Nonverbal Counting in Humans: The Psychophysics of Number Representation

In a nonverbal counting task derived from the animal literature, adult human subjects repeatedly attempted to produce target numbers of key presses at rates that made vocal or subvocal counting difficult or impossible. In a second task, they estimated the number of flashes in a rapid, randomly timed sequence. Congruent with the animal data, mean estimates in both tasks were proportional to target values, as was the variability in the estimates. Converging evidence makes it unlikely that subjects used verbal counting or time durations to perform these tasks. The results support the hypothesis that adult humans share with nonverbal animals a system for representing number by magnitudes that have scalar variability (a constant coefficient of variation). The mapping of numerical symbols to mental magnitudes provides a formal model of the underlying nonverbal meaning of the symbols (a model of numerical semantics).     

"Constructive" enumeration by chimpanzees (Pan troglodytes) on a computerized task

Two chimpanzees used a joystick to collect dots, one at a time, on a computer monitor (see video-clip in the electronic supplementary material), and then ended a trial when the number of dots collected was equal to the Arabic numeral presented for the trial. Both chimpanzees performed substantially and reliably above chance in collecting a quantity of dots equal to the target numeral, one chimpanzee for the numerals 1–7, and the second chimpanzee for the numerals 1–6. Errors that were made were seldom discrepant from the target by more than one dot quantity, and the perceptual process subitization was ruled out as an explanation for the performance. Additionally, analyses of trial duration data indicated that the chimpanzees were responding based on the numerosity of the constructed set rather than on the basis of temporal cues. The chimpanzees' decreasing performance with successively larger target numerals, however, appeared to be based on a continuous representation of magnitude rather than a discrete representation of number. Therefore, chimpanzee counting in this type of experimental task may be a process that represents magnitudes with scalar variability in that the memory for magnitudes associated with each numeral is imperfect and the variability of responses increases as a function of the numeral's value. Accepted after revision: 11 June 2001 Electronic Publication     

Nonverbal arithmetic in humans: light from noise

Animal and human data suggest the existence of a cross-species system of analog number representation (e.g., Cordes, Gelman, Gallistel, & Whalen, 2001; Meeck & Church, 1983), which may mediate the computation of statistical regularities in the environment (Gallistel, Gelman, & Cordes, 2006). However, evidence of arithmetic manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtraction, or reproduction of nonverbal counts. Variance analyses revealed that the magnitude of both inputs and answer contributed to the variability in the arithmetic responses, with operand variability dominating. Other contributing factors to the observed variability and implications for logarithmic versus scalar models of magnitude representation are discussed in light of these results.     

Operant modulation of low-level attributes of rule-governed behavior by nonverbal contingencies

A low-level and nonsalient attribute of behavior (i.e., speed of pressing) was subjected to differential nonverbal operant reinforcement when rules governed a high-level attribute of that behavior (i.e., counting by means of key presses). Unknown to the subjects, reinforcers depended on reduced (slow group) or increased (fast group) speed of pressing rather than on the correct number of presses. The reinforced attribute was modulated according to the arranged speed contingencies independently of the instructed task and independently of subjects' awareness of the critical contingency. A control group receiving random reinforcers demonstrated no systematic speed changes. Possible mechanisms related to behavior changes of this type were examined and discussed, and it was concluded that the behavior changes observed in this situation could be attributed to operant conditioning. The results substantiate the assumption that nonverbal operant contingencies may modulate low-level and nonsalient attributes of rule-governed behavior.     

“One-thousand<Emphasis Type="Italic">one</Emphasis> … one-thousand<Emphasis Type="Italic">two</Emphasis> …”: Chronometric counting violates the scalar property in interval timing

Medical College of Wisconsin, Milwaukee, Wisconsin Weber’s law applied to interval timing is called thescalar property. A hallmark of timing in the secondsto-minutes range, the scalar property is characterized by proportionality between the standard deviation of a response distribution and the duration being timed. In this temporal reproduction study, we assessed whether the scalar property was upheld when participants chronometrically counted three visually presented durations (8, 16, and 24 sec) as compared with explicitly timing durations without counting. Accuracy for timing and accuracy for counting were similar. However, whereas timing variability showed the scalar property, counting variability did not. Counting variability across intervals was accurately modeled by summing a random variable representing an individual count. A second experiment replicated the first and demonstrated that task differences were not due to presentation order or practice effects. The distinct psychophysical properties of counting and timing behaviors argue for greater attention to participant strategies in timing studies.     

The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: evidence for a segmented linear model

This experiment aimed to expand previous findings on the development of mental number representation. We tested the hypothesis that children's familiarity with numbers is directly reflected by the shape of their mental number line. This mental number line was expected to be linear as long as numbers lay within the range of numbers children were familiar with. Five- to 9-year-olds (N=78) estimated the positions of numbers on an external number line and additionally completed a counting assessment mirroring their familiarity with numbers. A segmented regression model consisting of two linear segments described number line estimations significantly better than a logarithmic or a simple linear model. Moreover, the change point between the two linear segments, indicating a change of discriminability between numbers, was significantly correlated with children's familiar number range. Findings are discussed in terms of the accumulator model, assuming a linear mental representation with scalar variability.     

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).     

What is the best and easiest method of preventing counting in different temporal tasks?

The aim of the present study was to determine the best and easiest method of suppressing spontaneous counting in a temporal judgment task. Three classic methods used to avoid counting—instructions not to count, articulatory suppression, and administration of an interference task—were tested in temporal generalization, bisection, and reproduction tasks with two duration ranges (1–4 and 2–8 s). All the three no-counting conditions prevented participants from counting, counting leading to estimates that were more accurate and less variable and to violations of the fundamental scalar property of timing. With regard to the differences between the no-counting conditions, the interference task distorted time perception more strongly and increased variability in temporal estimates to a greater extent than did articulatory suppression, as well as the no-counting instructions condition. In addition, articulatory suppression produced more noise in behavioral outcome than did the no-counting instruction condition. In sum, although all methods have disadvantages, the instructions not to count actually constitute the simplest and more efficient method of preventing counting in timing tasks. However, further studies must now concentrate on the role of explicit instructions in our experience of perception.     

Response acquisition under targeted percentile schedules: a continuing quandary for molar models of operant behavior.

The number of responses rats made in a "run" of consecutive left-lever presses, prior to a trial-ending right-lever press, was differentiated using a targeted percentile procedure. Under the nondifferential baseline, reinforcement was provided with a probability of .33 at the end of a trial, irrespective of the run on that trial. Most of the 30 subjects made short runs under these conditions, with the mean for the group around three. A targeted percentile schedule was next used to differentiate run length around the target value of 12. The current run was reinforced if it was nearer the target than 67% of those runs in the last 24 trials that were on the same side of the target as the current run. Programming reinforcement in this way held overall reinforcement probability per trial constant at .33 while providing reinforcement differentially with respect to runs more closely approximating the target of 12. The mean run for the group under this procedure increased to approximately 10. Runs approaching the target length were acquired even though differentiated responding produced the same probability of reinforcement per trial, decreased the probability of reinforcement per response, did not increase overall reinforcement rate, and generally substantially reduced it (i.e., in only a few instances did response rate increase sufficiently to compensate for the increase in the number of responses per trial). Models of behavior predicated solely on molar reinforcement contingencies all predict that runs should remain short throughout this experiment, because such runs promote both the most frequent reinforcement and the greatest reinforcement per press. To the contrary, 29 of 30 subjects emitted runs in the vicinity of the target, driving down reinforcement rate while greatly increasing the number of presses per pellet. These results illustrate the powerful effects of local reinforcement contingencies in changing behavior, and in doing so underscore a need for more dynamic quantitative formulations of operant behavior to supplement or supplant the currently prevalent static ones.     

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 successive differentiation of a lever displacement response

Maximum displacements of lever presses by rats were recorded under eight successively-smaller reinforcement zones (RZ). The largest RZ included displacements from 3° to 44°; the smallest, from 24° to 29°. As the RZ decreased, displacement distributions reflected a least-effort tendency: distributions peaked at the lower limit of RZ and most non-reinforced presses fell just below the lower limit. Successive distributions (a) differed significantly in shape, (b) showed reduced variability, and (c) indicated more presses and more presses per reinforcement. Prolonged training under the smallest RZ gave no improvement in performance.     

Classification without identification in visual search

Six subjects scanned displays of random consonants for a single target which was (a) another consonant; (b) a given number; or (c) any number. A second group of six subjects took part in three comparable conditions with number displays, and letters or numbers as targets. Scanning time for a number in a letter display or a letter in a number display was more rapid than scanning for a target drawn from the same set as the background. Several unpractised subjects, and all the subjects who practised the task, were able to scan as fast through letters for “any number” as for a specific number, or conversely through digits. The finding of different scanning rates for two precisely physically specified targets, depending on which class they were drawn from, runs counter to an explanation of high-speed scanning in terms of the operation of visual feature analysers. It is suggested that familiar categorization responses may be immediate and may provide the basis for the discrimination of relevant from irrelevant items in rapid visual scanning.     

One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.     

Sources of variability and systematic error in mouse timing behavior

In the peak procedure, starts and stops in responding bracket the target time at which food is expected. The variability in start and stop times is proportional to the target time (scalar variability), as is the systematic error in the mean center (scalar error). The authors investigated the source of the error and the variability, using head poking in the mouse, with target intervals of 5 s, 15 s, and 45 s, in the standard procedure, and in a variant with 3 different target intervals at 3 different locations in a single trial. The authors conclude that the systematic error is due to the asymmetric location of start and stop decision criteria, and the scalar variability derives primarily from sources other than memory.     

Consistency in interpretation of probabilistic phrases

It is occasionally claimed in both applied decision analysis and in basic research that people can better use and understand probabilistic opinions expressed by nonnumerical phrases, such as “unlikely” or “probably,” than by numbers. It is important for practical and theoretical reasons to evaluate this claim. The available literature indicates that there is large variability in the mapping of phrases to numbers, but provides no indication as to its cause. This study asks (a) whether the variability can be attributed to how people interpret the phrases per se, rather than to how they use the number scale and (b) whether the variability is due primarily to between-subject or to within-subject factors. In order to answer these questions, 32 subjects ranked and compared 19 probability phrases on each of three occasions. The results show that individuals have a relatively stable rank ordering of the phrases over time, but that different individuals have different rank orderings. Practical and methodological implications of these data are discussed.     

Levels of number knowledge during early childhood

Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children’s incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the ‘discontinuity’ account of number development in general and the ‘knower-levels’ account in particular.     

What dot clusters and bar graphs reveal: subitizing is fast counting and subtraction

In two studies, we found that dot enumeration tasks resulted in shallow-sloped response time (RT) functions for displays of 1-4 dots and steep-sloped functions for displays of 5-8 dots, replicating results implicating subitizing and counting processes for low and high ranges of dots, respectively. Extracting number from a specific type of bar graph within the same numerical range produced a shallow-sloped but scallop-shaped RT function. Factor analysis confirmed two independent subranges for dots, but all bar graph values defined a unitary factor. Significantly, factor scores and asymmetries both showed correlations of bar graph recognition to dot subitizing but not to dot counting, strongly suggesting that subitizing was used in both enumeration of low numbers of dots and bar graph recognition. According to these results, subitizing appears to be a nonverbal process operating flexibly in either additive or subtractive fashion on analog quantities having spatial extent, a conclusion consistent with a fast-counting model of subitizing but not with other models of the subitizing process.     

Short-term memory processes in counting

Four experiments examined the memory processes used to maintain location in a counting sequence. In Experiment 1, subjects who rapidly counted forward omitted many repeated-digit numbers (e.g., 77), as found previously with backward counting. Subjects in Experiment 2 counted backward with normal auditory feedback or with headphones through which white noise was channeled. In both cases, repeated-digit errors predominated, suggesting that the contents of short-term memory, rather than auditory sensory memory, are checked during counting. In Experiment 3, subjects silently wrote counting responses, and the omission errors resembled those in vocal counting. Repetition errors were also found and attributed to phonological recoding failures. Articulatory suppression in Experiment 4 greatly increased the number of repetition errors in the written counting task. A model of the counting process was proposed according to which subjects keep track of their location in the counting sequence by monitoring phonologically coded short-term memory representations of the numbers.     

A componential analysis of pacemaker-counter timing systems

Why does counting improve the accuracy of temporal judgments? Killeen and Weiss (1987) provided a formal answer to this question, and this article provides tests of their analysis. In Experiments 1 and 2, subjects responded on a telegraph key as they reproduced different intervals. Individual response rates remained constant for different target times, as predicted. The variance of reproductions was recovered from the weighted sum of the first and second moments of the component timing and counting processes. Variance in timing long intervals was mainly due to counting error, as predicted. In Experiments 3-5, unconstrained response rate was measured and subjects responded at (a) their unconstrained rate, (b) faster, or (c) slower. When subjects responded at the preferred rate, the accuracy of time judgment improved. Deviations in rates tended to increase the variability of temporal estimates. Implications for pacemaker-counter models of timing are discussed.