succeed on event-mapping tasks under more supportive conditions and provide insight into why event mapping is such a difficult task for young infants.