In grades K-4, the study of mathematics should include opportunities to make connections so that students can--

• link conceptual and procedural knowledge;
• relate various representations of concepts or procedures to one another;
• recognize relationships among different topics in mathematics;
• use mathematics in other curriculum areas;
• use mathematics in their daily lives.

Focus

This standard's purpose is to help children see how mathematical ideas are related. The mathematics curriculum is generally viewed as consisting of several discrete strands. As a result, computation, geometry, measurement, and problem solving tend to be taught in isolation. It is important that children connect ideas both among and within areas of mathematics. Without such connections, children must learn and remember too many isolated concepts and skills rather than recognizing general principles relevant to several areas. When mathematical ideas are also connected to everyday experiences, both in and out of school, children become aware of the usefulness of mathematics.

A classroom in which making connections is emphasized exhibits several notable characteristics. Ideas flow naturally from one lesson to another, rather than each lesson being restricted to a narrow objective. Lessons frequently extend over several days so that connections can be explored, discussed, and generalized. Once introduced, a topic is used throughout the mathematics program. Teachers seize opportunities that arise from classroom situations to relate different areas and uses of mathematics. Children are asked to compare and contrast concepts and procedures. They are helped to construct bridges between the concrete and the abstract and between different ways of representing a problem or concept. Learning and using mathematics are important aspects of the entire school curriculum.

Discussion

When children enter school, they have not segregated their learning into separate school subjects or topics within an academic area. Thus, it is particularly important to build on the wholeness of their perspective of the world and expand it to include more of the world of mathematics. This can be done in many ways, both within and outside the realm of mathematics.

Young children understand the underlying structure of many numerical problems and use counting to solve them. It is important to tie these conceptual ideas to more abstract procedures such as adding and subtracting. If conceptual understandings are linked to procedures, children will not perceive of mathematics as an arbitrary set of rules; will not need to learn or memorize as many procedures; and will have the foundation to apply, re-create, and invent new ones when needed. For example, if children are asked to fold paper and describe the process, they will understand why the procedure "multiplying the numerator and denominator by the same number" yields the same ratio in an equivalent fraction. See figure 4.1.

Fig. 4.1

Many concrete and pictorial models of concepts and procedures are available, and children need to create relationships among them and determine how each can be represented with symbols. For example, young children need to make the connection between seven toy cars, seven counters, seven tally marks, and the symbol 7. Older children need to understand the similarity between cutting a rectangle into four equal parts and sharing a bag of cookies among four friends and why the parts in each situation are called fourths. They need to see different representations of the same problem situation, as in figure 4.2.

Fig. 4.2

Children tend to think of mathematics as computation. One way to dispel this incorrect notion is to offer them more experiences with other topics; even so, unless connections are made, children will see mathematics as a collection of isolated topics. Only through extended exposure to integrated topics will children have a better chance of retaining the concepts and skills they are taught. For example, measurement situations should continually be part of the program, rather than introduced briefly in isolated lessons. The following activity integrates geometry with measurement. See figure 4.2a.

Fig. 4.2a

Cut a 12-by-16-cm rectangle on a diagonal as shown. What geometric shapes can you make? Which one has the shortest perimeter?

Similarly, addition practice can be placed in the context of measuring as children solve for the distances between cages at a zoo (see fig. 4.3).

Fig. 4.3

Another connection children can explore is that between solutions to open number sentences and graphing, as shown in figure 4.4.

Fig. 4.4

The K-4 program is rich with opportunities to use mathematics in other subject areas as well as to use other subjects in mathematics. This is especially true with science, but with a little imagination connections can be made to all areas. For example, the communication standard (Standard 2) calls for the integration of language arts as children write and discuss their experiences in mathematics. As children solve problems in mathematics classes, they can be learning about other countries and cultures. As children measure how far they can jump, they use mathematics in physical education. As children do art projects, they use geometry and measurement. See figure 4.5

Fig. 4.5

All too often, children come to believe that mathematics is an academic exercise that occurs only in schools, whereas solving problems outside of school is different. Many believe that it is not mathematics to explore the meaning of one-third by sharing a pitcher of milk equally among three people; to count on a clock face how long until it is time to go to a friend's house; or to figure 100 divided by 4 by thinking, "Four quarters make a dollar, so it's 25," or "It's 100 divided by 2 and 2 again." Mathematical methods exist to solve these problems in an efficient manner, but, at times, these are not as satisfactory as the informal ways. Students need to see when and how mathematics can be used, rather than be promised that someday they will use it.