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Year-long classroom teaching experiments in two predominantly Latino low-socioeconomicstatus (SES) urban classrooms (one English-speaking and one Spanish-speaking) sought to support first graders' thinking of 2-digit quantities as tens and ones. A model of a developmental sequence of conceptual structures for 2-digit numbers (the UDSSI triad model) is presented to describe children's thinking. By the end of the year, most of the children could accurately add and subtract 2-digit numbers that require trading (regrouping) by using drawings or objects and gave answers by using tens and ones on various tasks. Their performance was substantially above that reported in other studies for U.S. first graders of higher SES and for older U.S. children. Their responses looked more like those of East Asian children than of U.S. children in other studies.
The larger contexts for the problems addressed by this study are the considerably lower level of primary school mathematics learning by children in the United States compared with that of children in China, Japan, and Korea (Fuson & Kwon, 1992a, 1992b; Geary, Bow-Thomas, Liu & Siegler,1996; Miller,1990; Miller & Stigler,1987; Song & Ginsburg, 1987; Stephenson & Stigler, 1992; Stigler, Lee, & Stephenson, 1990) and the reform efforts of the National Council of Teachers of Mathematics (NCTM) to bring about mathematics classrooms appropriate for the future needs of society (NCTM,1989,1991). The special challenges are the particularly low levels of mathematics achievement by poor urban children, especially those who do not speak English or who have parents with low levels of education (e.g., Secada, 1992).
These issues were addressed by undertaking a year-long teaching experiment (or developmental research, Gravemeijer,1994a,1994b) in two first-grade classrooms in an urban school with a predominantly low-socioeconomic-status (SES) Latino population. The backgrounds of children in English-speaking and Spanish-speaking classrooms differ (e.g., children in the latter are more recent immigrants). For this reason, and to explore issues of language differences, we selected one classroom in each language (Spanish and English). We did not envision an experimental contrast between these two classes, but only a test of our approaches in both languages and with a range of Latino-background children. The mathematical foci were addition and subtraction of quantities expressible by two-digit numbers and place-value concepts (the conceptual relationships among number words, two-digit numerals, and quantities)....