and text. Each subsequent step assumes that children have had sufficient experiences with the topics in the previous step to learn the earlier content well. (See Box 5-1 for a discussion of what it means to learn something well.) However, many children can still learn the content at a given step without having fully mastered the previous content if they have sufficient time to learn and practice the more challenging content. Of course, some children have difficulty in learning certain kinds of mathematical concepts, and a few have really significant difficulties. But most children are capable of learning the foundational and achievable mathematics content specified in the learning steps outlined here.

In both the number and operations and the geometry and measurement core areas, children learn about the basic numerical or geometric concepts and objects (numbers, shapes), and they also relate those objects and compose/decompose (operate on) them. Therefore, each core area begins by discussing the basic objects and then moves to the relations and operations on them. In all of these, it is important to consider how children perceive, say, describe/discuss, and construct these objects, relations, and operations.

The development of the elements of the number core across ages is described first, and then the development of the relations and operations core

BOX 5-1

Learning Something Well

In most aspects of the number and the relations/operation core, children need a great deal of practice doing a task, even after they can do it correctly. The reasons for this vary a bit across different aspects, and no single word adequately captures this need, because the possible words often have somewhat different meanings for different people.

Overlearning can capture this meaning, but it is not a common word and might be taken to mean something learned beyond what is necessary rather than something learned beyond the initial level of correctness. Automaticity is a word with technical meaning in some psychological literature as meaning a level of performance at which one can also do something else. But to some people it carries only a sense of rote performance. Fluency is the term used by several previous committees, and we have therefore chosen to continue this usage. Fluency also carries for some a connotation of flexibility because a person knows something well enough to use it adaptively. We find this meaning useful as well as the usual meaning of doing something rapidly and relatively effortlessly. Research on reading in early childhood has recently used fluency only in the latter sense as measured by performance on standardized tests of reading, such as the Dynamic Indicators of Basic Early Literacy Skills (DIBELS). We do not mean fluency to be restricted to this rote sense. By fluent we mean accurate and (fairly) rapid and (relatively) effortlessly with a basis of understanding that can support flexible performance when needed.

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