Home | Chapter 10 |

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.

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.

The value of distinguishing between the role of the first and second factor may be open to dispute. I agree with the authors of

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.