Developing access to number magnitude: a study of the SNARC effect in 7- to 9-year-olds

The SNARC (spatial-numerical association of response codes) effect refers to the finding that small numbers facilitate left responses, whereas larger numbers facilitate right responses. The development of this spatial association was studied in 7-, 8-, and 9-year-olds, as well as in adults, using a task where number magnitude was essential to perform the task and another task where number magnitude was irrelevant. When number magnitude was essential, a SNARC effect was found in all age groups. But when number magnitude was irrelevant, a SNARC effect was found only in 9-year-olds and adults. These results are taken to suggest that (a) 7-year-olds represent number magnitudes in a way similar to that of adults and that (b) when perceiving Arabic numerals, children have developed automatic access to magnitude information by around 9 years of age.     

7～9岁儿童失言理解与执行功能的关系

为了说明学龄儿童心理理论与执行功能的关系,研究采用失言理解、威斯康星卡片分类和汉诺塔任务分别考察了90名7—9岁儿童的心理理论和执行功能。结果表明,失言理解与抑制-转换能力相关显著（r=0．34,P〈0．01）,但是与计划能力相关不显著（r=0．06,P〉0．05）。在控制年龄后,失言理解与抑制-转换能力的相关仍然显著（r=0．29,P〈0．05）。研究结果说明,学龄儿童心理理论与执行功能的相关模式与学龄前儿童完全一致,提示心理理论与执行功能的相关关系从学龄前延续到了学龄阶段。     

Feedback influences children's reasoning about math equivalence: A meta-analytic review

ABSTRACT

Decades of research have focused on children's reasoning about math equivalence problems for both practical and theoretical insights. Not only are math equivalence problems foundational in arithmetic and algebra, they also represent a class of problems on which children's thinking is resistant to change. Feedback is one instructional tool that can serve as a key trigger of cognitive change. In this paper, we review all experimental studies (N = 8) on the effects of feedback on children's (ages 6–11) understanding of math equivalence. Meta-analytic results indicate that feedback has positive effects for low-knowledge learners and negative effects for high-knowledge learners, and these effects are stronger for procedural outcomes than conceptual outcomes. Findings highlight the variable influences of feedback on math equivalence understanding and suggest that models of thinking and reasoning need to consider learner characteristics, learning outcomes, and learning materials, as well as the dynamic interactions among them.     

Individual differences in trajectories of arithmetical development in typically achieving 5- to 7-year-olds

The arithmetical performance of typically achieving 5- to 7-year-olds (N = 29) was measured at four 6-month intervals. The same seven tasks were used at each time point: exact calculation, story problems, approximate arithmetic, place value, calculation principles, forced retrieval, and written problems. Although group analysis showed mostly linear growth over the 18-month period, analysis of individual differences revealed a much more complex picture. Some children exhibited marked variation in performance across the seven tasks, including evidence of difficulty in some cases. Individual growth patterns also showed differences in developmental trajectories between children on each task and within children across tasks. The findings support the idea of the componential nature of arithmetical ability and underscore the need for further longitudinal research on typically achieving children and of careful consideration of individual differences.     

Event-related potential and behavioral measures of attention in 5-, 7-, and 9-year-olds

In 2 experiments, the development of auditory selective attention in children was assessed. The participants, aged 5, 7, and 9 years, responded to target stimuli in the left ear for 1 series (ignoring the standard stimuli) and in the right ear for the other series. In Experiment 1, event-related potentials (ERPs) evoked by the auditory stimuli were recorded from frontal, central, and parietal sites. The 9-year-olds showed a greater processing negativity (Nd) to the attended channel compared with the 5-year-olds. Both 7- and 9-year-olds showed significantly larger amplitudes for the P3 component of ERPs in the attended vs. ignored condition. Behaviorally, the 5-year-olds made fewer hits and more false alarms than did older children, and the 9-year-olds made significantly more false alarms than did the 7-year-olds. The results of Experiment 2 showed that the detriment in the performance of the 9-year-olds was a result of task parameters. The inability of 5-year-olds to attend selectively appears to involve problems with the inhibition of the processing of irrelevant information and with selection of the correct response.     

Situating math word problems: The story matters

In two experiments, we explored how the situation model of a math story problem impacts math problem performance. Participants completed multiplication story problems in which a set of objects was associated with or dissociated from a protagonist, making them more or less accessible in memory during answer retrieval. On the basis of previous findings that the sum of two numbers interferes with retrieval of their product, the number of objects in the math problem was either highly interfering (“9” for 4 3 5) or less interfering (“8” for 4 3 5) for multiplication retrieval in the problem. Participants made more errors in problem solving when highly interfering numerical content was associated with the protagonist and, thus, foregrounded. Moreover, the lower one’s working memory, the bigger this effect. In sum, small changes in the situation model of a math story problem can harm performance. These changes shift the balance of factors that influence math performance away from math knowledge and toward individual differences in general cognitive capacity.     

Interspersing easy math problems among challenging problems: detection of interspersal effects in whole-class applications

Interspersing easy problems among challenging problems has been shown to both increase students’ preference for completing math worksheets and the fluency with which they complete challenging problems. Although early research examining the interspersal procedure in mathematics was conducted with college-age students, there are a growing number of studies utilizing the interspersal procedure in elementary and secondary school settings. A particular concern with existing research however is whether the beneficial effects are maintained across multiple sessions and whether effects generalize when implemented with large groups of students. The current study addressed this concern by administering both control and interspersal math worksheets consisting of instructionally relevant problems to 75 fourth grade students across four sessions. Because students’ instructional levels differed, analyses were conducted separately for students identified as fluent and less fluent based upon student performance on the control worksheet administered during session 1. Dependent measures included digits correct and problems correct as well as students’ responses to four preference questions following each session. Although results differ from previous research in that students completed significantly more digits correct on control worksheets, explanations of these differences are examined and implications are discussed.     

An eye for relations: eye-tracking indicates long-term negative effects of operational thinking on understanding of math equivalence

Prior knowledge in the domain of mathematics can sometimes interfere with learning and performance in that domain. One of the best examples of this phenomenon is in students’ difficulties solving equations with operations on both sides of the equal sign. Elementary school children in the U.S. typically acquire incorrect, operational schemata rather than correct, relational schemata for interpreting equations. Researchers have argued that these operational schemata are never unlearned and can continue to affect performance for years to come, even after relational schemata are learned. In the present study, we investigated whether and how operational schemata negatively affect undergraduates’ performance on equations. We monitored the eye movements of 64 undergraduate students while they solved a set of equations that are typically used to assess children’s adherence to operational schemata (e.g., 3 + 4 + 5 = 3 + __). Participants did not perform at ceiling on these equations, particularly when under time pressure. Converging evidence from performance and eye movements showed that operational schemata are sometimes activated instead of relational schemata. Eye movement patterns reflective of the activation of relational schemata were specifically lacking when participants solved equations by adding up all the numbers or adding the numbers before the equal sign, but not when they used other types of incorrect strategies. These findings demonstrate that the negative effects of acquiring operational schemata extend far beyond elementary school.     

Analogical reasoning in 2-year-olds: The development of access and relational inference

Eighty-four 24- and 30-month-old children were tested with two analogy tasks: formal and problem solving. Experiment 1 included three Groups: relations specified, relations unspecified, and associative control (no exposure to base relations). In Experiment 2 the relation that linked that a- and b-terms in formal problems was explicitly shown in order to reduce relational inference demands. In Experiment 1, children responded systematically to the formal problems when given two chances to answer each question. In Experiment 2, children were systematic on their first responses. In the problem-solving task the rates of spontaneous analogies produced were similar to those seen in older children. However, unlike older children, 2-year-olds benefited minimally from prompts. We believe this may indicate that 2-year-olds have difficulty explicitly controlling their analogical activity. We conclude early analogical abilities may be limited by weak relational inference abilities and difficulty intentionally initiating access.     

Notational strategies for problem solving in 5- to 7-year-olds

The study focuses on the production and use of notations as a means to solve a memory problem. It reports results from two experiments. In the first, a memory task, 5- to 7-year-old children were asked to notate the contents of three identical boxes in order to later determine the identity and number of the objects inside the boxes once they had been closed and moved to a different position. Statistical analysis of the data revealed developmental trends in subjects' notational performance: prior to age seven their notations were generally ineffective in helping them to later determine the exact contents of the boxes. In order to rule out the possibility that their low performance on the memory task was due to their inability to represent information by means of a notation, we designed a second experiment, a representation task, in which subjects produced a notation to represent the same set of objects as in experiment 1. Results comparing performance on the two experiments (the memory and representation tasks) showed no differences at age five, statistical differences for identity at age six, and only marginal statistical differences for quantity at age seven, mainly due to the statistically significant age-trend in performance. Seven-year-olds' performance improved in both conditions, weakening the differences between the two conditions. These results suggest a metacognitive hypothesis that it is not until age seven that children begin to make the prospective analysis needed to solve the memory task.     

Curriculum-based measurement of math problem solving: A methodology and rationale for establishing equivalence of scores

The purpose of this article is to discuss curriculum-based measurement (CBM) as it is currently utilized in research and practice and to propose a new approach for developing measures to monitor the academic progress of students longitudinally. To accomplish this, we first describe CBM and provide several exemplars of CBM in reading and mathematics. Then, we present the research context for developing a set of seven curriculum-based measures for monitoring student progress in math problem solving. The rationale for and advantages of using statistical equating methodology are discussed. Details of the methodology as it was applied to the development of these math problem solving measures are provided.     

Assessing inference production in 5 to 7-year-olds: Benefits of using a pictorial task

IntroductionThis study focuses on the methodology used to assess children's inferential skills in a narrative comprehension context with children aged between 5 to 7 years.ObjectiveThe aim of this study is to test the validity of a pictorial task based on children's literature to probe children's ability to make inferences at different times during the listening of fictional stories.MethodThe task involves selecting one of three vignettes which best illustrates what is currently happening in the story (i.e., inferential vs. literal vs. off-topic). The choices expressed here are compared to those made during a preference task performed by the same students, which consists of selecting the vignette they like the most among each series presented in the comprehension task.ResultsFirst, based on the analysis of the data collected in the comprehension task, children most often select inferential vignettes. Second, the choices expressed in the comprehension task differ from those expressed in the preference task.ConclusionThese results confirm the relevance of using children's literature associated with a pictorial task to test children's inferential skills, even with preschoolers.     

Early maturity of face processing in children: Local and relational distinctiveness effects in 7-year-olds

It is still often claimed that face recognition in children matures relatively late (e.g., 10 years), supporting an “expertise” rather than an “innate” interpretation. However, a review indicates that many adult properties of face processing (inversion effects, first-order relational processing, second-order relational processing, race effects, distinctiveness effects on perception) are present at 6-7 years. A new experiment investigates distinctiveness effects on memory. We manipulated distinctiveness via both local alterations to an original face (thicker lips, bushier eyebrows), and second-order relational alterations (eyes closer together, mouth moved down; Leder & Bruce, 1998). We also varied orientation (upright, inverted). Seven-year-olds' memory showed adult patterns of sensitivity to all these manipulations. We conclude that face recognition matures relatively early, and that the focus of future studies must shift to much younger age groups.     

Inferring Sublexical Correspondences from Sight Vocabulary: Evidence from 6- and 7-year-olds

We report an experiment designed to investigate 6-to-7-year-old children's ability to acquire knowledge of sublexical correspondences between print and sound from their reading experience. A computer database containing the printed word vocabulary of children taking part in the experiment was compiled and used to devise stimuli controlled for grapheme-phoneme correspondence (GPC) frequency and rime neighbourhood consistency according to the children's reading experience. Knowledge of GPC rules and rime units was compared by asking children to read aloud three types of nonword varying in regularity of GPC and consistency of rime pronunciation. Results supported the view that children can acquire knowledge of both GPC rules and rime units from their reading experience. GPC rule strength affects the likelihood of a GPC response; rime consistency affects the likelihood of a rime response.     

Understanding linear and exponential growth: Searching for the roots in 6- to 9-year-olds

Previous studies have suggested that children as young as 9 years old have developed an understanding of non-linear growth processes prior to formal education. The present experiment aimed at investigating this competency in even younger samples (i.e., in kindergartners, first, and third graders, ages 6, 7 and 9, respectively). Children (N = 90) solved non-verbal inductive reasoning tasks by forecasting linear and exponential growth. While children of all ages forecasted linear growth adequately, exponential growth was also estimated remarkably well. Surprisingly, kindergartners and third graders showed similar high achievement concerning the magnitude and curve shape of forecasts, whereas first graders performed significantly worse. We concluded that primary knowledge of both linearity and non-linearity exists even in kindergartners. However, children's understanding is quite fragile, as their performance was strongly affected by task sequence: Children underestimated exponential growth when the previous task required a forecast of linear growth, and overestimated linear growth when the previous task required forecasting of exponential growth.     

The development of a human auditory localization response: a U-shaped function

Research during the past 10 years on the neonatal head-turn response to off-centred rattle sounds is reviewed, and various procedural and stimulus conditions that influence the probability of eliciting a correct response are identified. Also, the existence of a U-shaped developmental function is confirmed in a cross-sectional study of 104 infants between 3 days and 7 months of age. Neonates responded reliably, but slowly; the response decreased in frequency and magnitude between 1-3 months of age and increased again by 4-5 months of age. Speculation that this U-shaped function reflects a maturational shift in locus of control from subcortical to cortical structures was supported by the infants' responses to the presence effect (PE), which is thought to be cortically mediated. The PE was produced by playing the rattle sound through two loudspeakers with the output of one delayed by 5 ms, relative to the other; adults perceive only one sound at the leading loudspeaker. As predicted, neonates failed to respond to the PE, and the onset of correct PE responses corresponded closely to the upswing in the U-shaped function for SS responses. Other explanations for the temporary decline in orientation responses to sound are also discussed.     

Age-appropriate cues facilitate source-monitoring and reduce suggestibility in 3- to 7-year-olds

Providing cues to facilitate the recovery of source information can reduce postevent misinformation effects in adults, implying that errors in source-monitoring contribute to suggestibility (e.g., [Lindsay, D. S., & Johnson, M. K. (1989). The eyewitness suggestibility effect and memory for source. Memory & Cognition, 17, 349–358]). The present study investigated whether source-monitoring plays a similar role in children’s suggestibility. It also examined whether the accuracy of source judgements is dependent on the type of source task employed at test. After watching a film and listening to a misleading narrative, 3–4- and 6–7-year-olds (n = 116) were encouraged to attend to source memory at retrieval. This was achieved either via sequential “question pairs”, which are typically used in children’s source-monitoring research, or via a novel “posting-box” procedure, in which all source options were provided simultaneously. Performance elicited by each type of source task was compared with that evoked by old/new recognition procedures. Posting-box, but not question pair, source cues were effective at reducing the magnitude of the suggestibility effect, relative to that observed under recognition conditions. Furthermore, source question pairs provoked a bias to respond affirmatively for 3–4-year-olds. The findings imply that children’s suggestibility may be partially explained by sub-optimal use of intact source information, which may be activated by age-appropriate strategies at retrieval.     

Curriculum-based measurement of oral reading: Passage equivalence and probe-set development

Curriculum-based measurement of reading (CBM-R) is used to estimate oral reading fluency. Unlike many traditional published tests, CBM-R materials are often comprised of 20 to 30 alternate forms/passages. Historically, CBM-R assessment materials were sampled from curricular materials. Recent research has documented the potentially deleterious effects of poorly controlled alternate forms on CBM-R outcomes. The purpose of this study was to examine alternate procedures for the selection of passages that comprise CBM-R passage-sets. The study examined four procedures for the evaluation and selection of passages, including random sampling, Spache readability formula, mean level of performance evaluation, and Euclidean Distance evaluation. The latter two procedures relied on field testing and evaluation of student performance. Each of eighty-eight students in second- and third-grade were administered 50 CBM-R passages. Generalizability and dependability studies were used to examine students' performance on these passages and evaluate CBM-R passage selection procedures. Results provide support for the use of field testing methods (i.e., calculating performance means and Euclidean Distances) for passage selection. Implications are discussed for future research and practice.     

The cognitive foundations of reading and arithmetic skills in 7- to 10-year-olds

A range of possible predictors of arithmetic and reading were assessed in a large sample (N=162) of children between ages 7 years 5 months and 10 years 4 months. A confirmatory factor analysis of the predictors revealed a good fit to a model consisting of four latent variables (verbal ability, nonverbal ability, search speed, and phonological memory) and two manifest variables (digit comparison and phoneme deletion). A path analysis showed that digit comparison and verbal ability were unique predictors of variations in arithmetic skills, whereas phoneme deletion and verbal ability were unique predictors of variations in reading skills. These results confirm earlier findings that phoneme deletion ability appears to be a critical foundation for learning to read (decode). In addition, variations in the speed of accessing numerical quantity information appear to be a critical foundation for the development of arithmetic skills.     

Impaired prospective memory but intact episodic memory in intellectually average 7- to 9-year-olds born very preterm and/or very low birth weight

Relatively little is known about episodic memory (EM: memory for personally-experienced events) and prospective memory (PM: memory for intended actions) in children born very preterm (VP) or with very low birth weight (VLBW). This study evaluates EM and PM in mainstream-schooled 7- to 9-year-olds born VP (≤ 32 weeks) and/or VLBW (< 1500 g) and matches full-term children for comparison (n = 35 and n = 37, respectively). Additionally, participants were assessed for verbal and non-verbal ability, executive function (EF), and theory of mind (ToM). The results show that the VP/VLBW children were outperformed by the full-term children on the memory tests overall, with a significant univariate group difference in PM. Moreover, within the VP/VLBW group, the measures of PM, verbal ability and working memory all displayed reliable negative correlations with severity of neonatal illness. PM was found to be independent of EM and cognitive functioning, suggesting that this form of memory might constitute a domain of specific vulnerability for VP/VLBW children.