The assessment of students' mathematical disposition should seek information about their--

• confidence in using mathematics to solve problems, to communicate ideas, and to reason;
• flexibility in exploring mathematical ideas and trying alternative methods in solving problems;
• willingness to persevere in mathematical tasks;
• interest, curiosity, and inventiveness in doing mathematics;
• inclination to monitor and reflect on their own thinking and performance;
• valuing of the application of mathematics to situations arising in other disciplines and everyday experiences;
• appreciation of the role of mathematics in our culture and its value as a tool and as a language.

Focus

Learning mathematics extends beyond learning concepts, procedures, and their applications. It also includes developing a disposition toward mathematics and seeing mathematics as a powerful way for looking at situations. Disposition refers not simply to attitudes but to a tendency to think and to act in positive ways. Students' mathematical dispositions are manifested in the way they approach tasks--whether with confidence, willingness to explore alternatives, perseverance, and interest--and in their tendency to reflect on their own thinking. The assessment of mathematical knowledge includes evaluations of these indicators and students' appreciation of the role and value of mathematics.

This kind of information is best collected through informal observation of students as they participate in class discussions, attempt to solve problems, and work on various assignments individually or in groups. Such assessment procedures as attitude questionnaires fail to capture the full range of perceptions and beliefs that underlie students' dispositions.

Discussion

From their first encounter with shapes or numbers, children begin to form a conception of mathematics. Teachers implicitly provide information and structure experiences that form the basis of students' beliefs about mathematics. These beliefs exert a powerful influence on students' evaluation of their own ability, on their willingness to engage in mathematical tasks, and on their ultimate mathematical disposition.

Mathematical disposition is much more than a liking for mathematics. Students might like mathematics but not display the kinds of attitudes and thoughts identified by this standard. For example, students might like mathematics yet believe that problem solving is always finding one correct answer using the right way. These beliefs, in turn, influence their actions when they are faced with solving a problem. Although such students have a positive attitude toward mathematics, they are not exhibiting essential aspects of what we have termed mathematical disposition.

The assessment of students' dispositions requires information about their thinking and actions in a wide variety of situations and should consider all aspects of disposition and the degree to which they are exhibited. Disposition has many components, each of which a particular student exhibits to a greater or lesser extent. For example, a student might be very willing to try alternative methods of solving problems but be less inclined to reflect on the solutions. Another student might be fairly uninterested in routine exercises and yet work diligently to solve nonroutine problems. An adequate assessment of these students' disposition requires information on their willingness to engage in all aspects of solving problems, including learning through problem solving.

In the classroom students' dispositions are continuously reflected in how they ask and answer questions, work on problems, and approach learning new mathematics. As a result, teachers are in an excellent position to gather useful information for assessing disposition. In addition, teachers benefit from this assessment because it provides information for instructional planning. If an assessment indicates that most students in a class rarely attempt problems independently and frequently ask to be shown the solution method, a teacher can choose to reexamine classroom instruction to evaluate whether students are being encouraged to solve problems and develop alternative solutions. In short, the assessment of students' dispositions provides information about changes needed in instructional activities and classroom environments to promote the development of students' mathematical dispositions.

Because evidence of students' mathematical dispositions is apparent in every aspect of their mathematical activities, observation is a primary method of assessment. When presented with a problem, particularly one in a new and unfamiliar context, a student exhibits his or her mathematical disposition in a willingness to change strategies, reflect and analyze, and persist until a solution is identified. Students' disposition toward mathematics can be observed in class discussion. How willing are students to explain their point of view and defend that explanation? How tolerant are they of nontraditional procedures or solutions? Are they curious? Are they willing to ask, "What if . . .?" What kind of questions do they ask?

Although observation is the most obvious way of obtaining such information, students' written work, such as extended projects, homework assignments, and journals, as well as their oral presentations, offer valuable information about their mathematical dispositions. Such projects as individual or small-group presentations of problem solutions or proofs of theorems can serve as evidence of students' willingness to persevere at mathematical tasks and test alternative methods in solving problems.

Grades K-4

The assessment of mathematical disposition of students in the primary grades should focus on the number and quality of students' experiences. An inventory like the one in figure 10.1, when kept for a class or for individual students, can be used as a record of their experiences. This record can be used for determining if a full range of experiences has been offered and for reporting to parents, guardians, and other teachers the experiences of the students.

Fig. 10.1

Grades 5-8

In the middle grades, students' mathematical dispositions become more apparent in their daily work. A checklist of actions that demonstrate mathematical disposition can be used to help record students' progress. Such a checklist can be completed by a teacher while focusing on five students during a class period. Over six weeks, each student's mathematical disposition can be evaluated at least once. A sample checklist is given in figure 10.2.

Fig. 10.2

Grades 9-12

Essays and research papers are particularly useful for both developing and assessing students' appreciation of the role of mathematics in our culture. Judgments about the quality of the work should be based on its conceptual merit, thoroughness, and originality. Just as the assessment of writing includes a consideration of grammar, spelling, fluency of expression, force of ideas, and consistency of logic and narration, so mathematical assessment should include considerations of quality as well as accuracy. An innovative solution, a well-reasoned argument, and a willingness to go beyond the constraints of the task are of equal importance in obtaining a correct solution and should be acknowledged. Sample topics are the following:

• The role of geometry in our society today
• Mathematics and computers
• What if standard units of measures had not been created
• Changes in the applications of statistics in sports

No one of the preceding considerations across all grades will yield a definitive picture of a student's disposition toward mathematics. However, extended observations of students' efforts and interactions in a variety of mathematical contexts can give teachers the feedback and information necessary to adjust their instructional methods and encourage students' progress in attaining intellectual autonomy.

PROGRAM EVALUATION

The Program Evaluation Standards are an integral component of the overall vision of evaluation's primary role in guiding change. The vision includes program change so that all the pieces will fit together. This change will provide the foundation in grades K-4 for students to know mathematics through doing mathematics, will develop and expand the ideas in grades 5-8, and will lead to self-directed learning in grades 9-12. Evaluation can help determine a mathematics program's status in relation to the Curriculum Standards and ensure that the pieces fit together. It can indicate the steps that need to be taken so that a program aligns with the Standards. In a certain sense, the Program Evaluation Standards are a guide to creating a program that meets the challenge of the Standards.

One purpose of program evaluation is to obtain relevant and useful information for making decisions about curriculum and instruction. The assessment of students' capabilities, as described in the previous ten evaluation standards, is one means of collecting information about a program. The principles of general assessment apply as much to program evaluation as to student assessment. But there are other indicators of program quality. Does the curriculum include the range of topics and emphases described in the Standards? Is the program articulated from kindergarten through grade 12 so that students' knowledge and experiences build continually? Does prior mathematics study provide the conceptual underpinnings for additional study and for its application to real-world problems? Does instruction actively involve students in the investigation, exploration, and development of mathematical ideas and the relationships among them? Are calculators and computers used appropriately? Do all students have full access to the program? Do students know mathematics as an integrated whole?

Any program can be implemented by degrees. At one level, the language of the new program can be adopted while day-to-day instruction remains unchanged. At another level, minor changes in structure can be made by inserting a new unit into a course or slightly modifying the scope and sequence. At a third level, deep structural changes can be made that include altering how people think about a program, how mathematics is presented, and how students come to know mathematics. The Standards speaks to this third level of change. It is the role of program evaluation to facilitate this deep structural change.