The two preceding papers reported on the complexity of reforming mathematics education in one Midwestern state K-A elementary school. This paper focuses on teacher preparation for mathematics teaching and follows three preservice elementary teachers as they move through their university methods course and then into their practicum/student-teaching at another elementary school.
Current research studies and documents from the National Council of Teachers of Mathematics (NCTM) address the importance of the teacher's role in creating learning opportunity in mathematical situations (Tsuruda, 1994; Bright, Bowman & Vacc, 1999; NCTM, 2000). Also, some researchers focus on understanding preservice teachers' beliefs and practices and how those beliefs and practices transform in the classroom context (Pourdavood & Harrington, 1998; Portnoy, Graham, Berk, Gutmann & Rusch, 1999). This increasing body of literature has led to some interesting insights.
Tsurda (1994) described his changing beliefs and how these changes affected his practices. He had been a middle school teacher for seventeen years, teaching in mostly a conventional style. Not all of Tsuruda's students passed his class, but he believed that more did so than students of other teachers in his building, thanks to his gregarious nature. Tsuruda was concerned, however, because even the students who were earning passing grades in his classes were not mathematically powerful. They could not apply problem-solving strategies or use critical thinking processes.
Most of my students would have done no better than the nationwide sample of students asked this question: There are 12S sheep and 5 dogs in a flock.
How old is the shepherd? Three out of four students across the nation responded with a numerical answer, the most common being that the shepherd's age is twenty-five. (Tsuruda, 1994, p. 2)
According to Tsuruda, there had to be a better way of teaching and learning mathematics so that students could critically think about what was being asked and develop a solution that not only made sense but that they could explain.
Tsuruda recognized two aspects inherent to teaching and learning once his beliefs had been perturbed by participating in national organizations such as the National Council of Teachers of Mathematics, dialogues with his colleagues, and actively listening, observing, and reflecting on his classroom mathematical activities and events. He identified these two aspects as form and spirit. Form is the method, such as small-group cooperative learning, use of technology, using manipulatives, and incorporating alternative assessment. Spirit is the teacher's belief on how the above methods can be used meaningfully for creating learning opportunities for students. Therefore, according to Tsuruda (1994), teachers may use the form of change without internalizing those changes. In this sense the spirit of change actually relates to individual transformation from one set of beliefs to another. It is a paradigm shift.
Cohen (1990) examined a second grade teacher's evolving beliefs and practices. His research findings, similar to Tsuruda's work, suggest that although a teacher may state what s/he believes, it may not be what is displayed in practice. "She [Mrs. O, the second grade teacher] eagerly embraced change, rather than resisting it...But, Mrs. O seemed to treat new mathematical topics as though they were a part of traditional school mathematics" (p.312). Cohen asserted that Mrs. O stated she believed she allowed for more students' understanding of mathematical ideas by revising the curriculum to allow for more hands-on lessons (form). However, in the actual classroom setting her beliefs were not consistent with the researcher's observations of the classroom mathematical practices (spirit). Cohen observed Mrs. O teach using manipulatives, but the teacher-controlled activities made it difficult for students to make sense of mathematical activities. …