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Number Sense Growth in Kindergarten: A Longitudinal Investigation of Children at Risk for Mathematics Difficulties
 Child Development
, 2006
"... Number sense development of 411 middle and lowincome kindergartners (mean age 5.8 years) was examined over 4 time points while controlling for gender, age, and reading skill. Although lowincome children performed significantly worse than middleincome children at the end of kindergarten on all ta ..."
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Number sense development of 411 middle and lowincome kindergartners (mean age 5.8 years) was examined over 4 time points while controlling for gender, age, and reading skill. Although lowincome children performed significantly worse than middleincome children at the end of kindergarten on all tasks, both groups progressed at about the same rate. An exception was story problems, on which the lowincome group achieved at a slower rate; both income groups made comparable progress when the same problems were presented nonverbally with visual referents. Holding other predictors constant, there were small but reliable gender effects favoring boys on overall number sense performance as well as on nonverbal calculation. Using growth mixture modeling, 3 classes of growth trajectories in number sense emerged. Mathematics difficulties are widespread in the United States as well as in other industrialized nations. The consequences of such difficulties are serious and can be felt into adulthood (Dougherty, 2003; Murnane, Willett, & Levy, 1995). Low math achievement is especially pronounced in students from lowincome households (National Assessment of Educational Progress, 2004). Children with weaknesses in basic arithmetic may not develop the conceptual structures required to support the learning of advanced mathematics. Although competence in highlevel math serves as a gateway to a myriad careers in science and technology (Geary, 1994), many students never reach this stage. Some children gradually learn to avoid all things involving math and even develop math anxieties or phobias (Ash
SCADS: A model of children’s strategy choices and strategy discoveries
 Psychological Science
, 1998
"... Abstract—Preschoolers show surprising competence in choosing adaptively among alternative strategies and in discovering new approaches. The SCADS computer simulation illustrates how simple processes can generate this impressive competence. The model’s behavior parallels data on children’s addition i ..."
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Abstract—Preschoolers show surprising competence in choosing adaptively among alternative strategies and in discovering new approaches. The SCADS computer simulation illustrates how simple processes can generate this impressive competence. The model’s behavior parallels data on children’s addition in at least eight ways: It uses diverse strategies over prolonged periods of time, makes adaptive choices among strategies, discovers the same strategies as children, discovers strategies in the same sequence as children, makes discoveries without trial and error, makes discoveries without having experienced failure, narrowly generalizes new approaches, and generalizes more broadly following challenging problems. SCADS thus indicates plausible sources of young children’s surprising competence at strategy choice and strategy discovery. In the past 20 years, developmental psychologists have uncovered many surprising competencies in infants and young children. However,
Linguistic cues in the acquisition of number words
, 1997
"... Previous research has shown that children go through a stage in which they know that the number words each refer to a distinct numerosity, yet do not know which numerosity each number word picks out (Wynn, 1992). How do children attain this level of knowledge? We explore the possibility that particu ..."
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Previous research has shown that children go through a stage in which they know that the number words each refer to a distinct numerosity, yet do not know which numerosity each number word picks out (Wynn, 1992). How do children attain this level of knowledge? We explore the possibility that particular properties of how number words are used within sentences inform children of the semantic class to which they belong. An analysis of transcripts of the spontaneous speech of three one and twoyearold children and their parents (from the CHILDES database; MacWhinney & Snow, 1990) suggests that the relevant cues are available as input in parents ’ speech to children, and that children generally honour these properties of number words in their own speech. Implications of this proposal for word learning more generally are discussed.
Promoting broad and stable improvements in lowincome children’s numerical knowledge through playing number board games
 Child Development
, 2008
"... Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children’s numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased lowincome preschoolers ’ (mean age 5 5.4 years) pr ..."
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Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children’s numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased lowincome preschoolers ’ (mean age 5 5.4 years) proficiency on 4 diverse numerical tasks: numerical magnitude comparison, number line estimation, counting, and numeral identification. The gains remained 9 weeks later. Classmates who played an identical game, except for the squares varying in color rather than number, did not improve on any measure. Also as predicted, home experience playing number board games correlated positively with numerical knowledge. Thus, playing number board games with children from lowincome backgrounds may increase their numerical knowledge at the outset of school. Children vary greatly in the mathematical knowledge they possess when they enter school. These differences in initial mathematical knowledge appear to have large, longterm consequences. Proficiency in mathematics at the beginning of kindergarten is strongly predictive of mathematics achievement test scores years later: in elementary school, in middle school, and even in high school (Duncan et al., 2007; Stevenson & Newman, 1986). This pattern is consistent with the general finding that initial knowledge is positively related to learning (Bransford, Brown, & Cocking, 1999), but the relations in math are unusually strong and persistent. For example, they were considerably stronger than the relations between initial and subsequent reading proficiency in the same six longitudinal studies reviewed by Duncan et al. (2007; average standardized beta coefficients of.34 vs..16). Given the strong and persistent relation between early and later mathematical proficiency, it is especially unfortunate that preschoolers and kindergartners from lowincome families enter school with far less numerical knowledge than peers from more affluent families. Being clear on the locus of this gap is crucial for understanding it. On nonverbal numerical
Preschool children’s mathematical knowledge: The effect of teacher “math talk
 Developmental Psychology
, 2006
"... This study examined the relation between the amount of mathematical input in the speech of preschool or daycare teachers and the growth of children’s conventional mathematical knowledge over the school year. Three main findings emerged. First, there were marked individual differences in children’s ..."
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This study examined the relation between the amount of mathematical input in the speech of preschool or daycare teachers and the growth of children’s conventional mathematical knowledge over the school year. Three main findings emerged. First, there were marked individual differences in children’s conventional mathematical knowledge by 4 years of age that were associated with socioeconomic status. Second, there were dramatic differences in the amount of mathrelated talk teachers provided. Third, and most important, the amount of teachers ’ mathrelated talk was significantly related to the growth of preschoolers ’ conventional mathematical knowledge over the school year but was unrelated to their math knowledge at the start of the school year.
Playing linear number board games—but not circular ones —improves lowincome preschoolers’ numerical understanding
 Journal of Educational Psychology
, 2009
"... 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 th ..."
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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 lowincome 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.
An evolutionary perspective on learning disability in mathematics
 Developmental Neuropsychology
"... A distinction between potentially evolved, or biologicallyprimary forms of cognition, and the culturallyspecific, or biologicallysecondary forms of cognition that are built from primary systems is used to explore mathematical learning disability (MLD). Using this model, MLD could result from defi ..."
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A distinction between potentially evolved, or biologicallyprimary forms of cognition, and the culturallyspecific, or biologicallysecondary forms of cognition that are built from primary systems is used to explore mathematical learning disability (MLD). Using this model, MLD could result from deficits in the brain and cognitive systems that support biologicallyprimary mathematical competencies, or from the brain and cognitive systems that support the modification of primary systems for the creation of secondary knowledge and secondary cognitive competencies. The former include visuospatial longterm and working memory and the intraparietal sulcus, whereas the latter include the central executive component of working memory and the anterior cingulate cortex and lateral prefrontal cortex. Different forms of MLD are discussed as related to each of the cognitive and brain systems. When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarilynovel advances influence one’s ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturallyimportant knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for
Children’s Developing Mathematics in Collective Practices: A Framework for Analysis
 Journal of the Learning Sciences
, 2002
"... This article presents a cultural–developmental framework for the analysis of children’s mathematics in collective practices and illustrates the heuristic value of the framework through the analysis of videotaped episodes drawn from a middleschool classroom. The framework is presented in 2 related ..."
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This article presents a cultural–developmental framework for the analysis of children’s mathematics in collective practices and illustrates the heuristic value of the framework through the analysis of videotaped episodes drawn from a middleschool classroom. The framework is presented in 2 related parts. The first targets the children’s emerging mathematical goals in collective practices, with a particular focus on the complex role that artifacts play in children’s emerging goals. The second part focuses on children’s developing mathematics that takes form in their goaldirected activities: (a) Microgenetic analyses concern the process whereby children structure cultural forms like artifacts to serve particular functions as they accomplish emerging mathematical goals; (b) sociogenetic analyses concern the spread or travel of mathematical forms and associated functions within a community of individuals; and (c) ontogenetic analyses concern the interplay between the forms that children use and the functions that they serve over the course of children’s development. The analyses of the classroom episodes points to the promise (and limitations) of the framework as a method for furthering our understanding of the interplay between social and developmental processes in children’s mathematics. A growing body of classroombased research in mathematics education is concerned with understanding the role of artifacts in processes of teaching and learning. In her classroombased research, Ball (1993), for example, pointed to the way in which the use of one kind of artifact—the area of geometrical shapes—provides a representational context to explore properties of fractions. Lampert (1986) pointed to the utility of currency as support for teaching and learning about
How parent explanation changes what children learn from everyday scientific thinking
, 2007
"... Two studies examined how parent explanation changes what children learn from everyday shared scientific thinking. In Study 1, children between ages 3 and 8yearsold explored a novel task solo or with parents. Analyses of children's performance on a subsequent posttest compared three groups: c ..."
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Two studies examined how parent explanation changes what children learn from everyday shared scientific thinking. In Study 1, children between ages 3 and 8yearsold explored a novel task solo or with parents. Analyses of children's performance on a subsequent posttest compared three groups: children exploring with parents who spontaneously explained to them; children exploring with parents who did not explain; and children exploring solo. Children whose parents had explained were most likely to have a conceptual as opposed to procedural understanding of the task. Study 2 examined the causal effect of parent explanations on children's understanding by randomly assigning children to conditions in which they were or were not provided explanation while exploring a novel task with an adult. Children who heard explanations were more likely to switch from procedural to conceptual understanding. Results are discussed with respect to the role of everyday explanation in the development of children's scientific thinking.
Early Quantitative Abilities 780 Arithmetic in School 789 Mechanisms of Change 799
"... The nature of children’s numerical, arithmetical, and mathematical understanding and the mechanisms that underlie the development of this knowledge are at the center of an array of scientific, political, and educational debates. Scientific issues range from infants’ understanding of quantity and ari ..."
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The nature of children’s numerical, arithmetical, and mathematical understanding and the mechanisms that underlie the development of this knowledge are at the center of an array of scientific, political, and educational debates. Scientific issues range from infants’ understanding of quantity and arithmetic (Cohen & Marks, 2002; Starkey, 1992; Wynn, 1992a) to the processes that enable middleschool and highschool students to solve multistep arithmetical and algebraic word problems (Tronsky & Royer, 2002). The proposed mechanisms that underlie quantitative and mathematical knowledge range from inherent systems that have been designed through evolution to represent and process quantitative information (Geary, 1995; Spelke, 2000; Wynn, 1995) to general learning mechanisms that can operate on and generate arithmetical and mathematical knowledge but are not inherently quantitative (Newcombe, 2002). The wide range of competencies covered under the umbrella of children’s mathematical Preparation of this chapter was supported, in part, by grants