The assessment of students' ability to communicate mathematics should provide evidence that they can--

• express mathematical ideas by speaking, writing, demonstrating, and depicting them visually;
• understand, interpret, and evaluate mathematical ideas that are presented in written, oral, or visual forms;
• use mathematical vocabulary, notation, and structure to represent ideas, describe relationships, and model situations.

Focus

The Curriculum Standards present a dynamic view of the classroom environment. They demand a context in which students are actively engaged in developing mathematical knowledge by exploring, discussing, describing, and demonstrating. Integral to this social process is communication. Ideas are discussed, discoveries shared, conjectures confirmed, and knowledge acquired through talking, writing, speaking, listening, and reading. The very act of communicating clarifies thinking and forces students to engage in doing mathematics. As such, communication is essential to learning and knowing mathematics. But communicating mathematically presents unique difficulties for students. Mathematics is heavily based on the use of symbols and attaches specific, and sometimes different, meanings to common words. The result can be confusion and difficulty in expressing mathematical ideas. Traditional forms of testing cannot always identify such confusion and difficulty, and thus they ignore the social context of mathematics.

An assessment of students' ability to communicate mathematically should be directed at both the meanings they attach to the concepts and procedures of mathematics and their fluency in talking about, understanding, and evaluating ideas expressed in mathematics. The evaluation should include different forms of communication and should emphasize communication both among people and with various forms of technology. Assessment also must be sensitive to students' language development. As in any language, communication in mathematics means that one is able to use its vocabulary, notation, and structure to express and understand ideas and relationships. In this sense, communicating mathematics is integral to knowing and doing mathematics.

Discussion

Since communication is a social activity that takes place within a context, it should be assessed in a variety of situations. In assessment, as in teaching, teachers should be aware of how students express mathematical ideas and how they interpret the mathematical expressions of others. In assessing students' ability to communicate, teachers should pay attention to the clarity, precision, and appropriateness of the language used. In addition, students' ability to understand the written and oral communication of others is an important component of instruction and assessment.

Grades K-4

In the early grades, when students are introduced to mathematical vocabulary and notation, the assessment of their understanding of language is important as they form initial conceptions of the subject. In many situations, unfamiliar terms and phrases are encountered, and familiar terms are used in unfamiliar ways. For example, although children have often heard the question How many? the phrase How many more? suggests a quantitative relationship and might confuse them. At the same time, they are learning the meaning and correct use of symbols, some of which denote operations whereas others denote relationships. An assessment sensitive to this confusion will identify the possible causes of errors while it varies the format of the problems. A child who gives 12 as the answer to 5 + [_] = 7, for example, should be questioned further to determine if the equal sign is used as a cue for addition rather than as the symbol signifying that the quantities on each side are equivalent. In this example, the difficulty might lie in an understanding of the language rather than of the operation.

At these ages, the assessment of communication should occur informally in the context of instruction. Although children should be encouraged to verbalize their thoughts, asking for general explanations or explicit meanings of a concept or procedure can be threatening. Children at these ages are more comfortable describing these processes with specific examples, such as making a drawing of a number problem or demonstrating a procedure through multibase blocks. At the same time, they should be encouraged to verbalize their thinking so that their development of language, along with the concepts, can be monitored. Their willingness to participate in class discussion also should be noted. The degree to which children are comfortable in expressing their mathematical thinking and their flexibility in using various forms of communication are primary aspects of communication at this level.

It is also important that children be evaluated on how well they listen. In an environment in which children are encouraged to question and challenge, the teacher will find ample indications of how well children have interpreted and evaluated what has been said or presented. The posing of such specific questions as "How does Jim's solution differ from Susan's?" or such general ones as "Can you say that in your own words?" will reveal the extent to which a student has understood the message.

The following is an example of a task for assessing students' speaking and listening skills:

Two students each have a geoboard. One geoboard has a design, and the other is blank. Each student is seated so that he or she cannot see the other's geoboard. The student with the design gives the other student directions on how to reproduce the design. See figure 6.1.

Fig. 6.1. Arrangement of geoboards for assessing speaking and listening skills

A variation of this task is to let the second student ask questions. In an evaluation of students' communication skills, the accuracy of the reproduced design is one indication of their interchange. Other observations can focus on whether the student talked about the figure by breaking it into parts (vertical line, horizontal line, three units) or by identifying the whole figure (right triangle, square) and if the first student repeated an instruction in more than one way. The opportunity to observe the students' communication skills continues as the students compare the two designs and talk about their similarities and differences. Their vocabulary and if they described the design in more than one way should be noted.

Grades 5-8

In the middle grades, as students become more aware that the meanings of many mathematical terms and notations depend on their context, assessing the clarity of communication is vital. Because students tend to overgeneralize earlier meanings, special attention should be paid to their understanding of terms and operations. For example, previous experiences with whole numbers lead students to view the effects of multiplication as an increase in quantity. This conception must now be modified to include the effects of multiplication on fractions and negative numbers. Similar extensions of meaning occur with symbols-- for example, when a "-" sign that was originally encountered in subtraction is now used to denote quantities with magnitude and direction (e.g., 3 - (-4)).

As in the earlier grades, the assessment of students' ability to understand mathematical terms and concepts is best achieved through a natural extension of instructional activities. Such questions as "Why?" "What if?" and "How would you convince someone?" should be asked routinely to help students explain or justify their answers or conjectures. Here, the criterion for acceptability should be based more on the clarity of presentation than on the precision of mathematical vocabulary. Again, assessment should usually be oral and informal. When students accept the communication of ideas as a normal part of their lessons, they will be more willing to express them.

It also is important to assess how well students can interpret both the verbal and the visual communication (e.g., graphs) of others. Students should be able to interpret and extrapolate from graphs as well as read them. In turn, they should be able to discuss and defend their interpretations. For example, if one group of students used the box plots in figure 6.2 when comparing findings from a survey of the number of hours of TV watched each day by students in their school with the results of a national survey (see 5-8 Standard 2), another group can be asked to interpret the data. The ability of the second group to describe the central points and the distribution of the data and justify their conclusions can be assessed.

Fig. 6.2. Box plots of hours of TV watched by students for school and nation

Grades 9-12

At the secondary school level, students encounter more abstract ideas and experience in greater depth the formal language of mathematics. Assessment should be directed toward students' understanding of the mathematical language, its terms and syntax, as well as toward their appreciation of the role of rigor and precision in communicating mathematical ideas. Although informal observation continues to be important at the high school level, the increase in formal mathematical presentation requires new criteria for assessment. At this level, students' written work should be judged for its precision, clarity, and the appropriateness of the presentation. Students should be able to form multiple representations of ideas and relationships and recognize their relative appropriateness. However, expectations for the use of symbols should relate to the maturity of the students and the context of the task. At times, a symbolic expression might be required; in other contexts, a mixture of symbols and natural language might be adequate; in others, symbolism might be completely unwarranted.

It is important to note that students' ability to communicate mathematically can be assessed by having students write about mathematics. The written responses should be judged for accuracy, clarity, precision, and the proper use of mathematical terms and symbols. The following is a sample task:

Imagine you are talking to a student in your class on the telephone and want the student to draw some figures. The other student cannot see the figures. Write a set of directions so that the other student can draw the figure and graph exactly as shown in figure 6.3.

Fig. 6.3. Figures for writing a set of directions to reproduce drawings

Assessment should include more than judgments of written work. Although students' ability to understand mathematics texts or articles can be assessed through written summaries, discussion can be a more useful context for judging students' ability to function as active, critical participants in the reading or listening processes occurring during class or small-group discussions.

The assessment of students' ability to communicate through technology is also important. The increasing use of technology as a tool demands that students be able to use computers and software, such as spreadsheets and data-base programs, to structure and present information. The criterion for performance is whether students can communicate using technology. One way of assessing this ability is by determining if students can use a spreadsheet to simulate a situation and provide evidence for a conclusion. For example:

Using an electronic spreadsheet, demonstrate that if 1 gallon of deicer fluid is added with each fill-up of a fuel tank, a limit of 2 gallons of deicer in the tank at the fill-up will eventually be reached. Assume that the driver habitually fills the tank when it is half full.

A student who can adequately communicate through a computer can use one column to designate the number of fill-ups and a second column to report the amount of deicer. To generate a value in the second column, one uses the sum of a geometric series with the first term 1 and a ratio of 0.5. An example of a successful response to this task is given in figure 6.4 (Day and Scott 1987) on the next page.

Fig. 6.4. Spreadsheet simulation of repeated fill-ups of an automobile's fuel tank