Core multiplication in childhood

Abstract

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (Approximate Number System, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5–7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

“…One possible reason may be that they promote a different understanding of how proportional components should be scaled. Scaling can be defined as a process of transforming absolute magnitudes while conserving relational properties, and it is therefore an important aspect of proportional reasoning (Barth, Baron, Spelke, & Carey, 2009;Boyer & Levine, 2012;McCrink & Spelke, 2010). The importance of scaling for proportional reasoning is evident in everyday life, for instance when one wants to adjust the amounts of ingredients for a cake for 4 people to 6 people, or prepare the same concentrations of syrup-water mixtures in different jugs.…”

Learning to represent exact numbers

“…They can also be added, subtracted and multiplied (Barth et al, 2006;Dehaene, 1997Dehaene, , 2009McCrink & Spelke, 2010;Nieder & Dehaene, 2009) and individuals differ in the accuracy of their number estimation. For example, almost everyone (including human infants) can tell that a cloud of 200 dots is different from a cloud of 100, and most adults can tell the difference (without counting) between 100 and 130.…”

Symbolic Number Skills Predict Growth in Nonsymbolic Number Skills in Kindergarteners

“…This is an intuitive hypothesis, capitalizing on the broader notion that cultural skills must at some level co-opt extant, evolutionarily ancient, neural structures (Dehaene & Cohen, 2007). Moreover, many of the basic operations that form the basis of the SNS -such as relative quantity (greater/lesser), ordinality and even simple arithmetic -are available, at least in approximate form, to the AMS (e.g., Capaldi & Miller, 1988;Brannon et al, 2001;McCrink & Spelke, 2010Matthews et al, 2016). Furthermore, numerous studies have shown that the precision of an individual's AMS is predictive of SNS abilities (for a review and meta-analysis, see Chen & Li, 2014).…”

“…Recently, it has become apparent that humans and other species can do more than just compare nonverbal magnitudes, they can even perform simple, approximate arithmetic (such as sums and ratios; e.g., Capaldi & Miller, 1988;Brannon et al, 2001;McCrink & Spelke, 2010Matthews et al, 2016).…”