Magnitude knowledge: the common core of numerical development

Abstract

The integrated theory of numerical development posits that a central theme of numerical development from infancy to adulthood is progressive broadening of the types and ranges of numbers whose magnitudes are accurately represented. The process includes four overlapping trends: (1) representing increasingly precisely the magnitudes of non-symbolic numbers, (2) connecting small symbolic numbers to their non-symbolic referents, (3) extending understanding from smaller to larger whole numbers, and (4) accurately representing the magnitudes of rational numbers. The present review identifies substantial commonalities, as well as differences, in these four aspects of numerical development. With both whole and rational numbers, numerical magnitude knowledge is concurrently correlated with, longitudinally predictive of, and causally related to multiple aspects of mathematical understanding, including arithmetic and overall math achievement. Moreover, interventions focused on increasing numerical magnitude knowledge often generalize to other aspects of mathematics. The cognitive processes of association and analogy seem to play especially large roles in this development. Thus, acquisition of numerical magnitude knowledge can be seen as the common core of numerical development. Research highlights• The integrated theory of numerical development posits that the unifying theme of numerical development is children's growing understanding of numerical magnitudes.• The theory focuses on four aspects of numerical development: representing increasingly precisely the magnitudes of non-symbolic numbers, connecting small symbolic numbers to their non-symbolic referents, extending understanding from smaller to larger whole numbers, and accurately representing the magnitudes of rational numbers.• For both whole and rational numbers, knowledge of numerical magnitudes is correlated with, predictive of, and causally related to other crucial aspects of mathematics, including arithmetic and overall math achievement.• Interventions that emphasize acquisition of numerical magnitude knowledge have yielded a broad range of positive effects with both whole and rational numbers.

Narrowing the early mathematics gap: A play-based intervention to promote low-income preschoolers’ number skills

“…One explanation may be that including both numerals and non-symbolic quantities (circles) on the cards led children who played the War card game to focus more on the symbolic numeral information while making comparisons, largely ignoring the non-symbolic information. Theories of numerical development suggest that between 4-and 5-years old children begin to fully integrate their symbolic number knowledge with non-symbolic magnitude information (Siegler, 2016;Siegler & Lortie-Forgues, 2014;Siegler, Thompson, & Schneider, 2011), which may mean the symbolic magnitude information was particularly salient to our sample of predominately 4-and 5-year old children. In addition, the non-symbolic ordinality task showed only marginally significant pretest-posttest reliability and low inter-item reliability (Pearson's r(44) = .29, p = .051; inter-item reliability at first administration α = .24), which may be an indication that the children had trouble focusing on or understanding that specific task.…”

Numerosity representation is encoded in human subcortex

“…In the previous experiments, participants made quantity judgments on displays containing numerosities randomly and equiprobably drawn from a fixed set of numerosities (e.g., 8,11,16,22, or 32 in experiment 2). Thus, the probability of a specific response (e.g., choosing that the first array contains more dots) coincided with the numerical ratio and with the numerosity of the first array itself.…”

“…The ontogenetic emergence of numerical skills follows a reasonably well-established trajectory with an initial ability to approximate numerical quantity nonverbally, followed, over the course of early childhood, with the development of increasingly precise representations of numerical values, including a symbolic number system that allows children to conceive of numerical information as Arabic numerals or number words (14)(15)(16). The early ability to distinguish two nonsymbolic quantities, for example, in the context of a display of varying numbers of dots potentially forms the foundation for developing math abilities akin to the "number sense" (17).…”

Bridging psychological and educational research on rational number knowledge

“…Their basic claim for instruction is that "drawing the explicit analogy that fractions are like whole numbers in having magnitudes that can be ordered and represented on number lines may be helpful" (Siegler et al, 2011, p. 291). From our theoretical perspective, the account of rational number development offered by both research groups appears to downgrade the great discrepancy between natural and rational numbers with respect to early informal and formal experiences.Finally, both Moss and Case (1999) and Siegler and colleagues (Siegler, 2016; Fuchs et al, 2013Fuchs et al, , 2014Fuchs et al, , 2016 consistent with their accounts of numerical development, target students that are older than 8 years of age. On the contrary, we believe that rational number instruction should begin earlier.…”

“…For Moss and Case this similarity lies in the form of the two processes whereas for Siegler and colleagues the similarity lies in the instrumental role of number magnitude. For the framework theory, rational number knowledge is built on natural number knowledge but requires its gradual modification and reorganisation, a process that involves considerable conceptual changes.Moss and Case (1999) as well as Siegler and colleagues (Siegler, 2016; Siegler et al, 2011) agree that the development of natural number knowledge precedes the development of rational number knowledge. Moss and Case, however, argue in favour of building rational number knowledge on students' informal understandings of proportionality.…”