Learning without representational change: development of numerical estimation in individuals with Williams syndrome

Experience engenders learning, but not all learning involves representational change. In this paper, we provide a dramatic case study of the distinction between learning and representational change. Specifically, we examined long‐ and short‐term changes in representations of numeric magnitudes by asking individuals with Williams syndrome (WS) and typically developing (TD) children to estimate the position of numbers on a number line. As with TD children, accuracy of WS children's numerical estimates improved with age (Experiment 1) and feedback (Experiment 2). Both long‐ and short‐term changes in estimates of WS individuals, however, followed an atypical developmental trajectory: as TD children gained in age and experience, increases in accuracy were accompanied by a logarithmic‐to‐linear shift in estimates of numerical magnitudes, whereas in WS individuals, accuracy increased but logarithmic estimation patterns persisted well into adulthood and after extensive training. These findings suggest that development of numerical estimation in WS is both arrested and atypical.     

Developmental and individual differences in pure numerical estimation

The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed.     

The development of numerical estimation: evidence for multiple representations of numerical quantity

Abstract - We examined children's and adults' numerical estimation and the representations that gave rise to their estimates. The results were inconsistent with two prominent models of numerical representation: the logarithmic-ruler model, which proposes that people of all ages possess a single, logarithmically spaced representation of numbers, and the accumulator model, which proposes that people of all ages represent numbers as linearly increasing magnitudes with scalar variability. Instead, the data indicated that individual children possess multiple numerical representations; that with increasing age and numerical experience, they rely on appropriate representations increasingly often; and that the numerical context influences their choice of representation. The results, obtained with second graders, fourth graders, sixth graders, and adults who performed two estimation tasks in two numerical contexts, strongly suggest that one cause of children's difficulties with estimation is reliance on logarithmic representations of numerical magnitudes in situations in which accurate estimation requires reliance on linear representations.     

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.     

The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.     

Number games, magnitude representation, and basic number skills in preschoolers

The effect of 3 intervention board games (linear number, linear color, and nonlinear number) on young children's (mean age = 3.8 years) counting abilities, number naming, magnitude comprehension, accuracy in number-to-position estimation tasks, and best-fit numerical magnitude representations was examined. Pre- and posttest performance was compared following four 25-min intervention sessions. The linear number board game significantly improved children's performance in all posttest measures and facilitated a shift from a logarithmic to a linear representation of numerical magnitude, emphasizing the importance of spatial cues in estimation. Exposure to the number card games involving nonsymbolic magnitude judgments and association of symbolic and nonsymbolic quantities, but without any linear spatial cues, improved some aspects of children's basic number skills but not numerical estimation precision.     

Representational change and magnitude estimation: why young children can make more accurate salary comparisons than adults

Development of estimation has been ascribed to two sources: (1) a change from logarithmic to linear representations of number and (2) development of general mathematical skills. To test the representational change hypothesis, we gave children and adults a task in which an automatic, linear representation is less adaptive than the logarithmic representation: estimating the value of salaries given in fractional notation. The representational change hypothesis generated the surprising (and accurate) prediction that when estimating the magnitude of salaries given in fractional notation, young children would outperform adults, whereas when estimating the magnitude of the same salaries given in decimal notation, adults would outperform children.     

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.     

儿童数字估计的表征模式与发展

探讨中国儿童数字估计的表征模式与发展趋势。包括两个实验,均采用数字线估计任务,实验一以92名幼儿园、一年级及二年级儿童为被试,考察其在0~100范围的数字估计,结果显示,幼儿园儿童在数字估计更多地采用对数表征,而一二年级的儿童在数字估计中更多地采用线性表征;实验二以86名一、三、五年级儿童为被试,考察其在0~1000范围的数字估计,结果显示,一年级儿童有一半采用对数表征,另一半采用线性表征,而三五年级儿童大多采用线性表征。中国儿童的数字估计表现出与美国儿童相同的发展模式,都是由不精确的对数表征逐步向精确的线性表征发展;人的数表征有多种形式,即使在同一年龄阶段,也会因任务难度的不同而选择不同的表征模式。中国儿童精确数字估计能力的出现要早于美国儿童。     

Numerical estimation in blind subjects: evidence of the impact of blindness and its following experience

Vision was for a long time considered to be essential in the elaboration of the semantic numerical representation. However, early visual deprivation does not seem to preclude the development of a spatial continuum oriented from left to right to represent numbers (J. Castronovo & X. Seron, 2007; D. Szücs & V. Csépe, 2005). The authors investigated the impact of blindness and its following experience on a 3rd property of the mental number line: its obedience to Weber's law. A group of blind subjects and a group of sighted subjects were submitted to 2 numerical estimation tasks: (a) a keypress estimation task and (b) an auditory events estimation task. Blind and sighted subjects' performance obeyed Weber's law. However, blind subjects demonstrated better numerical estimation abilities than did sighted subjects, especially in contexts involving proprioception, indicating the existence of better mapping abilities between the symbolic representations of numbers and their corresponding magnitude representations, obeying Weber's law (e.g., J. S. Lipton & E. Spelke, 2005). These findings suggest that blindness and its following experience with numbers might result in better accuracy in numerical processing.     

The development of numerosity estimation: Evidence for a linear number representation early in life

Several studies investigating the development of approximate number representations used the number-to-position task and reported evidence for a shift from a logarithmic to a linear representation of numerical magnitude with increasing age. However, this interpretation as well as the number-to-position method itself has been questioned recently. The current study tested 5- and 8-year-old children on a newly established numerosity production task to examine developmental changes in number representations and to test the idea of a representational shift. Modelling of the children's numerical estimations revealed that responses of the 8-year-old children approximate a simple positive linear relation between estimated and actual numbers. Interestingly, however, the estimations of the 5-year-old children were best described by a bilinear model reflecting a relatively accurate linear representation of small numbers and no apparent magnitude knowledge for large numbers. Taken together, our findings provide no support for a shift of mental representations from a logarithmic to a linear metric but rather suggest that the range of number words which are appropriately conceptualised and represented by linear analogue magnitude codes expands during development.     

Numerical Estimation in Children for Both Positive and Negative Numbers

Numerical estimation has been used to study how children mentally represent numbers for many years (e.g., Siegler & Opfer, 2003). However, these studies have always presented children with positive numbers and positive number lines. Children’s mental representation of negative numbers has never been addressed. The present study tested children in the 2nd, 4th, and 6th grades to assess their mental representations of both positive and negative numbers using a standard numerical estimation task. We replicated the shift from a logarithmic to linear representation for positive numbers (0–1,000 scale) in that 2nd graders represented positive numbers logarithmically, but 4th and 6th graders represented the numbers linearly. Furthermore, children’s representation of negative numbers paralleled their representations of positive numbers and showed the same shift from a logarithmic representation at Grade 2 to linear representations at Grades 4 and 6. This is the first study to provide data on children’s representation of negative numbers, and the implications of these findings are discussed.     

Number estimation relies on a set of segmented objects

How do we estimate the number of objects in a set? Two types of visual representations might underlie this ability - an unsegmented visual image or a segmented collection of discrete objects. We manipulated whether individual objects were isolated from each other or grouped into pairs by irrelevant lines. If number estimation operates over an unsegmented image, then this manipulation should not affect estimates. But if number estimation relies on a segmented image, then grouping pairs of objects into single units should lead to lower estimates. In Experiment 1 participants underestimated the number of grouped objects, relative to disconnected objects in which the connecting lines were ‘broken’. Experiment 2 presents evidence that this segmentation process occurred broadly across the entire set of objects. In Experiment 3, a staircase procedure provides a quantitative measure of the underestimation effect. Experiment 4 shows that the strength of the grouping effect was equally strong for a single thin line, and the effect can be eliminated by a small break in the line. These results provide direct evidence that number estimation relies on a segmented input.     

Scaling occupational prestige by magnitude estimation and category rating methods: A comparison with the sensory domain

In a field study, models for magnitude estimation and for category ratings are applied to the scaling of occupational prestige. The two respective models provide sufficient conditions for magnitude estimates to yield logarithmic interval scales and for category ratings to lead to interval scales. Both models are found to hold reasonably well for the majority of respondents. As implied by a third model, the relation between magnitude estimation and category rating scales can well be described by a generalized power function. Although overall results do not favour one method over the other individual data analyses reveal substantial interindividual differences with respect to the capability of performing magnitude estimates and category ratings, respectively. The findings are compared to results recently found in psychophysical laboratory experiments, and it is concluded that the individual scale properties the two methods provide do not differ across the attitudinal and the sensory domains.     

Contextual effects on number-time interaction

Time perception has long been known to be affected by numerical representations. Recent studies further demonstrate that when participants estimate the duration of Arabic numbers, number magnitude, though task-irrelevant, biases duration judgment to produce underestimation for smaller numbers and overestimation for larger numbers. Such effects were found in the present study to be significantly reduced when a weight unit gram was suffixed to the numbers rendering the mental magnitude differences between different numbers less distinctive. The effects were enhanced when a different unit kilogram was suffixed to the numbers enlarging the perceived magnitude differences between different numbers. The results indicate that effects of number magnitude on duration estimation should not be attributed to the mathematical differences between numbers but to how the numbers are perceived to differ from each other in magnitude in specific contexts when they denote concrete items. The results also provide new evidence for the theoretical proposal of a common generalized magnitude system and indicate that the system must be extended to include other action-oriented magnitudes, such as weight.     

Asymmetries in the processing of Arabic digits and number words

Numbers can be represented as Arabic digits ("6") or as number words ("six"). The present study investigated potential processing differences between the two notational formats. In view of the previous finding (e.g., Potter & Faulconer, 1975) that objects are named slower, but semantically categorized faster, than corresponding words, it was investigated whether a similar interaction between stimulus format and task could be obtained with numbers. Experiment 1 established that number words were named faster than corresponding digits, but only if the two notation formats were presented in separate experimental blocks. Experiment 2 contrasted naming with a numerical magnitude judgment task and demonstrated an interaction between notation and task, with slower naming but faster magnitude judgment latencies for digits than for number words. These findings suggest that processing of the two notation formats is asymmetric, with digits gaining rapid access to numerical magnitude representations, but slower access to lexical codes, and the reverse for number words.     

Estimating national populations: cross‐cultural differences and availability effects

Estimates of national population were studied in two experiments. In Experiment 1, Canadian and Chinese undergraduates rated their knowledge of 112 countries and then estimated the population of each. In Experiment 2, Canadians rated their knowledge of 52 countries and then provided population estimates for these primed countries and for a comparable set of 52 unprimed countries. In Experiment 1, participants from both nations produced estimates that resembled those obtained from Americans in prior studies (Brown and Siegler, 1992 , 1993 , 1996 , 2001 ). However, there were several reliable cross‐national differences in performance which appear to reflect cross‐cultural differences in task‐relevant naive domain knowledge. In addition, both experiments produced findings consistent with the claim that availability‐based intuitions play an important role in this task. In Experiment 1, cross‐national differences in rated knowledge predicted cross‐national differences in estimated population; in Experiment 2, primed country names elicited larger population estimates than unprimed country names. We conclude by arguing for the general utility of this hybrid approach to real‐world estimation. Copyright © 2002 John Wiley & Sons, Ltd.     

Magnitude comparisons distort mental representations of magnitude

Many cognitive processes rely on representations of magnitude, yet these representations are often malleable (H. Helson, 1964; J. Huttenlocher, L. V. Hedges, & J. L. Vevea, 2000; A. Parducci, 1965). It is likely that factors that affect these representations in turn affect the psychological processes that rely on them. The authors conducted 4 experiments to investigate whether language-expressible magnitude comparisons distort mental representations of compared magnitudes. Participants compared magnitudes and estimated those magnitudes in a variety of tasks. Experiments 1 through 3 demonstrated systematic comparison-induced distortions. Experiment 4 demonstrated that comparison-induced distortions might account for the asymmetric dominance effect discussed in the decision-making literature. Potential effects of comparison-induced distortions on other psychological processes (e.g., density effects, order effects, body-size estimation, pain estimation, and consumer decision making) are discussed.     

Distractors interfere with numerical estimation in Chinese college students as a function of field-dependent or field-independent cognitive style

Field-dependent/independent groups' numerical estimation was assessed with respect to different distractors. Participants were 81 college students with field-dependent or field-independent cognitive styles. Cognitive style had a significant main effect on reaction without distractors. When the number of distractors was double the number of targets, the estimation accuracy of the field-dependent and field-independent groups differed. Under this double-distractors condition, both the field-dependent and field-independent participants used logarithmic representation more than linear representation in their numerical estimations, but no significant between-group differences were found.