Research Council, 2001a) recognized the difficulties of the number line representation for young children and recommended that its use begin at Grade 2 and not earlier.

The number line is particularly important when one wants to show parts of one whole, such as one-half. In early childhood materials, the term number line or mental number line often really means a number path, such as in the common early childhood games in which numbers are put on squares and children move along a numbered path. Such number paths are count models—each square is an object that can be counted—so these are appropriate for children from age 2 through Grade 1. Some research summarized in Chapter 3 did focus on children’s and adult’s use of the analog magnitude system to estimate large quantities or to say where specified larger numbers fell along a number line. Again, it is not clear, especially for children, whether they are using a mental number list or a number line; the crucial research issue is the change in the spacing of the numbers with age, and this could come either from children’s use of a mental number list or a number line. The use of number lines, such as in a ruler or a bar graph scale, is an important part of measurement and is discussed in Chapter 6. But for numbers, relations, and operations, physical and mental number word lists are the appropriate model.

Variability in Children’s Solution Methods

The focus of this chapter is on how children follow a learning path from age 2 to Grade 1 in learning important aspects of numbers, relations, and operations. We continually emphasize that there is variability within each age group in the numbers and concepts with which a given child can work. As summarized in Chapter 3, much of this variability stems from differences in opportunities to learn and to practice these competencies, and we stress how important it is to provide such opportunities to learn for all children. We close with a reminder that there is also variability within a given individual at a given time in the strategies the child will use for a given kind of task. Researchers through the years have shown that children’s strategy use is marked by variability both within and across children (e.g., Siegler, 1988; Siegler and Jenkins, 1989; Siegler and Shrager, 1984). Even on the same problem, a child might use one strategy at one point in the session, and another strategy at another point. As children gain proficiency, they gradually move to more mature and efficient strategies, rather than doing so all at once. The variability itself is thought to be an important engine of cognitive change. Similarly, as discussed above, accuracy can vary with effort, particularly with counting. The variability in the use of strategies within or across children can provide important opportunities to discuss different methods and extend understandings of all participants. The vari-



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