In grades K-4, the mathematics curriculum should develop whole number computation so that students can--

• model, explain, and develop reasonable proficiency with basic facts and algorithms;
• use a variety of mental computation and estimation techniques;
• use calculators in appropriate computational situations;
• select and use computation techniques appropriate to specific problems and determine whether the results are reasonable.

Focus

The purpose of computation is to solve problems. Thus, although computation is important in mathematics and in daily life, our technological age requires us to rethink how computation is done today. Almost all complex computation today is done by calculators and computers. In many daily situations, answers are computed mentally or estimates are sufficient, and paper-and-pencil algorithms are useful when the computation is reasonably straightforward. This standard addresses the importance of teaching children a variety of ways to compute, as well as the usefulness of calculators in solving problems containing large numbers or requiring complex computations. Related to this goal is the necessity of having reasonable expectations for proficiency with paper-and-pencil computation. Clearly, paper-and-pencil computation cannot continue to dominate the curriculum or there will be insufficient time for children to learn other, more important mathematics they need now and in the future.

By emphasizing underlying concepts, using physical materials to model procedures, linking the manipulation of materials to the steps of the procedures, and developing thinking patterns, teachers can help children master basic facts and algorithms and understand their usefulness and relevance to daily situations. This approach also promotes efficient learning of computational techniques and furthers the development of children's reasoning, mathematical insight, and confidence in their ability to do mathematics. Instruction also should emphasize a variety of ways to compute, the importance of checking whether computed results are reasonable, and the need to make appropriate decisions about how to compute in a problem situation. An awareness that computation is learned and used to attain some goal develops when problem situations and computations are explicitly linked throughout all aspects of work with computations.

Discussion

Strong evidence suggests that conceptual approaches to computation instruction result in good achievement, good retention, and a reduction in the amount of time children need to master computational skills. Furthermore, many of the errors children typically make are less prevalent.

Helping children develop thinking strategies for learning basic facts enables them to understand relationships and to reason mathematically. Figure 8.1 shows two examples.

Fig. 8.1

A developmental approach to computation fosters a problem-solving atmosphere in which children are actively involved in using materials, discussing their work, validating solutions, and raising questions. Placing computation in a problem-solving context motivates students to learn computational skills and serves as an impetus for the mastery of paper-and-pencil algorithms. The initial use of physical materials, such as base-ten blocks or bundling sticks, can be carefully connected to concrete models and, finally, to symbolic work. Figure 8.2 illustrates the connections that can be made between concrete materials and a paper-and-pencil algorithm.

Fig. 8.2

Mental computation and estimation offer exciting opportunities for making computation more dynamic and for developing insights into number relationships. Figure 8.3 illustrates several thinking patterns.

Fig. 8.3

Children need more time to explore and to invent alternative strategies for computing mentally. Both mental computation and estimation should be ongoing emphases that are integrated throughout all computational work. Estimation is discussed further in Standard 5.

The frequent use of calculators, mental computation, and estimation helps children develop a more realistic view of computation and enables them to be more flexible in their selection of computing methods. Calculators should be used to solve problems that require tedious calculations. Estimation and reasonableness of results need particular emphasis when students are using calculators. The following example illustrates how to design various problems so that students must check the reasonableness of their results once they have completed their work with a calculator.

Three fourth-grade teachers at Park City Elementary School decided to take all their students on a picnic. Mr. Clark spent $26.94 for refreshments. Since the three teachers wanted to share the cost of the picnic, Mr. Clark used his calculator to determine that each teacher should pay him $13.47. Is his answer reasonable? Explain.

After estimating, the students concluded that Mr. Clark was wrong because 27 divided by 3 is 9; thus, $9.00 is about what he should collect from each teacher, not $13.47.

In this example, estimation showed that the teacher's answer was in error.

Calculators also can be used as an effective instructional tool for teaching computational skills. For example:

Target Addition is a calculator game for reinforcing the recall of basic facts and mental arithmetic. After clearing the calculator's memory, two children select a target, such as 23, and take turns entering a number from 1 to 5. Each new sum is put into the memory by pressing the M+ key. A player who thinks the target number is in the memory just after his or her turn presses the memory-recall key to check.

Children should also be given many opportunities to decide whether they need an exact answer and how they will complete a computation. See figure 8.4.

Fig. 8.4

Rethinking the role of computation. The approach to computation taken in this standard requires educators to rethink traditional scope-and-sequence decisions. If they are to meet the comprehensive curricular goals articulated in the K-4 standards, for example, teachers must reduce the time and the emphasis they devote to computation and focus instead on the other mathematical topics and perspectives that are proposed.

Besides paper-and-pencil computation, children should learn when and how to use calculators and various mental arithmetic and estimation procedures. Calculators enable children to compute to solve problems beyond their paper-and-pencil skills. Mental computation and estimation techniques can be developed prior to, and in connection with, paper-and-pencil skills. It is inconsistent with the Standards to isolate paper-and-pencil procedures by focusing on them for an extended time prior to the introduction of other computing methods; this traditional practice suggests to children that computing means using paper-and-pencil methods.

Reasonable expectations for computation. Premature expectations for students' mastery of computational procedures not only cause poor initial learning and poor retention but also require that large amounts of instructional time be spent on teaching and reteaching basic skills. More important, the instructional focus centers on memorizing facts and rules for carrying out procedures rather than on the thoughtful use of operations and number relationships.

Children should master the basic facts of arithmetic that are essential components of fluency with paper-and-pencil and mental computation and with estimation. At the same time, however, mastery should not be expected too soon. Children will need many exploratory experiences and the time to identify relationships among numbers and efficient thinking strategies to derive the answers to unknown facts from known facts. Practice designed to improve speed and accuracy should be used, but only under the right conditions; that is, practice with a cluster of facts should be used only after children have developed an efficient way to derive the answers to those facts.

It is important for children to learn the sequence of steps--and the reasons for them--in the paper-and-pencil algorithms used widely in our culture. Thus, instruction should emphasize the meaningful development of these procedures, not speed of processing. The teaching of addition, subtraction, and multiplication algorithms should integrate renaming and no-renaming situations, and problems with remainders should be integrated throughout division. This approach is more efficient and eliminates some misconceptions that often occur.

Exploratory experiences in preparation for paper-and-pencil computation give children the opportunity to develop underlying concepts related to partitioning numbers, operating on the parts, and combining the results. Many such experiences can be provided in the context of using place-value materials, computing mentally, or performing computational estimation. Only after these ideas are carefully linked to paper-and-pencil procedures is it appropriate to devote time to developing proficiency by providing practice. Although the exploration of computation with larger numbers is appropriate, excessive amounts of time should not be devoted to proficiency.

Success is possible for almost all children when they receive careful instruction. Still, teachers should be sensitive to problems individual children might have and should be prepared to use a variety of methods to teach and assess computational knowledge.