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Descriptors:
Number Concepts; Feedback (Response); Disadvantaged Youth; Class Activities; Games; Group Activities; Teaching Methods; Low Income Groups; Mathematics; Mathematics Education; Mathematics Instruction; Preschool Children; Intervention; Minority Group Students; Outcome Measures; Scores; Pretests Posttests; Hierarchical Linear Modeling; Young Children; Early Childhood Education; Federal Programs

Abstract:
We examined whether a theoretically based number board game could be translated into a practical classroom activity that improves Head Start children's numerical knowledge. Playing the number board game as a small group learning activity promoted low-income children's number line estimation, magnitude comparison, numeral identification, and counting. Improvements were also found when a paraprofessional from the children's classroom played the game with the children. Observations of the game-playing sessions revealed that paraprofessionals adapted the feedback they provided to individual children's improving numerical knowledge over the game-playing sessions and that children remained engaged in the board game play after multiple sessions. These findings suggest that the linear number board game can be used effectively in the classroom context. (Contains 3 tables and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Numbers; Computation; Models; Prediction; Goodness of Fit; Statistical Analysis

Abstract:
Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Mathematics Achievement; Numbers; Achievement Tests; Arithmetic; Mathematics; Scores; Children

Abstract:
This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores, that fraction magnitude representations account for substantial variance in mathematics achievement test scores beyond that explained by fraction arithmetic proficiency, and that developing effective strategies plays a key role in improved knowledge of fractions. Theoretical and instructional implications are discussed. (Contains 6 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Low Income; Preschool Children; Comparative Analysis; Numeracy; Games; Middle Class; Learning Processes; Arithmetic; Intervention; School Readiness; Achievement Gap

Abstract:
We compared the learning from playing a linear number board game of preschoolers from middle-income backgrounds to the learning of preschoolers from low-income backgrounds. Playing this game produced greater learning by both groups than engaging in other numerical activities for the same amount of time. The benefits were present on number line estimation, magnitude comparison, numeral identification, and arithmetic learning. Children with less initial knowledge generally learned more, and children from low-income backgrounds learned at least as much, and on several measures more, than preschoolers from middle-income backgrounds with comparable initial knowledge. The findings suggest a class of intervention that might be especially effective for reducing the gap between low-income and middle-income children's knowledge when they enter school. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Class Activities; Learning Activities; Teaching Methods; Numbers; Misconceptions; Educational Practices; Mathematical Applications; Mathematical Concepts; Mathematics Activities; Mathematics Education; Number Concepts; Number Systems; Computation; Arithmetic; Problem Based Learning; Thinking Skills; Knowledge Base for Teaching; International Education

Abstract:
Students around the world have difficulties in learning about fractions. In many countries, the average student never gains a conceptual knowledge of fractions. This research guide provides suggestions for teachers and administrators looking to improve fraction instruction in their classrooms or schools. The recommendations are based on a synthesis of published research evidence produced by the United States Department of Education's Institute for Education Sciences, "Developing effective fractions instruction: A practice guide". The recommendations include a variety of classroom activities and teaching strategies, but all are focused on improving students' conceptual understanding of fractions. The guide starts with ideas for introducing fraction concepts in kindergarten and early elementary school, and continues with activities and teaching strategies designed to help older students understand fraction magnitudes and computational procedures involving fractions. It then examines ways of helping students use fractions to solve rate, ratio and proportion problems. The final recommendation suggests methods to increase teacher's conceptual knowledge of fractions. Teachers with a firm conceptual knowledge of fractions, along with knowledge of students' common errors and misconceptions, are essential for improving students' learning about fractions. Throughout the guide, the authors use the term "fraction" to encompass all of the ways of expressing rational numbers, including decimals, percentages and negative fractions. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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College Students; Community Colleges; Logical Thinking; Student Behavior; Tests; Universities; Comparative Analysis; Educational Strategies; Mathematics; Mathematics Instruction

Abstract:
We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However, atypical characteristics of the presented fractions might have provoked the use of atypical comparison strategies in that study. In our 3 experiments, university and community college students compared more balanced sets of single-digit and multi-digit fractions and consistently exhibited a logarithmic distance effect. Thus, adults used integrated, analog representations, akin to a mental number line, to compare fraction magnitudes. We interpret differences between the past and present findings in terms of different stimuli eliciting different solution strategies. (Contains 3 tables and 3 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Mathematics; Problem Solving; Young Children; Elementary Education; Instructional Effectiveness; Educational Practices; Guides; Mathematical Concepts; Comprehension; Cognitive Processes

Abstract:
This practice guide presents five recommendations intended to help educators improve students' understanding of, and problem-solving success with, fractions. Recommendations progress from proposals for how to build rudimentary understanding of fractions in young children; to ideas for helping older children understand the meaning of fractions and computations that involve fractions; to proposals intended to help students apply their understanding of fractions to solve problems involving ratios, rates, and proportions. Improving students' learning about fractions will require teachers' mastery of the subject and their ability to help students master it; therefore, a recommendation regarding teacher education also is included. Appendices include: (1) Postscript from the Institute of Education Sciences; (2) About the Authors; (3) Disclosure of Potential Conflicts of Interest; (4) Rationale for Evidence Ratings; and (5) Evidence Heuristic. A glossary and index are also provided. (Contains 5 tables, 10 figures, and 250 endnotes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Numeracy; Number Concepts; Arithmetic; Games; Low Income Groups; Preschool Children; Disadvantaged Youth; Mathematics Skills; Teaching Methods; Minority Group Children; Early Intervention; Educational Psychology

Abstract:
A theoretical analysis of the development of numerical representations indicated that playing linear number board games should enhance preschoolers' numerical knowledge and ability to acquire new numerical knowledge. The effect on knowledge of numerical magnitudes was predicted to be larger when the game was played with a linear board than with a circular board because of a more direct mapping between the linear board and the desired mental representation. As predicted, playing the linear board game for roughly 1 hr increased low-income preschoolers' proficiency on the 2 tasks that directly measured understanding of numerical magnitudes--numerical magnitude comparison and number line estimation--more than playing the game with a circular board or engaging in other numerical activities. Also as predicted, children who had played the linear number board game generated more correct answers and better quality errors in response to subsequent training on arithmetic problems, a task hypothesized to be influenced by knowledge of numerical magnitudes. Thus, playing linear number board games not only increases preschoolers' numerical knowledge but also helps them learn from future numerical experiences. (Contains 4 figures and 1 table.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Numbers; Cognitive Structures; Developmental Psychology; Information Processing; Child Development; Intervention; Adults

Abstract:
The relation between short-term and long-term change (also known as learning and development) has been of great interest throughout the history of developmental psychology. Werner and Vygotsky believed that the two involved basically similar progressions of qualitatively distinct knowledge states; behaviorists such as Kendler and Kendler believed that the two involved similar patterns of continuous growth; Piaget believed that the two were basically dissimilar, with only development involving qualitative reorganization of existing knowledge and acquisition of new cognitive structures. This article examines the viability of these three accounts in accounting for the development of numerical representations. A review of this literature indicated that Werner's and Vygotsky's position (and that of modern dynamic systems and information processing theorists) provided the most accurate account of the data. In particular, both changes over periods of years and changes within a single experimental session indicated that children progress from logarithmic to linear representations of numerical magnitudes, at times showing abrupt changes across a large range of numbers. The pattern occurs with representations of whole number magnitudes at different ages for different numerical ranges; thus, children progress from logarithmic to linear representations of the 0-100 range between kindergarten and second grade, whereas they make the same transition in the 0-1,000 range between second and fourth grade. Similar changes are seen on tasks involving fractions; these changes yield the paradoxical finding that young children at times estimate fractional magnitudes more accurately than adults do. Several different educational interventions based on this analysis of changes in numerical representations have yielded promising results. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Feedback (Response); Disadvantaged Youth; Learning Strategies; Pretests Posttests; Child Care Centers; Arithmetic; Mathematics Instruction; Games; Teaching Methods; Mathematics Skills; Mathematical Concepts; Numbers; Recall (Psychology); Preschool Children; Computation; Comparative Analysis

Abstract:
The present study focused on two main goals. One was to test the "representational mapping hypothesis": The greater the transparency of the mapping between physical materials and desired internal representations, the greater the learning of the desired internal representation. The implication of the representational mapping hypothesis in the present context is that if the desired internal representation of numerical magnitudes is a linear number line, then playing the number game with a linear board should promote greater learning of numerical magnitudes than playing the identical game with a circular board. A second major goal of this study was to test the prediction that forming a linear representation of numerical magnitudes should improve young children's ability to learn answers to arithmetic problems. Linear representations of numerical magnitudes seem likely to help children learn arithmetic because such representations maintain equal subjective spacing throughout the entire range of numbers, thus facilitating discrimination among answers to different problems. Consistent with this perspective, linearity of number line estimates is positively correlated with arithmetic proficiency among first through fourth graders (Booth & Siegler, 2006; 2008). Thus, playing the linear number board game was expected to produce greater subsequent ability to recall the answers to arithmetic problems following instruction in them; it also was expected to produce errors on the arithmetic problems that were closer to the correct answer. The research was conducted at seven Head Start centers and two child care centers in Pittsburgh, Pennsylvania. Participants were 88 preschoolers (56% female), ranging in age from 4 years 0 months to 5 years 5 months (M = 4 years 8 months, SD = 0.47). The data were consistent with both hypotheses. On the number line estimation task, linearity, slope, and accuracy were all greater on the posttest for those who played the game with the linear board than for those who played it with the circular board. Mean percent variance accounted for by the best fitting linear function for each child increased significantly from 14% to 39% among children who played the linear board game; it increased non-significantly from 15% to 21% among those who played the circular game. Differences between groups were not significant on the pretest but were significant on the posttest. On the magnitude comparison task, percent absolute error improved from pretest to posttest among children who played the game with the linear board (29% to 21%) and also among those who played the game with the circular board (29% to 26%). The results did not differ on the pretest, but children who played the linear game were more accurate on the posttest. Even more striking was the learning to learn effect in arithmetic: Children who earlier had played the linear board game learned more from subsequent practice and feedback on addition problems than children who earlier had played the circular board game (45% versus 30% correct posttest responses). Their errors were also closer to the correct answer than were the errors of children in the circular board game condition. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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