numbers (see sections below). However, the path from informal to formal knowledge is not necessarily a smooth one.

Impressive growth of numerical competence from age 2 to age 6 is stimulated by children’s learning of important cultural numerical tools: spoken number words, written number symbols, and cultural solution methods, like counting and matching. As shown by Wynn’s (1990, 1992b) research, the acquisition of the understanding of the cardinal meanings of number words is a protracted process. In a longitudinal study, Wynn found that it takes about a year for a child to move from succeeding in giving a set of “one” when requested to do so to being able to give the appropriate number for all numbers in his or her count list. The acquisition of such symbolic knowledge is important in promoting the abstractness of number concepts, that is, the concept of cardinality (that all sets of a given numerosity form an equivalence class). It is also important in promoting the exactness of number representations and the understanding of numerical relations, as only children who have acquired this knowledge understand that adding one item to a set means moving to the very next number in the count list (Sarnecka and Carey, 2008). The research concerning these cultural learning achievements is summarized in Chapter 5 in identifying foundational and achievable goals for teaching and learning. It is discussed in Chapter 4 as a major source of socioeconomic differences, connected to differential exposure to talk about number at home and at preschool.

DEVELOPMENT OF SPATIAL THINKING AND GEOMETRY

Spatial thinking, like numerical thinking, is a fundamental component of mathematics that has its roots in foundational skills that emerge early in life. Spatial thinking is critical to a variety of mathematical topics, including geometry, measurement, and part-whole relations (e.g., Ansari et al., 2003; Fennema and Sherman, 1977, 1978; Guay and McDaniel, 1977; Lean and Clements, 1981; Skolnick, Langbort, and Day, 1982; see Chapter 6, this volume). Spatial thinking has been found to be a significant predictor of achievement in mathematics and science, even controlling for overall verbal and mathematical skill (e.g., Clements and Sarama, 2007; Hedges and Chung, in preparation; Lean and Clements, 1981; Shea, Lubinski, and Benbow, 2001; Stewart, Leeson, and Wright, 1997; Wheatley, 1990). One reason that spatial thinking is predictive of mathematics and science achievement is because it provides a way to conceptualize relationships in a problem prior to solving it (Clements and Sarama, 2007).

The mental functions encompassed by spatial thinking include categorizing shapes and objects and encoding the categorical and metric relations among shapes and objects. Spatial thinking is also crucial in representing object transformations and the outcomes of these transformations (e.g.,



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