In grades 5-8, the study of mathematics should include opportunities to communicate so that students can--

• model situations using oral, written, concrete, pictorial, graphical, and algebraic methods;
• reflect on and clarify their own thinking about mathematical ideas and situations;
• develop common understandings of mathematical ideas, including the role of definitions;
• use the skills of reading, listening, and viewing to interpret and evaluate mathematical ideas;
• discuss mathematical ideas and make conjectures and convincing arguments;
• appreciate the value of mathematical notation and its role in the development of mathematical ideas.

Focus

The use of mathematics in other disciplines has increased dramatically, largely because of its power to represent and communicate ideas concisely. Society's increasing use of technology requires that students learn both to communicate with computers and to make use of their own individual power as a medium of communication. The ability to read, write, listen, think creatively, and communicate about problems will develop and deepen students' understanding of mathematics.

Middle school students should have many opportunities to use language to communicate their mathematical ideas. The communication process requires students to reach agreement about the meanings of words and to recognize the crucial importance of commonly shared definitions. Opportunities to explain, conjecture, and defend one's ideas orally and in writing can stimulate deeper understandings of concepts and principles. It is essential that mathematical concepts be firmly attached to the symbols that represent them; the need for symbolic representation arises out of the exploration of these concepts. In the process of discussing mathematical concepts and symbols, students become aware of the connections between them. Unless students frequently and explicitly discuss relationships between concepts and symbols, they are likely to view symbols as disparate objects to be memorized.

As students progress from grade 5 to grade 8, their ability to reason abstractly matures greatly. Concurrent with this enhanced ability to abstract common elements from situations, to conjecture, and to generalize--in short, to do mathematics--should come an increasing sophistication in the ability to communicate mathematics. But this development cannot occur without deliberate and careful acquisition of the language of mathematics.

Discussion

Communication involves the ability to read and write mathematics and to interpret meanings and ideas. Writing and talking about their thinking clarifies students' ideas and gives the teacher valuable information from which to make instructional decisions. Emphasizing communication in a mathematics class helps shift the classroom from an environment in which students are totally dependent on the teacher to one in which students assume more responsibility for validating their own thinking.

Teachers foster communication in mathematics by asking questions or posing problem situations that actively engage students, including situations that encourage students to create problems themselves. Small-group work, large-group discussions, and the presentation of individual and group reports--both written and oral--create an environment in which students can practice and refine their growing ability to communicate mathematical thought processes and strategies. Small groups provide a forum in which students ask questions, discuss ideas, make mistakes, learn to listen to others' ideas, offer constructive criticism, and summarize their discoveries in writing. Whole-class discussions enable students to pool and evaluate ideas, record data, share solution strategies, summarize collected data, invent notations, hypothesize, and construct simple arguments. For example, a teacher might present the class with the following situation:

A national magazine surveyed teenagers to determine the number of hours of TV they watched each day. How many hours do you think the magazine reported?

Students can discuss their predictions in small groups, write summaries of their group work or of their own ideas, share their predictions with the class, discuss their reasoning, and compare their predictions with the magazine's report. This exercise encourages students to evaluate the magazine report, discuss appropriate survey techniques, design and conduct their own survey and compare it with the national survey, prepare a written report, compare results from different groups, and evaluate their findings. As students refine their communication skills, they gain confidence in their ability to build convincing mathematical arguments.

Students' development of mathematical concepts and the language and symbols needed to describe and represent concepts is enhanced by carefully planned and orchestrated sequences of concrete situations that build a need for, and give meaning to, the symbols. These interconnections must be taught directly through varied examples and verbalization. For example, many students are intrigued by number tricks, such as that in figure 2.1: "Think of a number. Add 5 to it. Multiply the result by 2. Subtract 4. Divide by 2. Subtract the number you first thought of. I bet I can read your mind--your answer is 3." Modeling the situation with concrete materials, such as tiles and beans, builds one notion of variable and can lead to algebraic notation.

Fig. 2.1. Number trick

This example illustrates the role of written symbols in representing ideas, a concept that is developed throughout the middle school years. Students learn to use precise language in conjunction with the special symbol systems of mathematics, such as algebraic notation.

This standard is firmly tied to problem solving and reasoning. As students' mathematical language develops, so does their ability to reason about and solve problems. Moreover, problem-solving situations provide a setting for the development and extension of communication skills and reasoning ability. The following problem illustrates how students might share their approaches in solving problems:

The class is divided into small groups. Each group is given square pieces of grid paper and asked to make boxes by cutting out pieces from the corners. Each group is given 20 x 20 grid paper. See figure 2.2. Students cut and fold the paper to make boxes sized 18 x 18 x 1, 16 x 16 x 2, ... , 2 x 2 x 8. They are challenged to find a box that holds the maximum volume and to convince someone else that they have found the maximum. Groups are encouraged also to explore other grid sizes, such as 19 x 19 or 24 x 24.

Fig. 2.2. Building a grid-paper box

Some groups might decide not to limit themselves to boxes that are cut on the lines. Others might make a graph of the volume as compared to the height of the box. One group might decide to see what happens when they use the scraps left over from the corners. In a class discussion, students share their explicit findings, from which they eventually extrapolate generalizations. Students recognize that their solutions depend on the way in which they define the problem. These kinds of explorations provide opportunities for students to write about their ideas and the generalizations they have made.

Teachers' questioning techniques should help students construct connections among concepts, procedures, and approaches. Questions that limit answers to recitation of a single number, a simple yes or no, or a memorized procedure do not teach students the communication skills they will need. Consider instead the following exercises:

Give examples of a rectangle with four congruent sides; a parallelogram with four right angles; a trapezoid with two equal angles; a number between 1/3 and 1/2; a number with a repeating decimal representation; a jacket for a cube; an equation for a line that passes through the point (-1, 2).

These more open-ended problems can have several correct answers and can promote opportunities for students to write about their ideas, discuss interpretations, and expand their understandings.

An interchange occurs between common and mathematical language. Mathematical language builds on the existing structure and logic of common language and connects students' experiences and language to the mathematical world. Terms whose meanings change from one language to another must be addressed straightforwardly. For example, the use of such terms as improper fraction and right angle as mathematical descriptions can be misleading to students, who relate them to the common meanings of the words improper and right.

The NCTM position statement "Mathematics for Language Minority Students" (NCTM 1987) states that "cultural background or difficulties with the English language must not exclude any student from full participation in the school's mathematics program." Students whose primary language is not standard English may require special support to facilitate their learning of mathematics.

The ability to read, listen, think creatively, and communicate about problem situations, mathematical representations, and the validation of solutions will help students to develop and deepen their understanding of mathematics.