Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.     

The relationship between phonological processing and arithmetic in children with learning disabilities

Phonological processing skills have not only been shown to be important for reading skills, but also for arithmetic skills. Specifically, previous research in typically developing children has suggested that phonological processing skills may be more closely related to arithmetic problems that are solved through fact retrieval (e.g., remembering the solution from memory) than procedural computation (e.g., counting). However, the relationship between phonological processing and arithmetic in children with learning disabilities (LDs) has not been investigated. Yet, understanding these relationships in children with LDs is especially important because it can help elucidate the cognitive underpinnings of math difficulties, explain why reading and math disabilities frequently co-occur, and provide information on which cognitive skills to target for interventions. In 63 children with LDs, we examined the relationship between different phonological processing skills (phonemic awareness, phonological memory, and rapid serial naming) and arithmetic. We distinguished between arithmetic problems that tend to be solved with fact retrieval versus procedural computation to determine whether phonological processing skills are differentially related to these two arithmetic processes. We found that phonemic awareness, but not phonological memory or rapid serial naming, was related to arithmetic fact retrieval. We also found no association between any phonological processing skills and procedural computation. These results converge with prior research in typically developing children and suggest that phonemic awareness is also related to arithmetic fact retrieval in children with LD. These results raise the possibility that phonemic awareness training might improve both reading and arithmetic fact retrieval skills.

Research Highlights

• Relationships between phonological processing and various arithmetic skills were investigated in children with learning disabilities (LDs) for the first time.
• We found phonemic awareness was related to arithmetic involving fact retrieval, but not to arithmetic involving procedural computation in LDs.
• The results suggest that phonemic awareness is not only important to skilled reading, but also to some aspects of arithmetic.
• These results raise the question of whether intervention in phonemic awareness might improve arithmetic fact retrieval skills.

     

Associations of non‐symbolic and symbolic numerical magnitude processing with mathematical competence: a meta‐analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non‐symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta‐analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two‐level random‐effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non‐symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non‐symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school‐aged children and for adults.     

Early predictors of middle school fraction knowledge

Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.     

Better elementary number processing in higher skill arithmetic problem solvers: Evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Better elementary number processing in higher skill arithmetic problem solvers: evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Distinct Pathways From Parental Beliefs and Practices to Children’s Numeric Skills

It is imperative to identify contextual factors contributing to the development of early math skills, considering their role in later academic achievement. To pursue this goal, the present study investigated the paths connecting parental beliefs and practices during the preschool years to children’s numeric skills at the end of kindergarten (N = 98). Results were consistent with theoretical predictions of specific relations between particular types of parental input and different aspects of number knowledge. Direct math learning activities mediated the relation between parental beliefs and children’s number identification skills. Daily activities involving quantitative components mediated the relation between parental beliefs and children’s numerical magnitude understanding. Both types of activities predicted arithmetic skills that integrate the basic aspects of symbolic number knowledge. These findings contribute to developmental theory by specifying how characteristics of children’s environments are related to particular aspects of their development, which is critical for informing intervention work to improve early math skills.     

The Role of Executive Function in Arithmetic Problem-Solving Processes: A Study of Third Graders

This study investigated the roles of different executive function (EF) components (inhibition, shifting, and working memory) in 2-step arithmetic word problem solving. A sample of 139 children aged 8 years old and regularly attending the 3rd grade of primary school were tested on 6 EF tasks measuring different EF components, a reading task and a reading comprehension task, an arithmetic facts task evaluating basic knowledge of calculation, and three 2-step arithmetic word problems. Multiple hierarchical regression analyses were conducted to investigate the roles of the different EF components in the various phases of the problem-solving process. The results showed that EF affects the various phases of problem solving differently over and above calculation knowledge and reading abilities. The implications of these findings are discussed in relation to further understanding the role of cognitive skills in mathematical problem solving and in relation to instructional approaches that may increase children’s performance on 2-step arithmetic word problems.     

THE DEVELOPMENT OF KNOWLEDGE CONCERNING WRITTEN ARITHMETIC: A CROSS-CULTURAL STUDY

This study is concerned with the manner in which Western mathematics is assimilated by children raised in traditional African cultures. It was predicted that after an initial period of difficulty in learning school arithmetic, African children (Baoulé and Dioula from the Ivory Coast) acquire basic concepts, develop invented strategies, and make errors similar to those of American children. Further, it was predicted that Dioula children, since they are members of a commercial culture, perform at a somewhat higher level than the Baoulé, members of a farming culture which does not seem to stress computational activity. To investigate these hypotheses, American, Baoulé, and Dioula children at two age levels were given a variety of arithmetic problems involving basic skills (e.g., reading numbers), number knowledge (e.g., place value), and calculational abilities (e.g., written computation). The results showed that, in general, older African and American children (fifth- and sixth-graders) were quite similar in all respects. The only large and significant differences occurred in the case of younger children (second- and third-graders). At this level American children generally performed at a higher level than Africans, as predicted, and Dioula occasionally at a higher level than Baoulé, also as predicted. At the same time, qualitative aspects of African and American children's performance were similar: for example, both groups made the same types and proportions of errors in written calculations. The results generally support the hypotheses and can be explained in terms of both educational and cognitive factors.     

Atypical trajectories of number development: a neuroconstructivist perspective

Despite the fact that number deficits are as prevalent as literacy deficits, research on basic numerical skills lags seriously behind the successful studies identifying low-level deficits in dyslexia. We review current debates on number, discussing how the competing theories pertain to mathematical disabilities in normal children and numeracy deficits in genetic disorders. We stress the need to consider these issues within the framework of a developing system rather than from the neuropsychological perspective of focal damage. The earlier the exploration of atypical trajectories in very basic numerical skills, the better we will be able to chart their developmental impact on subsequent, higher-level arithmetic abilities.     

Literacy programs efficacy for developing children’s early reading skills in familiar language in Zambia

This study investigated the comparative efficacy of a phonics-based reading program and a language experience approach based literacy program to develop reading skills among Zambian early childhood school learners. The learners (n = 1 986; Grade 2 level; females = 50.1%) took either the phonics-based reading program (n = 1 593) or the alternative language experience approach based program (n = 393). They were all assessed for reading skills utilising the Early Grade Reading Assessment test (EGRA) in four languages (Cinyanja, Icibemba, Kiikaonde, and Silozi). Results suggest that learners in phonics-based literacy program were significantly better in letter-sound knowledge in all the four languages. Additionally, they were significantly better in reading skills (non-word reading, oral passage reading, and reading comprehension), yet only in Icibemba and Silozi, as compared to those who took the alternative program. Results reveal that children in the Primary Literacy Program (PLP) had significantly better performance in most reading skills than in the Primary Reading Program (PRP). However, the effect sizes were small or medium. The high floor effect in all reading-related measures is an indication that by following either PRP or the recently implemented PLP, most children do not acquire basic reading skill of the transparently written language they are familiar with. Instruction of the sounds of letters requires special attention where digital training tools (such as GraphoGame) may provide the most effective help to both teachers and children.     

Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.     

Examining the Effects of Skill Level and Reading Modality on Reading Comprehension

The purpose of this study was to examine the effects of reading skill and reading modality (oral versus silent) on reading comprehension. A normative sample of sixth-grade students (N = 74) read texts aloud and silently and then answered questions about what they read. Skill in word reading fluency was assessed by the Test of Word Reading Efficiency, Second Edition (Torgesen, Wagner, & Rashotte, 2012), and students were identified as either normal or at-risk readers based on those scores. A 2 (reading skill) X 2 (reading modality) mixed factorial ANOVA was conducted. Results revealed that both normal and at-risk readers demonstrated better comprehension of text read orally as compared to text read silently. The middle school curriculum requires independent silent reading, yet students may enter middle school without the literacy skills they need to be successful. These findings suggest that students transitioning to middle school may benefit from additional pedagogical support in silent reading comprehension.     

Typical development of quantity comparison in school-aged children

Although basic numerical skills have been widely studied in the last years, very few studies have undertaken a developmental approach. The present study evaluated the development of the magnitude comparison basic numerical ability, in children from first, third and sixth grades by means of the subject's response time in numerical tasks presented in symbolic and non-symbolic formats. The results showed a significant decrease on quantities processing speed as age increases, which suggests numerical skills tend to become automatic with instruction. The differences found, concerning the general achievement pattern in each school year, might express the maturational specificities of the numerical representation system through development.     

Curriculum-Based Measurement

Abstract

Curriculum-based measurement (CBM) has evolved as a reliable and valid method for measuring and monitoring student performance in basic academic skills. While the efficacy of CBM for assessing reading skills is not in question, issues remain regarding whether or not a difference exists between CBM probes derived directly from the instructional curriculum and generic probes. The current study extends previous research comparing the utility of two types of CBM reading probe materials. Both types of probes were administered to 13 second grade students twice weekly for 5 weeks. No significant differences were found between the two probe types' measurement of performance or progress over time, which suggests that school psychologists and educational professionals can use generic or curriculum-dependent probes in curriculum-based measurement.     

Neuropsychological significance of variations in patterns of academic performance: Verbal and visual-spatial abilities

Forty-five 9- to 14-year-old children with learning disabilities whose WISC Full Scale IQs fell within the range of 86–114 were divided into three groups on the basis of their patterns of reading, spelling, and arithmetic achievement. Group 1 was composed of children who were uniformly deficient in reading, spelling, and arithmetic; children in Group 2 were relatively adept at arithmetic as compared to their performance in reading and spelling; Group 3 was composed of children whose reading and spelling performances were average or above, but whose arithmetic performance was relatively deficient. The performances of these children on 16 dependent measures were compared. The performances of Groups 1 and 2 were superior to that of Group 3 on measures of visual-perceptual and visual-spatial abilities; Group 3 performed at a superior level to that of Groups 1 and 2 on measures of verbal and auditory-perceptual abilities. The results are discussed in terms of the relationships between varying patterns of academic abilities and patterns of brain-related behaviors, and the nature of the neuropsychological abilities that may limit performance on arithmetic calculation tasks.The research reported herein was supported, in part, under Grant #195 of the Ontario Mental Health Foundation. Funds were also supplied by Windsor Western Hospital Centre.     

Attention and reading skills

Attention plays a critical role in information processing. Its adequate functioning is required for correct development of complex cognitive abilities and regular scholastic progress. Children with attention deficits often have difficulties in reading, writing, and arithmetic. The present study investigated interactions among reading skills, overall scholastic performance as rated by teachers, and components of attention: visual reaction time, simple immediate span of attention, and selectivity. The sample was 98 students in the first and second years of public junior high school (age range 11-14 years, M = 12.6, SD = 1.2), i.e., with expected already well-established reading. Reading was evaluated using Comprehension, Accuracy, and Speed tests. Overall scholastic performance was obtained by means of teachers' ratings. Simple Reaction Time, Digit Span, and Color-Word Interference, included in a multitask computerized test, assessed attention. Analysis confirmed the hypothesis that the reading skills are strongly predictive of the Scholastic Assessment rated by the teachers. High scholastic ratings were correlated with Reading Speed and Accuracy rather than Reading Comprehension. Poor readers showed worse performances on the Digit Span test which measures simple immediate span of attention. Good and poor readers obtained a similar score on the Color-Word Interference task. This observation seems to contrast with the more common interpretation of this effect, suggesting that reading is an automatic process and, therefore, the semantic dimension overcomes the controlled perceptual one. According to other studies, an alternative explanation is suggested. In conclusion, present results confirm the hypothesis of a strong link among reading speed and accuracy, scholastic assessment as rated by teachers, simple immediate span of attention, and visual reaction time.     

阅读教学在大学英语中作用

英语阅读教学对大学生综合能力的培养有着至关重要的作用。由于英语阅读教学的现状,我们有必要对英语阅读教学进行研究并且提出阅读的策略,原则和阅读技巧,以提高学生的英语阅读能力。     

Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic.     

The effects of self-regulation training on the academic productivity of secondary students with learning problems

Four self-regulation skills were taught in a series of student-teacher conferences to eight learning disabled (LD) and three non-disabled (NLD) adolescents in an alternative high school. All participants had a prior history of school failure and low lesson-completion rates. The skills taught were behavior contracting, self-recording, self-monitoring, and self-reinforcement. Self-instructional materials in reading, writing, and math were used as the curriculum, and the dependent variables in the study were the number of lessons completed per student per day in the three academic areas. Multiple-baseline designs across students and across academic areas were employed to assess the effects of the self-regulation training and the students' application of the self-regulation skills on lesson completion. Results indicate that all participating LD and NLD students substantially increased their rate of lesson completion after the intervention. These increases occurred in the academic areas targeted by the students for the application of the self-regulation skills. Thus, self-regulation interventions hold promise for bringing at-risk students into contact with the school curriculum.