Relative Numerosity Research Articles

Relative numerosity discrimination in the pigeon: further tests of the linear-exponential-ratio model

This study tested a model of how animals discriminate the relative numerosity of stimuli in successive or sequential presentation tasks. In a discrete-trials procedure, pigeons were shown one light for nf times and then another for nl times. Next they received food for choosing the light that had occurred the least-number of times during the sample. At issue were (a) how performance varies with the interval between the two stimulus sets (the interblock interval) and the interval between the end of the sample and the beginning of the choice period (the retention interval); and (b) whether a simple mathematical model of the discrimination process could account for the data. The model assumed that the influence of a stimulus on choice increases linearly when the stimulus is presented, but decays exponentially when the stimulus is absent; choice probability is given by the ratio of the influence values of the two stimuli. The model also assumed that as the retention interval elapses there is an increasing probability that the ongoing discriminative process be disrupted and then the animal responds randomly. Results showed that increasing the interblock intervals reduced the probability of choosing the last stimulus of the sample as the least-frequent one. Increasing the retention interval reduced accuracy without inducing any stimulus bias. The model accounted well for the major trends in the data.

Aspectos bioecológicos del pejerrey Odontesthes bonariensis de la laguna de Monte

In 1997 and 1998, two limnological processes of high relevance occurred in the Monte laguna: the invasion of Potamageton pectinatlls and an extraordinary flood. The magnitude of theBe events affected the different components of the ecosystem on a greater or smaller scale. The aims of this paper are to know the present status of the fish community and the population ofpejerrey Odontesthes bonariensis, relating the results to the mentioned events. A limnologic sampling considering physical and chemical parameters of water, quantification of macrozooplankton, and structure of the fish community, was carried out. In the case of pejerrey, the cpue, the length-weight relationship, the K factor, cephalic index, health status, age and growth, relative numerosity, reproductive and feeding aspects were obtained. The physical-chemical composition of water is standard for this type of environment; the presence of organochlorates conforms a symptom of alarm. The zooplankton was abundant and object of a strong predation pressure by the fish. The present structure of the ichtyocenosis includes the carp, Cyprinus carpio. The captures of pejerrey presented a marked decrease of the intermediate lengths, corresponding to 3+ and 4 + age, which would be the annual classes most affected by the mentioned events. The causes of this could be an increase of mortality because of greater predation pressure of larvae and juvenile)lsh and passive or active migration. Thus, the present population is conformed by specimens 0[0+, 1 + and 2+ age, and those of a gt'eater size which survived the flood and the presence of P pectinatus.

L and M cone relative numerosity and red-green opponency from fovea to midperiphery in the human retina.

The relative numerosity of the long-wavelength-sensitive (L) and middle-wavelength-sensitive (M) cones and the red-green color appearance, as assessed by means of unique yellow, are stable from fovea to midperiphery (+/- 28 deg nasotemporal). As foveal tests decrease in size, unique yellow progressively shifts toward longer wavelengths, favoring a model of red-green opponency carried by cells whose centers receive input from either L or M cones and whose surrounds receive mixed contributions from both. Individual differences in unique yellow over a 20-nm range and the relative numerosity of L and M cones can be linked by means of this model, suggesting that the relative number of L and M cones is a factor that regulates individual variations in red-green color appearance.

The spatial arrangement of the L and M cones in the central fovea of the living human eye

Experiments designed to estimate the placement of L and M cones in fovea centralis of the living human eye are presented. Hyperacuity performances for two observers were measured for the full and the separate L and M cone submosaics using 2-dot chromatic stimuli on cone-selective adapting backgrounds. Simulated performances, based on an ideal observer model, were generated for all possible mosaics by varying L and M cone relative numerosity and spatial configuration. The best match between the simulated and measured performances determined the solution mosaic. Each observer's solution mosaic contained more L than M cones, randomly arrayed as assessed by statistical tests.

Discrimination of relative numerosity and stimulus mixture by pigeons with comparable tasks.

Discrimination of relative numerosity and stimulus mixture by pigeons with comparable tasks.

Discrimination of relative numerosity and stimulus mixture by pigeons with comparable tasks.

Arrays of small squares of 2 colors were presented in various proportions to pigeons on a video screen. Birds pecked differentially at the left or right side of the screen to obtain grain. In Experiment 1, pecking at 1 side was correct when more blue than red elements were Proportions of responses to the 2 locations reflected the proportions of elements in an orderly manner and were little affected by alterations in spacing or size of elements. When red Experiment 2, 1 side of the screen was correct when uniform red or blue arrays were presented; the other was correct for mixed arrays. Orderly gradients of response location reflected degree of stimulus mixture. Good transfer was obtained with green and blue elements. These results support the robust nature of discriminations of emergent properties of complex arrays when stimuli are equally associated with reinforcement and when response location, and not response rate, indicates stimulus control.

Relative numerosity as a dimension of stimulus control: The peak shift

Pigeons were trained to discriminate between arrays containing equal numbers of two different elements as S+, and arrays which contained more elements of one kind than the other as S−. They were then tested with the full range of the proportions of the two elements. This resulted in behavioral contrast and peak shift, as the pigeons responded more to arrays containing more positive than negative elements than they did to the positive training arrays. These findings were obtained with elements that differed in color (blue vs. red dots) and with elements that differed in orientation (horizontal vs. vertical rectangles). The results indicate that the stimulus control exerted by the derived dimension of relative numerosity involves the same processes as the fundamental dimensions that characterize simple stimulus elements.

Class inclusion, the conjunction fallacy, and other cognitive illusions

Class inclusion, the conjunction fallacy, and other phenomena are presented to illustrate cognitive illusions, that reasoning can be misdirected so that errors of judgment violate known principles. Errors in class inclusion are traced to accessing and implementing logical knowledge, rather than logical deficits, memory overload, or misleading questions. It is argued that superordinate-set cuing does not afford subset comparisons, but, rather, facilitates implementation of logical knowledge. According to fuzzy-trace theory, successful performance involves applying the cardinality principle, but also suppressing solutions based on judgments of relative numerosity, in which horizontal relationships between classes are apt to usurp the role of vertical ones.

Sequential stimuli and relative numerosity discriminations in pigeons.

Pigeons were trained in symbolic matching-to-sample procedures. The sample stimuli were sequences of red and blue flashes of light. If more blue flashes appeared in a sequence, then one side key was correct and provided occasional reinforcement, and the other side key produced a short blackout. If there were more red flashes, these contingencies were reversed. The number of flashes, their ordering, flash durations, and gaps between flashes were varied in 3 experiments. The proportion of «more blue» responses was an orderly function of the proportion of blue stimuli. Varying the gaps between flashes showed that both temporal and ordinal locations were important variables. Increasing the duration of flashes increases their salience. Mechanisms for discriminating the relative numerosity of successive stimuli are discussed

Occupancy model of perceived numerosity

Observers saw 234 different pairs of stochastically organized dot patterns and indicated which of the two patterns appeared to be more numerous. All of the data can be accounted for by supposing that the choice of the more numerous pattern is based on the determination of the occupancy indices of both patterns. Each dot is posited to have an impact upon its neighborhood in a constant occupancy radius R. The area of the stimulus plane occupied collectively by all dots provides a basis for judging relative numerosity; the pattern with the larger occupancy value is chosen as more numerous. The occupancy model, besides providing a general explanation of known numerosity illusions in strictly quantitative terms, can explain some puzzling aspects of numerosity perception.

Size invariance in visual number discrimination

This study deals with the observer's ability to discriminate the numerosity of two random dot-patterns irrespective of their relative size. One of these two patterns was a reference one that was always composed of 32 dots randomly distributed within a K x K invisible square window (K = 1.92 degrees). The second one was the test pattern with one of the five magnifications (K = 0.64 degrees, 1.28 degrees, 1.92 degrees, 2.56 degrees, 3.20 degrees) and the relative number of dots varied on 11 levels (N = -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, or 15 dots). The observer's task was to indicate which of the two patterns contained more dots. The results show that the stimulus size, as an irrelevant stimulus attribute, can be ignored in the judgements about relative numerosity. This means that the perceived numerosity is size invariant, at least for a 1.6-times magnification and a 3-times reduction of the test pattern. The size invariance observed constrains the range of potential models, since the perceived numerosity can be identified only by means of a feature of the stimulus that will remain invariant after any change in the absolute stimulus size.

Sequential stimuli and relative numerosity discriminations in pigeons.

Sequential stimuli and relative numerosity discriminations in pigeons.

Discrimination of relative numerosity by pigeons

In three experiments, pigeons were trained to discriminate between uniform arrays of two elements that differed in color, form, or size. They were then tested with arrays that contained different proportions of the two elements on these dimensions. In all cases, orderly discrimination gradients reflected these proportions. The discrimination readily transferred to new arrays with similar stimuli, but with different total numbers of elements. In Experiment 4, the pigeons were taught to discriminate between two groups of categorical stimuli: pictures of birds and pictures of flowers. A test with different proportions of each again produced a gradient based on relative numerosity. Experiment 5 demonstrated transfer of stimulus control on the numerosity dimension when pigeons were trained with one set of instances from two categories, and then were tested with new instances from the same categories.

Infants' understanding of numerical transformations

This research explored infants' understanding of transformations that change the numerosity of a set. Do infants know how the insertion or deletion of a single object affects the relative numerosity of two sets? Four age groups (14‐, 18‐, 24‐ and 28‐month‐olds) were tested on problems involving sets whose numerosities were never greater than 2. The children were encouraged to find the set with more objects, after watching the experimenter cover up two sets and then transform one of them by performing either an insertion or a deletion. On the start‐equal problems, the two sets had the same number of objects until the transformation was performed, so that when an object was inserted the transformed set always had more and when an object was deleted the untransformed set always had more. On the start‐unequal problems, the initial numerosities of the two sets were varied so that these relations between the type of transformation that was performed and the relative numerosities of the two sets no longer held. One set initially had two objects in it and the other set had none. Insertions were performed on the smaller set, so that it still had less than the untransformed set; and deletions were performed on the larger set, so that it still had more than the untransformed set. Performance improved significantly across the four ages tested. However, every age group showed some understanding of the transformations. These results add to the growing body of evidence for an understanding of numerical relationships in infancy.

Why preschoolers are reluctant to count spontaneously

Two experiments are reported which examine children's counting and its role in reasoning about the relative numerosity of two arrays. In the first experiment, children's number judgements were compared under different conditions designed to evaluate the importance of three different cues to number—length and density of rows, small number perception and counting. Children were found to count very rarely unless specifically asked to do so. Experiment 2 investigated some possible reasons why children who count readily in some situations are reluctant to count spontaneously in this number judgement task. Spontaneous counting in 4‐year‐olds increased in one condition only: when they were given feedback as to the correctness of their previous judgements. This feedback showed that basing judgements on number as counted was always correct whereas length and density judgements were only sometimes correct. Preschoolers' preference for length as a cue to number may therefore be due to their belief that length is a more reliable cue than counting, rather than to their ignorance about the link between counting and numerical reasoning.

Relative numerosity judgments by infants

Relative numerosity judgments by infants

Effects of Perceptual Variations on Number Comparison Tasks

Three types of number-comparison tasks were administered to 502 children (in the age range between four years, and six years and ten months). The tasks consisted of two-choice tasks and multiple-choice in which the density of the elements (dots) was manipulated, i.e., in one condition the elements stood far apart and for the other condition the distance between the elements was small, but within each of these conditions the elements were evenly spaced. The multiple-choice tasks were divided into tasks with constant target size (i.e. the area of all the elements of each set was identical) and tasks without constant target size. It was shown that two-choice tasks were easier than multiple-choice tasks controlling for chance responding. The density manipulation produced a strong effect, i.e., tasks in which the elements were placed closely together were more difficult than tasks in which the elements were spaced far apart. Multiple-choice tasks with constant target size were easier than tasks without constant target size. This is a noteworthy finding, because the opposite result might be expected. Since in tasks without constant target size a correct judgment could be based on either the relative numerosity or on redundant non-numerical cues (e.g. amount of target size, brightness, background area)), these tasks might be considered easier than tasks with constant target size in which the influence of these cues was reduced. However, our results can be interpreted that by using constant target size another cue has been introduced, because an accurate judgment can be made now by simply relying on the size of the target.

Dimensions of number invariance

The magic paradigm was used to examine preschoolers’ abilities to make judgments requiring a number-related invariance concept. Findings from previous, similar studies have suggested number invariance schemes in 3- and 4-year-olds. The stimuli in those investigations, however, contained a redundancy of number and relative numerosity. When these dimensions were separated in the present study, fewer children exhibited conservation-like performance than in the redundant-cue task and, contrary to previous findings, most did not justify their responses with reference to number. The results were consistent with the suggestion that preschoolers do make use of invariance concepts in the magic paradigm, but that these need not always represent number invariance per se.

Bases of young children's judgments about more

While the young child's difficulties in correctly comprehending the term less are well known, it appears less widely recognized that young children may also encounter difficulties with the term more. The present study shows that in a task which requires judgments about more to be based on the relative numerosity of sets, 3–4-year-old children may base their judgments on other parameters, such as the extent to which the sets are homogenous with respect to the color of their components. The implications of such findings are briefly considered.

Amount of information and polarity of attraction

In two experiments, subjects given larger sets of information judged others more extremely than did subjects given smaller sets. This set-size effect emerged only when subjects were made aware of relative numerosity of stimuli. The condition of awareness helps reconcile previous disparities between studies in which subjects judge one-size vs. multiple-sized sets. It is suggested that increased confidence underlies the polarizing effect of larger set size and that confidence requires knowledge that one could have received less information. This knowledge is often lacking when subjects are given only one set size.