Number games, magnitude representation, and basic number skills in preschoolers.

The effect of 3 intervention board games (linear number, linear color, and nonlinear number) on young children's (mean age = 3.8 years) counting abilities, number naming, magnitude comprehension, accuracy in number-to-position estimation tasks, and best-fit numerical magnitude representations was examined. Pre- and posttest performance was compared following four 25-min intervention sessions. The linear number board game significantly improved children's performance in all posttest measures and facilitated a shift from a logarithmic to a linear representation of numerical magnitude, emphasizing the importance of spatial cues in estimation. Exposure to the number card games involving nonsymbolic magnitude judgments and association of symbolic and nonsymbolic quantities, but without any linear spatial cues, improved some aspects of children's basic number skills but not numerical estimation precision. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding.

A theoretical analysis of the development of numerical representations indicated that playing linear number board games should enhance preschoolers’ numerical knowledge and ability to acquire new numerical knowledge. The effect on knowledge of numerical magnitudes was predicted to be larger when the game was played with a linear board than with a circular board because of a more direct mapping between the linear board and the desired mental representation. As predicted, playing the linear board game for roughly 1 hr increased low-income preschoolers’ proficiency on the 2 tasks that directly measured understanding of numerical magnitudes—numerical magnitude comparison and number line estimation—more than playing the game with a circular board or engaging in other numerical activities. Also as predicted, children who had played the linear number board game generated more correct answers and better quality errors in response to subsequent training on arithmetic problems, a task hypothesized to be influenced by knowledge of numerical magnitudes. Thus, playing linear number board games not only increases preschoolers’ numerical knowledge but also helps them learn from future numerical experiences. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Arithmetic Split Effects Reflect Strategy Selection: An Adult Age Comparative Study in Addition Comparison and Verification Tasks.

We tested whether split effects in arithmetic (i.e., better performance on large-split problems, like 3 + 8 = 16, than on small-split problems, like 3 + 8 = 12) reflect decision processing or strategy selection. To achieve this end, we tested performance of younger and older adults, matched on arithmetic skills, on two arithmetic tasks: the addition/number comparison task (e.g., 4 + 8 ? 13; which item is the larger?) and in the inequality verification task (e.g., 4 + 8     

Access to information in working memory: Exploring the focus of attention.

Participants memorized briefly presented sets of digits, a subset of which had to be accessed as input for arithmetic tasks (the active set), whereas another subset had to be remembered independently of the concurrent task (the passive set). Latencies for arithmetic operations were a function of the setsize of active but not passive sets. Object-switch costs were observed when successive operations were applied to different digits within an active set. Participants took 2 s to encode a passive set so that it did not affect processing latencies (Experiment 2). The results support a model distinguishing 3 states of representations in working memory: the activated part of long-term memory, a capacity limited region of direct access, and a focus of attention. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Differential calculation abilities in young children from middle- and low-income families.

Examined the performance of 42 middle- and 42 low-income kindergarten children on arithmetic calculations presented in a nonverbal format as well as in 3 different verbal formats. On the nonverbal task, the child was shown an initial set of disks, which was then hidden with a cover. The set was transformed by adding or removing disks. After the transformation, the child's task was to construct an array of disks that contained the same number of disks as in the final hidden set. A significant interaction between income level and task format was obtained. Although middle-income children performed better than low-income children on each of the verbal calculation tasks, the 2 income groups did not differ in performance on the nonverbal calculation task. The findings suggest that the nonverbal task format is less sensitive to socioeconomic variation than are the verbal task formats. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Go with the flow: How to master a nonlinear multiple-cue judgment task.

The authors examined the cognitive processes that participants use in linear and nonlinear multiple-cue judgment tasks, hypothesizing that people are unable to use explicit cue abstraction in a nonlinear task, instead turning to exemplar memory. Experiment 1 confirmed that people are unable to use cue abstraction in nonlinear tasks but failed to confirm the hypothesized, spontaneous shift to exemplar memory. Instead, the participants appeared to be trapped in persistent and futile attempts to abstract the cue-criterion relations. Only after being instructed to rely on exemplar memory in Experiment 2 did they master the nonlinear task. The results suggest that adaptive shifts of representation need not occur spontaneously and that analytical thought may sometimes harm performance in nonlinear tasks. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The Genevan and Cattell-Horn conceptions of intelligence compared: Early implementation of numerical solution aids.

Highlights the similarities and differences of the Genevan and Cattell-Horn theories of intelligence and reports an investigation of the relation of operative level and set of performance on tasks indexing children's knowledge of correspondence relations. 105 children (aged 4 yrs to 7 yrs 11 mo) completed counting, instructional set, conservation of number, and static numerical comparison tasks. Findings indicate that performance on quantitative comparision task reflecting Ss' understanding of correspondence relations was highly related to operative level and that Ss' capacity to implement solution aids in making quantitative comparison was moderated by their level of operative development. While findings lend support to the Genevan theory, this theory tends to neglect issues pertaining to localized functioning such as the question of relative efficacy of rival solution approaches. In contrast, the Cattell-Horn theory emphasizes that well-learned knowledge-producing skills constitute potential solution aids and that children differ in what solution aids they learn to implement. (44 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The organization of arithmetic facts in memory: evidence from a brain-damaged patient

We report a single case study of a brain-damaged patient with impaired arithmetic performance. Three principal findings are presented: First, in a task involving production of answers to simple arithmetic problems, the patient's performance was far better for subtraction than for addition or multiplication. Second, in all arithmetic operations performance was generally much better for problems potentially solvable by rule (e.g., 5 + 0) than for problems requiring retrieval of specific facts (e.g., 5 + 3). Third, the dissociation between subtraction and the other arithmetic operations obtained in the production task was not observed in a verification task. The implications of these findings for claims concerning the organization of stored arithmetic facts are discussed.     

Team reflexivity, development of shared task representations, and the use of distributed information in group decision making.

Decision-making groups often exchange and integrate distributed information to a lesser extent than is desirable for high-quality decisions. One important reason for this lies in group members’ understanding of the decision task—their task representations—specifically the extent to which they understand the importance of exchange and integration of information. The authors hypothesized that a group’s development of a (shared) understanding of the information elaboration requirements of their task is influenced by collective reflection on the task. When not all group members initially realize the importance of information elaboration, team reflexivity increases the degree to which the group understands the importance of information elaboration. In an experiment, the authors showed that team reflection fostered the development of task representations emphasizing information elaboration and subsequent information elaboration and decision quality. When all members initially already held representations emphasizing information elaboration, team reflection promoted elaboration and performance to a lesser degree. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Temporal Construal Effects on Abstract and Concrete Thinking: Consequences for Insight and Creative Cognition.

Six studies investigate whether and how distant future time perspective facilitates abstract thinking and impedes concrete thinking by altering the level at which mental representations are construed. In Experiments 1-3, participants who envisioned their lives and imagined themselves engaging in a task 1 year later as opposed to the next day subsequently performed better on a series of insight tasks. In Experiments 4 and 5 a distal perspective was found to improve creative generation of abstract solutions. Moreover, Experiment 5 demonstrated a similar effect with temporal distance manipulated indirectly, by making participants imagine their lives in general a year from now versus tomorrow prior to performance. In Experiment 6, distant time perspective undermined rather than enhanced analytical problem solving. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A generalized fraction: An entity smaller than one on the mental number line.

The representation of fractions in long-term memory (LTM) was investigated by examining the automatic processing of such numbers in a physical comparison task, and their intentional processing in a numerical comparison task. The size congruity effect (SiCE) served as a marker of automatic processing and consequently as an indicator of the access to the primitives of numerical representation in LTM. Mixed pairs composed of a natural number and a fraction showed both a SiCE and a distance effect. The SiCE for mixed pairs was stable across relative sizes of natural numbers compared to the fraction digits (Experiment 4). However, comparing pairs of fractions revealed a strong influence of fractional components: An inverse SiCE was found for pairs of unit fractions (Experiment 1), while no SiCE was found for pairs of non-unit fractions (Experiments 2–3). This leads to the conclusions that: (1) there are no unique representations of distinct fraction values in LTM, and (2) there is a representation of a “generalized fraction” as an “entity smaller than one” that emerges from the notational structure common to all fractions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives.

Children's failure on equivalence problems (e.g., 5 + 4 = 7 + __) is believed to be the result of misunderstanding the equal sign and has been tested using symbolic problems (including "="). For Study 1 (N = 48), we designed a nonsymbolic method for presenting equivalence problems to determine whether Grade 2 children's difficulty is due to the presence of symbols or to a more fundamental misunderstanding of equivalence. Children's superior performance on nonsymbolic versus symbolic problems suggests that children fail to map their understanding of equivalence onto problems presented with the symbols of arithmetic. For Study 2 (N = 32), we implemented a within-subject design to assess whether experience with nonsymbolic problems would facilitate performance on symbolic problems. This hypothesis was confirmed. Exposure to nonsymbolic problems may have enabled children to map their successful concepts and strategies to symbolic equivalence problems. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

An Encoding-Complex Approach to Numerical Cognition in Chinese-English Bilinguals.

We present a model of the cognitive architecture of basic numerical skills in adult Chinese-English bilinguals. The model is based on data reported by Campbell, Kanz, and Xue (1999) and combines Dehaene and Cohen's triple-code theory with Campbell and Clark's encoding-complex approach to modeling number processing. Participants were required to name, add or multiply Arabic or Mandarin numerals and to respond in English or Chinese. They also performed magnitude comparisons on pairs of Arabic or Mandarin numerals. The proposed model of their performance on this set of tasks assumes 1) that number processing is modular with respect to representational code (e.g., visual, visuo-spatial, verbal) rather than with respect to numerical function, 2) task-specific communication between representational codes is interactive rather than additive, and 3) memory for arithmetic facts is at least partially language- based and our Chinese-English bilinguals possessed both Chinese and English-language number-fact representations. We provide new analyses of the arithmetic data and a review of research on the role of language in simple arithmetic to substantiate our claims about linguistic codes for number-fact memory. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Lessons from arithmetic: Influences on the selection of problem-solving procedures.

Mental arithmetic involves an intriguing interaction between (a) memory and problem-solving processes that are domain independent and (b) solution procedures and factual knowledge that are specific to the domain of arithmetic. Relations between domain-independent and domain-specific knowledge are of central concern to many developmental, cognitive, and instructional psychologists, and consequently the performance of children and adults on mental arithmetic tasks have come under increasing scrutiny. Despite this common interest, in recent years the focus of research on adults has diverged somewhat from that of research on children. Recent research with children has concentrated on the selection process, the process by which one of several available procedures is selected for use on specific problems. In contrast, it is commonly assumed that adults always or nearly always use fact retrieval to answer simple arithmetic problems. Thus research with adults has been focused primarily on processes and knowledge structures involved in the retrieval of arithmetic facts, and the selection process has been virtually ignored. This article describes two studies which help to highlight the value of focusing on the selection process as a common and dynamic feature in the performance of children and adults. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Executive control of cognitive processes in task switching.

In 4 experiments, participants alternated between different tasks or performed the same task repeatedly. The tasks for 2 of the experiments required responding to geometric objects in terms of alternative classification rules, and the tasks for the other 2 experiments required solving arithmetic problems in terms of alternative numerical operations. Performance was measured as a function of whether the tasks were familiar or unfamiliar, the rules were simple or complex, and visual cues were present or absent about which tasks should be performed. Task alternation yielded switching-time costs that increased with rule complexity but decreased with task cuing. These factor effects were additive, supporting a model of executive control that has goal-shifting and rule-activation stages for task switching. It appears that rule activation takes more time for switching from familiar to unfamiliar tasks than for switching in the opposite direction. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Spontaneous representation of number in cotton-top tamarins ( Saguinus oedipus).

The "violation of expectancy" looking-time methodology has proven a powerful tool for exploring prelinguistic mental representations in human infants as well as in nonhuman primates. Four studies applying this methodology to the question of spontaneous number representations in cotton-top tamarins (Saguinus oedipus) are reported here. Monkeys were shown 1?+?1 events in which objects were placed behind a screen, 1 by 1. The screen was removed, revealing consistent (2 objects) and inconsistent (1, 3, or 1 large object twice the mass of original object) outcomes. In all studies, monkeys looked longer at the inconsistent than at the consistent outcome. When the monkeys view a 1?+?1 operation, they expect exactly 2 objects. It is likely that these numerical representations are spontaneously available to a variety of primate species and could provide a foundation on which humans' number sense was constructed over evolution and development. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"What counts in the development of young children's number knowledge?": Correction to Levine et al. (2010).

Reports an error in "What counts in the development of young children's number knowledge" by Susan C. Levine, Linda Whealton Suriyakham, Meredith L. Rowe, Janellen Huttenlocher and Elizabeth A. Gunderson (Developmental Psychology, 2010[Sep], Vol 46[5], 1309-1319). A coding error resulted in incorrect item-level data being reported on the point-to-x task (not the children‘s overall performance on this task) in Table 2 and in the section of the Results headed Point-to-X Task Performance (second column, p. 1314). In the first paragraph in the section, the correct average score for knowledge of cardinal meanings of the number words. In the second paragraph in the section, there is an example illustrating children’s greater performance on items involving a target and a distractor that were one digit apart. An additional adjustment in the second paragraph involves the finding that children performed better when at least one of two choice sets was a small number (1–3) than when both choice sets were greater than or equal to 4. More information for the corrections and the corrected table are given in the erratum. (The following abstract of the original article appeared in record 2010-17955-026.) Prior studies indicate that children vary widely in their mathematical knowledge by the time they enter preschool and that this variation predicts levels of achievement in elementary school. In a longitudinal study of a diverse sample of 44 preschool children, we examined the extent to which their understanding of the cardinal meanings of the number words (e.g., knowing that the word “four” refers to sets with 4 items) is predicted by the “number talk” they hear from their primary caregiver in the early home environment. Results from 5 visits showed substantial variation in parents' number talk to children between the ages of 14 and 30 months. Moreover, this variation predicted children's knowledge of the cardinal meanings of number words at 46 months, even when socioeconomic status and other measures of parent and child talk were controlled. These findings suggest that encouraging parents to talk about number with their toddlers, and providing them with effective ways to do so, may positively impact children's school achievement. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The effects of recent practice on task switching.

Four experiments investigated the effect of recent selective practice on the cost of switching between 2 tasks afforded by letter-digit pairs: alphabet arithmetic and shape comparison. Experiments 1 and 2 found a greater cost associated with switching to the more recently practiced task: evidence that task-set inertia contributes to switching costs. Experiment 3 found this effect to be limited to trials on which a recently trained stimulus followed another such stimulus: a result problematic for all current theories of task-set priming. Experiment 4 showed that the effect of recent practice was eliminated by active preparation for a task switch: It appears that endogenous task-set preparation reduces the effects of task-set inertia. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A longitudinal study of children’s performance on simple multiplication and division problems.

The authors investigated the performance on simple multiplication and division problems of 8-year-old children longitudinally to determine the developmental trajectories of both operations. Twice a year, during 2 consecutive school years, children performed a multiplication and division verification task and a number-matching task. All effects that were observed in multiplication performance (problem size, 5, and tie effect and Tie × Size interaction) were also observed in division performance. The developmental trajectories of these effects are described. The authors observed strong developmental parallels between both operations. These results are in line with strongly interconnected memory networks for multiplication and division facts, at least in young children. The results of the number-matching task showed that the interference effect developed differently for multiplication and division, indicating that automatic activation spreading from division operands to division answers is not at work in children of that age. (PsycINFO Database Record (c) 2010 APA, all rights reserved)