Two birds in the hand: Concurrent and switching cognitive flexibility in preschoolers.

Cognitive flexibility, the ability to think of something in more than one way, has been studied in preschoolers from two different approaches. Within executive function research, most studies operationalize cognitive flexibility using sequential tasks in which children must think of a specific stimulus first in one way and then in another way (switching cognitive flexibility). In contrast, Piagetian multiplicative classification tasks also require cognitive flexibility by asking children to consider multiple dimensions of a single stimulus at the same time. Piagetian tasks are typically inductive and require abstraction skills. Although research with Piagetian tasks suggests that children are unable to consider multiple dimensions simultaneously until they reach the concrete operational stage, there is some evidence that preschoolers can coordinate two dimensions simultaneously in deductive tasks. In this study, we examined a deductive version of a matrix completion task in which preschoolers (N = 102) were asked to consider two dimensions of a given stimulus simultaneously. We show that 4- and 5-year-olds can succeed on this task and compare children's performance on the matrix completion task with their performance on a widely used switching cognitive flexibility task, the Dimensional Change Card Sort. We discuss the implications of relating the two aspects of cognitive flexibility development.

Conveying symbolic relations: Children's ability to evaluate and create informative legends.

Two studies were conducted to examine 4- to 6-year-old children's understanding of how to convey symbol-referent relations using legends. Study 1 investigated children's ability to evaluate legends in terms of whether or not they clearly convey information (N = 74). In this study, 41% of children were successful, with performance uniquely differentiated by sensitivity to ambiguity and executive function. Study 2 investigated children's ability to create informative legends (N = 115), with 39% being successful. Nearly half of those who were unsuccessful improved after exposure to exemplars (relative to only 9% in the baseline group). Sensitivity to ambiguity uniquely differentiated their ability to create a legend and improve after exposure. These studies provide insight into children's developing understanding of how symbol meanings are effectively conveyed and the contributions of other cognitive factors.

Preschoolers' Saving Behavior: The Role of Planning and Self-Control.

Planning and self-control were examined in relation to preschoolers' (41- to 74-months) saving behavior. Employing a marble run paradigm, 54 children participated in two trials in which they could use their marbles immediately on a less desirable run, or save for a more desirable run. Twenty-nine children received the opportunity to create a budget. On Trial 1, children in the budgeting condition saved significantly more than did children in the control condition, and their planning ability related to saving (after controlling for age and language). Those who consistently budgeted at least one marble for the more desirable run were more likely to save. Control children's performance improved across trials, with no between-condition differences on Trial 2. Self-control was not related to saving.

Theory of mind and preschoolers' understanding of misdeed and politeness lies.

We told ninety-nine 4- and 5-year-olds stories in which speakers told lies and truths in two contexts: those told to deny a transgression (misdeeds) and those told to spare another's feelings (politeness). Participants identified each statement as a lie or as the truth, morally judged it as good or bad, and decided whether or not to assign punishment to the speaker. All children received measures of first- and second-order false-belief understanding. Although 4-year-olds were above chance in their identification accuracy, they did not differentiate between lies and truths in their moral judgments or punishments. Five-year-olds outperformed 4-year-olds in their identification accuracy, morally judged lies more negatively than truths, and assigned speakers of lies more punishment than those who told truths. Five-year-olds also treated lies about misdeeds more negatively than politeness lies in their moral judgments and punishments. Children's identification of both lies and truths related to their first-order false-belief understanding, while their moral judgments and punishment of lies related to their second-order false-belief understanding. These findings suggest that different considerations are made when children reason about the conceptual and moral differences between lies and truths. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Measuring on the go: Response to Morra, Panesi, Traverso, and Usai.

Morra, Panesi, Traverso, and Usai's (Journal of Experimental Child Psychology, 2017, Vol. 167, pp. 246-258) effort to clarify theoretical models and nomenclature confusion surrounding young children's executive functions development is laudable and important. In this article, we address some of the points these authors raised regarding our previous article (Journal of Experimental Child Psychology, 2017, Vol. 159, pp. 199-218). Although we agree that the Multidimensional Card Selection Task makes working memory demands, it goes beyond working memory to measure concurrent cognitive flexibility in preschoolers. Using this task will allow researchers to fine-tune our models of cognitive flexibility and executive functions development.

The Multidimensional Card Selection Task: A new way to measure concurrent cognitive flexibility in preschoolers.

Most executive function research examining preschoolers' cognitive flexibility, the ability to think about something in more than one way, has focused on preschoolers' facility for sequentially switching their attention from one dimension to another (e.g., sorting bivalent cards first by color and then by shape). We know very little about preschoolers' ability to coordinate more than one dimension simultaneously (concurrent cognitive flexibility). Here we report on a new task, the Multidimensional Card Selection Task, which was designed to measure children's ability to consider two dimensions, and then three dimensions, concurrently (e.g., shape and size, and then shape, size, and color). More than half of the preschoolers in our sample of 107 (50 3-year-olds and 57 4-year-olds) could coordinate three dimensions simultaneously and consistently across three test trials. Furthermore, performance on the Multidimensional Card Selection Task was related, but not identical, to performance on other cognitive tasks, including a widely used measure of switching cognitive flexibility (the Dimensional Change Card Sort). The Multidimensional Card Selection Task provides a new way to measure concurrent cognitive flexibility in preschoolers, and opens another avenue for exploring the emergence of early cognitive flexibility development.

Refining the quantitative pathway of the Pathways to Mathematics model.

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to children's mathematical development. We attempted to refine the quantitative pathway by combining children's (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

Charting the role of the number line in mathematical development.

Individuals who do well in mathematics and science also often have good spatial skills. However, the predictive direction of links between spatial abilities and mathematical learning has not been firmly established, especially for young children. In the present research, we addressed this issue using a sample from a longitudinal data set that spanned 4 years and which includes measures of mathematical performance and various cognitive skills, including spatial ability. Children were tested once in each of 4 years (Time 1, 2, 3, and 4). At Time 3 and 4, 101 children (in Grades 2, 3, or 4 at Time 3) completed mathematical measures including (a) a number line task (0-1000), (b) arithmetic, and (c) number system knowledge. Measures of spatial ability were collected at Time 1, 2, or 3. As expected, spatial ability was correlated with all of the mathematical measures at Time 3 and 4, and predicted growth in number line performance from Time 3 to Time 4. However, spatial ability did not predict growth in either arithmetic or in number system knowledge. Path analyses were used to test whether number line performance at Time 3 was predictive of arithmetic and number system knowledge at Time 4 or whether the reverse patterns were dominant. Contrary to the prediction that the number line is an important causal construct that facilitates learning arithmetic, no evidence was found that number line performance predicted growth in calculation more than calculation predicted number line growth. However, number system knowledge at Time 3 was predictive of number line performance at Time 4, independently of spatial ability. These results provide useful information about which aspects of growth in mathematical performance are (and are not) related to spatial ability and clarify the relations between number line performance and measures of arithmetic and number system knowledge.

When this means that: the role of working memory and inhibitory control in children's understanding of representations.

We investigated cognitive skills that contribute to 4-year-olds' understanding of representations. In our main task, children used representations on a perspective line drawing to find stickers hidden in a model room. To compare the contributions made by various cognitive skills with children's understanding of different types of representations, we manipulated the resemblance between the representations and their referents. Our results indicate that when representations are iconic (i.e., look like their referents), children have very little difficulty with the task. Controlling for performance on this baseline version of the task, we found that specific cognitive skills are differentially predictive of performance when using arbitrary and conflicting representations (i.e., symbols). When the representation was arbitrarily linked to the sticker, performance was related to phonological and visuospatial working memory. When the representation matched the color of an alternate sticker (thereby conflicting with the desired sticker), performance was related to phonological working memory and inhibitory control. We discuss the role that different cognitive skills play in representational understanding as a function of the nature of the representation-referent relation.

The role of executive attention in the acquisition of mathematical skills for children in Grades 2 through 4.

We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in children's acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8-5). Children in Grades 2 and 3 (N=157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1 year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1 year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in children's acquisition of novel procedures and the development of automatic access to arithmetic facts.

Thinking about representations: the case of opaque contexts.

To address the question of whether young children are differentially sensitive to referential opacity, an advanced Theory of Mind skill, we assessed 4-, 6-, and 8-year-olds on three types of opaque contexts: epistemic, quotational, and intentional. Children's performance improved as a function of age and varied significantly by opacity type. Performance was best for epistemic opacity (using the verb "know") and poorest for intentional opacity (using "mean to"), with quotational opacity (using "said") falling in-between. Importantly, the current results suggest that children's sensitivity to referential opacity is not an all-or-nothing competence but rather one that varies across the contexts examined. Possible reasons for this variability are discussed.

Pathways to mathematics: longitudinal predictors of performance.

A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

Knowledge of counting principles: how relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.

What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2.

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.