are reviewed in chapter 5.) In both reading and mathematics, some children enter school without the knowledge and experience that school instruction presumes they possess. In both domains, there is evidence that early intervention can prevent full-blown problems in school.4

For both reading and mathematics, children’s performance at the end of elementary school is an important predictor of their ultimate educational success. If they have not mastered certain basic skills, they can expect problems throughout their schooling and later. Research on reading indicates that all but a very small number of children can learn to read proficiently, though they may learn at different rates and may require different amounts and types of instructional support. Furthermore, experiences in pre-kindergarten and the early elementary grades serve as a crucial foundation for students’ emerging proficiency. Similar observations can be made for mathematics.

For example, nearly all second graders might be expected to make a useful drawing of the situation portrayed in an arithmetic word problem as a step toward solving it. Representing numbers by means of a drawing is a task that few children find difficult. Other tasks, however, depend much more heavily on children’s knowledge and experience. For example, in Roman numerals, the value of V is five regardless of where it is located in the numeral, whether IV, VI, or VII. The Hindu-Arabic numerals used in everyday life are different; a digit’s value depends on the place it occupies. For example, the 5 in 115 denotes five, whereas in 151 it denotes fifty, and in 511, five hundred. Also, a special symbol, 0, is used to hold a place that would otherwise be unoccupied. Although adults may view this place-value system as simple and straightforward, it is actually quite sophisticated and challenging to learn (see chapters 5 and 6).

To make progress in school mathematics, children must understand Hindu-Arabic numerals and be able to use them fluently. But the children in, say, a second-grade class can be expected to differ considerably in the rate at which they grasp place value. It is a complex system of representation that functions almost like a foreign language that a child is learning to use and simultaneously using to learn other things. Much of school mathematics has this mutually dependent quality. Abstractions at one level are used to develop abstractions at a higher level, and abstractions at a higher level are used to gain insights into abstractions at a lower level.

To ensure that students having reading difficulties get prompt and effective assistance outside the regular school program, the reading community has developed a variety of intervention programs designed to address the problems students are having and to bring them back rapidly into the regular



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