Is thirty-two three tens and two ones? The embedded structure of cardinal numbers

Abstract

The acquisition and representation of natural numbers have been a central topic in cognitive science. However, a key question in this topic about how humans acquire the capacity to understand that numbers make 'infinite use of finite means' (or that numbers are generative) has been left unanswered. Here, we test the hypothesis that children's understanding of the syntactic rules for building complex numerals-or numerical syntax-is a crucial foundation for the acquisition of number concepts. In two independent studies, we assessed children's understanding of numerical syntax by probing their knowledge about the embedded structure of cardinal numbers using a novel task called Give-a-number Base-10 (Give-N10). In Give-N10, children were asked to give a large number of items (e.g., 32 items) from a pool that is organized in sets of ten items. Children's knowledge about the embedded structure of numbers (e.g., knowing that thirty-two items are composed of three tens and two ones) was assessed from their ability to use those sets. Study 1 tested English-speaking 5-to 10-year-olds and revealed that children's understanding of the embedded structure of numbers emerges relatively late in development (several months into kindergarten), beyond when they are capable of making a semantic induction over a local sequence of numbers. Moreover, performance in Give-N10 was predicted by children's counting fluency even when school grade was controlled for, suggesting that children's understanding of the embedded structure of numbers is founded on their earlier knowledge about the syntactic regularities in the counting sequence. In Study 2, this association was tested again in monolingual Korean kindergarteners (5-6 years), as we aimed to test the same effect in a language with a highly regular numeral system. It replicated the association between Give-N10 performance and counting fluency, and it also demonstrated that Koreanspeaking children understand the embedded structure of cardinal numbers earlier in the acquisition path than English-speaking peers, suggesting that the knowledge of numerical syntax governs children's understanding of the generative properties of numbers. Based on these observations and our theoretical analysis of the literature, we propose that the syntax for building complex numerals represents the structure of numerical thinking.

Syntactic theory of mathematical expressions

“…Such syntax is in line with the syntactic structures of number words; for example, the phrase structure twenty-seven can be analyzed as [[two -ty] seven] (Hurford, 2007;Guerrero and Park, 2023). Indeed, children's understanding of the structural relationship of multi-digit numbers is dependent on their native languages (Guerrero et al, 2020). Furthermore, Hung et al (2015) reported the neural correlate of syntactically complex multi-digits in the left IFG in a previous fMRI study.…”

“…Despite the frequent use of mathematical expressions as examples of context-free grammar (CFG;Hopcroft et al, 2006; see Supplementary Information for a CFG-based formal description), theoretical research that adequately describes their syntax in a manner consistent with that of theoretical linguistics remain absent. Although several attempts are made to conduct the theoretical analyses of the natural number system (Watanabe, 2017;Hiraiwa, 2017) and multi-digit numbers (Di Sciullo, 2015;Guerrero et al, 2020;Hurford, 1987Hurford, , 2007, the interpretation of arithmetic expressions and advanced algebraic equations from a linguistic perspective is largely unknown.…”

Complex Cardinal Numerals and the Strong Minimalist Thesis

“…The hypothesis that unbounded Merge yields arithmetic is abandoned in Chomsky (2018). An evolutionary argument to this effect is discussed in Casares (2019) [22], and Guerrero et al (2020) [23] brings experimental support from language acquisition. See also Roeper (2022) [24] for discussion.…”

Thirty years of the Give-N task: A systematic review, reflections, and recommendations

“…Additionally, Give-N methodology has been expanded with clever variations to test other mathematical cognition theories, such as a Give-N task requesting specific objects ("one person, two animals, and three pieces of furniture") (Sorariutta, Hannula-Sormunen & Silvén, 2017), Give-N tasks with ordinal instead of cardinal responses ("give me the third toy") (Colomé & Noël, 2012;Meyer, Barbiers & Weerman, 2018), Give-N tasks requesting numbers with digits, finger-counting or sets of objects instead of number words (Möhring et al, 2019;Merkley, 2015;Nicoladis, Pika, Marentette, 2010;Salsa & Martí, 2015), a Give-N task prompting place value ("give me twenty-three" with tens and units) (Guerrero et al, 2020) and a Give-N task requesting quantifiers instead of number words ("give me some toys") Sullivan, Bale & Barner, 2018). The Give-N task has proven a versatile research tool beyond just assessment of cardinality acquisition and will likely continue to grow in relevance.…”