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Learning Engineering Applies the Learning Sciences
Published in Jim Goodell, Janet Kolodner, Learning Engineering Toolkit, 2023
Jim Goodell, Janet Kolodner, Aaron Kessler
Teachers may also help Mia and her classmates expand their mental models by using a number line as shown in Figure 2.4 (above). First the positive side of a number line is used to help children understand whole numbers and fractional quantities. Later the negative side is used to introduce the concept of negative numbers. The number line introduces fractional quantities and then negative numbers in a way that makes sense alongside well-established concepts of positive whole number quantities from playing with and counting blocks. With the number line model, addition and subtraction of positive integers still works, but children can begin to grasp more advanced concepts like negative numbers and fractions.
Estimation in the primary mathematics curricula of the United Kingdom: Ambivalent expectations of an essential competence
Published in International Journal of Mathematical Education in Science and Technology, 2022
Paul Andrews, Constantinos Xenofontos, Judy Sayers
Importantly, from the perspective of this paper, number line estimation competence is a strong predictor of both later mathematical learning difficulties (Andersson & Östergren, 2012; Wong et al., 2017) and mathematical achievement across all ages of compulsory school (Fuchs et al., 2010; Schneider et al., 2018; Simms et al., 2016), to the extent that ‘number line estimation at age 16 was significantly related to mathematics … at each age, beyond variance explained by other cognitive abilities at that age’ (Tosto et al., 2017, p. 1934). In particular, whole number line estimation competence has been implicated in children’s arithmetical development across the years of primary education (Dietrich et al., 2016; Fuchs et al., 2010; Träff, 2013) in ways suggesting that number line estimation and arithmetical competence may be reciprocally related (Friso-van den Bos et al., 2015). Number line estimation accuracy is a predictor of fractions knowledge in general (Bailey et al., 2014; Hansen et al., 2015; Vukovic et al., 2014), particularly in the upper primary and lower secondary age ranges (Fazio et al., 2014; Torbeyns et al., 2015; Van Hoof et al., 2017). Finally, the ability to estimate accurately the position of fractions on the number line is a strong predictor of algebraic readiness (Booth & Newton, 2012) and equation solving competence (Booth et al., 2014). Further, decimal number line estimation competence is a better predictor of algebraic competence than either integer or fraction number line estimation (DeWolf et al., 2015).
Different intuitive rules in mathematical thinking
Published in International Journal of Mathematical Education in Science and Technology, 2022
Comparing intervals in one-dimension, which appear as line segments in the number line, has been intuitively done by comparing their lengths. Perhaps, it is difficult to imagine something else. This is despite the fact that, based on the Cantorian set theory and the one-to-one criterion, every two intervals, as the subsets of the real number, are equivalent. It is enough to define a bijection mapping between them. Although from a mathematical point of view it can be proved by this method that any two intervals are in one-to-one correspondence, but intuitively the problem may still remain. In the continuous states, which there is no large variety of representations, geometric representation could also be useful. Regarding two bounded intervals with different lengths, the following geometric representation (Figure 4) could create a more deeply understanding of one-to-one correspondence than bijection mapping (quoted from Fischbein, 2001, p. 311):
Using performance-based warranties to influence consumer purchase decisions
Published in The Engineering Economist, 2020
Clay Koschnick, Joseph C. Hartman
Let be the probability density function defining the operating costs of a product of age a (such that where N is the useful life of the product) at time t (such that where T is the time total time horizon under consideration). Then let be the lowest possible operating cost and be the largest possible operating cost. We assume that all costs are positive and finite so the range of must be a finite segment of the positive real number line. The expected operating costs for an asset of age a at time t are