Learning the generative principles of a symbol system from limited examples

Comment: A role of Language in Infant Emotion Concept Acquisition

“…Further, there are many irregularities in the symbolic number system, and infants seldom see or hear three-digit number words (e.g., "518"; Fuson & Kwon, 1992;Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010). Nevertheless, by observing the multiple predictive patterns in the surface structure of numbers (Yuan, Xiang, Crandall, & Smith, 2020), preschoolers learn to map spoken multidigit number words to their written forms (Mix, Prather, Smith, & Stockton, 2014;Yuan, Prather, Mix, & Smith, 2019). In sum, children learn from complex, irregular input (i.e., language, symbolic numbers) without supervised instruction, in order to make predictions about their environment (Saffran, 2020).…”

The First Step to Learning Place Value: A Role for Physical Models?

“…From the perspective of the Structure Mapping Theory, relations are found through aligning arrays with the same relational structure and do not necessarily require that the aligned arrays include physical models. Thus, the alignment of number names (one symbolic form) to written forms (another symbolic form)-with no physical model-could be sufficient for an early learner (Mix et al, 2019;Yuan et al, 2020). Multi-digit number names and the written forms have distinct surface forms, but the same underlying relational structure as shown in Figure 2.…”

“…Foremost, children's later success in mastering place value may benefit from exposure to multi-digit numbers during the preschool and kindergarten years. Accumulating evidence with large and nationally representative samples (Byrge et al, 2014;Mix et al, 2014;Yuan et al, 2019;Yuan et al, 2020) indicates that many young children, before formal schooling, are building partial (not perfect but often correct) knowledge about how relative magnitudes are represented by place and the naming conventions for 3-and 4-digit numbers, and that this early knowledge is a strong predictor to later school learning of mathematics (Mix et al, Under review). The present result further suggests that early familiarity with these structures is learnable by preschool children with just a few trials-through mapping number names to their written forms, and/or to simple physical analogies about relative magnitudes-embedded in a game-like context without explicit teaching of the precise base-10 principles (Yuan et al, 2020).…”

“…Children first learn about the place relations that structure multidigit numbers before formal school instruction on place value. This early understanding does not include precise knowledge of the different quantities represented by the places nor their multiplicative relations to each other (Byrge et al, 2014;Mix et al, 2014;Yuan et al, 2019;Yuan et al, 2020). Instead, it is an "approximate" understanding that multi-digit numbers are made up of places that represent different relative magnitudes ordered from left to right.…”

“…Critically, this early approximate knowledge strongly predicts later success in learning and using base-10 principles (Mix et al, Under review), suggesting that approximate understanding is a useful step to more explicit correct understanding. Current evidence suggests further that young children acquire this approximate understanding through experience with the correspondences between spoken and written number names, e.g., learning that the "hundred" and the "-ty" in "two hundred thirty two" mark the places in the string "232" as signifying different amounts (Mix et al, Under review;Yuan et al, 2020). Multiple experiments (Byrge et al, 2014;Mix et al, 2014;Mix et al, 2017;Yuan et al, 2019) have shown that this partial knowledge enables preschool and kindergarten children to map unfamiliar number names to written forms for 3-and 4-digit numbers, to judge the relative magnitudes of 3-and 4-digit numbers, and to write multi-digit numbers given the spoken name (albeit sometimes with meaningful and interesting errors) (Mix et al, 2014).…”