Assessing Students Beliefs about Mathematics.

The beliefs that students and teachers hold about mathematics have been well-documented in the research literature in recent years. (e.g., Cooney, 1985; Frank, 1988, 1990; Garofalo, 1989a, 1989b; Schoenfeld, 1987; Thompson, 1984, 1985) The research has shown that some beliefs are quite salient across various populations. These commonly held beliefs include the following: • Mathematics is computation. • Mathematics problems should be solved in less than five minutes or else there is something wrong with either the problem or the student. • The goal of doing a mathematics problem is to obtain the correct answer. • In the teaching-learning process, the student is passive and the teacher is active. (Frank, 1988) It is generally agreed that these beliefs are not “healthy” in that they are not conducive to the type of mathematics teaching and learning envisioned in the Curriculum and Evaluation Standards for School Mathematics [Standards] (NCTM, 1989). There appears to be a cyclic relationship between beliefs and learning. Students’ learning experiences are likely to contribute to their beliefs about what it means to learn mathematics. In turn, students’ beliefs about mathematics are likely to influence how they approach new mathematical experiences. According to the Standards, “[Students’] 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.” (NCTM, 1989, p. 233) This apparent relationship between beliefs and learning raises the issue of how the cycle of influence can be broken. The type of mathematics teaching and learning envisioned in the Standards can provide mathematical experiences that will enrich students’ beliefs about mathematics. Thus, mathematical experiences provide one place where intervention can occur; however, it may also be advantageous to intervene at the other point in the cycle, namely students’ beliefs. The Standards suggest that the assessment of students’ beliefs about mathematics is an important component of the overall assessment of students’ mathematical knowledge. Beliefs are addressed in the tenth standard of the evaluation section, which deals with assessing mathematical disposition. Mathematical disposition is defined to include students’ beliefs about mathematics. It is recommended that teachers use informal discussions and observations to assess students’ mathematical beliefs (NCTM, 1989). Although teachers’ awareness of students’ mathematical beliefs is important, it may be equally important for students to be aware of their own beliefs toward mathematics. One medium for bringing students’ beliefs to a conscious level is open-ended questions. As students ponder their responses to such questions, some of their beliefs about mathematics will be revealed. As groups of students discuss their responses to these questions, some students’ beliefs will likely be challenged, leading to an examination of these beliefs and their origins, and, possibly, to the modification of these beliefs. This article presents some open-ended questions that can be used to address students’ beliefs about mathematics. These questions have been used by the author with elementary, junior high, and senior high school students; preservice and inservice elementary, junior high, and senior high school teachers; and graduate students in mathematics education. The questions have been culled from a variety of sources and do not represent original ideas of the author. Each question is followed by a summary of typical responses from the aforementioned groups. The responses from the various populations were strikingly similar, which is not surprising since the beliefs held by these groups are generally quite similar. In some cases, possible origins of the belief or possible avenues for further discussion are included.