Evidence against “absolute” scaling

Zwislocki and Goodman (1980) argue that there exists an “absolute coupling” of numbers to sensation magnitudes and conclude that, when subjects are left unconstrained by a designated stimulus-number pair, they may use this “absolute” scale. The purpose of the studies reported here was to test whether Zwislocki and Goodman’s (1980) absolute scaling procedure reduces contextual effects due to variations in the stimulus spacing. It was found that magnitude estimations vary as a function of the stimulus spacing, regardless of whether subjects are instructed to use a standard and modulus, and, furthermore, that category ratings yield effects of the stimulus spacing comparable to those obtained with magnitude estimations. It is argued that removing the so-called constraints of a standard and modulus does not yield an “absolute” scale of sensation. The absolute scaling procedure increases response variability and thereby lowers the power of a test for contextual effects.     

Negative numbers are generated in the mind

The goal of the present study was to disentangle two possible representations of negative numbers--the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from--100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.     

Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production

Stevens's direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens's central axioms of commutativity and multiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens's (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

Individual differences in the development of sensation seeking and impulsivity during adolescence: further evidence for a dual systems model

Consistent with social neuroscience perspectives on adolescent development, previous cross-sectional research has found diverging mean age-related trends for sensation seeking and impulsivity during adolescence. The present study uses longitudinal data on 7,640 youth from the National Longitudinal Study of Youth Children and Young Adults, a nationally representative sample assessed biennially from 1994 to 2006. Latent growth curve models were used to investigate mean age-related changes in self-reports of impulsivity and sensation seeking from ages 12 to 24 years, as well individual differences in these changes. Three novel findings are reported. First, impulsivity and sensation seeking showed diverging patterns of longitudinal change at the population level. Second, there was substantial person-to-person variation in the magnitudes of developmental change in both impulsivity and sensation seeking, with some teenagers showing rapid changes as they matured and others maintaining relatively constant levels with age. Finally, the correlation between age-related changes in impulsivity and sensation seeking was modest and not significant. Together, these results constitute the first support for the dual systems model of adolescent development to derive from longitudinal behavioral data.     

Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production

Stevens’s direct scaling methods rest on the assumption that subjects are capable of reporting or producing ratios of sensation magnitudes. Only recently, however, did an axiomatization proposed by Narens (1996) specify necessary conditions for this assumption that may be put to an empirical test. In the present investigation, Narens’s central axioms ofcommutativity andmultiplicativity were evaluated by having subjects produce loudness ratios. It turned out that the adjustments were consistent with the commutativity condition; multiplicativity (the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), however, was violated in a significant number of cases. According to Narens’s (1996) axiomatization, this outcome implies that although in principle a ratio scale of loudness exists, the numbers used by subjects to describe sensation ratios may not be taken at face value.     

Magnitude estimation and magnitude production: stimulus frequency effects on magnitudes of lingual vibrotactile sensation

The methods of magnitude estimation and magnitude production were employed to investigate the effects of stimulus frequency on supra-threshold lingual-vibrotactile sensation-magnitude functions. The method of magnitude estimation was used to obtain numerical judgments of sensation magnitudes for nine stimulus intensities presented to the anterior dorsum of the tongue. The vibrotactile stimulus frequencies employed for 10 subjects (M age = 21.1 yr.) were 100, 250, and 400 Hz. The numerical responses obtained during the magnitude-estimation task were in turn used as stimuli to obtain magnitude-production values for the same three vibrotactile stimulus frequencies. The results appeared to present two suggestions. First, the effects of stimulus frequency on lingual vibrotactile-sensation magnitudes may be dependent on the psychophysical method used in any particular experiment. Second, lingual-vibrotactile magnitude-estimation scales may demonstrate asymptotic growth functions above about 25 dB sensation level. The limitation in the growth of sensation magnitude occurred for all three vibrotactile stimulus frequencies employed.     

A mental number line in human newborns

Humans represent numbers on a mental number line with smaller numbers on the left and larger numbers on the right side. A left‐to‐right oriented spatial–numerical association, (SNA), has been demonstrated in animals and infants. However, the possibility that SNA is learnt by early exposure to caregivers’ directional biases is still open. We conducted two experiments: in Experiment 1, we tested whether SNA is present at birth and in Experiment 2, we studied whether it depends on the relative rather than the absolute magnitude of numerousness. Fifty‐five‐hour‐old newborns, once habituated to a number (12), spontaneously associated a smaller number (4) with the left and a larger number (36) with the right side (Experiment 1). SNA in neonates is not absolute but relative. The same number (12) was associated with the left side rather than the right side whenever the previously experienced number was larger (36) rather than smaller (4) (Experiment 2). Control on continuous physical variables showed that the effect is specific of discrete magnitudes. These results constitute strong evidence that in our species SNA originates from pre‐linguistic and biological precursors in the brain.     

Non-verbal numerical cognition: from reals to integers

Data on numerical processing by verbal (human) and non-verbal (animal and human) subjects are integrated by the hypothesis that a non-verbal counting process represents discrete (countable) quantities by means of magnitudes with scalar variability. These appear to be identical to the magnitudes that represent continuous (uncountable) quantities such as duration. The magnitudes representing countable quantity are generated by a discrete incrementing process, which defines next magnitudes and yields a discrete ordering. In the case of continuous quantities, the continuous accumulation process does not define next magnitudes, so the ordering is also continuous ('dense'). The magnitudes representing both countable and uncountable quantity are arithmetically combined in, for example, the computation of the income to be expected from a foraging patch. Thus, on the hypothesis presented here, the primitive machinery for arithmetic processing works with real numbers (magnitudes).     

Effects of non-symbolic numerical information suggest the existence of magnitude-space synesthesia

In number-space synesthesia, numbers are visualized in spatially defined arrays. In a recent study (Gertner et al. in Cortex, doi: 10.1016/j.cortex.2012.03.019 , 2012), we found that the size congruency effect (SiCE) for physical judgments (i.e., comparing numbers' physical sizes while ignoring their numerical values, for example, 8) was modulated by the spatial position of the presented numbers. Surprisingly, we found that the neutral condition, which is comprise solely of physical sizes (e.g., 3), was affected as well. This pattern gave rise to the idea that number-space synesthesia might entail not only discrete, ordered, meaningful symbols (i.e., Arabic numbers) but also continuous non-symbolic magnitudes (i.e., sizes, length, luminance, etc.). We tested this idea by assessing the performance of two number-space synesthetes and 12 matched controls in 3 comparative judgment tasks involving symbolic and non-symbolic stimuli: (1) Arabic numbers, (2) dot clusters, and (3) sizes of squares. The spatial position of the presented stimuli was manipulated to be compatible or incompatible with respect to the synesthetic number-space perceptions. Results revealed that for synesthetes, but not for controls, non-symbolic magnitudes (dot clusters) as well as symbolic magnitudes (i.e., Arabic numbers) interacted with space. Our study suggests that number-space synesthetes might have a general magnitude-space association that is not restricted to concrete symbolic stimuli. These findings support recent theories on the perception and evaluation of sizes in numerical cognition.     

Factors producing and factors not producing time errors: An experiment with loudness comparisons

Pairs of 1-sec, 1,000-Hz tones, with interstimulus intervals of 1.5 sec, were judged by 60 subjects in categories of “louder,” “softer,” and “equal.” The judgments referred to the first tone in the pair for half of the subjects and to the second tone for the other half. Perceived loudness differences were scaled by a Thurstonian method. The SPL of the standard tone alternated between 50 and 70 dB in one experimental series and between 30 and 50 dB in the other. Time errors (TEs) were consistently positive (first tone overestimated relative to second) at the lower SPL and negative at the higher SPL. This “classical” effect of stimulus level on TE was thus shown to depend upon the relative, rather than the absolute, level of stimulation. The judgment mode was of very little consequence, which strongly contradicts TE theories that emphasize response-bias effects. The quantitative results are interpreted in terms of a general successive-comparison model employing the concepts of adaptation and differential weighting of sensation magnitudes.     

The number–time interaction depends on relative magnitude in the suprasecond range

Numerical representations influence temporal processing. Previous studies have consistently shown that larger numbers are perceived to last longer than smaller ones. However, whether this effect is modulated by the absolute or relative magnitudes of the numbers has yet to be fully understood. Here, participants observed single- and double-digit Arabic numerals in separate experimental blocks and reproduced stimulus duration of 600 or 1200 ms. Our results replicated previous findings that the duration of larger numbers was reproduced longer than that of smaller numbers within each digit set. Although the effect of numerical magnitude across single- and double-digit numerals was found when the numerals were presented for 600 ms, the difference was negligible when they were presented for 1200 ms, suggesting that relative magnitude is an important factor in the number–time interaction in the suprasecond range. These results suggest that contextual influence on number–time interaction may depend on the actual stimulus duration.     

Spatial numerical associations in preschoolers

ABSTRACT

Three-to-five-year-old French children were asked to add or remove objects to or from linear displays. The hypothesis of a universal tendency to represent increasing number magnitudes from left to right led to predict a majority of manipulations at the right end of the rows, whatever children's hand laterality. Conversely, if numbers are not inherently associated with space, children were expected to favour laterality-consistent manipulations. The results showed a strong tendency to operate on the right end of the rows in right-handers, but no preference in left-handers. These findings suggest that the task elicited a left-to-right oriented representation of magnitudes that counteracted laterality-related responses in left-handed children. The young age of children and the lack of a developmental trend towards right preference weaken the hypothesis of a cultural origin of this oriented representation. The possibility that our results are due to weaker brain lateralisation in left-handers compared to right-handers is addressed in Discussion section.     

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial–numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.     

Examining the validity of numerical ratios in loudness fractionation

Direct psychophysical scaling procedures presuppose that observers are able to directly relate a numerical value to the sensation magnitude experienced. This assumption is based on fundamental conditions (specified by Luce, 2002), which were evaluated experimentally. The participants' task was to adjust the loudness of a 1-kHz tone so that it reached a certain prespecified fraction of the loudness of a reference tone. The results of the first experiment suggest that the listeners were indeed able to make adjustments on a ratio scale level. It was not possible, however, to interpret the nominal fractions used in the task as "true" scientific numbers. Thus, Stevens's (1956, 1975) fundamental assumption that an observer can directly assess the sensation magnitude a stimulus elicits did not hold. In the second experiment, the possibility of establishing a specific, strictly increasing transformation function that related the overt numerals to the latent mathematical numbers was investigated. The results indicate that this was not possible for the majority of the 7 participants.     

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.     

Stevens vs Fechner: A plea for dismissal of the case

Fechner's Law is based on the assumption that jnd's can be concatenated in order to get a measure of sensation. Stevens contended this assumption; he argued that measurement of sensation by measuring jnd's is ‘indirect’. His ‘direct’ methods, however, appear to be much less direct than originally assumed. Especially Stevens' (implicit) assumption that subjects use numbers on an absolute scale appears to be unwarranted. Ratio scaling is therefore nothing more than cross-modality matching between the number scale and another modality. This view has two consequences. Firstly, it is quite natural to find power functions by cross-modality matching, even when Fechner's Law holds for both modalities. Secondly, if Stevens' power law is true, there will be an overestimation of all exponents by a factor 2.5. All typical power functions will be concave downward after correction for this effect. The discussion boils down to a discrimination between two quite similar curve families: log functions and power functions with an exponent below 0.6. The practical significance of this discussion is doubted.     

Children’s representation of symbolic magnitude: The development of the priming distance effect

The comparison distance effect (CDE), whereby discriminating between two numbers that are far apart is easier than discriminating between two numbers that are close, has been considered as an important indicator of how people represent magnitudes internally. However, the underlying mechanism of this CDE is still unclear. We tried to shed further light on how people represent magnitudes by using priming. Adults have been shown to exhibit a priming distance effect (PDE), whereby numbers are processed faster when they are preceded by a close number than when they are preceded by a more distant number. Surprisingly, there are no studies available that have investigated this effect in children. The current study examined this effect in typically developing first, third, and fifth graders and in adults. Our findings revealed that the PDE already occurs in first graders and remains stable across development. This study also documents the usefulness of number priming in children, making it an interesting tool for future research.     

Time errors are perceptual

Summary Methods for the measurement of time-errors (TEs) in the comparison of successive stimulus magnitudes are discussed. Combining a Thurstonian scaling method with the assumption of a fixed subjective width of the equal category, independent of stimulus level, a ratio scale for subjective differences within pairs of successive stimuli is derived. In a tone duration comparison experiment, with the TE defined in the terms of these subjective duration differences, data from four experimental groups were compared, the groups using different modes of judging and responding. Only minor effects of this factor were found, and hence it is concluded that the TE is a true perceptual phenomenon rather than an effect of response bias, criterion bias, or mediating verbal responses to the absolute level of stimulation. The quantitative results are interpreted in terms of a general model for the comparison of successive stimuli, employing the concepts of adaptation and differential weighting of sensation magnitudes.This investigation was supported by grants to the author from the University of Stockholm and from the Swedish Office of Administrative Rationalization and Economy (for computer time), and by grants to Mats Björkman and Hannes Eisler from the Swedish Council for Social Science Research, whose free consultation service at the Department of Statistics, University of Stockholm, also benefited the author.     

Absolute identification by relative judgment

In unidimensional absolute identification tasks, participants identify stimuli that vary along a single dimension. Performance is surprisingly poor compared with discrimination of the same stimuli. Existing models assume that identification is achieved using long-term representations of absolute magnitudes. The authors propose an alternative relative judgment model (RJM) in which the elemental perceptual units are representations of the differences between current and previous stimuli. These differences are used, together with the previous feedback, to respond. Without using long-term representations of absolute magnitudes, the RJM accounts for (a) information transmission limits, (b) bowed serial position effects, and (c) sequential effects, where responses are biased toward immediately preceding stimuli but away from more distant stimuli (assimilation and contrast).     

Fractions but not negative numbers are represented on the mental number line

The present study is the first to directly compare numerical representations of positive numbers, negative numbers and unit fractions. The results show that negative numbers and unit fractions were not represented in the same way. Distance effects were found when positive numbers were compared with fractions but not when they were compared with negative numbers, thus suggesting that unit fractions but not negative numbers were represented on the number line with positive numbers. As indicated by the semantic congruity effect, negative numbers were perceived to be small, positive numbers were perceived as large, while unit fractions were perceived neither as large nor small. Comparisons between negative numbers were faster than between unit fractions, possibly due to the smaller differences between the holistic magnitudes of the unit fractions. Finally, comparing unit fractions to 1 was faster than comparing them to 0, consistent with the idea that unit fractions are perceived as entities smaller than 1 (Kallai & Tzelgov, 2009). The results are consistent with the idea of a mental division between numbers that represent a quantity (positive numbers and unit fractions) and those that do not (negative numbers).