Connections Standard for Grades 3–5

Two big ideas that recur throughout the study of mathematics in grades 3–5 were elaborated on in the introduction at the beginning of the chapter: equivalence and multiplicative reasoning. Each should receive major emphasis at this level, in part because each is connected to so many topics studied in grades 3–5. For example, students learn that a fraction has an equivalent decimal representation, that the area of a right triangle is equal to half of the area of a related rectangle, that 150 centimeters is the same as 1.5 meters, and that the likelihood of getting heads when flipping a coin is the same as the likelihood of rolling an even number on a number cube. Some equivalences are not obvious to students and thus prompt further exploration to understand "why." As equivalence continues to emerge in the study of different mathematical content areas, it fosters a sense of unity and connectedness in the study of mathematics. Likewise, as students solve problems as diverse as counting the possible combinations of shirts and shorts in a wardrobe and measuring the area of a rectangle, they begin to see and use a similar multiplicative structure in both situations. Their work in developing computational algorithms highlights properties of multiplication that they can model geometrically, reason about, and express in general terms. Thus, multiplicative structures connect ideas from number, algebra, and geometry. Equivalence and multiplicative reasoning help students see that mathematics is not a set of isolated topics but rather a web of closely connected ideas.

Real-world contexts provide opportunities for students to connect what they are learning to their own environment. Students' experiences at home, at school, and in their community provide contexts for worthwhile mathematical tasks. For example, ideas of position and direction » such as those used in walking from one place to another can be used to develop the geometric idea of using coordinates to describe a location. In a fourth-grade class, students could make a map on a coordinate system of the various routes they use to walk to school. With the map, they could determine and compare the distances traveled. Everyday experiences can also be the source of data. In a fifth-grade classroom, students may want to investigate questions about after-school activities. How many students participate in such activities? What are the activities? How frequently do they participate? Is the level of participation consistent across the year? Is there a way to describe the class on the basis of their activities? Encouraging students to ask questions and to use mathematical approaches to find answers helps them see the value of mathematics and also motivates them to study new mathematical ideas.

There are connections within mathematics, and mathematics is also connected to, and used within, other disciplines. Building on these connections provides opportunities to enrich the learning in both areas. For example, in a social studies unit, a fifth-grade class might discuss the population and area of selected states. They can investigate which states are most and least crowded. By using almanacs, Web-based databases, and maps, they can collect data and construct charts to summarize the information. Once the information is collected, they will need to determine how to consider both area and population in order to judge crowdedness. Such discussions could lead to an informal consideration of population density and land use.

In grades 3–5, students should be developing the important processes needed for scientific inquiry and for mathematical problem solving—inferring, measuring, communicating, classifying, and predicting. The kinds of investigations that enable students to build these processes often include significant mathematics as well as science. It is important that teachers stimulate discussion about both the mathematics and the science ideas that emerge from the investigations, whether they occur in a science lesson or a mathematics lesson. For example, students might study the evaporation of liquid from an open container. How does the volume of liquid in the container change over time? From which type of jar does 100 cubic centimeters of water evaporate faster—one with a large opening or one with a small opening? Figure 5.36 (Goldberg 1997, p. 2) shows the results of an experiment to examine this question. The table shows the volume of water in each jar over a five-day period. Is there a pattern in the data? If so, what are some ways to describe the pattern? How many days will it take for all the water in jar 1 to evaporate? A discussion about why the water evaporates faster from a wider container and what might happen if certain conditions are altered—for example, if a fan is left blowing on the containers—integrates concepts of both mathematics and science.

Fig. 5.36. Results of an evaporation experiment

Students at all grade levels and in every domain of science should have the opportunity to use scientific inquiry and develop the ability to think and act in ways associated with inquiry, including asking questions, planning and conducting investigations, using appropriate tools and techniques to gather data, thinking critically and logically about relationships between evidence and explanations, constructing and analyzing alternative explanations, and communicating scientific arguments.

At times, opportunities for mathematical investigations arise spontaneously in class. For example, after a fifth-grade class spent some time learning about environmental issues, a question arose as to whether the water fountain was an efficient way of getting water to students. The class formulated a plan to respond to the question. This included estimating how much water the fountain released during a "typical" turn at the fountain and how much water was actually consumed. In a situation like this, the teacher plays an important role in helping students understand and think about the scientific and mathematical topics that this investigation evokes. »

Although the teacher's role includes being alert and responsive to unexpected opportunities, it is also important that teachers plan ahead to integrate mathematics into other subject areas and experiences that students will have during the year. Consider, for example, the following episode, adapted from Russell, Schifter, and Bastable (1999).