Field dependence/independence cognitive style and problem posing: an investigation with sixth grade students

Field dependence/independence cognitive style was found to relate to general academic achievement and specific areas of mathematics; in the majority of studies, field-independent students were found to be superior to field-dependent students. The present study investigated the relationship between field dependence/independence cognitive style and problem-posing ability among sixth grade students. The 94 students’ sample was clustered into three groups, according to the cognitive-style field dependence/independence (field dependents, field mixed and field independents). The results suggest that field-independent participants outperformed field-mixed and field-dependent ones in both problem-posing ability and the complexity of the problems posed. It was also found that the content of the task influenced the differences between the three groups of students as regards the ability in problem posing and the complexity of the problems posed; while in the first (informal context), second and fourth task differences were found between the groups, in the third task (formal context), no differences were evident. The results are interpreted in consideration to the relevant literature findings. Some educational implications are discussed and directions for future research are drawn.     

Academic and Cognitive Characteristics of Persistent Mathematics Difficulty from First Through Fourth Grade

This longitudinal study investigated the academic and cognitive characteristics of persistent mathematics difficulty (MD-p) from first to fourth grade. Ninety-nine children were retrospectively categorized into one of three groups: MD-p, transient mathematics difficulty (MD-t), or typically developing. MD-p was defined as persistently low mathematics achievement (≤25th percentile) in at least 2 years from first to fourth grade. The results indicated that the MD-p group was more likely than other groups to have deficits in calculation, practical problem solving, number facts, and reading. In terms of cognitive characteristics, MD-p was specifically characterized by deficits in math concepts and phonological decoding, though there was some evidence for the involvement of working memory, processing speed, and numerical reasoning.     

Predicting First‐Grade Math Achievement from Developmental Number Sense Trajectories

Number sense development was tracked from the beginning of kindergarten through the middle of first grade, over six time points. Children (n= 277) were then assessed on general math achievement at the end of first grade. Number sense performance in kindergarten, as well as number sense growth, accounted for 66 percent of the variance in first‐grade math achievement. Background characteristics of income status, gender, age, and reading ability did not add explanatory variance over and above growth in number sense. Even at the beginning of kindergarten, number sense was highly correlated with end of first‐grade math achievement (r= 0.70). Clarifying the observed slope effect, general growth mixture modeling showed that children who started kindergarten with low number sense but made moderate gains by the middle of kindergarten had higher first‐grade math achievement than children who started out with similarly low number sense with flat growth. The majority of children in the low/flat growth class were from low‐income families. The findings indicate that screening early number sense development is useful for identifying children who will face later math difficulties or disabilities.     

教师变量对小学四年级数学成绩的影响:一个增值性研究

以房山区42所小学的1238名四年级学生和42名数学教师为研究对象,以四年级下学期的数学测验成绩为因变量,四年级上学期的数学测验成绩作为协变量加入两水平线性模型中建立基础增值模型.在控制学生背景变量的条件下,考察教师的特征变量、教学方式和教师培训三个方面对小学生四年级数学学业成绩增长的影响.结果显示,教师的特征变量中,教师的性别、年龄、教龄和学历专业对学生学业成绩的增长无显著影响,而教师的职称和最终受教育水平则对学生学业成绩的增长有显著影响;与教学相关的变量中,是否参加过新课程培训这一变量对学生学业成绩的增长有显著影响.     

Early Math Trajectories: Low‐Income Children's Mathematics Knowledge From Ages 4 to 11

Early mathematics knowledge is a strong predictor of later academic achievement, but children from low‐income families enter school with weak mathematics knowledge. An early math trajectories model is proposed and evaluated within a longitudinal study of 517 low‐income American children from ages 4 to 11. This model includes a broad range of math topics, as well as potential pathways from preschool to middle grades mathematics achievement. In preschool, nonsymbolic quantity, counting, and patterning knowledge predicted fifth‐grade mathematics achievement. By the end of first grade, symbolic mapping, calculation, and patterning knowledge were the important predictors. Furthermore, the first‐grade predictors mediated the relation between preschool math knowledge and fifth‐grade mathematics achievement. Findings support the early math trajectories model among low‐income children.     

The importance of number sense to mathematics achievement in first and third grades

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial, and memory), number sense made a unique and meaningful contribution to the variance in mathematics achievement at both first and third grades. Furthermore, the strength of the predictions did not weaken over time. Number sense was most strongly related to the ability to solve applied mathematics problems presented in various contexts. The number sense screen taps important intermediate skills that should be considered in the development of early mathematics assessments and interventions.     

Differentiation of Children's Competence Beliefs and Achievement Values,and their Relations to Children's Performance and Choice of Different Activities

ABSTRACT

Results of two studies of children's competence beliefs and achievement values for mathematics and reading are summarized. Approximately 1700 children and adolescents participated in the studies; the participants were in first through 12th grades. The studies were based on an expectancy — value model of achievement choice proposed by Eccles et al. (1983). Results indicated that children's competence beliefs and achievement task values are distinctive belief systems, even in first grade children. During the elementary school years, the strength of relations between children's competence beliefs and adult evaluations of children's competence increased across grade. Children and adolescents’ competence beliefs predict their mathematics performance, whereas their task values predict their intentions to continue taking mathematics.     

Estimating the co-development of executive functions and math achievement throughout the elementary grades using a cross-lagged panel model with fixed effects

Executive functioning (EF) is associated with children’s math skill development, both concurrently and longitudinally. However, it is not known how components of EF might be related to mathematics skills and vice versa over the course of elementary school. The present study addresses this issue by investigating relations between math achievement and two key components of EF -- working memory (WM) and cognitive flexibility (CF) -- from kindergarten to 5th grade, using the large-scale nationally representative dataset (N = 18,174) from the Early Childhood Longitudinal Study-Kindergarten (ECLS-K: 2011). Results from cross-lagged panel models with fixed effects support a transactional theoretical model, demonstrating a long-term reciprocal relationship between WM and math achievement from kindergarten to 5th grade and between CF and math achievement from 2nd grade to 5th grade. However, we found that reciprocal relations decrease as children grow older, suggesting that their math achievement relies less on EF and more on prior math knowledge over time.     

Motivational beliefs and cognitive processes in mathematics achievement, analyzed in the context of cultural differences: a Korean elementary school example

The relations among students’ motivational beliefs, cognitive processes, and academic achievement were investigated. A 51-item questionnaire together with a mathematics achievement test was administered to 459 fifth graders in Korean elementary school mathematics classrooms. Results indicated that, in general, students’ cognitive processes related closely to competence beliefs, task values, and achievement goals, and more importantly their success or failure in mathematics achievement was closely linked to competence beliefs, performance-avoidance goals, and persistence strategies. Positive evidence of performance-approach goals was observed in math learning relative to task goals. As expected, performance-avoidance goals turned out to be detrimental to students’ math learning. These findings are generally congruent with the motivational theories and support the position that students should be encouraged to adopt task goals and actively involve themselves in math class activities. However, it also behooves us to recognize the potential benefits of performance-approach goals in different cultural contexts, such as the Korean elementary school math classrooms.     

国产干涉条纹视力计预测白内障术后视力的临床研究

探讨国产干涉条纹视力计对预测白内障术后视力的准确性及其影响因素。方法:收集105例(105眼)白内障患者,年龄为69.89±10.04岁(30~92岁),术前分别用德国Heine视网膜视力计和国产干涉条纹视力计检测患者的术后预测视力,与术后1月最佳矫正视力比较,进行统计学分析。国产干涉条纹视力计和德国Heine视网膜视力计测量的预测视力分别为0.13±0.04和0.17±0.11,术后1月的最佳矫正视力为0.07±0.03,三者之间两两比较,有显著性差异(P<0.05),国产干涉条纹视力计与术后最佳矫正视力之间的差异较小。国产干涉条纹视力计预测视力符合率为66.67%(两者之间相差≤2行)。晶状体重度混浊和合并眼部疾病如青光眼、黄斑前膜、糖尿病性眼底病变等是其影响因素。国产干涉条纹视力计能有效预测白内障术后视力,但在重度白内障和合并眼底其他疾病的患者应结合相关临床检查做综合性评估。     

Number line estimation from kindergarten to grade 2: A longitudinal study

A bulk of evidence supports the association of number line estimations using Arabic digits and dots with math learning. Surprisingly few studies have been conducted to explore the relationship between estimations using number words and mathematics. The present study expands previous findings by investigating estimations in three formats (Arabic digits, dots and number), adding language as predictor and by focusing at timed and untimed math learning. A sample of 132 children was followed from kindergarten till grade 2. Results reveal variability in estimation accuracy and errors declining with age and instruction in all children. In addition, our findings suggest that Arabic numerals have a more linear distribution than number words. Moreover, our findings suggest that language explains variation in kindergarten but not in evolution and, more in particular, untimed math achievement can be predicted by number line estimation. Implications for assessment, prediction of math learning and instruction are discussed.     

The role of numerical and non-numerical ordering abilities in mathematics and reading in middle childhood

This study investigated whether different types of ordering skills were related to mathematics achievement in children (n = 100) in middle childhood, after the effects of age, socio-economic status, IQ, and processing speed were taken into account. The relations between ordering skills and magnitude processing were also investigated, as well as the possibility that some of the shared variance between math and reading is explained by ordering abilities. The ordering tasks included the ordering of familiar numerical and non-numerical sequences, a parental report of children’s everyday ordering skills, and an order working memory task. Three magnitude processing tasks (symbolic and non-symbolic comparison and number line estimation), were also administered, as well as measures of inhibition and spatial working memory. From this set of measures, number ordering, order working memory and number line estimation emerged as the most important predictors of mathematics skills. We found that number ordering mediated the effect of both symbolic and non-symbolic comparison skills on mathematics, further confirming that this task captures some essential skills related to mathematics. Additionally, order working memory mediated the effect of both number comparison and reading skills on math. Finally, whereas non-symbolic comparison and number line estimation are considered important indicators of magnitude processing skills, there was no relationship between these abilities, but there was a correlation between each of these abilities and reading skills, with number line estimation also mediating the effect of reading skills on math. These novel findings could contribute to a better understanding of the basic processes underlying math ability, and why math and reading are strongly related in typical populations and in children with learning difficulties.     

Effects of cooperation on spelling achievement at three age levels (Grades 2, 4, and 6)

In a field experiment among second, fourth and sixth grade pupils from 27 primary schools, two instruction methods were compared. At 13 schools (control condition) pupils did spelling exercises as usual, i.e. individually. At 14 schools pupils worked in pairs; each pupil was asked to check a partners’ work and discuss mistakes. At grade 2 cooperation led to less off-task behaviour when pupils did spelling assignments of a more complex type. Nevertheless we found a negative effect on achievement. At grade 4 cooperation led to less off-task behaviour for complex assignments and an increase in pupil’s effort as rated by their teachers as well. These findings were accompanied by a weak effect on achievement. Cooperation did not affect effort nor achievement of pupils from grade 6. At none of three grade levels we found indications of better reasoning strategies as a consequence of pair learning. The differential effect of cooperation at the three age levels may be explained in terms of complexity of learning subject and of lesson type.     

Evaluating the structure of the Mathematics Teacher Questionnaire: A measure of exposure to mathematics instructional practices and content

Psychometric properties of 24 items from the fifth grade Early Childhood Longitudinal Study-Kindergarten Cohort Mathematics Teacher Questionnaire were investigated in a sample of 5,181 participants. These items asked teachers to report how often they had their classroom students engage in different mathematics content, skills and instructional practices. Exploratory factor analyses were employed to evaluate the underlying factor structure of the selected items. Results suggest that these items effectively measure three distinct latent constructs that are specific to mathematics instruction: Instructional Practices for Teaching Mathematics, Developing Number Sense, and Beyond Number Sense. These results support the use of this instrument as a guide for research and policy, or to inform teacher curriculum decisions.     

Teacher Locus of Control,Teacher Behavior,and Student Behavior as Determinants of Student Achievement

Abstract

The relationship between teacher locus of control (A), teacher behavior (B), student behavior (C), and student achievement (D) was investigated. It was predicted that internal teachers would produce higher achieving students by maintaining a controlled learning environment, thereby engaging students in more appropriate on-task behavior. The first part of the study found modest correlations between I-E scores of 44 fourth grade teachers and student achievement in reading, language, and math. In the second part, the behavior of a subsample of 17 teachers and their students was observed. Although the complete A-B-C-D link was not obtained, several parts of the model did relate significantly.     

Development of the Metacognitive Skills of Prediction and Evaluation in Children With or Without Math Disability

Metacognition refers to knowledge about one's own cognition. The present study was designed to assess metacognitive skills that either precede or follow task engagement, rather than the processes that occur during a task. Specifically, we examined prediction and evaluation skills among children with (n= 17) or without (n= 179) mathematics learning disability (MLD), from grades 2 to 4. Children were asked to predict which of several math problems they could solve correctly; later, they were asked to solve those problems. They were asked to evaluate whether their solution to each of another set of problems was correct. Children's ability to evaluate their answers to math problems improved from grade 2 to grade 3, whereas there was no change over time in the children's ability to predict which problems they could solve correctly. Children with MLD were less accurate than children without MLD in evaluating both their correct and incorrect solutions, and they were less accurate at predicting which problems they could solve correctly. However, children with MLD were as accurate as their peers in correctly predicting that they could not solve specific math problems. The findings have implications for the usefulness of children's self‐review during mathematics problem solving.     

The relation between classroom management and mathematics achievement: A multilevel structural equation model

The impact of teachers on their students’ academic achievement continues to be an area of inquiry. One area not fully explored is the relation between teachers’ behavior and classroom management (CM) skills, student motivation, and student achievement. We examined these relations using a multi‐level structural equation model. Data included Behavior Management subscale scores of the Classroom Assessment Scoring System for 247 fourth and fifth grade teachers, indicators of academic motivation, and end‐of‐year state standardized mathematics assessment results for 4,847 students from the National Center of Teacher Effectiveness Main Study. The results indicate teachers’ CM skills have a significant direct effect on student motivation and a significant indirect effect on students’ math achievement, suggesting that students in classrooms of teachers with better CM are more motivated to learn math and have higher math achievement scores. This underscores the importance of helping teachers increase their use of evidence‐based classroom management techniques.     

Verbal counting skill predicts later math performance and difficulties in middle school

This study examined the role of verbal counting skill as an early predictor of math performance and difficulties (at or below −1.5 standard deviation in basic math skills) in middle school. The role of fourth-grade level arithmetical skills (i.e., calculation fluency, multi-digit arithmetic i.e. procedural calculation, and word problem solving) as mediators was also investigated. The participants included 207 children in central Finland who were studied from kindergarten to the seventh grade. Path modeling showed that verbal counting in kindergarten is a strong predictor for basic math performance in seventh grade, explaining even 52% of the variance in these skills after controlling for the mothers’ education levels. This association between early verbal counting skill and basic math performance was partly mediated through fourth-grade procedural calculation and word problem solving skills. Furthermore, verbal counting had an unique predictive relation to middle school math performance above and beyond the basic arithmetical and problem solving skills in fourth grade. Poor kindergarten verbal counting skill was a significant indicator for later difficulties in mathematics.     

Examining the dynamics of mathematics anxiety,perceived cost,and achievement: A control-value theory approach

Concerns about the influence of students’ perceived negative consequences of engagement in a task (i.e., cost) on their emotions, motivation, and cognition have increased in the last decade. The use of longitudinal models is needed to provide new insights into the role of perceived cost in mathematics learning. Grounded in the control-value theory, this study examined cross-lagged relations of mathematics anxiety, perceived cost, and mathematics achievement. The participants (N = 335) reported their mathematics anxiety and perceived cost four times during Grades 7 and 8, and their mathematics grades were attained from their school records. Cross-lagged panel model analysis revealed evidence of a long-term positive reciprocal relationship between mathematics anxiety and effort/emotional cost, a gradually diminished relationship between effort/emotional cost and mathematics performance, and a positive achievement to anxiety link during the transition between grade levels. Moreover, mathematics performance is a distal predictor of mathematics anxiety through effort/emotional cost rather than a proximal predictor or an outcome of anxiety. This study also clarified the distinction in the central role of effort/emotional versus opportunity cost in the interrelatedness of mathematics anxiety and performance, where the latter failed to demonstrate significant paths. Specific timing for interventions was discerned. Early cost prevention interventions along with considerations of academic achievement to alleviate both anxiety and perceived effort/emotional are highlighted as crucial for a positive high school mathematics experience.     

A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability

This study investigated students' mathematics achievement, estimation ability, use of estimation strategies, and academic self-perception. Students with learning disabilities (LD), average achievers, and intellectually gifted students (N = 135) in fourth, sixth, and eighth grade participated in the study. They were assessed to determine their mathematics achievement, ability to estimate discrete quantities, knowledge and use of estimation strategies, and perception of academic competence. The results indicated that the students with LD performed significantly lower than their peers on the math achievement measures, as expected, but viewed themselves to be as academically competent as the average achievers did. Students with LD and average achievers scored significantly lower than gifted students on all estimation measures, but they differed significantly from one another only on the estimation strategy use measure. Interestingly, even gifted students did not seem to have a well-developed understanding of estimation and, like the other students, did poorly on the first estimation measure. The accuracy of their estimates seemed to improve, however, when students were asked open-ended questions about the strategies they used to arrive at their estimates. Although students with LD did not differ from average achievers in their estimation accuracy, they used significantly fewer effective estimation strategies. Implications for instruction are discussed.