In grades 5-8, the mathematics curriculum should include the investigation of mathematical connections so that students can--

• see mathematics as an integrated whole;
• explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations;
• use a mathematical idea to further their understanding of other mathematical ideas;
• apply mathematical thinking and modeling to solve problems that arise in other disciplines, such as art, music, psychology, science, and business;
• value the role of mathematics in our culture and society.

Focus

For many students, mathematics in the middle grades has far too often simply repeated or extended much of the computational work covered in the earlier grades. The intent of this standard is to help students broaden their perspective, to view mathematics as an integrated whole rather than as an isolated set of topics, and to acknowledge its relevance and usefulness both in and out of school. Mathematics instruction at the 5-8 level should prepare students for expanded and deeper study in high school through exploration of the interconnections among mathematical ideas.

Students should have many opportunities to observe the interaction of mathematics with other school subjects and with everyday society. To accomplish this, mathematics teachers must seek and gain the active participation of teachers of other disciplines in exploring mathematical ideas through problems that arise in their classes. This integration of mathematics into contexts that give its symbols and processes practical meaning is an overarching goal of all the standards. It allows students to see how one mathematical idea can help them understand others, and it illustrates the subject's usefulness in solving problems, describing and modeling real-world phenomena, and communicating complex thoughts and information in a concise and precise manner. Different representations of problems serve as different lenses through which students interpret the problems and the solutions. If students are to become mathematically powerful, they must be flexible enough to approach situations in a variety of ways and recognize the relationships among different points of view.

This standard precedes those that discuss specific mathematical content in order to stress the importance of viewing the standards as an integrated whole, not as a list of content areas. If the standards are interpreted as a listing of topics to be covered sequentially, it is likely that in most classrooms there will be insufficient time to cover them all. Instead, implementation of the standards should be organized in such a way that several goals will be addressed simultaneously.

Discussion

Connections among various mathematical topics can be drawn in many ways. When a student makes and describes the translation in geometry "20 to the right" and follows it with a second translation, "45 to the left," the result is fundamentally identical to adding integers. Various interpretations of fractions can illustrate connections to measurement, ratios, and ideas in algebra. As students study one topic, relationships to other topics can be highlighted and applied. Connections also can emerge as students do mathematics. The development and exploration of patterns in Pascal's triangle, for example, can be used to illustrate relationships among counting, exponents, algebra, geometric patterns, probability, and number theory. Although these connections should not be made in a formal way at the 5-8 level, teachers can foster an informal familiarity with them through problem solving: "Can you find the triangular numbers in the Pascal triangle? Why do they appear? Does Pascal's triangle have anything to do with the probability of getting two heads in three flips of a coin? Why?"

This persistent attention to recognizing and drawing connections among topics will instill in students an expectation that the ideas they learn are useful in solving other problems and exploring other mathematical concepts. For example, a knowledge of area can help them in understanding the operations on fractions, representing statistical data, solving proportion problems, finding factors and probabilities, and exploring the meanings of algebraic expressions. As they learn new ideas or solve new problems, students enrich their own thought processes and skills by drawing on previously developed ideas; this ability to integrate ideas and concepts fosters students' confidence in their own thinking as well as in their skills of communication. Curriculum materials can foster an attitude in students that will encourage them to look for connections, but teachers must also look for opportunities to help students make mathematical connections.

Technology is useful for identifying connections. For example, in investigating computer-generated spirals of line segments, students might find that the size of the initial angle, which they enter into the computer, is related to the "shape" of the spiral; for example, an angle of 90 degrees generates a "square" spiral. Because the computer allows students to enter countless values and immediately see the resulting geometric shape, they might find it both interesting and rewarding to investigate interrelationships between number and geometry: An angle of 72 degrees gives a five-sided spiral, and 2 x 72 = 144 degrees gives a five-sided star.

Varied problem settings are a means by which students can highlight and build mathematical connections. A lesson on measurement can be an occasion for students to formulate and solve problems while exploring geometric, measurement, and algebraic ideas. Consider the following situation with pattern blocks (fig. 4.1):

Fig. 4.1. Situation A

Describe the pattern. What questions can we ask about the pattern?

This situation and that in figure 4.2 allow students to make conjectures and to convince others that their ideas are valid. In addition, questions about the area (number of squares) or the perimeter of the tenth or the nth term in the patterns are likely to arise.

Fig. 4.2. Situation B

We can represent the changing area (the number of triangles) or perimeter by a number sequence, a verbal description, or an algebraic rule. If we want to compare the growth of the perimeter and the number of triangles needed, we can make a graph on the same axes. The graph for situation B is shown in figure 4.3.

Fig. 4.3. Pattern growth

This type of interconnected pattern work can lead to surprises for both teachers and students. Students often identify patterns very different from those the teacher had anticipated. This creative aspect of making sense of mathematics is a real confidence builder for students. Such situations also allow the class to explore various ways to model the relationships mathematically and determine how these models are alike and how each can highlight a different aspect of the problem. The more rapid growth of the area in situation B can be seen from the numbers, but the graph shows the change in a more dynamic way. One of the most important connections students in grades 5-8 should understand is this relationship between data in tables, algebraic generalizations, and graphical representations.

Many opportunities to show the connections between mathematics and other disciplines are missed in school. Mathematics arises not only in science but in other subjects as well. In social studies, for example, the study of maps is an excellent time to also study scaling and its relation to the concepts of similarity, ratio, and proportion. A topic such as measurement has implications for social studies, science, home economics, industrial technology, and physical education and is increasingly important to teachers of these subjects.

"Connected" mathematics should not be disconnected from students' daily lives. For example, although the "handshake" problem can be used to show connections between triangular numbers and the diagonals of a polygon, classroom discussion might focus on reasons why airlines use "hub cities" to map routes among the cities they serve. As students in grades 5-8 become aware of the world around them, probability and statistics become increasingly important connections between the real world and the mathematics classroom. Weather forecasting, scientific experiments, advertising claims, chance events, and economic trends are but a few of the areas in which students can investigate the role of mathematics in our society. Statistics offer students insights into problems of social equity. Perspective, proportion, and the golden ratio are ways of learning mathematics in the context of art and design. Whatever the context, a vital role of mathematics education is to instill in students an attitude of inquiry and investigation and a sensitivity to the many interrelationships between formal mathematics and the real world.