Non-verbal number acuity correlates with symbolic mathematics achievement: but only in children

The process by which adults develop competence in symbolic mathematics tasks is poorly understood. Nonhuman animals, human infants, and human adults all form nonverbal representations of the approximate numerosity of arrays of dots and are capable of using these representations to perform basic mathematical operations. Several researchers have speculated that individual differences in the acuity of such nonverbal number representations provide the basis for individual differences in symbolic mathematical competence. Specifically, prior research has found that 14-year-old children’s ability to rapidly compare the numerosities of two sets of colored dots is correlated with their mathematics achievements at ages 5–11. In the present study, we demonstrated that although when measured concurrently the same relationship holds in children, it does not hold in adults. We conclude that the association between nonverbal number acuity and mathematics achievement changes with age and that nonverbal number representations do not hold the key to explaining the wide variety of mathematical performance levels in adults.     

The development of numerical estimation: evidence against a representational shift

How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number‐line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children’s estimation performance are identified and discussed. This work demonstrates that proportion‐judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.     

The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming

The development of number processing is generally studied by examining the performance on basic number tasks (comparison task, same-different judgment, and priming task). Using these tasks, so-called numerical distance effects are obtained. All these effects are generally explained by assuming a magnitude representation related to a mental number line: magnitudes are represented from left to right with partially overlapping representations for nearby numbers. In this study, we compared the performance of adults on these different tasks using non-symbolic stimuli. First, we investigated whether the effects obtained in these behavioral tasks are reliable. Second, we examined the relation between the three different effects. The results showed that the observed effects in the case of the comparison task and the same-different task proved to be reliable. The numerical distance effect obtained in the priming task, however, was not reliable. In addition, a correlation was found between the distance effects in the comparison task and the same-different task. The priming distance effect did not correlate with the other two effects. These results suggest important differences between distance effects obtained under automatic and intentional task instructions regarding the use of them as indices of mathematical ability.     

The effect of external representations on numeric tasks

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: The comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect is discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations, and the behaviour in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical cognition should consider external representations as an essential component.     

Effects of aging on interference control in selective attention and working memory

Working memory decay in advanced age has been attributed to a concurrent decrease in the ability to control interference. The present study contrasted a form of interference control in selective attention that acts upon the perception of external stimuli (access) with another form that operates on internal representations in working memory (deletion), in order to determine both of their effects on working memory efficiency in younger and older adults. Additionally, we compared memory performance under these access and deletion functions to performance in their respective control conditions. The results indicated that memory accuracy improved in both age groups from the access functions, but that only young adults benefited from the deletion functions. In addition, intrusion effects in the deletion condition were larger in older than in younger adults. The ability to control the irrelevant perception- and memory-elicited interference did not decline in general with advancing age; rather, the control mechanisms that operate on internal memory representations declined specifically.     

Adult age differences in subjective and objective measures of strategy use on a sequentially cued prediction task

Differences in strategy use are thought to underlie age-related performance deficits on many learning and decision-making tasks. Recently, age-related differences in learning to make predictions were reported on the Triplets Prediction Task (TPT). Notably, deficits appeared early in training and continued with experience. To assess if age differences were due to early strategy use, neural networks were used to objectively assess the strategies implemented by participants during Session 1. Then, the relationship between these strategies and performance was examined. Results revealed that older adults were more likely to implement a disadvantageous strategy early in learning, and this led to poorer task performance. Importantly, the relationship between age and task performance was partially mediated by early strategy use, suggesting that early strategy selection played a role in the lower quality of predictions in older adults.     

Children’s representation of abstract relations in relational/array match-to-sample tasks

Five experiments compared preschool children’s performance to that of adults and of non-human animals on match to sample tasks involving 2-item or 16-item arrays that varied according to their composition of same or different items (Array Match-to-Sample, AMTS). They establish that, like non-human animals in most studies, 3- and 4-year-olds fail 2-item AMTS (the classic relational match to sample task introduced into the literature by Premack, 1983), and that robust success is not observed until age 6. They also establish that 3-year-olds, like non-human animal species, succeed only when they are able to encode stimuli in terms of entropy, a property of an array (namely its internal variability), rather than relations among the individuals in the array (same vs. different), whereas adults solve both 2-item and 16-item AMTS on the basis of the relations same and different. As in the case of non-human animals, the acuity of 3- and 4-year-olds’ representation of entropy is insufficient to solve the 2-item same-different AMTS task. At age 4, behavior begins to contrast with that of non-human species. On 16-item AMTS, a subgroup of 4-year-olds induce a categorical rule matching all-same arrays to all-same arrays, while matching other arrays (mixed arrays of same and different items) to all-different arrays. These children tend to justify their choices using the words “same” and “different.” By age 4 a number of our participants succeed at 2-item AMTS, also justifying their choices by explicit verbal appeals using words for same and different. Taken together these results suggest that the recruitment of the relational representations corresponding to the meaning of these words contributes to the better performance over the preschool years at solving array match-to-sample tasks.     

The effects of age and task context on Stroop task performance

In this study, we investigated the impact of age and task context on Stroop task performance, using error scores, response latencies, and process dissociation estimates (e.g., Lindsay & Jacoby, 1994). Across three experiments, the findings showed that although older adults were able to evaluate Stroop task demands and modify their representations of task context in response to this knowledge, they were less able to maintain and update these representations on a trial-by-trial basis in tasks with high stimulus uncertainty or ambiguity. Moreover, although there was no age-related decline in the ability to modulate print color information, older adults were consistently less able to control the activation of conflicting word information. Together, these findings suggest that whereas age differences in the Stroop task may be magnified under conditions that promote transient failures to maintain task context, the primary source of these differences seems to be a more enduring decline in the efficiency of processes that are responsible for suppressing the activation of irrelevant lexical information.     

Aging neuromodulation impairs associative binding: a neurocomputational account

Relative to young adults, older adults are particularly impaired in episodic memory tasks requiring associative binding of separate components into compound episodes, such as tasks requiring item-context and item-item binding. This associative-binding deficit has been attributed to senescent changes in frontal-hippocampal circuitry but has not been formally linked to impaired neuromodulation involving this circuitry. Previous neurocomputational work showed that impaired neuromodulation could result in less distinct neurocognitive representations. Here we extend this computational principle to simulate aging-related deficits in associative binding. As expected, networks with simulated deficiency in neuromodulation resulted in less distinct internal representations than did networks simulating the processing and performance of young adults, and were also more impaired under task conditions that required associative binding. The findings suggest that senescent changes in neuromodulatory mechanisms may play a basic role in aging-related impairment in associative binding by reducing the efficacy of distributed conjunctive coding.     

The orientation of young and older adults' mental representations of their home town with familiar and new landmarks

Familiarity with an environment produces refined mental representations in adults of all ages, but it is not clear whether these representations tend to have a north-up orientation and whether familiarity facilitates the learning of new spatial information, especially in ageing. Thirty-two young and 32 older adults studied a map of their home town that included familiar and new fictitious landmarks, then performed pointing tasks, some aligned with the cardinal points south–north (SN), and others counter-aligned, north–south (NS). A measure of visuo-spatial working memory (VSWM) and a questionnaire on pleasure in exploring were also administered. The results showed that performance was better when pointing SN than NS (alignment effect), whereas pointing performance for familiar landmarks was similar for SN and NS alignments (no alignment effect). No interaction involving age was found. VSWM emerged as a significant predictor of pointing performance. Spatial mental representations of familiar environments are elaborate and flexible as regards familiar landmarks, in both young and older adults; and a familiar layout does not seem to enhance older adults' spatial memory as regards new landmarks.     

Imagery and Motor Processes — When Are They Connected? The Mental Rotation of Body Parts in Development

Motor influences on the mental transformation of body parts have been observed in both children and adults. Previous findings indicated that these influences were more pronounced in children than in adults, suggesting a stronger link between motor processes and imagery in children. The present series of two experiments casts doubt on the general validity of such an interpretation. Kindergartners' (aged 5–6 years), first graders' (aged 7 years), and adults' performance in the mental rotation of pictures of body parts was monitored for influences of internal representations of motor constraints (motor effect). In both experiments, evidence for mental rotation was obtained for each group. Unexpectedly, kindergarten boys made significantly more errors than kindergarten girls. A motor effect was only found in the second experiment, where it was least pronounced in the youngest age group. Our results suggest that mental transformations of body parts do not necessarily involve motor processes and that embodiment may become stronger with development rather than weaker with certain tasks.     

Aging affects word-stem completion and recognition but not category generation

In this study, we investigated the effect of aging on two implicit memory tasks, word-stem completion and category generation, and on explicit recognition. We compared the performance of young and older adults on these implicit memory tasks with those of explicit recognition. We expected better performance of young than older adults in the explicit memory task and similar priming in both implicit memory tasks. The results showed that young adults performed better than older adults in the recognition task. Moreover, both age groups showed priming in the implicit memory tasks, although priming was greater in young adults compared to older adults in the word-stem completion memory task, whereas both age groups showed similar levels of priming in the category generation task. The present results showed dissociations as a function of age not only between the explicit and the implicit tasks but also between the implicit tasks.     

Orthographic error monitoring in old age: lexical and sublexical availability during perception and production

Previous research suggests that older adults suffer declines in producing accurate spellings but retain the ability to accurately detect misspellings. The preservation of perception in the face of impaired production has been used to support a model of aging in which age impairs access to linguistic representations under specific circumstances, while representations themselves remain intact. The current research tests two predictions of this Transmission Deficit Hypothesis (TDH): first, that the differential effect of age on perception and production occurs when tasks are equated on response requirements and underlying representations, and second, that both word and spelling frequency interact to determine the effect of age on performance. Results of two error monitoring tasks supported the predictions of the TDH, demonstrating age-related production impairments that interacted with both word and spelling frequency, but no impairment of older adults' spelling perception, even for low frequency words or spellings.     

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.     

Age-related differences and similarities in dual-task interference

Differences between younger adults (mean age, 20.7 years) and older adults (mean age, 72.7 years) in dual-task performance were examined in 7 experiments in which the overlap between 2 simple tasks was systematically varied. The results were better fit by a task-switching model in which age was assumed to produce generalized slowing than by a shared-capacity model in which age was assumed to reduce processing resources. The functional architecture of task processing appears the same in younger and older adults. There was no evidence for a specific impairment in the ability of older adults to manage simultaneous tasks. There was evidence for both input and output interference, which may be greater in older adults.     

Age differences in prospective memory for everyday life intentions: A diary approach

The age benefit found in many naturalistic prospective memory (PM) tasks has been taken as evidence that PM performance in real life may be spared from aging. However, this conclusion lacks empirical confirmation. Hence, the aim of the present study was to examine possible age differences in the content of everyday PM intentions and their performance. Everyday PM was assessed in young and older adults using a diary approach. Results confirmed a general age benefit for real-life PM tasks. Importantly, this finding was qualified by revealing that the benefit only held true for specific types of intentions such as health and social intentions. Further, moderation analyses showed that the relationships between cognitive functioning and everyday PM were different for young and older adults. While better inhibition, short-term and long-term memory were related with successful PM performance in the young, this was not the case in the older adults. The present findings suggest that the age benefit found in naturalistic experimenter-given tasks extends to real-life PM performance, but may differ depending on the type of intention. Furthermore, cognitive functioning predicts performance in the young, but not in the older adults.     

Language, space, and the development of cognitive flexibility in humans: the case of two spatial memory tasks

Prior experiments have shown that young children, like adult rats, rely mainly on information about the macroscopic shape of the environment to reorient themselves, whereas human adults rely more flexibly on combinations of spatial and non-spatial landmark information. Adult rats have also been shown to exhibit a striking limitation in another spatial memory task, movable object search, again a limitation not shown by human adults. The present experiments explored the developmental change in humans leading to more flexible, human adult-like performance on these two tasks. Experiment 1 identified the age range of 5-7 years as the time the developmental change for reorientation occurs. Experiment 2 employed a multiple regression approach to determine that among several candidate measures, only a specific language production measure, the production of phrases specifying exactly the information needed to solve the task like adults, correlated with the reorientation performance of children in this age range. Experiment 3 revealed that similar language production abilities were associated with more flexible moving object search task performance. These results, in combination with findings with human adults, suggest that language production skills play a causal role in allowing older humans to construct novel representations rapidly, which can then be used to transcend the limits of phylogenetically older cognitive processes.     

Rehearsal strategies during motor-sequence learning in old age: execution vs motor imagery

Motor imagery and action-based rehearsal were compared during motor sequence-learning by young adults (M = 25 yr., SD = 3) and aged adults (M = 63 yr., SD = 7). General accuracy of aged adults was lower than that of young adults (F(1,28) = 7.37, p =.01) even though working-memory capacity was equivalent in the two groups. Motor imagery and rehearsal by action increased accuracy in both age groups, compared with minimization of opportunity for rehearsal (F(1,28) = 30.95, p < .001), but no interaction was found with age group, which suggests that young and aged adults were equally capable of motor imagery and action-based rehearsal. It was assumed that differences in performance between young and aged participants related to the formation of mental representations of sequences and integration of new elements into these representations rather than the capacity for motor imagery or rehearsal by action per se. The current study was exploratory and involved a relatively small sample of 15 participants per age group. Caution must be taken when considering the results.     

Preschoolers’ nonsymbolic arithmetic with large sets: Is addition more accurate than subtraction?

Adult and developing humans share with other animals analog magnitude representations of number that support nonsymbolic arithmetic with large sets. This experiment tested the hypothesis that such representations may be more accurate for addition than for subtraction in children as young as 3½ years of age. In these tasks, the experimenter hid two equal sets of cookies, visibly added to or subtracted from the sets, and then asked 3½-year-olds which set had more cookies. Initial set size was either large (7 or 9) or very large (18 or 30), and the final sets differed by either a high proportion (ratio of 1:2) or a low proportion (difference of 1 cookie). Children’s addition performance exceeded chance, as well as their subtraction performance, across set sizes and proportions, whereas subtraction performance did not exceed chance. Arithmetic performance was also independent of counting ability. Addition performance was remarkably accurate when ratios between outcomes were close to 1, in contrast to previous findings. Interpretations for the asymmetry between addition and subtraction are discussed with respect to the nature of representations for nonsymbolic arithmetic with large sets.     

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.