In their article, “From ‘Sense of Number’ to ‘Sense of Magnitude’: The Role of Continuous Magnitudes in Numerical Cognition,” Leibovich et al. challenge the prevailing view that non-symbolic number sense (e.g., sensing “sixness” in a group of six objects in the same way one might sense color [Dehaene Reference Dehaene1997]) is innate, that detection of numerosity is distinct from detection of continuous magnitude. Rather, the researchers argue that infants are hardwired to sense continuous magnitudes (e.g., objects' areas, contour lengths, spacing) and that this information supports perception of numerosity. Non-symbolic number sense “develops from understanding the correlation between numerosity and continuous magnitudes” (sect. 8, para. 6). For example, infants might learn through their everyday experiences that more objects typically take up more space. However, in cases where this rule is violated (e.g., a line of three long toy trucks that is longer than a line of five short trucks), general cognitive control or executive function may be needed to inhibit a response based on the expected correlation of continuous magnitude and numerosity. In the present commentary, we discuss Liebovich et al.'s viewpoint in light of the integrative theory of numerical development (Siegler & Lortie-Forgues Reference Siegler and Lortie-Forgues2014), which was not considered in the article, along with implications for understanding mathematics disabilities.
The integrative theory “proposes that the continuing growth of understanding magnitudes provides a unifying theme for numerical development” (Siegler & Lortie-Forgues Reference Siegler and Lortie-Forgues2014, p. 144). A mental number line, Siegler and Lortie-Forgues contend, coordinates knowledge of different forms of magnitude ranging from non-symbolic continuous quantities and numerosities to symbolic whole numbers, fractions, and decimals. Arguably, a mental number line could be grounded in core knowledge of continuous magnitudes, which allows children to perceive non-symbolic numerosities and eventually symbolic numbers. However, we believe that the argument that an innate sense of non-symbolic magnitude is more fundamental than non-symbolic number sense is less important than is the notion that an understanding of how discrete and continuous quantities are related is critical for constructing the beginnings of a mental number line. According to the integrative theory, the construction of a mental number line structures mathematics learning, and early experiences with both continuous magnitudes and discrete objects in the external world shape children's understandings of quantity right from the start.
Leibovich et al. present implications of their view for understanding mathematics disabilities, including dyscalculia, as well as mathematics disabilities that co-occur with other conditions (e.g., dyslexia or attention deficits). The possibility that children at risk for mathematics disabilities have trouble grasping the correlation between continuous magnitudes and numerosities is intriguing. Previous work has shown that core deficits in understanding numerical magnitudes in symbolic contexts underpin mathematics disabilities (e.g., Butterworth Reference Butterworth1999; Reference Butterworth2005; Butterworth & Reigosa-Crespo Reference Butterworth, Reigosa-Crespo, Berch and Mazzocco2007; Landerl et al. Reference Landerl, Bevan and Butterworth2004). For example, children with dyscalculia perform much more poorly than do their typically achieving peers when asked to identify which of two numerals is larger and to relate quantities to their symbols (Butterworth & Reigosa-Crespo Reference Butterworth, Reigosa-Crespo, Berch and Mazzocco2007; Landerl et al. Reference Landerl, Bevan and Butterworth2004; Rousselle & Noel Reference Rousselle and Noel2007). Moreover, ability to estimate the placement of numbers on a number line strongly predicts whether children will go on to struggle in mathematics (Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Resnick et al. Reference Resnick, Jordan, Hansen, Rajan, Rodrigues, Siegler and Fuchs2016). It is possible children's difficulties with symbolic representations of quantity may be at least partly rooted in the detection of non-symbolic continuous magnitudes. For example, Matthews et al. (Reference Matthews and Lewis2016) showed that mathematics competencies that depend on an understanding of fractions are predicted by non-symbolic processing of ratio information, which is inherently continuous.
Nonetheless, we generally argue against the “domain-general account of dyscalculia” discussed by Leibovich et al. Although general cognitive competencies partially explain why some children struggle with mathematics (Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Rousselle & Noel Reference Rousselle and Noel2007), basic weaknesses in understanding numerical magnitudes have been shown to be the more definitive characteristic of mathematics disabilities (Clarke & Shinn Reference Clarke and Shinn2004; Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Mazzocco & Thompson Reference Mazzocco and Thompson2005). We recognize, however, that children's core difficulties with numerical understanding are exacerbated by weaknesses in executive functioning related to inhibition and set shifting, which constrain children's later numerical development (e.g., Hassinger-Das et al. Reference Hassinger-Das, Jordan, Glutting, Irwin and Dyson2014). Being able to focus on number while ignoring irrelevant information helps children master foundational number skills more quickly. Preschoolers' “spontaneous focusing on numerosity” (SFON) predicts rational number understanding 6 years later (McMullen et al., Reference McMullen, Hannula-Sormunen and Lehtinen2015). Cognitive control may be especially important for learning fractions, where a larger number in a fraction's denominator may not always correspond to its magnitude (e.g., 2/6 is smaller than 2/3). In fact, attention emerges as a strong and unique predictor of fraction learning among a constellation of variables (Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Rinne et al. Reference Rinne, Ye and Jordan2017). Not surprisingly, attention deficits frequently co-occur with mathematics disabilities (Zentall et al. Reference Zentall, Smith, Lee and Wieczorek1994). Recognizing the correlation between continuous magnitudes and numerosities may be an important cue for determining numerical magnitude, but it is likely just one cue among a number of others, and an inability to use such cues in general likely reflects a domain-specific core deficit in processing magnitudes numerically.
Finally, the term number sense has been used by some to indicate a fundamental ability that has been interpreted as being relatively impervious to change. A similar interpretation might be made for “magnitude sense.” However, we and others define number sense more broadly to include core symbolic knowledge of number and numerical magnitudes or relations (Jordan & Dyson Reference Jordan, Dyson and Henik2016; National Research Council Reference Cross, Woods and Schweingruber2009). A growing body of experimental evidence – at least at the symbolic level, which is most closely related to learning mathematics – indicates that number sense can be developed in all or most children (Frye et al. Reference Frye, Baroody, Burchinal, Carver, Jordan and McDowell2013; Jordan & Dyson Reference Jordan, Dyson and Henik2016). Rather than separating continuous magnitude sense from number sense, as Leibovich et al. propose, it might be more useful to view these understandings as interacting along a continuum of numerical development.
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In their article, “From ‘Sense of Number’ to ‘Sense of Magnitude’: The Role of Continuous Magnitudes in Numerical Cognition,” Leibovich et al. challenge the prevailing view that non-symbolic number sense (e.g., sensing “sixness” in a group of six objects in the same way one might sense color [Dehaene Reference Dehaene1997]) is innate, that detection of numerosity is distinct from detection of continuous magnitude. Rather, the researchers argue that infants are hardwired to sense continuous magnitudes (e.g., objects' areas, contour lengths, spacing) and that this information supports perception of numerosity. Non-symbolic number sense “develops from understanding the correlation between numerosity and continuous magnitudes” (sect. 8, para. 6). For example, infants might learn through their everyday experiences that more objects typically take up more space. However, in cases where this rule is violated (e.g., a line of three long toy trucks that is longer than a line of five short trucks), general cognitive control or executive function may be needed to inhibit a response based on the expected correlation of continuous magnitude and numerosity. In the present commentary, we discuss Liebovich et al.'s viewpoint in light of the integrative theory of numerical development (Siegler & Lortie-Forgues Reference Siegler and Lortie-Forgues2014), which was not considered in the article, along with implications for understanding mathematics disabilities.
The integrative theory “proposes that the continuing growth of understanding magnitudes provides a unifying theme for numerical development” (Siegler & Lortie-Forgues Reference Siegler and Lortie-Forgues2014, p. 144). A mental number line, Siegler and Lortie-Forgues contend, coordinates knowledge of different forms of magnitude ranging from non-symbolic continuous quantities and numerosities to symbolic whole numbers, fractions, and decimals. Arguably, a mental number line could be grounded in core knowledge of continuous magnitudes, which allows children to perceive non-symbolic numerosities and eventually symbolic numbers. However, we believe that the argument that an innate sense of non-symbolic magnitude is more fundamental than non-symbolic number sense is less important than is the notion that an understanding of how discrete and continuous quantities are related is critical for constructing the beginnings of a mental number line. According to the integrative theory, the construction of a mental number line structures mathematics learning, and early experiences with both continuous magnitudes and discrete objects in the external world shape children's understandings of quantity right from the start.
Leibovich et al. present implications of their view for understanding mathematics disabilities, including dyscalculia, as well as mathematics disabilities that co-occur with other conditions (e.g., dyslexia or attention deficits). The possibility that children at risk for mathematics disabilities have trouble grasping the correlation between continuous magnitudes and numerosities is intriguing. Previous work has shown that core deficits in understanding numerical magnitudes in symbolic contexts underpin mathematics disabilities (e.g., Butterworth Reference Butterworth1999; Reference Butterworth2005; Butterworth & Reigosa-Crespo Reference Butterworth, Reigosa-Crespo, Berch and Mazzocco2007; Landerl et al. Reference Landerl, Bevan and Butterworth2004). For example, children with dyscalculia perform much more poorly than do their typically achieving peers when asked to identify which of two numerals is larger and to relate quantities to their symbols (Butterworth & Reigosa-Crespo Reference Butterworth, Reigosa-Crespo, Berch and Mazzocco2007; Landerl et al. Reference Landerl, Bevan and Butterworth2004; Rousselle & Noel Reference Rousselle and Noel2007). Moreover, ability to estimate the placement of numbers on a number line strongly predicts whether children will go on to struggle in mathematics (Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Resnick et al. Reference Resnick, Jordan, Hansen, Rajan, Rodrigues, Siegler and Fuchs2016). It is possible children's difficulties with symbolic representations of quantity may be at least partly rooted in the detection of non-symbolic continuous magnitudes. For example, Matthews et al. (Reference Matthews and Lewis2016) showed that mathematics competencies that depend on an understanding of fractions are predicted by non-symbolic processing of ratio information, which is inherently continuous.
Nonetheless, we generally argue against the “domain-general account of dyscalculia” discussed by Leibovich et al. Although general cognitive competencies partially explain why some children struggle with mathematics (Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Rousselle & Noel Reference Rousselle and Noel2007), basic weaknesses in understanding numerical magnitudes have been shown to be the more definitive characteristic of mathematics disabilities (Clarke & Shinn Reference Clarke and Shinn2004; Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Mazzocco & Thompson Reference Mazzocco and Thompson2005). We recognize, however, that children's core difficulties with numerical understanding are exacerbated by weaknesses in executive functioning related to inhibition and set shifting, which constrain children's later numerical development (e.g., Hassinger-Das et al. Reference Hassinger-Das, Jordan, Glutting, Irwin and Dyson2014). Being able to focus on number while ignoring irrelevant information helps children master foundational number skills more quickly. Preschoolers' “spontaneous focusing on numerosity” (SFON) predicts rational number understanding 6 years later (McMullen et al., Reference McMullen, Hannula-Sormunen and Lehtinen2015). Cognitive control may be especially important for learning fractions, where a larger number in a fraction's denominator may not always correspond to its magnitude (e.g., 2/6 is smaller than 2/3). In fact, attention emerges as a strong and unique predictor of fraction learning among a constellation of variables (Hansen et al. Reference Hansen, Jordan, Fernandez, Siegler, Fuchs, Gersten and Micklos2015; Jordan et al. Reference Jordan, Hansen, Fuchs, Siegler, Gersten and Micklos2013; Rinne et al. Reference Rinne, Ye and Jordan2017). Not surprisingly, attention deficits frequently co-occur with mathematics disabilities (Zentall et al. Reference Zentall, Smith, Lee and Wieczorek1994). Recognizing the correlation between continuous magnitudes and numerosities may be an important cue for determining numerical magnitude, but it is likely just one cue among a number of others, and an inability to use such cues in general likely reflects a domain-specific core deficit in processing magnitudes numerically.
Finally, the term number sense has been used by some to indicate a fundamental ability that has been interpreted as being relatively impervious to change. A similar interpretation might be made for “magnitude sense.” However, we and others define number sense more broadly to include core symbolic knowledge of number and numerical magnitudes or relations (Jordan & Dyson Reference Jordan, Dyson and Henik2016; National Research Council Reference Cross, Woods and Schweingruber2009). A growing body of experimental evidence – at least at the symbolic level, which is most closely related to learning mathematics – indicates that number sense can be developed in all or most children (Frye et al. Reference Frye, Baroody, Burchinal, Carver, Jordan and McDowell2013; Jordan & Dyson Reference Jordan, Dyson and Henik2016). Rather than separating continuous magnitude sense from number sense, as Leibovich et al. propose, it might be more useful to view these understandings as interacting along a continuum of numerical development.