The standards and principles of the National Council of Teachers of Mathematics (NCTM) (2000) emphasize the importance of understanding numerical magnitude as the pre-K2 students endeavor to make sense of numbers. “Number magnitude is an essential feature for algorithms . Use of the algorithms not only depends on the automatic recall of number facts but also reinforces it. Far too many middle and high school students lack the ability to accurately compare the magnitudes of such numbers." NCTM also acknowledge the estimation proficiency and precision provide solid foundation for future mathematics success not only for individual differences of computational skills (Gilmore, McCarthy, & Spelke, 2007; Laski & Siegler, 2007), but also closely related to individual differences in arithmetic, several types of estimation, magnitude comparison, numerical categorization, and overall mathematics achievement test scores (Booth & Siegler, 2008; Ramani & Siegler, 2008b). Strong number sense also includes the ability to estimate the order of magnitudes to support computational skills. Therefore, mathematics instruction has to foster the development of numerical representation that draw from robust understanding of numerical magnitudes. Furthermore, recent studies have shown that children’s understanding of numerical magnitudes is a strong predictor of later mathematics performance (Siegler & Booth, 2008).
NMAP (2008) has also addressed the importance of understanding numerical magnitudes. It reported that understanding number magnitude is essential to acquiring algorithms of addition and subtraction in middle and high schools. Students lacking this ability are unable to compare magnitudes of two or ...
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...quantity as precisely as ten. Early experiments suggest that preschoolers have to depend on a hypothesized number line to understand number magnitudes. That is, they could only comprehend numerical magnitudes by referring to their positions on the number line (Dehaene, 1993, 1998). For instance to understand the number 8 on the 0−10 number line, the preschoolers would position it on the left side of number 9, but on the right side of number 7. In terms of the left-to-right direction of the 0−10 number line, the preschoolers could understand that number 8 is larger than number 7, but smaller than number 9. For the young children who have not received formal education, the mental representations of a number line have been found to be an effective way to make sense of diverse and abstract concepts of numbers (De Smedt, Verschaffel, & Ghesquiere, 2009; Wynn 1992).
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