Quantity Judgments Research Articles

Split decisions: Problems in the interpretation of results from commissurotomized subjects

It has been widely accepted that there is little if any interhemispheric transfer of perceptual information in subjects following section of the forebrain commissures, although there may be diffuse transfer of emotional and connotative information, and an ability to direct spatial attention between hemispheres. However Sergent has recently shown that split-brained subjects can make rapid and accurate perceptual judgments based on visual stimuli presented simultaneously to the two visual fields. These include judgments of alignment, judgments of relative quantity, and higher-order judgments such as lexical decisions about letter strings straddling the visual fields. However it is premature to conclude that these tasks are accomplished by subcortical transfer of perceptual information; at least some of them can be explained in terms of strategies based on information available to just one hemisphere, or on the transfer of rudimentary information, perhaps accomplished through cross-cueing.

How quantity judgment changes as the number of cues increases: An analytical framework and review.

People frequently must make judgments of quantities. This article concerns how increasing the number of cues supporting such judgments affects accuracy as well as the judgment process itself. Research on these issues is sparse. This seems partly due to ambiguity about what it means to say that quantity judgments are accurate. Thus, a major focus of the article is on integrating 2 disparate approaches to analyzing quantity judgment, lens model analysis from psychology and mean squared error analysis from economic forecasting

How quantity judgment changes as the number of cues increases: An analytical framework and review.

How quantity judgment changes as the number of cues increases: An analytical framework and review.

Judgments of quantity and conservation of quantity: The area of a rectangle

This study examines the relationships between perceptive judgment of quantity and conservation judgment of quantity with regard to the concept of area. Twenty children who passed an area conservation task (mean age = 8) and 20 children who failed it (mean age = 6.7) were administered an area judgment task. This task was a replication of Anderson and Cuneo's study (1978, Journal of Experimental Psychology: General, 107, 335–378). The conserving children generally appeared to apply an additive rule (the height and width rule) observed by Anderson and Cuneo in 5 year olds. The nonconserving children in contrast generally presented patterns suggesting centration on one of the two dimensions. The implications for Anderson's and Piaget's conceptions of conservation development are discussed.

Probability judgment accuracy: China, Japan, and the United States

Subjects in China, Japan, and the United States reported probability judgments. In Study 1, Chinese and American subjects indicated degrees of certainty about their answers to general-knowledge questions with discrete alternatives, e.g., whether potatoes grow better in warm or in cool climates. In Study 2, Japanese subjects made similar discrete-alternative assessments. In Study 3, subjects in China and the United States reported probability distribution judgments for various quantities, e.g., the maximum temperature on a specified day. Judgment accuracy was evaluated overall and with respect to several underlying accuracy dimensions. The overall quality of discrete-alternative judgments was indistinguishable among the subjects from the three countries. The accuracy component patterns of the Japanese and American subjects were essentially the same. However, the Chinese subjects achieved the common overall accuracy level very differently. On some accuracy dimensions, e.g., calibration, the American and Japanese subjects' judgments were superior. On others, e.g., discrimination, the assessments of the Chinese subjects excelled. Results for quantity judgments were similar to those for discrete-alternative judgments, although there were notable differences. Potential explanations and implications are discussed.

"Perceptual and memorial constructs in children's judgments of quantity: A law of across-representation invariance": Correction to Wolf and Algom (1987).

"Perceptual and memorial constructs in children's judgments of quantity: A law of across-representation invariance": Correction to Wolf and Algom (1987).

Are the Operations the Same in Traditional and Nontraditional Conservation Tasks?

This study evaluated the relation of performance on consetvation tasks (Piaget, 1952) to judgments of quantity on tasks without initial equivalence. Subjects (29 girls and 29 boys) were drawn from 5 kindergarten classes in metropolitan New York City. They were of normal to above average intelligence, with a mean of 5 yr., 3 mo. (SD = 4 mo.). Children were assessed on conservation of number and liquid tasks, each containing two items. Two other tasks for number and liquid were then used without the initial equivalence of traditional conservation tasks. Each child was instructed to find out which container had more in it or whether the containers had the same amount. The number problem consisted of two transparent plastic boxes, one with 8 checkers, the other with 8 plastic pieces varying in size, shape, and color. For the liquid problem, the experimenter screened a plastic champagne glass and an opaque paper cup while she poured 3 oz. of colored liquid in each, then removed the screen. Near the two containers of liquid were a measuring cup, another opaque paper cup of the same size, and two identical clear plastic cups. The experimenter recorded what the child said and did. Scoring of conservation used Gelman's (1969) criteria. On traditional and nontraditional tasks of conservation of number, performance was not closely associated (+ = .21, p > -05). Thirty-one of the 58 children gave nonconserving responses on the traditional tasks. Of these, 12 (39%) succeeded on the number with initial nonequivalence. In most cases, the determinations were based on one-to-one correspondence or counting. Of the 27 children classified as partial or full conservers on number, only 16 (59%) determined that the quantities were equivalent. There was a significant association of performance on both types of liquid tasks (+ = .53, p < .001). All 6 conservers on the traditional conservation of liquid tasks were able to conserve when there was initial nonequivalence of liquid. Eleven of the nonconservers (21%) determined that the quantities were the same and provided sufficient verbal justification to classify them as conservers on this task. In addition, 9 nonconservers (17%) said the two quantities were the same though they could not adequately articulate their reasons. Are the schemata accessed in evaluating quantities with initial equivalence and initial nonequivalence the same? Because there is no relationship between performance on the two types of numerical tasks, it appears that different schemata may be used. One possibility is that the operatory suuctures of novice and expert conservers actually differ (Resnick & Gelman, 1984).

Perceptual and memorial constructs in children's judgments of quantity: A law of across-representation invariance.

Children at three different ages made judgments of physically presented (perceptual estimation) or symbolically represented (memorial estimation) rectangles. Height and width were integrated according to different, age-dependent algebraic rules. Memorial data obeyed the same integration rules that operated in the original perceptual judgments even when younger children and older children used completely different combination models. Valuation operations were the same in perception and memory for the youngest group (6-year-olds) but became discriminably different at older ages (for the 8- and 10-year olds). Three additional experiments on judgments of volume, liquid quantity, and visual length yielded strong cross-validation support for the general invariance claim (with respect to integration rule theory) but less strong support for the specific invariance claim (with respect to valuation function for the 6-year-old subjects). Results are interpreted as demonstrating lawful and long-enduring ecological constraints on internal representation.

Perceptual and memorial constructs in children's judgments of quantity: A law of across-representation invariance.

Perceptual and memorial constructs in children's judgments of quantity: A law of across-representation invariance.

Accessing Concrete Operational Reasoning Strategies in Preoperational Children

ABSTRACT This study addresses three questions: (a) Are the cognitive skills that underlie games of make-believe play-specific or are they of a more general cognitive order? (b) Is the reported prevalence of identity explanations on the conservation training tasks a function of the standard conservation inquiry, or is it an indication that the identity explanation is a more primitive cognitive precursor of concrete operational reversibility? (c) Does SES affect the ability to formulate the logical explanations required for the conservation task? Following conservation pretests, 64 nonconserving 4 and 5 year olds were assigned to four different conditions: Group 1 participated in pretense play, Group 2 was exposed to a correct conserving judgment of quantity, Groups 3 and 4 received only conservation posttests. The conservation posttests administered to Groups 1, 2, and 3 deviated from the usual standard procedures by providing the subjects with a correct judgment and a set of specific questions designed to...

Children's judgments of numerical quantity: A new view of early quantification

Three experiments investigated the role of length, density, and number dimensions in children's numerosity judgments of linear arrays of beads. The number dimension is physically defined by a length × density rule. Judgments based, even in part, on the number dimension would show signs of a length × density rule. Experiment 1 examined numerosity judgments of large arrays. Results showed that judgments by 3- and 4-year-olds obeyed a length + density rule, providing cogent evidence for a general adding strategy in young children's judgments of quantity. The physical length × density rule emerged gradually with increasing age. Experiment 2 showed that numerosity judgments of small arrays by 3- and 4-year-olds obeyed a length × density rule, indicating response to the number dimension. This result was expected and thereby validated the linearity of the response scale at the youngest ages. Experiment 3 verified the integration rules for individual 3- and 4-year-olds. The integration rules were interpreted in terms of Piaget's stages of the development of quantification. No evidence for Piaget's initial stage was found. This leads to a new view of early quantification which grants young children the ability to integrate stimulus dimensions.

A General Strategy for Quantity Judgments: The Height + Width Rule

Previous work has found that 5-year-olds judge the area of rectangles by a height width rule. Present results indicated that 3- and 4-year-olds also judge area by a height width rule. This adding rule was interpreted in terms of a general-purpose judgmental strategy that young children employ to make quantitative judgments. Both the use of a rating scale and the use of an adding rule in judgment indicate an impressive level of quantitative ability at a very early age.

A General Strategy for Quantity Judgments: The Height + Width Rule

A General Strategy for Quantity Judgments: The Height + Width Rule

Comment on Anderson and Cuneo's "The height + width rule in children's judgments of quantity."

Comment on Anderson and Cuneo's "The height + width rule in children's judgments of quantity."

The height + width rule in children's judgments of quantity.

SUMMARY Judgments of area of rectangles by S-year-old children obeyed an adding rule: height + width. This was surprising because simple perception of area ought to follow the physical height X width rule that was in fact obtained with older children. For the younger children, however, the height + width rule was reliable across a variety of experimental manipulations in eight experiments. This height + width rule was interpreted in terms of a general-purpose integration rule and a general metric sense. Young children lack a clear, adult conception of specific quantities such as area. They utilize whatever quantitative cues seem relevant and integrate them by a general-purpose adding rule to arrive at their quantitative judgment. A single general metric sense, which is present at least by age 3, mediates the overt expression of quantitative judgments of many different concepts, both physical and social. In collateral work, 5-year-olds judged amount of liquid in a glass by a heightonly rule. This result was expected from Piagetian theory of centration, which says that young children do not integrate information, but instead base their judgment on a single salient aspect of the stimulus. But this centration interpretation did not hold for the rectangles mentioned above. It was suggested, therefore, that the height-only rule represented transfer from daily drinking experience, not true centration. This hypothesis was supported by experiments showing that judgments of amount of wax in translucent cylinders obeyed the height-only rule when the cylinders were inside of glasses, but obeyed the height + width rule when they stood alone. These results, together with related studies of information integration in children, show that centration is not a pervasive characteristic of the young child's thinking. On the contrary, ydling children can exhibit quite sophisticated integration strategies. It is suggested that the concept of centration has resulted largely from inadequate methodology and that the methods of information integration theory provide a more analytical and correct picture of cognitive development. These results also lead to a new interpretation of conservation. It is suggested that conservation is derivative from object invariance and variegated specific experiential factors. The Piagetian concept of compensation is not causal in the development of conservation, but instead compensation arises from an attempt to articulate a verbal justification for a prior concept of conservation. In this view, conservation develops gradually as an expectancy of object invariance. The methods of integration theory and functional measurement can be used to trace out the course of the development of conservation.

Comment on Anderson and Cuneo's "The height?+?width rule in children's judgments of quantity."

Comment on Anderson and Cuneo's "The height?+?width rule in children's judgments of quantity."

The height?+?width rule in children's judgments of quantity.

The height?+?width rule in children's judgments of quantity.

ABSOLUTE JUDGMENTS OF DISCRETE QUANTITIES RANDOMLY DISTRIBUTED OVER TIME.

ABSOLUTE JUDGMENTS OF DISCRETE QUANTITIES RANDOMLY DISTRIBUTED OVER TIME.

A note on Piaget's concept of conservation.

In his discourse on the development of the number concept Piaget (x) gives central emphasis to the concept of conservation as an essential principle that forms the basis for a framework of numerical reasoning. He describes conservation as developing in three stages: an initial stage in which perceptual factors exclusively determine the judgment of quantity, an intermediate stage of transition when perceptual as well as conservation considerations influence the judgment, and finally a last stage of complete conservation. These stages are, according to Piaget, one manifestation of a general trend from a perceptual-intuitive to an operational orientation, which characterizes the development of conceptual thinking. Some of the most thoughtful experimental analyses of Piaget's theoretical speculations regarding the development of number concept have been recently conducted by Wohlwill (3, 4). In an attempt to gain some further understanding of the factors that contribute to the development of the principle of conservation, Wohlwill and Lowe (4) report a study in which they evaluated the relative contribution of three forms of specific training on the development of conservational thinking. The three experimental conditions of training were: (a) Reinforced Practice-S determined the number of objects immediately before and immediately after their spatial arrangement (in terms of their spread in a horizontal line) had been changed; (b) Addition and Subtraction-S was trained in observing the effects of addition or subtraction of one object from a larger aggregate on the determination of the number of such objects after their spatial arrangement had been changed; and (c) Dissociation-S received practice in counting an aggregate of objects

Grading, A Study in Semantics

The first thing to realize about grading as a psychological process is that it precedes measurement and counting. Judgments of the type “A is larger than B” or “This can contains less milk than that” are made long before it is possible to say, e.g., “A is twice as large as B” or “A has a volume of 25 cubic feet, B a volume of 20 cubic feet, therefore A is larger than B by 5 cubic feet,” or “This can contains a quart of milk, that one 3 quarts of milk, therefore the former has less milk in it.” In other words, judgments of quantity in terms of units of measure or in terms of number always presuppose, explicitly or implicitly, preliminary judgments of grading. The term four means something only when it is known to refer to a number which is “less than” certain others, say five, six, seven, arranged in an ordered series of relative mores and lesses, and “more than” certain others, say one, two, three, arranged in an ordered series of relative mores and lesses. Similarly, a foot as a unit of linear measure has no meaning whatever unless it is known to be more than some other stretch, say an inch, and less than a third stretch, say a yard.