数学焦虑影响儿童数学任务表现的作用机制探析

研究设计了心算加法和图形旋转/平移两类数学任务,同时采用《儿童数学焦虑问卷》、工作记忆和抑制任务,以探讨数学焦虑影响儿童数学任务表现的作用机制.213名小学五年级儿童参与研究,其中89人参与正式实验.结果发现:(1)数学焦虑与数学成绩呈显著负相关;(2)高数学焦虑组在工作记忆和两类数学任务中的得分均显著低于低数学焦虑组,而两组在抑制任务指标上差异不显著;(3)言语工作记忆和视空间工作记忆是数学焦虑影响心算加法任务的完全中介变量;视空间工作记忆是数学焦虑影响图形旋转/平移任务的完全中介变量,言语工作记忆是二者关系的调节变量.研究结果支持了加工效能理论.     

Contributions of linguistic,quantitative, and spatial attention skills to young children's math versus reading: Same,different, or both?

The study used Bayesian and Frequentist methods to investigate whether the roles of linguistic, quantitative, and spatial attention skills are distinct in children's acquisition of reading and math. A sample of 175 Chinese kindergarteners was tested with measures of linguistic skills (phonological awareness and phonological memory), quantitative knowledge (number line task, symbolic digit comparison, and non-symbolic number estimation), spatial attention skills (visual span, mental rotation, and visual search), word reading, and calculation. After statistically controlling for age and nonverbal intelligence, phonological awareness and digit comparison performance explained unique variance in both math and reading. Moreover, number line estimation was specifically important for math, while phonological memory was specifically essential for reading. These findings highlight the possibility of developing early screening tools with different cognitive measures for children at risk of learning disabilities in reading and/or math.     

Girls’ Spatial Skills and Arithmetic Strategies in First Grade as Predictors of Fifth-Grade Analytical Math Reasoning

This study investigated longitudinal pathways leading from early spatial skills in first-grade girls to their fifth-grade analytical math reasoning abilities (N = 138). First-grade assessments included spatial skills, verbal skills, addition/subtraction skills, and frequency of choice of a decomposition or retrieval strategy on the addition/subtraction problems. In fifth grade, girls were given an arithmetic fluency test, a mental rotation spatial task, and a numeric and algebra math reasoning test. Using structural equation modeling, the estimated path model accounted for 87% of the variance in math reasoning. First-grade spatial skills had a direct pathway to fifth-grade math reasoning as well as an indirect pathway through first-grade decomposition strategy use. The total effect of first-grade spatial skills was significantly higher in predicting fifth-grade math reasoning than all other predictors. First-grade decomposition strategy use had the second strongest total effect, while retrieval strategy use did not predict fifth-grade math reasoning. It was first-grade spatial skills (not fifth-grade) that directly predicted fifth-grade math reasoning. Consequently, the results support the importance of early spatial skills in predicting later math. As expected, decomposition strategy use in first grade was linked to fifth-grade math reasoning indirectly through first-grade arithmetic accuracy and fifth-grade arithmetic fluency. However, frequency of first-grade decomposition use also showed a direct pathway to fifth-grade arithmetic reasoning, again stressing the importance of these early cognitive processes on later math reasoning.     

Symbolic control of visual attention: Semantic constraints on the spatial distribution of attention

Humans routinely use spatial language to control the spatial distribution of attention. In so doing, spatial information may be communicated from one individual to another across opposing frames of reference, which in turn can lead to inconsistent mappings between symbols and directions (or locations). These inconsistencies may have important implications for the symbolic control of attention because they can be translated into differences in cue validity, a manipulation that is known to influence the focus of attention. This differential validity hypothesis was tested in Experiment 1 by comparing spatial word cues that were predicted to have high learned spatial validity (“above/below”) and low learned spatial validity (“left/right”). Consistent with this prediction, when two measures of selective attention were used, the results indicated that attention was less focused in response to “left/right” cues than in response to “above/below” cues, even when the actual validity of each of the cues was equal. In addition, Experiment 2 predicted that spatial words such as “left/right” would have lower spatial validity than would other directional symbols that specify direction along the horizontal axis, such as “←/→” cues. The results were also consistent with this hypothesis. Altogether, the present findings demonstrate important semantic-based constraints on the spatial distribution of attention.     

DSM-IV-Defined Inattention and Sluggish Cognitive Tempo: Independent and Interactive Relations to Neuropsychological Factors and Comorbidity

The aim of the present study was to investigate the independent relations of DSM-IV-defined inattention and behaviors characteristic of sluggish cognitive tempo (SCT) to neuropsychological factors and problem behaviors often comorbid with attention deficit/hyperactivity disorder (ADHD). By controlling for symptoms of DSM-IV-defined inattention, unique relations to SCT could be ascertained. Additionally, interactive relations of DSM-IV-defined inattention and SCT were of interest. A community-based sample of school children (N = 209; the higher end of the ADHD-symptom range was oversampled) completed neuropsychological tasks designed to measure executive function (EF), sustained attention, and state regulation. Behavioral symptoms were measured using parental and teacher ratings of the DSM-IV criteria for ADHD and Oppositional Defiant Disorder (ODD). The results showed that these two domains of inattention, DSM-IV-defined inattention and SCT, have neuropsychological processes and comorbid behavioral problems in common. However, when controlling for the overlap, DSM-IV-defined inattention was uniquely related to EF and state regulation, while SCT was uniquely related to sustained attention. In addition, the results showed an interactive relation of DSM-IV-defined inattention and SCT to ODD. Findings from the present study support the notion that DSM-IV-defined inattention constitutes a somewhat heterogeneous condition. Such results can further our theoretical understanding of the neuropsychological impairments and comorbid behavioral problems associated with ADHD symptoms.     

Formal notations are diagrams: Evidence from a production task

Although a general sense of the magnitude, quantity, or numerosity of objects is common in both untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but well-specified situations--to behave formally, that is--are skills unique to people trained in symbolic notations. These symbolic notations typically employ complex, hierarchically embedded structures, which all extant analyses assume are constructed by concatenative, rule-based processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notations--in short, that formal notations are a kind of diagram. We examined self-generated productions in the domains of handwritten arithmetic expressions and typewritten statements in a formal logic. In both tasks, we found substantial evidence for spatial representational schemes even in these highly symbolic domains.     

Moving attention along the mental number line in preschool age: Study of the operational momentum in 3‐ to 5‐year‐old children's non‐symbolic arithmetic

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3–5 years old were tested with non‐symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give‐a‐number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial‐directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial‐directional and magnitude OM. This undermines the canonical version of the number line‐based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial‐numerical associations in numerical development are proposed.     

Cognitive and behavioral attention in children with math difficulties

Cognitive assessments and behavioral ratings of attention were used to examine the relation of inattention to math performance in children. Third grade students with math difficulties (MD; n?=?17) and math and reading difficulties (MDRD; n?=?35) were administered the Attentional Network Test (ANT), as well as achievement and intelligence measures. Strengths and Weaknesses of ADHD-Symptoms and Normal-Behavior-IV (SWAN-IV) Inattention ratings were collected from teachers. Two comparison groups were also recruited: a typically achieving group (n?=?23) and a group that responded to a math-tutoring intervention (responders; n?=?54). On the ANT, children with MD and MDRD did not perform significantly different than typically achieving children or responders on measures of alerting and orienting attention and executive control. All subgroups did demonstrate performance patterns that were expected on the ANT. However, performance across blocks of the task was inconsistent, suggesting poor reliability. There were no relations between ANT performance and SWAN-IV behavioral inattention scores, though behavioral ratings of inattention correlated significantly with math performance. Children with MD and MDRD may have more difficulty with distraction and attention to detail in contextual situations, as opposed to impulsive responding in these settings. The lack of relation between cognitive attention and math performance may suggest that either the ANT does not assess the relevant attention constructs associated with math difficulties or may parallel studies of attention deficit/hyperactivity disorder (ADHD) in which cognitive and behavioral assessments are weakly related.     

The interacting effect of load and space on visual selective attention

In the study of visual attention, two major determinants of our ability to ignore distracting information have been isolated, namely, (1) the spatial separation from the focus of attention and (2) perceptual load. This study manipulated both factors using a dual-task adaptation of the flanker paradigm (Eriksen & Hoffman, 1973). It showed that (1) although attention followed a gradient profile under low perceptual load it followed a Mexican-hat profile under high perceptual load, consistent with the idea that increasing load focuses spatial attention; and (2) increasing perceptual load did not improve overall selectivity: Though selectivity improved at near separations, it was impaired at far ones. Load and spatial separation exert interacting effects.     

Children’s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children’s predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.     

Going the distance: Extra-symbolic contributions to the symbolic control of spatial attention

Over the past 30 years, researchers have used symbolic cues such as spatial words and arrows to direct observers' attention to the location of target objects that appear at a single, fixed distance in simple visual environments; however, they have done so by using symbolic cues that provide only partial—directional—information about the spatial location of target objects. Thus, it remains unclear whether observers can only shift their attention broadly in the cued direction in response to these cues or whether they can shift their attention to the specific location of the target by combining an expectation about direction (derived from apprehension of the directional cue) with an expectation about distance (derived from some extra-symbolic cue). The results of two experiments showed that observers do shift their attention to relatively coarse spatial locations by combining expectancies about direction and distance, and that the extent to which observers do so is modulated by the presence of visible markers and the expected distance of the target. These findings are important because they provide the first evidence that extra-symbolic information about distance contributes to the symbolic control of spatial attention.     

Approximate number sense,symbolic number processing,or number–space mappings: What underlies mathematics achievement?

In this study, the performance of typically developing 6- to 8-year-old children on an approximate number discrimination task, a symbolic comparison task, and a symbolic and nonsymbolic number line estimation task was examined. For the first time, children’s performances on these basic cognitive number processing tasks were explicitly contrasted to investigate which of them is the best predictor of their future mathematical abilities. Math achievement was measured with a timed arithmetic test and with a general curriculum-based math test to address the additional question of whether the predictive association between the basic numerical abilities and mathematics achievement is dependent on which math test is used. Results revealed that performance on both mathematics achievement tests was best predicted by how well children compared digits. In addition, an association between performance on the symbolic number line estimation task and math achievement scores for the general curriculum-based math test measuring a broader spectrum of skills was found. Together, these results emphasize the importance of learning experiences with symbols for later math abilities.     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

Working Memory in Nonsymbolic Approximate Arithmetic Processing: A Dual‐Task Study With Preschoolers

Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a dual‐task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic.     

Competence and confusion: How stereotype threat can make you a bad judge of your competence

Women tend to have competence doubts for masculine‐stereotyped domains (e.g., math), whereas men tend to think they can handle both feminine‐stereotyped and masculine‐stereotyped domains equally well. We suggest that perhaps women's more frequent experience with stereotype threat can partly explain why. Our results showed that when stereotype threat was primed in high school students (n = 244), there was no relationship between their performance on an academic test (the SweSAT) and their assessment of their performance (how well they did), whereas in a non‐stereotype threat condition, there was a medium‐sized relationship. The effect was similar for both men and women primed with stereotype threat. The results imply that stereotype threat undermines performance assessments.     

Grounding spatial language in the motor system: Reciprocal interactions between spatial semantics and orienting

Shifts of attention that are directed by spatial words are thought to be mediated by a spatial frame of reference. Previous studies have documented a selection cost when visual attention is directed along the left/right axis relative to the above/below axis due to the greater ambiguity of the horizontal endpoints. The present study investigated whether the horizontal endpoints may be more fully disambiguated when “left/right” cues are first enacted by nondivisible, more discrete, modes of orienting such as directional motor responses. This issue was investigated using a repetition priming paradigm in which the effects of prior enactments were measured on the subsequent conceptual control of visual attention. The results of four experiments showed that grounding spatial language in the motor system can enhance the focus of visual attention by disambiguating the endpoints of the horizontal axis. In so doing, the present study extends previous research demonstrating that spatial concepts can influence the spatial distribution of attention by demonstrating that different modes of orienting can also influence the semantics of space. Implications for the premotor theory of attention and grounded theories of cognition are also discussed.     

Number sense at 12 months predicts 4-year-olds’ maths skills

Preverbal infants spontaneously represent the number of objects in collections. Is this ‘sense of number’ (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood.

Research Highlights

• In the literature there is a lively debate about the correlation between the ANS and maths skills.
• We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants).
• The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills.
• The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.

     

Association between individual differences in non-symbolic number acuity and math performance: A meta-analysis

Many recent studies have examined the association between number acuity, which is the ability to rapidly and non-symbolically estimate the quantity of items appearing in a scene, and symbolic math performance. However, various contradictory results have been reported. To comprehensively evaluate the association between number acuity and symbolic math performance, we conduct a meta-analysis to synthesize the results observed in previous studies. First, a meta-analysis of cross-sectional studies (36 samples, N = 4705) revealed a significant positive correlation between these skills (r = 0.20, 95% CI = [0.14, 0.26]); the association remained after considering other potential moderators (e.g., whether general cognitive abilities were controlled). Moreover, a meta-analysis of longitudinal studies revealed 1) that number acuity may prospectively predict later math performance (r = 0.24, 95% CI = [0.11, 0.37]; 6 samples) and 2) that number acuity is retrospectively correlated to early math performance as well (r = 0.17, 95% CI = [0.07, 0.26]; 5 samples). In summary, these pieces of evidence demonstrate a moderate but statistically significant association between number acuity and math performance. Based on the estimated effect sizes, power analyses were conducted, which suggested that many previous studies were underpowered due to small sample sizes. This may account for the disparity between findings in the literature, at least in part. Finally, the theoretical and practical implications of our meta-analytic findings are presented, and future research questions are discussed.     

The relationship between numerical mapping abilities,maths achievement and socioeconomic status in 4- and 5-year-old children

Background

Early numeracy skills are associated with academic and life-long outcomes. Children from low-income backgrounds typically have poorer maths outcomes, and their learning can already be disadvantaged before they begin formal schooling. Understanding the relationship between the skills that support the acquisition of early maths skills could scaffold maths learning and improve life chances.

Aims

The present study aimed to examine how the ability of children from different SES backgrounds to map between symbolic (Arabic numerals) and non-symbolic (dot arrays) at two difficulty ratios related to their math performance.

Sample

Participants were 398 children in their first year of formal schooling (Mean age = 60 months), and 75% were from low SES backgrounds.

Method

The children completed symbolic to non-symbolic and non-symbolic to symbolic mapping tasks at two difficulty ratios (1:2; 2:3) plus standardized maths tasks.

Results

The results showed that all the children performed better for symbolic to non-symbolic mapping and when the ratio was 1:2. Mapping task performance was significantly related to maths task achievement, but low-SES children showed significantly lower performance on all tasks.

Conclusion

The results suggest that mapping tasks could be a useful way to identify children at risk of low maths attainment.     

A new form of human spatial attention: Automated symbolic orienting

The control of human attention is typically conceptualized either in terms of exogenous automatic processes that are driven by external sensory stimulation or endogenous strategic processes that are driven by internal expectancies about events in the environment. However, this classic dichotomy has struggled to explain a wealth of new data demonstrating that behaviourally and biologically relevant visual stimuli, like arrow and eye direction, elicit shifts of spatial attention that on the one hand, appear exogenous, and on the other hand, endogenous. To address this issue, we used a double-cueing task that combined arrows with classic cues known to invoke either exogenous or endogenous orienting. Our data suggest that behaviourally relevant directional cues, like arrows, engage a new form of cortically mediated orienting—automated symbolic orienting—that operates independent of, and in parallel with, the two classic forms of exogenous and endogenous spatial attention.