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Article Excerpt
Abstract. Concern about poor mathematics achievement in U.S. schools has increased in recent years. In part, poor achievement may be attributed to a lack of attention to early instruction and missed opportunities to build on young children's early understanding of mathematics. This study examined the development and feasibility testing of a kindergarten mathematics curriculum designed to focus on the development of early number sense, geometry, measurement, and mathematics vocabulary. A mixed-model analysis of covariance, using pretest score as a covariate, was used to determine the effect of the experimental curriculum on student achievement on a standardized measure of early mathematics. Achievement results as well as implementation fidelity and teacher satisfaction suggest that further empirical research on the efficacy of the curriculum is warranted.
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The advancement and perfection of mathematics are ultimately connected with the prosperity of the state.
--Napoleon Bonaparte
Mathematics achievement is clearly important to a productive society. This long-held belief, along with recent attention to accountability and high standards, has led to serious concern about the state of mathematics education in the United States. The relatively low levels of mathematics performance of American students compared to national standards as well as in students around the world (National Research Council [NRC], 2001) have stimulated calls for a significant overhaul of mathematics education (e.g., Klein et al., 2005). Persistent problems in mathematics achievement are particularly troubling, given that the achievement gap faced by students from low-income (SES) and minority backgrounds as well as students with disabilities is significant and represents a growing number of students in public school districts (National Assessment of Educational Progress [NAEP], 2005). On the eve of the release of the National Mathematics Panel report, concern about student achievement in mathematics and an increased recognition that mathematics skill will play a significant role in life opportunities and outcomes in our society highlight the need for "[A]ll young Americans ... to think mathematically, and ... think mathematically to learn" (NRC, 2001, p. 16).
The problem of flagging achievement in mathematics in the United States may be attributed to any number of key issues that have been thoroughly discussed in the literature, including poorly prepared teachers (Hill & Ball, 2004; Ma, 1999); low expectations and poorly conceived standards (Chard & Kame'enui, 1995; Romberg & Kaput, 1999); insufficient and traditional instruction (Battista, 1999; Geist, 2000; McKnight & Schmidt, 1989); and mathematics anxiety (Furner & Berman, 2003).
Another contributor to later mathematics difficulties may simply be a missed opportunity to develop young children's mathematical understanding early. Research in developmental psychology indicates that infants demonstrate early skills in subitizing, recognizing when the number of objects or sounds changes after being habituated to a first number (Wynn, 1990, 1992). Moreover, Xu and Spelke (2000) have demonstrated that infants can perceive quantity differences in large arrays. Instruction in mathematics such as that offered to students in pre-K and kindergarten classrooms should be designed to take advantage of these already emerging skills (Clements, 2004). However, there is little empirical evidence in the research literature on instructional programs designed to teach students early number sense and to develop it more formally into early arithmetic skills in the elementary grades.
One potential approach to improving math achievement is the delivery of effective core instruction to all students in the early primary grades to lay a sound foundation for mathematical understanding and prevent early difficulties in mathematics (Clarke, Baker, & Chard, 2007). Preventing academic difficulties through focused early instruction is garnering increased attention in both general education and special education circles (Fuson, Smith, & LoCicero, 1997; Griffin, 2004). Consistent findings illustrate that remediating academic problems once they have emerged becomes increasingly difficult the longer the problems remain unresolved and content expectations grow in later grades. Fortunately, recent efforts have moved to research focused on identifying critical variables that predict which students may be at risk for later academic difficulties (Chard et al., 2005; Clarke & Shinn, 2004) and on preventing these difficulties before they become persistent problems (Fuchs et al., 2005).
Traditional mathematics instruction in early primary classrooms frequently consists of activities guided in part by students' interests and may be described as episodic rather than systematic. Popular instructional programs provide teachers with daily activities that build on students' knowledge of their environment, but are often not linked to a strong, logical sequence of instruction. In addition, many of these programs have emphasized authentic childhood activities that may or may not result in students' mastery of key concepts and skills. While intended to make mathematics more meaningful for children, these authentic activities ignored the importance of skill development to conceptual understanding (Wu, 1999).
We set out to develop a mathematics instructional program to support early mathematics development for all students in kindergarten, the Early Learning in Mathematics (ELM) program. A detailed description of the program and its conceptual foundation follows.
Early Learning in Mathematics (ELM) Program
The ELM program was designed to specifically enhance students' number sense. Number sense is an emerging construct (Dehaene, 1997) that refers broadly to a child's fluidity and flexibility in using and manipulating numbers, an almost intuitive sense of what numbers mean, and an ability to perform mental mathematics and look at the world and make what, in essence, boils down to quantitative comparisons without difficulty (Berch, 1998; Gersten & Chard, 1999).
While children may be born with a predisposition for making quantitative distinctions, an inability to develop a refined understanding of number has been implicated as a key predictor of later mathematics difficulties. Often, those who teach mathematics to young children, as well as those who develop curricula for teaching numbers and basic arithmetic concepts to kindergartners, fail to fully take into account that children develop, or fail to develop, number sense.
Some children acquire this conceptual structure informally. That is, they acquire number sense over time without intense formal instruction. These children acquire number sense before they begin kindergarten, either in preschool or familial settings that provide multiple and ongoing opportunities for developing quantitative thinking and analysis built on their early understanding of numbers. Other children who have not had these opportunities in preschool or at home require formal explicit instruction to develop this understanding because waiting for them to learn "on their own" the way many middle-class children seem to do takes too long, and the chance that these at-risk students will fall dangerously behind their peers before they develop it is too risky (Bruer, 1997; Griffin, 1998).
Mathematics differences among children in kindergarten are dramatic, if perhaps difficult to notice, unless...
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