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This chapter has expanded significantly from previous editions. It reflects my current thinking about place value as a topic that must be integrated significantly with a flexible approach to computation rather than one that precedes computation. This connection is one that you are unlikely to find in any other books.

There is no doubt that place value provides the conceptual foundation for all aspects of whole-number and decimal computation. However, the manner in which we develop place value should be predicated on the type of computation we wish to emphasize. The traditional approach to place value—reflected in the first portion of this chapter—focuses on how groupings of ten are recorded in each individual position. This understanding is critical for understanding the traditional algorithms. This digit-orientation to place value fails to prepare students for more flexible approaches to computation often referred to as "invented strategies." And ironically, use of the traditional algorithms tends to focus on the digits and actually obscures the concepts on which the algorithms are based.

Because I am increasingly interested in development of flexible computation strategies, I believe that a significant focus of place-value development should be based on the patterns and relationships in the number system. These activities (beginning on p. 201) might be called computational activities. The perspective taken here is that students can and should explore these activities before developing any computational algorithms. By engaging in these pre-computational activities they are actually learning about place value and developing computational flexibility at the same time. Many of these activities have migrated from the computation chapter in previous editions. I want to emphasize to teachers that there is no need to segregate place value development from computation but rather these topics should be integrated.

If you accept this view of place value development, you will very likely expand the time you spend on this chapter and possibly decrease the time spent on computation. For K-3 teachers, I believe that the full chapter is important. For teachers of the upper grades, I believe that the Patterns and Relationships section is the most important.

The traditional base-ten and place-value ideas are summarized in Figure 12.3 (T-70). The connection of these ideas to children's meaningful counting by ones is the first goal of traditional place-value concept development.

Models for base-ten concepts play a major role in the development of these ideas. I argue that the only materials that truly model base-ten concepts are proportional models. Colored counters, the abacus, and even money are all excluded from my list of valuable models. A more important distinction in the early development is between groupable and pre-grouped models, since there is real evidence that children do not see all of these the same way. The display of models in Figure 12.4 includes the little ten-frame cards which are significant in that they always illustrate the distance to the next grouping of ten.