In grades K-4, the mathematics curriculum should include concepts of addition, subtraction, multiplication, and division of whole numbers so that students can--

• develop meaning for the operations by modeling and discussing a rich variety of problem situations;
• relate the mathematical language and symbolism of operations to problem situations and informal language;
• recognize that a wide variety of problem structures can be represented by a single operation;
• develop operation sense.

Focus

Understanding the fundamental operations of addition, subtraction, multiplication, and division is central to knowing mathematics. One essential component of what it means to understand an operation is recognizing conditions in real-world situations that indicate that the operation would be useful in those situations. Other components include building an awareness of models and the properties of an operation, seeing relationships among operations, and acquiring insight into the effects of an operation on a pair of numbers. These four components are aspects of operation sense. Children with good operation sense are able to apply operations meaningfully and with flexibility. Operation sense interacts with number sense and enables students to make thoughtful decisions about the reasonableness of results. Furthermore, operation sense provides a framework for the conceptual development of mental and written computational procedures.

Instruction on the meaning of operations focuses on concepts and relationships rather than on computation, which is the focus of Standard 8. Children need extensive informal experience with problem situations and language prior to explicit instruction and symbolic work with the operations. Thus, informal experiences with all four operations should begin in kindergarten and continue through grade 4. Instruction should help children connect their intuitions and informal language to operations, including the mathematical language and symbols of each operation. Children should encounter the four basic operations in a wide variety of problem structures. For example, in addition to problems with joining and separating structures, teachers should provide problems involving comparing and equalizing. Time devoted to conceptual development provides meaning and context to subsequent work on computational skills.

Discussion

When most children enter school, they can use objects and counting to solve many kinds of problems. Class discussion of a wide variety of problems prepares students for explicit instruction on operations (the models, language, and symbols associated with an operation). Examples might include those in figure 7.1.

Fig. 7.1

Children draw upon their insight and intuition to represent and discuss these problems. This work helps link operations to many types of situations. Even after an operation has been introduced, this emphasis on linking it to appropriate situations should continue. For example, children might draw pictures and then tell or write stories about the equation as 18 ÷ 6 = [_]. This kind of activity emphasizes connections between mathematics and the real world and encourages children to recognize and use a variety of situations and problem structures.

Word problems also should be used to help children increase their recognition of the relationship between a single operation and problems with different structures. Although children initially might solve the following problem by drawing a picture, they should also see that because it involves separating a whole into equal parts, it can be solved using division.

Twenty-eight children are going on a picnic. Four children can ride in each car. How many cars are needed?

Connecting problem structures to operations should be emphasized throughout grades K-4 for both one-step and appropriate two-step problems. For example, multiplication is most commonly linked to the process of combining equal groups. Children also need to see that it relates to array, "times as many," and "combination" (e.g., three blouses, four skirts, how many outfits?] situations. An example of combination situations is finding the number of outfits that can be made with two blouses and three skirts.(Fig. 7.1a)

Fig. 7.1a

The language of basic operations, such as the terms addend, sum, difference, factor, multiple, product, and quotient, can be introduced and used informally in work with operations. The notions of factors and multiples can prompt interesting explorations. Children can find the factors of a number using tiles or graph paper. This can lead to an investigation of numbers that have only two factors (prime numbers) and numbers with two equal factors (square numbers)(Fig. 7.1b).

Fig. 7.1b

Multiples of a number can be shaded on hundred charts. Children can then find numbers that are multiples of 2 and multiples of 3, and thus be introduced to the concept of common multiples. Calculators can be useful in exploring multiples of a number through repeated addition. After children become familiar with finding multiples of 3 (3, 6, 9,...), they can find how many threes make 30 using repeated addition and predict how many threes make 60. The calculator is used to check their predictions. Since work with concepts of operations does not emphasize the computing of answers, calculators are a valuable tool.

Properties of an operation, a key component of operation sense, also can be explored. Children note that reversing the order of two addends does not change the sum, and they use this to solve 2 + 19 by starting with 19 and counting two more. With graph paper, they can see that 3 x 7 can be found by adding 3 fives and 3 twos. See figure 7.2. Naming properties is not necessary.

Fig. 7.2

Operation sense also involves relationships between operations. Addition and subtraction are related; for addition you find the whole, and for subtraction you find a part (see fig. 7.3).

Fig. 7.3

Multiplication and division also have an inverse relationship. The relationships between addition and multiplication and between subtraction and division should be investigated.

Operation sense also involves acquiring insight and intuition about the effects of operations on two numbers. Adding 5 to 25, for example, produces a far smaller change in size than multiplying 25 by 5. Children should sense that the sum of two numbers, each of which is greater than 50, must be greater than 100. They can explore the effect of increasing one addend by 1 and decreasing the other addend by 1, and compare this to the corresponding results in multiplication:

Understandings of the relationships between operations can be used to extend work with equations. Children with a solid understanding of operations will be able to apply this knowledge to solve such equations as 15 + [_] = 25, [_] - 15 = 15, and [_] x 25 = 50.