Pizzas or no pizzas: An advantage of word problems in fraction arithmetic?

Fractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties by making fraction arithmetic less abstract. The current study therefore investigated to what extent students (106 sixth graders, 187 seventh graders, and 192 eighth graders) perform better on fraction arithmetic problems presented as word problems compared to these problems presented symbolically. Results showed that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts. However, in addition, subtraction, and multiplication of two fractions, symbolic problems were easier. There were no performance differences by students’ grade, but higher conceptual fraction knowledge was associated with higher fraction arithmetic performance. Taken together this study showed that situating fraction arithmetic in a realistic setting may support or hinder performance, dependent on the problem demands.     

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.     

Patterns in children's ability to recall explicit,implicit and metaphorical information

The purpose of this study was to examine the effects of grade level and recall task type on children's memory for explicit, implicit and metaphorical information following reading. Forty second and 40 fifth grade subjects balanced as to gender read four prose passages and were randomly assigned to cued and free recall conditions. A MANOVA produced significant main effects for grade level and task type on each of three dependent variables; EXPLICIT, IMPLICIT and METAPHOR recall measures. As predicted, fifth graders fared better than second graders and the cued condition exceeded free recall. Of greater significance was the finding that second graders were able to engage in metaphorical reasoning when the task was appropriately structured. The results suggest that providing externally generated, structured probes can greatly enhance children's reading recall at both grade levels and can enable children to demonstrate metaphorical comprehension prior to the age at which it spontaneously appears. In addition, significantly different patterns of recalled information were found within the two task conditions, patterns that remained stable across age groups.     

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children’s numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.     

Radical sensitivity is the key to understanding Chinese character acquisition in children

This study investigated Chinese children’s development of sensitivity to positional (orthographic), phonological, and semantic cues of radicals in encoding novel Chinese characters. A newly designed picture-novel character mapping task, along with nonverbal reasoning ability, vocabulary, and Chinese character recognition were administered to 198 kindergartners, 172 second graders and 165 fifth graders. Children’s strategies in using positional, phonological, and semantic cues of radicals varied across grades. The higher the children’s grade level, the more commonly children used semantic and positional cues of radicals. Regression analyses showed that the contribution of semantic radical awareness for explaining Chinese character reading increased as children’s grade increased, whereas the contribution of positional regularity awareness decreased. These findings suggest that learning Chinese characters involves a transition from a sound- and position-based approach to a meaning-based approach.     

Mathematics Achievement of Children in China and the United States

First and fifth graders in Beijing and Chicago were given a battery of mathematics test. Whether tested with problems requiring solely computation or with ones requiring application of knowledge about mathematics, American children's performance was consistently inferior to that of Chinese children. Interviews with American children suggested that they like mathematics, believe they are doing well in mathematics, and do not perceive mathematics as a difficult subject. American children's poor performance appears to be attributable, in part, to low motivation for devoting more attention to mathematics. Low standards held by American parents for academic achievement and lower interest in teaching mathematics by American teachers appear to contribute to American children's poor performance.     

Beliefs and Achievement: A Study of Black, White, and Hispanic Children

School achievement among black, white, and Hispanic elementary school children was investigated, and efforts were made to study the beliefs about academic achievement of the children and their mothers. A total of approximately 3,000 first, third, and fifth graders enrolled in 20 schools in the Chicago metropolitan area were given achievement tests in mathematics and reading. Black and Hispanic children performed at a significantly lower level than white children, but at fifth grade ethnic differences in mathematics scores were no longer significant when mothers' education was statistically controlled. This was not the case in reading, where differences were found after controlling for the effects of mothers' education. Interviews with subsamples of approximately 1,000 mothers and children revealed greater emphasis on and concern about education among minority families than among white families. Black and Hispanic children and mothers evaluated the children and their academic abilities highly; they were positive about education and held high expectations about the children's future prospects for education. Mothers of minority children and teachers in minority schools believed more strongly than white mothers and teachers in the value of homework, competency testing, and a longer school day as means of improving children's education.     

Analytic character learning in Chinese children: evidence from associative pseudocharacter learning paradigm

The present study investigated the children’s capabilities of utilizing analytic strategies in Chinese character learning using an associative pseudocharacter learning paradigm. The participants were 54 Chinese primary school children (26?second graders and 28 fifth graders) who completed a pseudocharacter learning task that was followed by a generalization test. Repeated measures ANOVA results showed a significant main effect of block and grade in pseudocharacter learning. One-sample t-test results indicated that generalization performance was significantly better than chance level in both grade levels. Findings from this study suggested that Chinese primary school children were able to identify print–sound–meaning regularities that were not explicitly taught and apply those rules to learning new pseudocharacters. They contribute to a comprehensive understanding of the character learning processes in Chinese children when encountering new characters.     

Attention,Scholastic Achievement and Timing of Lessons

Previous research has revealed that lower grade elementary school pupils who differ in their ability to learn to read also differ with respect to the optimal time of day for reading achievement. The present study strove to determine the hours of optimal attention in older pupils who varied in their apparent aptitude for mathematics. A sample of 204 fifth graders and 202 tenth graders were divided into three groups (high, average and low ability) on the basis of their mean yearly mathematics achievement scores. The subjects were asked to report on their attention levels throughout the day. The findings showed that fifth graders' attention levels were especially high in the afternoon, whereas tenth graders reported increased concentration during the morning hours. Among tenth graders, but not among fifth graders, there was a significant interaction between mathematics achievement and attention levels at various times of day. The practical implications of these findings for the planning of school schedules are discussed.     

Reasoning about structure and function: Children's conceptions of gears

Twenty-three second graders and 20 fifth graders were interviewed about how gears move on a gearboard and work in commonplace machines. Questions focused on transmission of motion; direction, plane, and speed of turning; and mechanical advantage. Several children believed that meshed gears turn in the same direction and at the same speed. Many second graders provided very incomplete explanations of transmission of motion. Most children confused mechanical advantage with speed. Yet as the interview proceeded, several fifth graders generalized conceptions about transmission of motion into a rule about turning direction. They increasingly justified their ideas about gear speed by referring to ratio. Children's reasoning became more general, formal, and mathematical as problem complexity increased, suggesting that mathematical forms of reasoning may develop when they provide a clear advantage over simple causal generalizations. © 1998 John Wiley & Sons, Inc. J Res Sci Teach 35: 3-25, 1998.     

Development of disjunctive reasoning and the use of MAYBE in fifth through eighth graders

Approximately one-half of the fifth through eighth graders in a school district (n = 164) were randomly selected to be administered a group test of disjunctive reasoning containing 48 inclusive and exclusive items varying in content of the premises (symbolic, object, and human), and affirmation or negation of the conclusion. Using an analysis of variance for repeated measures it was found that performance improved until seventh grade. Eighth graders scored similar to sixth graders. There was a main effect for negative, with negative conclusions producing more correct answers. Further, there were significant first order interactions for Disjunctive by Negation, Content by Grade, and Negation by Content. Since either “YES” or “NO” were the only correct answers, and “MAYBE” was always wrong, contrast of the MAYBE responses to other wrong answers revealed an increasing tendency to use MAYBE among older subjects. Implications were discussed in relation to cognitive developmental theory and educational practices.     

Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance

This study explored mathematics anxiety in a longitudinal sample of 113 children followed from second to third grade. We examined how mathematics anxiety related to different types of mathematical performance concurrently and longitudinally and whether the relations between mathematics anxiety and mathematical performance differed as a function of working memory. Concurrent analyses indicated that mathematics anxiety represents a unique source of individual differences in children’s calculation skills and mathematical applications, but not in children’s geometric reasoning. Furthermore, we found that higher levels of mathematics anxiety in second grade predicted lower gains in children’s mathematical applications between second and third grade, but only for children with higher levels of working memory. Overall, our results indicate that mathematics anxiety is an important construct to consider when examining sources of individual differences in young children’s mathematical performance. Furthermore, our findings suggest that mathematics anxiety may affect how some children use working memory resources to learn mathematical applications.     

Socio-cultural variation in reading comprehension development among fifth graders in Peru

In this study, the socio-cultural variation in reading comprehension development was examined in 331 fifth graders from schools in Lima, Peru. Reading comprehension was measured using an adaptation of the PIRLS Reading Literacy test. The fifth graders?? reading comprehension results, measured over the course of fifth grade, were related to the development of word decoding, vocabulary, and motivation for reading. Children??s development in these domains was related to their gender, intellectual maturity, home literacy climate, and socio-economic status. Structural Equation Modelling showed that the development of reading comprehension was influenced by the children??s ability to decode words, their vocabulary, and reading motivation. Furthermore, gender and intellectual maturity as well as children??s home literacy climate and socio-economic status appeared to substantially predict reading comprehension development, directly or indirectly. More than half of the variance in reading comprehension by the end of the fifth grade could be explained based on these predictor variables.     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

The role of teacher-related factors in mathematics skills between children attending Estonian-speaking and Russian-speaking schools

This study examined the extent to which teaching practices and teaching experience predict mathematics skills of children’s in the third through fifth grade by focusing on three cognitive components: (1) knowing facts and procedures, (2) applying this knowledge and (3) using reasoning. This study was longitudinal. The data was collected from 1810 students and 90 teachers attending Estonian- and Russian-speaking schools. The students’ were tested twice, at the beginning of the third and at the beginning of the fifth grade. The teachers were asked to complete a questionnaire assessing their teaching practices. The results revealed that teachers from Estonian-speaking and Russian-speaking schools’ classrooms differed in their use of teaching practices and teaching experience. Teachers in Russian-speaking schools were slightly more experienced, and they reported using more teacher-centred practices compared to teachers in Estonian-speaking schools. The study showed different effects on the outcomes of the children attending classrooms where the study language was Estonian and Russian in solving arithmetic word problems when accounting for the teachers’ use of different learner-centred teaching practices in the classroom.     

Study of fractions in elementary mathematics textbooks from Finland and Taiwan

This study examines the different ways in which fractions are dealt with in two elementary mathematics textbook series: Kang Hsuan (KH) in Taiwan and Laskutaito in Finland. The results showed that the total number of fraction questions in Laskutaito is higher than that in KH and that Laskutaito highlights the basic definition of fractions to help children develop higher levels of reasoning. Fraction questions in Laskutaito were presented with a number of pictorial representations to clarify the concepts. Laskutaito includes more symbolic representation of questions. Moreover, the questions posed in the Laskutaito reflect real-life situations, whereas the questions posed in KH do not connect with real life. The implications of this study are discussed and suggestions for future studies are made in this paper.     

Contexts of achievement: a study of American, Chinese, and Japanese children

The major purpose of this study was to attempt to understand some of the reasons for the high academic achievement of Chinese and Japanese children compared to American children. The study was conducted with first and fifth graders attending elementary schools in the Minneapolis metropolitan area, Taipei (Taiwan), and Sendai (Japan). 1,440 children (240 first graders and 240 fifth graders in each city) were selected as target subjects in the study. The children were selected from 20 classrooms at each grade in each city and constituted a representative sample of children from these classrooms. In a follow-up study, first graders were studied again when they were in the fifth grade. The children were tested with achievement tests in reading and mathematics constructed specifically for this study, the children and their mothers were interviewed, the children's teachers filled out a questionnaire, and interviews were held with the principals of the schools attended by the children. In the follow-up study, achievement tests were administered, and the children and their mothers were interviewed. Background information about the children's everyday lives revealed much greater attention to academic activities among Chinese and Japanese than among American children. Members of the three cultures differed significantly in terms of parents' interest in their child's academic achievement, involvement of the family in the child's education, standards and expectations of parents concerning their child's academic achievement, and parents' and children's beliefs about the relative influence of effort and ability on academic achievement. Whereas children's academic achievement did not appear to be a central concern of American mothers, Chinese and Japanese mothers viewed this as their child's most important pursuit. Once the child entered elementary school, Chinese and Japanese families mobilized themselves to assist the child and to provide an environment conducive to achievement. American mothers appeared to be less interested in their child's academic achievement than in the child's general cognitive development; they attempted to provide experiences that fostered cognitive growth rather than academic excellence. Chinese and Japanese mothers held higher standards for their children's achievement than American mothers and gave more realistic evaluations of their child's academic, cognitive, and personality characteristics. American mothers overestimated their child's abilities and expressed greater satisfaction with their child's accomplishments than the Chinese and Japanese mothers. In describing bases of children's academic achievement, Chinese and Japanese mothers stressed the importance of hard work to a greater degree than American mothers, and American mothers gave greater emphasis to innate ability than did Chinese and Japanese mothers.     

Examining the reciprocal relations of mathematics anxiety to quantitative reasoning and number knowledge in Chinese children

This study aimed to examine the longitudinal relations of mathematics anxiety to quantitative reasoning and number knowledge in Chinese children. Three hundred and sixteen 6-year-old Chinese children in Hong Kong participated in two waves of assessments, eight months apart. Cross-lagged panel analyses showed that prior quantitative reasoning and number knowledge predicted lower mathematics anxiety, even after the effects of gender, mothers’ educational levels, and general anxiety were taken into account. However, earlier mathematics anxiety did not predict later quantitative reasoning and number knowledge. Our findings were consistent with the Deficit Theory, which postulates that mathematics anxiety comes from poor mathematical competence but not vice versa. We also found a reciprocal association between quantitative reasoning and number knowledge, in which initial quantitative reasoning had a stronger prediction on later number knowledge. Taken together with previous research, this result highlights the importance of quantitative reasoning in children’s mathematics learning and its role in mathematics education.     

Roles of Abductive Reasoning and Prior Belief in Children’s Generation of Hypotheses about Pendulum Motion

The purpose of the present study was to test the hypothesis that student’s abductive reasoning skills play an important role in the generation of hypotheses on pendulum motion tasks. To test the hypothesis, a hypothesis-generating test on pendulum motion, and a prior-belief test about pendulum motion were developed and administered to a sample of 5th grade children. A significant number of subjects who have prior belief about the length to alter pendulum motion failed to apply their prior belief to generate a hypothesis on a swing task. These results suggest that students’ failure in hypothesis generation was related to abductive reasoning ability, rather than simple lack of prior belief. This study, then, supports the notion that abductive reasoning ability beyond prior belief plays an important role in the process of hypothesis generation. This study suggests that science education should provide teaching about abductive reasoning as well as scientific declarative knowledge for developing children’s hypothesis-generation skills.     

First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain

Our study investigated children’s knowledge of multiplicative reasoning (multiplication and division) at the end of Grade 1, just before the start of formal instruction on multiplicative reasoning in Grade 2. A large sample of children (N = 1176) was assessed in a relatively formal test setting, using an online test with 28 multiplicative problems of different types. On average, the children correctly answered more than half (58%) of the problems, including several bare number problems. This indicates that before formal instruction on multiplicative reasoning, children already have a considerable amount of knowledge in this domain, which teachers can build on when teaching them formal multiplication and division. Using analysis of variance and cross-classified multilevel regression analysis, we identified several predictors of children’s pre-instructional multiplicative knowledge. With respect to the characteristics of the multiplicative problems, we found that the problems were easiest to solve when they included a picture involving countable objects, and when the multiplicative situation was of the equal groups semantic structure (e.g., 3 boxes of 4 cookies). Regarding student characteristics, pre-instructional multiplicative knowledge was higher for children with higher-educated parents. Finally, the mathematics textbook used in school appeared to have influenced children’s pre-instructional multiplicative knowledge.