The Experts below are selected from a list of 105945 Experts worldwide ranked by ideXlab platform
K Zeger - One of the best experts on this subject based on the ideXlab platform.
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Network routing capacity
IEEE Transactions on Information Theory, 2006Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every Rational Number in (0, 1] is the routing capacity of some solvable network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
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network routing capacity
International Symposium on Information Theory, 2005Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every non-negative Rational Number is the routing capacity of some network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
Erno Lehtinen - One of the best experts on this subject based on the ideXlab platform.
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spontaneous focusing on quantitative relations as a predictor of Rational Number and algebra knowledge
Contemporary Educational Psychology, 2017Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Erno LehtinenAbstract:Abstract Spontaneous Focusing On quantitative Relations (SFOR) has been found to predict the development of Rational Number conceptual knowledge in primary students. Additionally, Rational Number knowledge has been shown to be related to later algebra knowledge. However, it is not yet clear: (a) the relative consistency of SFOR across multiple measurement points, (b) how SFOR tendency and Rational Number knowledge are inter-related across multiple time points, and (c) if SFOR tendency also predicts algebra knowledge. A sample of 140 third to fifth graders were followed over a four-year period and completed measures of SFOR tendency, Rational Number conceptual knowledge, and algebra knowledge. Results revealed that the SFOR was relatively consistent over a one-year period, suggesting that SFOR is not entirely context-dependent, but a more generalizable tendency. SFOR tendency was in a reciprocal relation with Rational Number conceptual knowledge, each being uniquely predictive of the other over a four-year period. Finally, SFOR tendency predicted algebra knowledge three-years later, even after taking into account non-verbal intelligence and Rational Number knowledge. The results of the present study provide further evidence that individual differences in SFOR tendency may have an important role in the development of mathematical knowledge, including Rational Numbers and algebra.
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spontaneous focusing on quantitative relations as a predictor of the development of Rational Number conceptual knowledge
Journal of Educational Psychology, 2016Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Eero Laakkonen, Erno LehtinenAbstract:Many people have serious difficulties in understanding Rational Numbers, limiting their ability to interpret and make use of them in modern daily life. This also leads to later difficulties in learning more advanced mathematical content. In this study, novel tasks are used to measure 263 late primary school students’ spontaneous focusing on quantitative relations, in situations that are not explicitly mathematical. Even after controlling for a Number of known predictors of Rational Number knowledge, spontaneous focusing on quantitative relations is found to have a strong impact on students’ learning of Rational Number conceptual knowledge. This finding opens new possibilities for developing pedagogical solutions to one of the most difficult challenges of mathematics education. The findings suggest that students’ own focusing tendency and self-initiated practice may have on important role in the long-term development of complex cognitive skills. (PsycINFO Database Record (c) 2016 APA, all rights reserved)
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the relation between learners spontaneous focusing on quantitative relations and their Rational Number knowledge
Studia Psychologica, 2016Co-Authors: Jo Van Hoof, Tine Degrande, Jake Mcmullen, Minna M Hannulasormunen, Erno Lehtinen, Lieven Verschaffel, Wim Van DoorenAbstract:IntroductionRational Number KnowledgeThere is a broad agreement in the literature that a good understanding of Rational Numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction Number line estimation, 0-5 fraction Number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded that fifth graders' Rational Number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, whole Number knowledge, family income, and family education.Despite the critical importance of a good Rational Number knowledge, a large body of literature reported that children and even adults have a lot of difficulties dealing with various aspects of Rational Numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for Rational Numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with Rational Number tasks are often - at least in part - attributed to the "natural Number bias" (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of "whole Number bias"), which is the tendency to inappropriately use natural Number properties in Rational Numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to Rational Numbers in the classroom, they have already formed an idea of what a Number is. This idea is based on their experiences (both in daily life and in school) with natural Numbers. Once the learners are then instructed about Rational Numbers, the properties of natural Numbers are not always applicable anymore, leading to problems and misconceptions with Rational Numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in Rational Number tasks where reasoning purely in terms of natural Numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in Rational Number tasks where reasoning in terms of natural Numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural Number bias reports three main aspects that elicit such systematic errors. The first aspect relates to the density of the set of Rational Numbers. While natural Numbers are characterized by a discrete structure (one can always indicate which Number follows a given Number; for example after 13 comes 14), Rational Numbers are characterized by a dense structure (you cannot say which Number comes next, because between any two given Rational Numbers are always infinitely many other Rational Numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of Rational Numbers. Research indicates that errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural Numbers, "longer decimals are larger, shorter decimals are smaller", or "that a fraction's numerical value always increases when its denominator, numerator, or both increase" (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …
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modeling the developmental trajectories of Rational Number concept s
Learning and Instruction, 2015Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Eero Laakkonen, Erno LehtinenAbstract:Abstract The present study focuses on the development of two sub-concepts necessary for a complete mathematical understanding of Rational Numbers, a) representations of the magnitudes of Rational Numbers and b) the density of Rational Numbers. While difficulties with Rational Number concepts have been seen in students' of all ages, including educated adults, little is known about the developmental trajectories of the separate sub-concepts. We measured 10- to 12-year-old students' conceptual knowledge of Rational Numbers at three time points over a one-year period and estimated models of their conceptual knowledge using latent variable mixture models. Knowledge of magnitude representations is necessary, but not sufficient, for knowledge of density concepts. A Latent Transition Analysis indicated that few students displayed sustained understanding of Rational Numbers, particularly concepts of density. Results confirm difficulties with Rational Number conceptual change and suggest that latent variable mixture models can be useful in documenting these processes.
J Cannons - One of the best experts on this subject based on the ideXlab platform.
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Network routing capacity
IEEE Transactions on Information Theory, 2006Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every Rational Number in (0, 1] is the routing capacity of some solvable network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
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network routing capacity
International Symposium on Information Theory, 2005Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every non-negative Rational Number is the routing capacity of some network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
Jake Mcmullen - One of the best experts on this subject based on the ideXlab platform.
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spontaneous focusing on quantitative relations as a predictor of Rational Number and algebra knowledge
Contemporary Educational Psychology, 2017Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Erno LehtinenAbstract:Abstract Spontaneous Focusing On quantitative Relations (SFOR) has been found to predict the development of Rational Number conceptual knowledge in primary students. Additionally, Rational Number knowledge has been shown to be related to later algebra knowledge. However, it is not yet clear: (a) the relative consistency of SFOR across multiple measurement points, (b) how SFOR tendency and Rational Number knowledge are inter-related across multiple time points, and (c) if SFOR tendency also predicts algebra knowledge. A sample of 140 third to fifth graders were followed over a four-year period and completed measures of SFOR tendency, Rational Number conceptual knowledge, and algebra knowledge. Results revealed that the SFOR was relatively consistent over a one-year period, suggesting that SFOR is not entirely context-dependent, but a more generalizable tendency. SFOR tendency was in a reciprocal relation with Rational Number conceptual knowledge, each being uniquely predictive of the other over a four-year period. Finally, SFOR tendency predicted algebra knowledge three-years later, even after taking into account non-verbal intelligence and Rational Number knowledge. The results of the present study provide further evidence that individual differences in SFOR tendency may have an important role in the development of mathematical knowledge, including Rational Numbers and algebra.
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spontaneous focusing on quantitative relations as a predictor of the development of Rational Number conceptual knowledge
Journal of Educational Psychology, 2016Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Eero Laakkonen, Erno LehtinenAbstract:Many people have serious difficulties in understanding Rational Numbers, limiting their ability to interpret and make use of them in modern daily life. This also leads to later difficulties in learning more advanced mathematical content. In this study, novel tasks are used to measure 263 late primary school students’ spontaneous focusing on quantitative relations, in situations that are not explicitly mathematical. Even after controlling for a Number of known predictors of Rational Number knowledge, spontaneous focusing on quantitative relations is found to have a strong impact on students’ learning of Rational Number conceptual knowledge. This finding opens new possibilities for developing pedagogical solutions to one of the most difficult challenges of mathematics education. The findings suggest that students’ own focusing tendency and self-initiated practice may have on important role in the long-term development of complex cognitive skills. (PsycINFO Database Record (c) 2016 APA, all rights reserved)
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the relation between learners spontaneous focusing on quantitative relations and their Rational Number knowledge
Studia Psychologica, 2016Co-Authors: Jo Van Hoof, Tine Degrande, Jake Mcmullen, Minna M Hannulasormunen, Erno Lehtinen, Lieven Verschaffel, Wim Van DoorenAbstract:IntroductionRational Number KnowledgeThere is a broad agreement in the literature that a good understanding of Rational Numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction Number line estimation, 0-5 fraction Number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded that fifth graders' Rational Number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, whole Number knowledge, family income, and family education.Despite the critical importance of a good Rational Number knowledge, a large body of literature reported that children and even adults have a lot of difficulties dealing with various aspects of Rational Numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for Rational Numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with Rational Number tasks are often - at least in part - attributed to the "natural Number bias" (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of "whole Number bias"), which is the tendency to inappropriately use natural Number properties in Rational Numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to Rational Numbers in the classroom, they have already formed an idea of what a Number is. This idea is based on their experiences (both in daily life and in school) with natural Numbers. Once the learners are then instructed about Rational Numbers, the properties of natural Numbers are not always applicable anymore, leading to problems and misconceptions with Rational Numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in Rational Number tasks where reasoning purely in terms of natural Numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in Rational Number tasks where reasoning in terms of natural Numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural Number bias reports three main aspects that elicit such systematic errors. The first aspect relates to the density of the set of Rational Numbers. While natural Numbers are characterized by a discrete structure (one can always indicate which Number follows a given Number; for example after 13 comes 14), Rational Numbers are characterized by a dense structure (you cannot say which Number comes next, because between any two given Rational Numbers are always infinitely many other Rational Numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of Rational Numbers. Research indicates that errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural Numbers, "longer decimals are larger, shorter decimals are smaller", or "that a fraction's numerical value always increases when its denominator, numerator, or both increase" (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …
-
modeling the developmental trajectories of Rational Number concept s
Learning and Instruction, 2015Co-Authors: Jake Mcmullen, Minna M Hannulasormunen, Eero Laakkonen, Erno LehtinenAbstract:Abstract The present study focuses on the development of two sub-concepts necessary for a complete mathematical understanding of Rational Numbers, a) representations of the magnitudes of Rational Numbers and b) the density of Rational Numbers. While difficulties with Rational Number concepts have been seen in students' of all ages, including educated adults, little is known about the developmental trajectories of the separate sub-concepts. We measured 10- to 12-year-old students' conceptual knowledge of Rational Numbers at three time points over a one-year period and estimated models of their conceptual knowledge using latent variable mixture models. Knowledge of magnitude representations is necessary, but not sufficient, for knowledge of density concepts. A Latent Transition Analysis indicated that few students displayed sustained understanding of Rational Numbers, particularly concepts of density. Results confirm difficulties with Rational Number conceptual change and suggest that latent variable mixture models can be useful in documenting these processes.
Chris Freiling - One of the best experts on this subject based on the ideXlab platform.
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Network routing capacity
IEEE Transactions on Information Theory, 2006Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every Rational Number in (0, 1] is the routing capacity of some solvable network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
-
network routing capacity
International Symposium on Information Theory, 2005Co-Authors: J Cannons, Randall Dougherty, Chris Freiling, K ZegerAbstract:We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and Rational, we present an algorithm for its computation, and we prove that every non-negative Rational Number is the routing capacity of some network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used
