| In recent decades, we have come to accept that children's appropriation of written numbers is not automatic or simple. Various studies of children's use of notation point to several different types of notational practice possibly linked to stages in the understanding of place value in base-ten (Alvarado, 2002; Brizuela, 2004; Scheuer et al., 2000; Seron & Fayol, 1994).;The project described in this dissertation explores the connections between three external representations of the number system: written numbers, oral numbers, and numbers represented through base-ten valued tokens.;In a semi-longitudinal approach, twenty-seven first grade students were interviewed individually regarding their productions of numbers in these three external systems of representation. The following year, twenty-six second-grade participants, twenty-one of whom were part of the original first-grade sample were again interviewed. A second cohort of eighteen Kindergarten children from the same school was also interviewed. The school these children attend is in an urban suburb of Boston, Massachusetts, in the USA that is ethnically, racially, and socio-economically diverse.;The main research questions of the study reported in this dissertation are: (1) Are there differences in how children represent (through production and interpretation): (a) a number in writing; (b) a number orally; and (c) a number with valued tokens? (2) How are these three modes of representation related to one another within children?;These questions were considered through three lenses: (1) General trends amongst each method of representation. (2) Associations between conventional representations and particular grade levels and number magnitudes. (3) Relationships between conventional and unconventional responses amongst all children and within each grade.;Results show that there are specific and identifiable progressions that children go through in the appropriation of all three of these number systems. Different types of unconventional responses were associated with different grade levels. Moreover, the system of representation that was most likely to incur a conventional representation differed by both grade level and number magnitude.;Additionally, some types of unconventional responses involving the overuse of zero may act as predictors in future performance, where responses with fewer zeros in first grade are likely to predict more conventional performance in second grade than responses with more zeros. |
