Young Children Intuitively Divide Before They Recognize the Division Symbol

Abstract

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Compositionality and constituent structure in the analogue mind

“…2006). Qu and colleagues (2021) recently went further, finding that children of a similar age group can even use their ANS to perform approximate multiplications (see also McCrink & Spelke 2010), with Szkudlarek and colleagues (2022) also finding a related capacity for division (see also McCrink & Spelke 2016).…”

Rational number representation by the approximate number system

“…Evidence for this system is found in many creatures, including fish (e.g., Agrillo, Dadda, Serena, & Bisazza, 2008), rats (e.g., Platt & Johnson, 1971), pigeons (e.g., Emmerton, Lohmann, & Niemann, 1997), monkeys (e.g., Brannon & Terrace, 1998), human infants (e.g., Xu & Spelke, 2000), pre-numerate human children (e.g., Mix, Huttenlocher, & Levine, 2002), human adults whose language lacks precise number words (Pica, Lemer, Izard, & Dehaene, 2004), as well as human adults with a formal math education (e.g., Barth, Kanwisher, & Spelke, 2003). In addition, the system is found to support a diverse range of numerical computationsfor instance, ordinal comparisons (Temple & Posner, 1998), the ability to identify two collections as equinumerous (Barth et al, 2003), number estimations (Cordes, Gelman, Gallistel, & Whalen, 2001), as well as addition (McCrink & Wynn, 2004), subtraction (Barth et al, 2006), multiplication (McCrink & Spelke, 2010;Qu, Szkudlarek, & Brannon, 2021) and division operations (McCrink & Spelke, 2016;Szkudlarek, Zhang, DeWind, & Brannon, 2022). Consequently, an orthodox view in cognitive science is that the ANS is widespread in nature, operates independently of natural language, and enables humans and other organisms to process number throughout the lifespan, albeit approximately and in accord with Weber's Law.…”

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children

“…Individuals also can perform operations on ANS representations, transforming or manipulating ANS representations in the face of real-world changes to quantities, and this ability is already present in infancy and childhood ( McCrink and Wynn, 2004 ; Barth et al, 2006 ; Booth and Siegler, 2008 ; McCrink and Spelke, 2010 , 2016 ; Kibbe and Feigenson, 2015 , 2017 ; Qu et al, 2021 ; Szkudlarek et al, 2022 ). For example, children who observe a set of dots move behind an occluder, and then observe a second set of dots move behind the same occluder, can update their representation of the quantity behind the occluder, effectively summing over their ANS representations of the sets ( Barth et al, 2006 ).…”