Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The integrated theory of numerical development posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of the magnitudes to which they refer, and this magnitude understanding is central to general mathematical competence. We investigated relations among fraction magnitude understanding, arithmetic and general mathematical abilities in countries differing in educational practices: U.S., China and Belgium. Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders' fraction magnitude understanding was positively related to their general mathematical achievement in all countries, and this relation remained significant after controlling for fraction arithmetic knowledge in almost all combinations of country and age group. These findings suggest that instructional interventions should target learners' interpretation of fractions as magnitudes, e.g., by practicing translating fractions into positions on number lines.     

A comparative study of variations in arithmetic fluency between Norwegian and Finnish third graders

ABSTRACT

The study aimed to investigate variations in addition and subtraction fluency by observing grade three students in Norway (n = 253, M_age = 8.38 y.) and Finland (n = 209, M_age = 9.35 y.) while controlling for their age and non-verbal reasoning. Gender differences were also examined. The focus of the study was on the performance of the low-achieving (LA) students in comparison to the typically achieving (TA) group, not neglecting differences in how early educational support was organised across the two countries. Two-minute speed tests in both addition and subtraction within the 1–20 number range were used to assess fluency. The Finnish students outperformed students in the Norwegian sample both in addition and subtraction fluency. There were more Norwegian students in the LA group (i.e. performance at or below the 25th percentile) in both addition (37.9% vs. 20.1%) and subtraction (39.1% vs. 15.8%). In comparison to the TA students, the LA students made more errors and skipped over more arithmetic tasks in an attempt to solve them. Observed differences are discussed in relation to both country characteristics concerning early mathematics education and early educational support.     

多媒体教学绩效的二维有向图模型

针对目前高校中不同课程的多媒体教学绩效差异问题,提出一个二维有向图模型,用于描述采用不同多媒体教学手段与课程教学绩效的关系,目的是找出产生多媒体教学绩效的因素。利用二维有向图的三维关联矩阵计算课程教学绩效,并以课程为对象举例说明二维有向图的应用。     

Young girls’ spatial and arithmetic performance: The mediating role of maternal supportive interactions during joint spatial problem solving

The present study addresses girls’ (6–7-year-olds; N = 162) early spatial and arithmetic skills within the context of learning environments provided by their mothers. The study was designed to determine the relationship between maternal supportive interactions on a joint origami spatial problem-solving task and their first grade daughters’ spatial and arithmetic skills. During home visits the mothers and daughters were videotaped jointly solving origami tasks with maternal supportive interactions assessed through ratings of maternal stimulation of cognitive development and maternal quality of assistance; the girls were separately assessed in school on spatial and arithmetic skills. Using structural equation modeling, the main findings were (1) maternal supportive interactions on a mother–child origami task mediated the relation between mothers’ spatial skills/educational level and their daughters’ spatial skills and (2) their daughters’ spatial skills in turn mediated the relation between quality of maternal supportive spatial interactions and the daughters’ arithmetic achievement. The present findings indicate the importance of early maternal supportive interactions relating to spatial problem solving for girls’ spatial and arithmetic achievement. Furthermore, all pathways linking girls’ home environments and arithmetic skills were mediated through the girls’ spatial skills, suggesting that for young girls, development of early spatial skills may be important for effective arithmetic learning.     

Inhibitory control and visuospatial working memory contribute to 5-year-old children's use of quantitative inversion

This study examined the hypothesis that general cognitive resources moderated 5-year-old children's performance differences between the Concrete Identical and the Pure Quantity conditions on inversion problems (a + b – b) but not on standard problems (a + b – c). Study 1 (N = 104) showed that children who experienced higher visuospatial working memory burden performed significantly poorer in solving the inversion problems in the Pure Quantity condition than in the Concrete Identical condition, whereas those who experienced lower working memory burden showed no such difference. Study 2 (N = 194) demonstrated that children with lower levels of inhibitory control solved significantly fewer inversion problems in the Pure Quantity condition than in the Concrete Identical condition, whereas no such difference was found in children with higher levels of inhibitory control. These findings suggest that inhibitory control and visuospatial working memory may support children's use of quantitative inversion.     

Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary

When solving word problems, many children encounter difficulties in making sense of the information and integrate it into a meaningful schema. This is the fundamental phase on which subsequent problem solution depends. To better understand the processing underlying this fundamental phase, this study examined the roles of schema construction and knowledge of mathematical vocabularies in word problem solving. The participants were 139 Chinese third graders studying in Hong Kong. Path analysis showed that there were two kinds of pathways to word problem solving: language-related and number-related. In particular, reading fluency was related to word problem solving in two mediated language-related pathways: one via schema construction, the other via knowledge of mathematical vocabularies. In the number-related pathway, arithmetic concept was related to word problem solving via knowledge of mathematical vocabularies. These findings highlight the specific roles of schema construction and mathematical vocabulary in word problem solving, thereby providing useful implications of how best to support children in understanding and integrating the information from the problem.     

Mastery of the counting principles in toddlers: A crucial step in the development of budding arithmetic abilities?

Counting abilities have been described as determinative precursors for a good development of later arithmetic abilities. Mastery of the stable order, the one–one-correspondence and the cardinality principles can be seen as essential features for the development of counting abilities. Mastery of the counting principles in kindergarten was assessed in a large group of children with a broad range of arithmetic abilities (N = 423). Not all children mastered the counting principles by the end of kindergarten. Mastery of the counting principles in kindergarten was predictive for arithmetic abilities one year later in first grade, especially for scores on arithmetic achievement tests. Children sharing a common educational background tend to have more similar scores on arithmetic tests, yet the importance of mastery of the essential counting principles in the prediction of later arithmetic achievement was the same for all classrooms.     

轻度认知障碍患者单位数加法的ERP早成分研究

目的研究轻度认知障碍患者单位数加法的心算效率及事件相关电位（ERP）早成分改变。方法被试者分为正常老年（对照）组和轻度认知障碍（MCI）组,每组16名,均为右利手。使用事件相关电位技术,采集单位数加法计算过程的脑电信号,比较两组行为数据和算式早成分（N1,P1,N170,P2）的波幅和潜伏期的差异。结果MCI组的反应时间长于对照组,差异有显著性（P〈0．01）,正确率MCI组低于对照组,差异有显著性（P〈0．01）;MCI组与对照组比较,颞枕区N170以及额中央区P2潜伏期延长（P〈0．01）,波幅增大（P〈0.05）-MCI组左侧颞枕区N170波幅（-1．70uV）低于正常组（-2．96uV）,差异具有显著性（P〈0．01）;MCI组右侧颞枕区N170波幅（-3．85uV）高于对照纽（-0．75uV）,差异具有显著性（P〈0．01）。结论MCI患者的单位数心算效率降低,计算的正确率降低,同时提示大脑在算式早期加工阶段出现异常。     

Kindergarten children's symbolic number comparison skills relates to 1st grade mathematics achievement: Evidence from a two-minute paper-and-pencil test

Basic numerical skills provide an important foundation for the learning of mathematics. Thus, it is critical that researchers and educators have access to valid and reliable ways of assessing young children's numerical skills. The purpose of this study was to evaluate the concurrent, predictive, and incremental validity of a two-minute paper-and-pencil measure of children's symbolic (Arabic numerals) and non-symbolic (dot arrays) comparison skills. A sample of kindergarten children (M_age = 5.86, N = 439) were assessed on the measure along with a number line estimation task, a measure of arithmetic, and several control measures. Results indicated that performance on the symbolic comparison task explained unique variance in children's arithmetic performance in kindergarten. Longitudinal analyses demonstrated that both symbolic comparison and number line estimation in kindergarten were independent predictors of 1st grade mathematics achievement. However, only symbolic comparison remained a unique predictor once language skills and processing speed were taken into account. These results suggest that a two-minute paper-and-pencil measure of children's symbolic number comparison is a reliable predictor of children's early mathematics performance.     

Updating Working Memory and arithmetical attainment in school

Here we wished to determine how the sub-components of Working Memory (Phonological-Loop and Central Executive) influence children's arithmetical development. Specifically, we aimed at distinguishing between Working Memory inhibition and updating processes within the Central Executive, and the domain-specificity (words and numbers) of both subcomponents in a population of children with low attainment in arithmetic and their age matched typically-attaining controls. We show that both groups were similar for phonological loop abilities, while Working Memory updating demonstrated a domain-specific modulation related to the level of children's arithmetical performance. Moreover, inhibition processes interacted with domain-specificity and arithmetical attainment. These results are particularly relevant to the diagnostic assessment of arithmetical ability and should be considered in existing tests of arithmetical development.