Children's number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting children's learning trajectories in elementary school mathematics. Keywords number competence; mathematics achievement; longitudinal studiesPoor achievement in mathematics is a major concern in U.S. schools. Advanced study in high school mathematics is a pillar for success in college science (Sadler & Tai, 2007) and serves as a gateway for a wide range of vocations in science, technology, engineering, and mathematics disciplines. Unfortunately, many U.S. students do not develop the foundations for learning advanced mathematics. Mathematics difficulties are cumulative and worsen with time. For example, difficulties with whole numbers are obstacles to learning fractions, and difficulties with fractions lead to failure in algebra (National Mathematics Advisory Panel, 2008). Mirroring U.S. national trends, recent data from the state of Delaware showed approximately 25% of students failed to meet state mathematics standards in third grade, with © 2009 American Psychological AssociationCorrespondence concerning this article should be addressed to Nancy C. Jordan, School of Education, University of Delaware, Newark, DE 19716. njordan@udel National statistics in the United States reveal persistent income disparities in mathematics achievement across grade levels, with low-income students scoring substantially lower than middle-income students (National Assessment of Educational Progress, 2007). Yet, lowincome youth who acquire strong mathematical skills by the end of middle school are much more likely to graduate from college than their peers who do not acquire such skills (National Mathematics Advisory Panel, 2008). Early Number CompetenceThere is growing consensus that many mathematics difficulties in elementary school can be traced to weaknesses in basic whole number competencies, that is, in understanding the meaning of numbers and number relationships (e.g.,...

Number sense development of 411 middle‐ and low‐income kindergartners (mean age 5.8 years) was examined over 4 time points while controlling for gender, age, and reading skill. Although low‐income children performed significantly worse than middle‐income 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 low‐income 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.

This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.

Mathematical competencies of 180 children were examined at 4 points between 2nd and 3rd grades (age range between 7 and 9 years). Children were initially classified into one of 4 groups: math difficulties but normal reading (MD only), math and reading difficulties (MD-RD), reading difficulties but normal math (RD only), and normal achievement in math and reading (NA). The groups did not differ significantly in rate of development. However, at the end of 3rd grade the MD only group performed better than the MD-RD group in problem solving but not in calculation. The NA and RD only groups performed better than the MD-RD group in most areas. Deficiencies in fact mastery and calculation fluency, in particular, are defining features of MD, with or without RD.

Number sense development was tracked from the beginning of kindergarten through the middle of first grade, over six time points. Children (n = 277) were then assessed on general math achievement at the end of first grade. Number sense performance in kindergarten, as well as number sense growth, accounted for 66 percent of the variance in first-grade math achievement. Background characteristics of income status, gender, age, and reading ability did not add explanatory variance over and above growth in number sense. Even at the beginning of kindergarten, number sense was highly correlated with end of first-grade math achievement (r = 0.70). Clarifying the observed slope effect, general growth mixture modeling showed that children who started kindergarten with low number sense but made moderate gains by the middle of kindergarten had higher first-grade math achievement than children who started out with similarly low number sense with flat growth. The majority of children in the low/flat growth class were from low-income families. The findings indicate that screening early number sense development is useful for identifying children who will face later math difficulties or disabilities.

The performance of 210 2nd graders in different areas of mathematical cognition was examined. Children were divided into 4 achievement groups: children with difficulties in mathematics but not in reading (MD-only), children with difficulties in both mathematics and reading (MD/RD), children with difficulties in reading but not in mathematics, and children with normal achievement. Although both MD groups performed worse than normally achieving groups in most areas of mathematical cognition, the MD-only group showed an advantage over the MD/RD group in exact calculation of arithmetic combinations and in problem solving. The 2 groups did not differ in approximate arithmetic and understanding of place value and written computation. Children with MD-only seem to be superior to children with MD/RD in areas that may be mediated by language but not in ones that rely on numerical magnitudes, visuospatial processing, and automaticity.

Children with poor arithmetic fact mastery (n = 45) at the end of third grade were compared to gradelevel peers with good arithmetic fact mastery (n = 60) in competencies related to reading and mathematics. Children were assessed longitudinally across second and third grades. When predictor variables such as IQ were held constant, the poor fact mastery and good fact mastery groups performed at about the same level and progressed at a comparable rate on math story problems and on broad reading achievement. The groups also progressed at a comparable rate on broad math achievement, although children with poor fact mastery performed at a significantly lower level. Children with poor fact mastery showed remarkably little growth on timed number facts during the study period, despite normal growth in other areas of mathematics. Deficits in fact mastery are highly persistent and appear to be independent of reading and language abilities. Keywords Mathematics; Learning disabilities; Cognitive developmentYoung children show uneven patterns of competencies in mathematics. In particular, some children cannot master basic arithmetic facts despite relatively strong problem-solving skills (Hanich, Jordan, Kaplan, & Dick, 2001;Jordan & Hanich, 2000;Jordan & Montani, 1997). Deficits related to mastery, or fast retrieval, of arithmetic facts is a key characteristic of children with mathematics difficulties (MD) who are good readers (Geary, Hamson, & Hoard, 2000Hanich et al., 2001; Jordan, Hanich, & Kaplan, in press). In contrast, children with both MD and reading difficulties (RD) are characterized by weaknesses in problem solving as well as in arithmetic fact mastery.Speed of processing numerical information appears to be a particular problem for children with MD only. Jordan and Montani (1997) presented children with MD only and children with MD/ RD with number facts and math story problems, both in timed and in untimed conditions. All the problems involved addition or subtraction operations. On the timed condition, children were required to answer orally presented problems within 3 s. Relative to children with normal mathematics achievement, children with MD only were strong on untimed conditions but not on timed conditions. When given enough time, MD only children successfully represented the problems and employed verbal counting methods. Children with MD/RD performed worse than normally achieving children on untimed as well as on timed tasks. They inaccurately represented problems and made numerous counting mistakes, even in the absence of time constraints.*Corresponding author. Fax: 1-302-831-0241. njordan@udel.edu (N.C. Jordan). NIH Public Access NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptIt has been suggested that number fact deficits are associated with general weaknesses in processing speed (Bull & Johnston, 1997) and in linguistic processes related to representing phonological information and retrieving information from long-term semantic memory (Ashcraft, 1992;Geary, 1993). If nu...

No abstract