ing a rich picture of the step-by step manner in which children typically construct knowledge of whole numbers can help teachers create knowledge-centered classrooms and learning pathways that fit children’s spontaneous development.

BUILDING ON CHILDREN’S CURRENT UNDERSTANDINGS

What number knowledge do children have when they start preschool around the age of 4? As every preschool teacher knows, the answer varies widely from one child to the next. Although this variation does not disappear as children progress through the primary grades, teachers are still responsible for teaching a whole classroom of children, as well as every child within it, and for setting learning objectives for their grade level. It can be a great help to teachers, therefore, to have some idea of the range of understandings they can expect for children at their grade level and, equally important, to be aware of the mistakes, misunderstandings, and partial understandings that are also typical for children at this age level.

To obtain a portrait of these age-level understandings, we can consider the knowledge children typically demonstrate at each age level between ages 4 and 8 when asked the series of oral questions provided on the Number Knowledge test (see Box 6-3). The test is included here for discussion purposes, but teachers who wish to use it to determine their student’s current level of understanding can do so.

Before we start, a few features of the Number Knowledge test deserve mention. First, because this instrument has been called a test in the developmental research literature, the name has been preserved in this chapter. However, this instrument differs from school tests in many ways. It is administered individually, and the questions are presented orally. Although right and wrong answers are noted, children’s reasoning is equally important, and prompts to elicit this reasoning (e.g., How do you know? How did you figure that out?) are always provided on a subset of items on the test, especially when children’s thinking and/or strategy use is not obvious when they are solving the problems posed. For these reasons, the “test” is better thought of as a tool or as a set of questions teachers can use to elicit children’s conceptions about number and quantity and to gain a better understanding of the strategies children have available to solve number problems. When used at the beginning (and end) of the school year, it provides a good picture of children’s entering (and exit) knowledge. It also provides a model for the ongoing, formative assessments that are conducted throughout the school year in assessment-centered classrooms.

Second, as shown in Box 6-3, the test is divided into three levels, with a preliminary (warm-up) question. The numbers associated with each level



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