Beyond Classroom Description: Methods of Understanding Reflection and Beliefs in Mathematics Teaching

Article excerpt

In the present study I examine two methods of unpacking teacher's mathematics-content/pedagogy beliefs and their instructional decisionmaking The study took place over the course of a school year. For each the two participants, I observed & videotaped 10 classroom mathematics lessons, and held 6 follow-up dialogues (three of each type) of more than 60 minutes each. I then used data from the Video Reflection and Theory Reflection methods and qualitatively analyzed each for meaningful patterns and richness of response. Results indicated that both methods were useful for varied purposes in obtaining teacher belief data.

How have teachers incorporated recent reform efforts, particularly those in mathematics education (NCTM, 1989, 1991; NRC, 1989; Grouws, 1992)? Beliefs about mathematics influence how a teacher teaches mathematics. As teaching beliefs in general move from a behaviorist orientation to more constructivist and humane, are individual teacher's beliefs changing in this direction also? Are such belief changes carried into the classroom? Beliefs are not readily observable and are often tacit to the teacher, him/herself. Indirect methods of discovering and furthering teacher beliefs are needed. The study analyzes two reflective forms of inquiry that attempt to go beyond the observable level to understand two teachers` beliefs about teaching and learning mathematics.

Change and redefinition in teaching require reflection and a personal questioning of the underlying beliefs that drive practice (Schon, 1983). These actions, common in the assessment process of students may be new to the teacher as a form of professional self-study.

The study analyzed two different methods of reflection in accomplishing the following: (1) as a setting for deep reflection on beliefs and practices in mathematics, (2) as a context for teacher change in either belief or practice, and (3) as a personalized approach to professional development. The first method is referred to as The Video Reflection Method and the second, The Theory Reflection Method. Both will be described in detail later in the article.

Background

Reforms in mathematics education are dependent on the teacher in the classroom. However, many teachers' beliefs about mathematics and what learning mathematics entails are incompatible with the reform effort. Such beliefs not only hinder the progress of school reform but also result in an enacted curriculum "that is seriously damaging the mathematical health of our children." (Battista, 1994, p.462)

The need for reform and its success stemming from a change in traditional teacher beliefs suggests novel approaches to collecting, refocusing, and analyzing data. Duckworth (1986; 1987) used elicitation methods during actual science investigations in order to learn and change teachers beliefs about teaching science. Cobb, Yackel, & Wood (1992) worked intensively with one second grade teacher in the area of mathematics discussing classroom practice, challenging established beliefs. and providing her with actual constructivist lesson plans. Thompson's (1984) classic study of three teachers' conceptions of mathematics and how those conceptions effected their teaching practice employed the use of questionnaire, interview, and observation methods1.

Fenstermacher ( 1986, 1993) pursued the question of change in beliefs by drawing on Green's (1971, 1976) notions of the tendency of the mind to group ideas into "clusters" with internal but not necessarily external consistency. Through dialogue with an "Other" beliefs were brought out and examined in relation to each another, thus forcing the consistency issue. This process Fenstermacher calls a "practical argument." By applying these ideas to a contemporary analysis of teaching reading, Richardson i 1994) suggested a method to draw out and affect a teacher's beliefs through the use of video tape reflection and intense dialogue between the teacher and the staff developer. …