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If you have not read the "main ideas" section for Chapter 12 in this book, you should probably do so now. Pages 201 through 209 in Chapter 12 of the text are filled with activities that might be called readiness for flexible or invented computational strategies. In the previous edition many of these activities were found in this chapter. With this edition I am trying to emphasize the value of integrating the development of place value with the development of invented strategies.

In this chapter, I have tried to show a natural progression of strategies from the most primitive direct modeling (counts by ones to use of base-ten models), to invented strategies (supported by written work or done mentally) and finally the traditional algorithms. The chapter includes sufficient detail to help students with traditional algorithms but situates them last in the development of each operation. See the quotation concerning "computational fluency" from Principles and Standards (T-75). I believe this should be a key agenda for teaching computation.

Mental strategies are not distinguished from those that are supported with jottings on paper. In the past there have been hard lines separating mental computation and the traditional pencil-and-paper algorithms. Mental computation is often linked - I believe inappropriately - with computational estimation. Further, strategies that students might begin with naturally that are often inefficient have been ignored all together.

Invented Strategies

The thrust of this chapter is supported by the research cited on page 218. Students can and do invent useful strategies for computation. With proper guidance and the time to work through these ideas, a given classroom may easily develop a variety of strategies for any one of the operations. Some of these can be done mentally by some children. They all enhance students' understanding of place value. They are generally adequate for most real-world purposes and for typical high-stakes tests. Once students have developed one or more methods, the traditional algorithm can also be discussed and in a much more meaningful manner than had the invention period taken place.

Transparency T-76 can be used to help contrast invented strategies with the traditional algorithms. These ideas will be new to students an you will need to use several examples to help them see what this discussion is about. (Also on pp. 218-219. The benefits of invented strategies are listed on T-77 and on p. 219.

When the traditional algorithms appear in the classroom by way of siblings or parents, the text suggests that these methods should also be examined and included as one of a variety of choices. I promote a firm rule: If you use it, you must understand why it works and be able to explain it. It is also useful for your students to see how quite frequently the traditional algorithms are not as efficient as invented strategies.

General Suggestions for Developing Invented Strategies

Four general suggestions are given for helping students develop invented strategies:
1. Integrate computation with place value development (the activities found in Chapter 12, pp. 201-209).
2. Use story problems to encourage useful strategies.
3. Use the three-part lesson structure (problem-based lessons with an emphasis on sharing student strategies).
4. Record students' ideas so that all can benefit.

These are outlined on T-78 and might be useful for a general discussion.

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