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

Exploring the linguistic complexity of third-grade numerical literacy

Reading numbers aloud, a central aspect of numerical literacy, is a challenging skill to acquire, but the origins of this difficulty remain poorly understood. To investigate this matter, we examined the performance of 127 third- and fourth-grade children who read aloud, in Hebrew, numbers with 2-5 digits. We found several key observations. First, we observed a substantial variation among the 3rd graders-7% and 59% errors in the top and bottom deciles, respectively. Second, the task difficulty stemmed from syntactic processing: Most errors were distortions of the number's syntax, as opposed to digit substitutions or transpositions, and the main factor affecting a specific number's difficulty was not its magnitude, as is commonly assumed, but rather its syntactic structure. Third, number reading performance was not predicted by a school-like task that assessed syntactic-conceptual knowledge of the decimal system structure, but rather by knowledge of specific syntactic-verbal rules, suggesting that the syntactic-verbal knowledge is separate from the syntactic-conceptual knowledge. Last, there was a double dissociation between 4-digit numbers and 5-digit numbers, which in Hebrew have completely different syntactic structures: Half of the children showed a significant advantage in one number length compared to the other, with equal numbers of children preferring either length. This indicates that the different syntactic-verbal rules are learned relatively independently of each other, with little or no generalization from one rule to another. In light of these findings, we propose that schools should specifically teach number reading, with focus on specific syntactic-verbal rules.

Syntactic theory of mathematical expressions

Mathematical expressions consist of recursive combinations of numbers, variables, and operators. According to theoretical linguists, the syntactic mechanisms of natural language also provide a basis for mathematics. To date, however, no theoretically rigorous investigation has been conducted to support such arguments. Therefore, this study uses a methodology based on theoretical linguistics to analyze the syntactic properties of mathematical expressions. Through a review of recent behavioral and neuroimaging studies on mathematical syntax, we report several inconsistencies with theoretical linguistics, such as the use of ternary structures. To address these, we propose that a syntactic category called Applicative plays a central role in analyzing mathematical expressions with seemingly ternary structures by combining binary structures. Besides basic arithmetic expressions, we also examine algebraic equations and complex expressions such as integral and differential calculi. This study is the first attempt at building a comprehensive framework for analyzing the syntactic structures of mathematical expressions.

What children's number naming errors tell us about early understanding of multidigit numbers

This investigation examined the nature and development of a foundational symbolic numeric skill-number identification-focusing on children's emerging knowledge of multidigit numbers. Two studies were conducted with Russian preschoolers. Study 1 (N = 350; 51-77 months of age) investigated age-related changes in the accuracy of number naming and in the types of errors children produced. The errors fell into distinct categories: syntactic (structural errors such as naming each digit separately without using place-value markers) and lexical (nonstructural errors such as replacing the name of a digit with the name of another digit). Number reading accuracy improved with age, primarily due to a decreased frequency of syntactic errors. Boys made fewer syntactic errors than girls. Study 2 (N = 110; 61-74 months of age) showed that accuracy of naming double-digit numbers was related to conceptual understanding of the base-10 numeric structure. The frequency of syntactic errors in number naming was negatively associated with the use of base-10 representations, whereas lexical errors were not related to children's ability to represent base-10 number structure. Implications for understanding children's mathematics trajectories are discussed.