Evaluation of the teaching of mathematics should be based on information from a variety of sources including

the teacher's goals and expectations for student learning;

the teacher's plans for achieving these goals;

the teacher's portfolio, consisting of a sample of lesson plans, student activities and materials, and means of assessing students' understanding of mathematics;

analyses of multiple episodes of classroom teaching;

the teacher's analysis of classroom teaching;

evidence of students' understanding of, and disposition to do, mathematics.

Any evaluation of teaching should be based on multiple observations and a variety of types of data not on a single observation or a single source or type of information. A teacher's goals and expectations for student learning should be part of the information used in the evaluation process and should be discussed with the teacher prior to classroom observations. It is important that any evaluation maintain a longitudinal and cyclical orientation by considering goals previously established and those looking to the future.

Evaluation should be based on observations of the teacher teaching mathematics in a variety of contexts that is, at different grade levels, with a variety of students, and across mathematical topics. Such variety provides a sound basis for evaluating a teacher's expertise in using various teaching methods. Some visitations should be on consecutive days to understand better the continuity of classroom events. As suggested in Standard 1, these sources of information can be used either by a teacher engaged in self-assessment or by colleagues and supervisors working collegially with the teacher.

Clearly it is unreasonable to expect that all possible teaching situations for a given teacher can be observed. Accordingly, a sample of a teacher's lesson plans, student activities and materials, projects, and student assessment techniques that the teacher has gathered over a period of time should supplement observations of the teaching process. The portfolio should consist of an ongoing collection of sample materials that provide information about "life in this teacher's classroom." The portfolio can provide a basis for self-analysis or analysis and discussion with peers or supervisors.

A teacher's own analysis of the teaching process could provide valuable information about what the teacher is intending to accomplish and could also provide a basis for improvement.

Evidence of students' understanding of, and disposition to do, mathematics should provide a source of information about teaching, but it should not be the only source of information. Furthermore, learning and disposition should be considered with respect to every student; that is, increasing the learning of mathematics and promoting a disposition to do mathematics for some students at the expense of neglecting other students is not appropriate. Finally, an assessment of students' learning and disposition to do mathematics should be based on the full range of mathematical activity described in the NCTM Curriculum and Evaluation Standards for School Mathematics; it should not be based on a narrow range of specified objectives.

Vignettes

In the conference before the observation, the principal and teacher confer about the teacher's plans.

The principal is forming an opinion on how the teacher teaches, but the source of information is limited.

3.1 Sandi Olson, the principal at Westside Middle School, is visiting Ben Bede's seventh-grade mathematics class. Before observing the class, she had talked with Ben about his plans for the class. She notes that he is always well organized. However, her first observation during the third week of school left her with the impression that his teaching rarely goes beyond what is presented in the textbook. At least the examples and activities he used were from the textbook. Still, she is impressed with Ben's ability to describe individual students and to diagnose individual learning problems. He seems to know his students very well especially since it is only October.

Ben begins class by covering the homework on changing written statements to simple equations. After answering all the questions, Ben introduces the new material involving writing equations based on data presented in a table using the examples in the text. He has the students work a few problems and then assigns problems from the text. The balance of the period is spent on students doing the homework assignment.

The principal is forming conclusions based on the observation of one class.

Mrs. Olson makes a note for their planned after-school conference to encourage Mr. Bede to be more creative, since the lesson was very routine. On the checklist, she is contemplating giving Ben high marks for organization, knowledge of the content, and general manner in conducting the class. She observes that the room is attractively arranged with interesting bulletin board suggesting some degree of creativity. She is considering giving him a below-average rating on his ability to facilitate his students' mathematical disposition or to encourage problem solving or communication in the classroom. She is generally unimpressed with the development of new material.

The teacher is helping students make connections between mathematics and natural phenomena.

The connection between the arrangement of the leaves and the Fibonacci sequence is noted.

The teacher will point out other examples of the Fibonacci sequence.

The principal notes the teacher's efforts to make mathematical connections and to help students' develop a disposition to do mathematics.

The principal realizes that evaluation of teaching should be based on observing more than a single class. She will also use the teacher's analysis of the lesson.

A teacher expresses a concern regarding the evaluation process. His mathematics supervisor shares the concern and is willing to work collegially with the teachers to help resolve the problem.

About this time Mrs. Olson is paged to come to the office to meet with the superintendent. She had forgotten the appointment, since she had become so fascinated with Ben's lesson. His efforts to make mathematics interesting and the students' obvious excitement has not gone unnoticed by Mrs. Olson. She is sorry that she is late for the appointment, but she is so very pleased that she stayed a few minutes to observe Ben's second-period class. What a difference it will make in her evaluation of his teaching. She is looking forward to discussing the two lessons with Mr. Bede after school and benefiting from his analysis.

3.2 Doug Reid teaches first-year algebra at Jefferson High School. He voices concern to his mathematics supervisor, Yolanda Hernandez, that his total evaluation last year was based on only two classroom observations. Yolanda is concerned about this also, but she explains that her responsibilities make it virtually impossible for her to visit each teacher more than two or three times a year. During the conversation, she indicates a willingness to base her evaluation on information provided by the teachers, if they can suggest a reasonable way of doing this.

A teacher's goals and plans should be sources of information for evaluation.

The portfolios provide information about the teaching process to which the supervisor can react and provide suggestions for improvement.

The principal is supportive of the new approach.

Yolanda indicates that the portfolios are helpful in keeping her in touch with what the teachers are doing. Actually, she is surprised that they provide as much information as they do. She notes, for example, that Doug's algebra tests tend to overemphasize computational algorithms. She offers several alternative items that he can use in subsequent testing. She likes his warm-up activities and compliments him on the way he begins the algebra classes.

At a faculty meeting later that month, the principal compliments the department for its innovative approach to improving instruction. She is particularly pleased that the teachers and the supervisor have worked together in a professional way in developing procedures for evaluation.

The first three evaluation standards are based on the assumptions stated at the beginning of this section. The process of evaluation should reflect that the overall intent is to improve instruction, that it should be a dynamic and continual process, that teachers should be an integral part of that process, and that because of the complexity of teaching, it should involve a variety of sources of information gathered in various ways. The standards emphasize that teachers should be encouraged and supported to engage in self-analysis and to work with colleagues in improving their teaching. When evaluation involves supervisors or administrators, their relationships with teachers should be collegial with the intent to improve instruction.

The following five standards focus on what should be observed during the evaluation process. They assume that observers, including teachers themselves, should have a framework from which to observe classroom activities. These standards provide that framework. The framework is emphasized in the vision of teaching presented in the first section of this volume and is rooted in the Curriculum and Evaluation Standards for School Mathematics. The standards involve both mathematical content and processes.