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 valuereflected in the first portion of this chapterfocuses 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.
Early Development of Base-Ten and Place-Value Concepts: The Place-Value Triangle
An effort is made at the outset of the chapter to illustrate the count-by-ones approach to number that most children develop as early as kindergarten. Even those children who count quantities to 100 and can read and write these numbers, are very likely using a count-by-ones concept to understand these quantities. Children working in traditional textbooks easily learn about sticks and cubes as tens and ones without having as much as a clue about the quantities these ideas represent. The fear is that desired behaviors such as counting the tens and ones pieces and writing the amount shown, or marking the number in the tens place, or even using base-ten models in assorted trading games and activities, can all be exhibited with only surface-level understanding.
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.
Patterns and Relationships
This is the section of the chapter that was referred to earlier. It includes an emphasis on the hundreds chart and a variety of activities that technically involve addition and subtraction of two-digit and even three-digit numbers. My own bias is that you may well want to explore nearly every activity in this section and perhaps spend less time on the earlier portion of the chapter.
Working with Money
This is a completely new to this edition of the book. Working with money has long frustrated primary teachers, principally due to the fact that the money activities in traditional books requires children to understand patterns and relationships in place value and to be able to add numbers mentally while looking at coins rather than at numbers. The traditional curriculum does not provide an adequate foundation for these skills. The discussion and activities in this section are designed to bridge this gap.
Number Sense and Large Numbers
The activities under these headings are much more reflective and open-ended than those that come earlier in the chapter. We need to find more ways to build base-ten ideas into estimation, data collection, measurement, and connections to the world outside of classrooms. That these activities come late in the chapter should not diminish their importance.