By Fennell, Francis; Rowan, Tom
Teaching Children Mathematics , Vol. 7, No. 5
Representation is more than a process; it is a way of teaching and learning mathematics.
Vignette: Four beginning third graders were working to solve a story problem: Anna had 63 stickers. She gave some to Tricia. Then Anna had 27 stickers. How many stickers did Anna give to Tricia? Brett said, "Well, 27 and 3 make 30, 30 more makes 60, and 3 more makes 63, so it's 3 + 30 + 3 = 36." Stacey wrote her solution on her paper as 60 - 20 = 40. She crossed out 40 and wrote 30 in its place; next she wrote 13 - 7 = 6, then the difference of 36. Heather, using Unifix cubes, gathered 6 ten-sized rods and 3 single cubes, took 2 away, then broke 1 of the remaining 4 rods into ones and took 7 away. She had 36 left. Maria added 4 to 63 to make 67, then added 4 to 27 to make 31. She said that seeing the difference of 36 was easy. These four students, all in the same grade at the same school, used different strategies and representations to solve the problem.
What Is Representation?
Representation is one of five mathematical processes presented in Principles and Standards for School Mathematics (NCTM 2000). These processes may be thought of as the "filter" through which the five Content Standards are developed. They should guide the ways in which we teach mathematics and support students' learning.
Teachers and students easily recognize the importance of the mathematics content topics in a particular grade or grade band but are less likely to recognize and understand the role of the mathematical processes. In the twelve years since the publication of Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), curriculum developers, textbook publishers, and teachers have ignored the fact that content and process are intertwined. Students cannot solve problems without knowing the mathematical content conveyed in those problems, nor can they solve problems without using mathematical processes.
The Process Standards provide essential support for learning mathematics content. The writers of Principles and Standards reaffirmed the importance of the four Process Standards--Problem Solving, Reasoning, Connections, and Communication--articulated in the NCTM's 1989 Curriculum and Evaluation Standards. They also acknowledge the central importance of the process of mathematical representation. In the 1989 Standards document, representation was discussed as part of the communication standard. Recognizing the important role that it plays as both a tool for communication and a tool for thinking (see fig. 1), representation is given more prominence in the updated Principles and Standards.
In the opening vignette of this article, the third graders used representations to model the action of a story problem that they understood. Their representations helped them solve the problem and share their thinking with others. The students used mental mathematics, paper and pencil, linking cubes, and pictures to represent their actions as they solved the problem. Each representation gave the student who used it a means for understanding and thinking through the problem. Such representations are essential in enabling children to analyze problems and find ways to solve them. Note that the representations used by the children clearly grew out of their own thinking. This aspect is an essential component of good representations--they should represent how the children are thinking about a problem. In some instances, children are taught to use concrete materials as the only way to solve a problem, and such materials may come to replace the child's thinking rather than represent it. As a result, the materials ma y actually interfere with learning or, at the least, become an alternative way of solving problems rather than a pathway to understanding mathematics.
When students are able to represent a problem or mathematical situation in a way that is meaningful to them, the problem or situation becomes more accessible. …