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The properties of equality, meanings of the operations, and operation properties are a significant portion of the content of algebraic thinking as described in Chapter 15, Algebraic Thinking: Generalizations, Patterns, and Functions. You may want to look at Chapter 15 before embarking on this chapter. Some mathematics educators begin with algebraic thinking rather than number and operations and use that as a framework for these chapters.

To view addition and subtraction in the early grades as simply "put together" and subtraction is "take away" is unfortunately all too common in traditional texts. Later, by the time multiplication and division are encountered, teachers no longer require students to model and to think through or analyze the problems they encounter.

This chapter encourages the use of two different approaches to developing operation meaning: the use of word problems and the use of models. While some would argue for one approach over the other, the view taken here is similar to that developed by the CGI group. That is, the best approach to meaning is through word problems. The models are offered as a more directed approach primarily to connect symbolism to the operations or to emphasize important properties such as commutivity.

I strongly encourage you to look at the detailed table of contents for this chapter listing all of the first-level headings. This will help give you a feel for how the chapter is organized. While this may sound like a simple topic, I have consistently found it to be challenging for teachers.

Additive and Multiplicative Structures

Within the chapter you will find a parallel development of additive problems (addition and subtraction) and multiplicative problems (multiplication and division). For each category, story problems of all variations found in the research on this topic are compared to a general schematic model. This information is strictly for teachers and is almost always new and challenging for them.

Teaching the Meanings of the Operations

For both additive and multiplicative structures, this section first discusses the use of contextual problems and then model-based problems. (Fosnot and Dolk distinguish contextual problems from more sterile "story problems.") Selected properties of the operations are also discussed. I only addressed those properties that seemed most useful to students as they expand their understanding of the operations and computation. Note that is the operation properties that are the focus of generalizations made in algebra.

Addition and Subtraction Issues

An important idea to develop is to help students connect the concepts of addition and subtraction. I explicitly suggest not using the phrase "take away" for subtraction but rather "minus" or "subtract." Similarly, addition is not defined as a join action and subtraction as remove. The reason is that there are joining action problems that can be classified as subtraction and remove problems that require addition. This is one of the values of seeing the full 14 types of additive problems.

By way of example, if a first or second grade problem involves a missing part, the syntax may suggest 4 + [ ] = 9 (Join, Change unknown) . But that is a subtraction situation and with large numbers, subtraction must be used to solve the equation in the computational form: 9 - 4 = [ ]. How do you get children to recognize the equivalence of these equations? These and other subtle questions are generally beyond the novice or pre-service teacher.

As students begin to write equations, there is an unfortunate but extremely strong tendency to see the equal sign as a symbol that must precede the answer. The correct meaning of the equal sign is then very difficult to develop. Here is another place where you may want to make explicit connections to the ideas around the equal sign developed in Chapter 15.

The most important idea in my mind is to view subtraction as missing part. That is, subtraction is used to name or find a part of the whole when the whole and one of the parts is known. This definition plays an enormous role in the development of subtraction facts via a think-addition strategy. When seeing 9 - 6, the idea of thinking "6 and what make 9?" is quite foreign to most teachers. Work on this issue is well worth it, now and in the following chapter on basic fact mastery.

Multiplication and Division Issues

The most difficult issue for teachers is to distinguish between the two different concepts of division - those generally referred to as partition (fair shares) and measurement (repeated subtraction). I have never thought it terribly important that teachers can name division situations as either partition or measurement but that they be able to understand and distinguish the two situations. With some care, the two schematic models can help. As suggested by the Pause and Reflect on p. 153, it is also helpful for teachers to match each of the story problems with the schematics and to model them using counters.

The value of distinguishing between the role of the first and second factor may be open to dispute. I agree with the authors of Investigations that there is no value at all in worrying children over this convention. That there is a distinction is more important than which factor stands for the number of sets. This issue becomes clearer when trying to distinguish between partition and measurement forms of multiplication. I have found it most successful to almost ignore completely the terms "measurement" and "partition" or any synonyms. Rather, I want students to do things with models, and write a multiplication and a division equation to go with the model. Then they are able to say, for example, "it was the size of the set (or number of sets) that was unknown." The array is a model that is especially important for multiplicative structures and I like to be sure that teachers both see this model and its values (especially for the commutative property) but also contrast this with the number line and set models.

With the multiplicative structures the word stories pose additional difficulties for students. Those involving equal sets are fine when discrete objects are involved. However, this same structure applies to prices, speeds, measurement conversions, and others. The research suggests that children do not well understand rates. My experience says that many elementary teachers do not understand them either. Word stories with a multiplicative comparison structure tend to be a bit easier.

Another issue worthy of attention is the handling of remainders. You will find that this is also something that teachers have never considered.

Issues Across All Operations

The text addresses three additional issues. First, how do you help students in the upper grades know what operation to use? The simple approaches suggested for younger children do not apply to problems with large numbers, decimals and fractions. Second, what is the value of a key word approach? Almost none. In fact, such an approach has several negative effects. Finally, multi-step problems pose a special difficulty. Some useful suggestions based on research are offered for each of these issues.

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