Adult Nonconservation of Numerical Equivalence

Cognition, 1979

on relevant experience.

Cognitive Development, 1988

Child Development, 2003

Two experiments were conducted to examine whether and how 4-and 5-year-olds learn to distinguish determinate from indeterminate evidence. Children were asked to decide whether various patterns of evidence were sufficient to reach unambiguous conclusions. This study replicated the finding that young children tend to use a strategy that, although generally successful, fails on evidence patterns in which a single positive instance co-occurs with an unexplored source of evidence. Experiment 1 demonstrated that this positive-capture strategy is deeply entrenched, even in a meaningful, pragmatic context. With a microgenetic design, Experiment 2 revealed that young children are capable of replacing the positive-capture strategy with a correct strategy when they are exposed to various analogous tasks in several training sessions.

Journal of Experimental Psychology: General, 1998

Evolutionary approaches to judgment under uncertainty have led to new data showing that untutored subject reliably produce judgments that conform to may principles of probability theory when (a) they are asked to compute a frequency instead of the probability of a single event, and (b) the relevant information is expressed as frequencies. But are the frequency-computation systems implicated in these experiments better at operating over some kinds of input than others? Principles of object perception and principles of adaptive design led us to propose the individuation hypothesis: that these systems are designed to produce well-calibrated statistical inferences when they operate over representations of “whole” objects, events, and locations. In a series of experiments on Bayesian reasoning, we show that human performance can be systematically improved or degraded by varying whether a correct solution requires one to compute hit and false-alarm rates over “natural” units, such as whole objects, as opposed to inseparable aspects, views, and other parsings that violate evolved principles of object construal.

A word problem is more difficult to solve when the minimum number of different operations to reach the correct solution is large, when it is of a different type than a problem preceding it, when the indexed complexity of its most complex sentence is great, when there are a large number of words in the problem, and when a conversion of units (as from days to weeks) is required. These variables of problem difficulty were determined to be significant in an experiment using 16 disadvantaged sixth-grade students, who were given access to a computer-based teletype. Variables that did not make a significant contribution to the regression analysis were: the "verbal-clue" variable, the "order" variable, and the "steps" variable. (MF)

Journal of Experimental Child Psychology, 2012

The current experiments examined the role of scale factor in children's proportional reasoning. Experiment 1 used a choice task and Experiment 2 used a production task to examine the abilities of kindergartners through fourth-graders to match equivalent, visually depicted proportional relations. The findings of both experiments show that accuracy decreased as the scaling magnitude between the equivalent proportions increased. In addition, children's errors showed that the cost of scaling proportional relations is symmetrical for problems that involve scaling up and scaling down. These findings indicate that scaling has a cognitive cost that results in decreasing performance with increasing scaling magnitude. These scale factor effects are consistent with children's use of intuitive strategies to solve proportional reasoning problems that may be important in scaffolding more formal mathematical understanding of proportional relations.

Developmental Psychology, 2008

Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10-to 12-years of age, but can solve parallel problems involving continuous quantities by 6-years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through fourth-graders in Study 1, and to first-and third-graders in Study 2. Both studies involved four between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously and in Study 2 target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems.

Cognitive Psychology, 1988

Word problems are notoriously difficult to solve. We suggest that much of the difliculty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate final questions to incomplete word problems, and (3) modeling performance patterns using a computer simulation. Solution performance was found to be systematically related to recall and question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error patterns were successfully simulated by manipulating a computer model's language comprehension strategies, as opposed to its knowledge of logical set relations. o 1st~ Academic Press, Inc.

Psychological Research, 2008

Adult's simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-diVerence variables, such as working-memory span. The current study investigates (a) whether the eVects of problem size, operation, and working-memory span on children's simple-arithmetic strategy use are equal to those observed in adults, and (b) how these eVects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year-old children. Results showed that the problem-size eVect in children results from the same strategic performance diVerences as in adults (i.e., sizerelated diVerences in strategy selection, retrieval eYciency, and procedural eYciency). Operation-related eVects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more eYcient strategy execution in addition, and more pronounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The diVerences and similarities across children's and adult's strategic performance and the relevance of arithmetic models are discussed.

Developmental Psychology, 2008

Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10-to 12-years of age, but can solve parallel problems involving continuous quantities by 6-years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through fourth-graders in Study 1, and to first-and third-graders in Study 2. Both studies involved four between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously and in Study 2 target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems.