Instructional programs from prekindergarten through grade 12 should enable all students to— create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; use representations to model and interpret physical, social, and mathematical phenomena.

In grades 3–5, students need to develop and use a variety of representations of mathematical ideas to model problem situations, to investigate mathematical relationships, and to justify or disprove conjectures. They should use informal representations, such as drawings, to highlight various features of problems; they should use physical models to represent and understand ideas such as multiplication and place value. They should also learn to use equations, charts, and graphs to model and solve problems. These representations serve as tools for thinking about and solving problems. They also help students communicate their thinking to others. Students in these grades will use both external models—ones that they can build, change, and inspect—as well as mental images.

What should representation look like in grades 3 through 5?

Students in grades 3–5 should continue to develop the habit of representing problems and ideas to support and extend their reasoning. Such representations help to portray, clarify, or extend a mathematical idea by focusing on essential features. Students represent ideas when they create a table of data about weather patterns, when they describe in words or with a picture the important features of an object such as a cylinder, or when they translate aspects of a problem into an equation. Good representations fulfill a dual role: they are tools for thinking and instruments for communicating. Consider the following problem:

What happens to the area of a rectangle if the lengths of its sides are doubled?

p. 206

Students who represent the problem in some way are more likely to see important relationships than those who consider the problem without a representation. One student's initial response to the problem was that the new rectangle would be twice the size of the first rectangle. Her thinking might have stopped there, but another student questioned her answer, prompting her to think more deeply. She decided she needed a picture to help her think about the problem. Her drawing (see fig. 5.38) helped her consider the complexity of the problem more carefully and showed her that the new rectangle is not only bigger but that it is four times bigger than the original rectangle. It was also a way to show her answer and to justify it to others. »

 Fig. 5.38. A student's representation of the results of doubling the lengths of the sides of a rectangle

Students will have learned about, and begun to use, many symbolic and graphical representations (e.g., numerals, equals sign, and bar graphs) in the primary grades. In grades 3–5, students should create representations that are more detailed and accurate than is expected in the primary grades. Their repertoire of symbols, tools, and conventional notation should expand and be clearly connected to concepts as they are explored. For example, in representing algebraic and numerical relationships, students should become comfortable using equations and understanding the equals sign as a balance point in the equation. Many students who have only seen equations with an arithmetic expression on the left side of the equation and a call for the numerical answer on the right side, such as 630 = , don't understand that equations may have several symbols on each side, as in 256 = 345.

Students in grades 3–5 should also become familiar with technological tools such as dynamic geometry software and spreadsheets. They should learn to set up a simple spreadsheet (see fig. 5.39) and use it to pose and solve problems, examine data, and investigate patterns. For example, a fourth-grade class could keep track of the daily temperature and other features of the weather for the whole year and consider questions such as these: What month is coldest? What would we tell a visitor to expect for weather in October? After two months, they might find that they are having difficulty managing and ordering the quantity of data they have collected. By entering the data in a spreadsheet, they can easily see and select the data they want, compare certain columns, or graph particular aspects of the data. They can conveniently find the median temperature for February or calculate the total amount of rainfall for April. In January, if the class notices that temperature alone is no longer giving them enough information, they can add a column for wind chill to get a more accurate summary of the weather they are experiencing.

 Fig. 5.39. A simple spreadsheet can be used to organize and examine data, pose and solve problems, and investigate patterns.

p. 207

Learning to interpret, use, and construct useful representations needs careful and deliberate attention in the classroom. Teaching forms of representation (e.g., graphs or equations) as ends in themselves is not productive. Rather, representations should be portrayed as useful tools » for building understanding, for communicating information, and for demonstrating reasoning (Greeno and Hall 1997). Students should become flexible in choosing and creating representations—standard or nonstandard, physical models or mental images—that fit the purpose at hand. They should also have many opportunities to consider the advantages and limitations of the various representations they use.

What should be the teacher's role in developing representation in grades 3 through 5?

Learning to record or represent thinking in an organized way, both in solving a problem and in sharing a solution, is an acquired skill for many students. Teachers can and should emphasize the importance of representing mathematical ideas in a variety of ways. Modeling this process as they work through a problem with the class is one way to stimulate students to use and analyze representations. Talking through why some representations are more effective than others in a particular situation gives prominence to the process and helps students critique aspects of their representations. Teachers can strategically choose student representations that will be fruitful for the whole class to discuss. For example, consider the following question, which a third-grade class might explore:

Are there more even or odd products in the multiplication table shown in figure 5.41? Explain why.

Students may initially generate many examples to formulate an answer, as illustrated in figure 5.40. Other students may use a multiplication table to organize their work, as illustrated in figure 5.41. Organizing the work in this way highlights patterns that support students in thinking more systematically about the problem.

 Fig. 5.40. An exploration of odd and even numbers in the multiplication table

 Fig. 5.41. Using a multiplication table to solve an "odd and even numbers" problem

p. 208

Each representation reveals a different way of thinking about the problem. Giving attention to the different methods as well as to the different representations will help students see the power of viewing a problem from different perspectives. Observing how different students select and use representations also gives the teacher assessment information about what aspects of the problem they notice and how they reason about the patterns and regularities revealed in their representations. »

As students discuss their ideas and begin to develop conjectures based on representations of the problem, the teacher might want to represent the students' thinking in other ways in order to support and extend their ideas. For example, when students notice that an even number multiplied by an even number always produces an even result, the teacher might record this idea as "eveneven = even." This representation serves as a summary of the students' thinking. It suggests a way to record the generalization and may prompt students to look for other generalizations of the same type.

Some students will need explicit help in representing problems. Although in the rectangle problem (fig. 5.38), the student quickly decided on a representation that was effective in showing the important relationships, many students need support in constructing pictures, graphs, tables, and other representations. If they have many opportunities for using, developing, comparing, and analyzing a variety of representations, students will become competent in selecting what they need for a particular problem.

As students work with a variety of representations, teachers need to observe carefully how they understand and use them. Representations do not "show" the mathematics to the students. Rather, the students need to work with each representation extensively in many contexts as well as move between representations in order to understand how they can use a representation to model mathematical ideas and relationships.

p. 209

By listening carefully to students' ideas and helping them select and organize representations that will show their thinking, teachers can help students develop the inclination and skills to model problems effectively, to clarify their own understanding of a problem, and to use representations to communicate effectively with one another. »