The computational algorithms for fractions continue to be introduced to children far too early in school. While estimation with fraction computation can and probably should begin as early as the fourth grade, there seems to be little reason for children to learn the pencil-and-paper algorithms before perhaps the seventh grade. But that is my personal opinion and certainly the teachers with whom I work cannot adopt my views when their school systems say otherwise. At the same time, It is useful for teachers to have an informed perspective on curricular issues such as this one. For that reason, it seems that a discussion of the issue of when and how much fraction computation is important. It is equally important to stress the value of informal approaches to each of the algorithms.
Guidelines for Developing Fraction Computation Strategies
Four ideas recur throughout the chapter. These are outlined on T-96 and discussed on page 317 of the book.
Present simple tasks in context. What minimal research exists on developing computational strategies suggests that students will make much more meaning out of the task when it is structured in a story problem. I find that many problems with fractions are pretty unusual or not realistic, but they do serve the purpose. Connect the ideas to those used for whole numbers. The meanings of the operations do not change from whole numbers to fractions. This is especially important for multiplication and division. By middle grades, the large number of rules that can accumulate in mathematics, many of which are related to fractions, becomes overwhelming. Children need to see that not every idea is completely new and disconnected. Here is an ideal opportunity to make that point. Capitalize on the value of estimation and informal methods. To consider about how large an answer might be does two things: First, it provides a rough check on whatever result is obtained. Second, and more importantly, it will cause the student to think about the meaning of the operation. Use models in the initial exploration of the operations. Often, the informal approaches will not be directly related to the algorithms that are to be developed. Continue to use models in the directed development of the algorithms and require that students be able to justify and explain the algorithms with the use of models.
Addition and Subtraction
The most important idea here is to help students see that when they "get a common denominator" what they are really doing is re-writing the original problem with easier notation. While clear for the mathematically literate, this is not at all clear to children and is not always obvious to pre-service teachers. The student work in Figure 17.1 is significant because the drawings that these fifth-grade students offer (their ideas) do not match the symbolic common denominator algorithms that they attempt to use.
It is important to explore this operation using a variety of models and to stress an oral interpretation of what is being done. For example, if the rectangular region model were the only one used to model the operation, it is easy to develop a mindless drawing algorithm: "Draw vertical lines and shade one of the fractions. Draw horizontal lines and shade the other faction. The region that is double shaded is the answer." While correct, there is virtually no rationale. Nor does the approach transfer to a set model or even to pie pieces. Is that really meaningful?
The discussion in the text describes a series of problem types based on whether one factor or the other is a whole number and on the relationship between the factors. Do not expect your teachers to memorize these in sequence - the book will be available for reference. Nor should you expect to see these problem types attended to in regular textbooks. However, if you are interested in students developing their own methods, these distinctions are useful.
As Liping Ma suggests, you can expect that the teachers almost certainly will not be able to offer a word story problem for a fraction divided by a fraction or even a whole number divided by a fraction. An informal modeling of these problems is equally difficult, especially if you suggest a partitioning approach. To help with exploring partition concepts with fractional divisors, the emphasis should be on the question, "How much is one?" See the discussion on pp. 326-328. Within the measurement context, whole number quotients are more easily understood. However, remainders will cause almost as much difficulty as the partition problems.
If this whole area of fraction division is so difficult, even for teachers, a good question to ask is, "How will children be able to deal with these ideas?" Certainly the answer involves the need for a lot of time and exploration, discussion among students, and a significant delay in rushing to the algorithm. The other side of this argument concerns the question, How important is this anyhow? If you choose to explore both partition and measurement tasks informally, it will be time consuming and you may not even get to the algorithm.
As with any algorithm, this one must be developed on a single concept and then applied to all contextual situations. The book presents both the common-denominator method and the invert-and-multiply algorithm with arguments on both sides of each. If your agenda in this chapter is to help teachers with a straightforward approach to teaching the algorithms (rather than a lot of informal exploration and discussion) then I would argue strongly that the common-denominator algorithm is the one to work on.