What does it mean for a learner to "bring" something to a situation? A metaphor, it conveys the idea that learners do not arrive empty-handed (or empty-headed), but that they come instead with ideas, understandings, ways of thinking, inclinations, and habits. Recent research on student learning in a variety of domains suggests that pupils' prior knowledge and beliefs — what they bring — shapes the ways in which they make sense of new ideas (see, for example, Anderson, 1984; Confrey, 1987; Davis, 1983; diSessa, 1982; Posner, Strike, Hewson, and Gertzog, 1982; Roth, 1985; Schoenfeld, 1983). Confrey (1987), tracing the recent history of this perspective on learning, describes it as reflecting "a basic rejection of a tabula rasa approach to learning" and, as being founded instead on the assumption that "existing knowledge serves as both a filter and a catalyst to the acquisition of new ideas" .
Although teacher educators talk about this constructivist perspective on learning in their courses, it rarely influences what they do with their students. The professional preparation of teachers instead seems often to be based on the assumption that preservice teachers simply lack the knowledge they need and that the purpose of professional preparation is to fill them up with or give them the necessary knowledge and skills (Zarinnia, Lamon, & Romberg, in press). Debates about teacher education, when they occur, center on what knowledge is of most worth, on methods, or on the sequence and structure of professional education (e.g., Holmes Group, 1986; Sarason, Davidson, & Blatt, 1962; Smith, 1980). Feiman-Nemser (1983) argues that teacher educators "tend to underestimate the pervasive effects" of school and culture on prospective teachers. She suggests that the ideas they bring to teacher education are rarely challenged; that, instead, these ideas are generally ignored and, as a result, teachers tend to maintain their preconceptions: "Formal training does not mark a separation between the perceptions of naive laypersons and the informed judgments
of professionals" (p. 153). Because teaching is a familiar cultural activity, the notion that it requires specialized knowledge or perspectives is not commonplace (Buchmann, 1987b;
Jackson, 1986). Still, the "teaching knowledge" derived from everyday experience with teachers and classrooms is at sometimes insufficient and at other times misleading as preparation for teaching. It is a central and difficult task of teacher education to help prospective teachers to move from commonsense to pedagogical ways of thinking (Feiman-Nemser & Buchmann, in press). Although evidence exists that formal teacher education is a weak intervention compared to the potency of teacher candidates' prior experiences, there has been little research on what teacher candidates do or do not learn from teacher education. The Knowledge Use in Learning to Teach project, conducted by Sharon Feiman-Nemser and her colleagues at the Institute for Research on Teaching, is one exception. Between 1982 and 1984, researchers (myself included) followed seven prospective elementary teachers through two years of undergraduate teacher education, observing and documenting courses and field experiences and interviewing the teacher candidates about what they were learning. We focused on how they made sense of their courses and field experiences, and traced changes in their knowledge, skills, and dispositions over the two years of their professional program. Several analyses revealed how powerfully prospective teachers' preconceptions could shape what they learned from their teacher education programs.
One example from this work shows the way in which Janice, one of the teacher candidates, made sense of Jean Anyon's (1981) "Social Class and School Knowledge," an article that analyzes and critiques the variation in curriculum across schools serving different socioeconomic populations. The article describes four urban and suburban schools and the differences in the social studies curricula among these four schools. Janice's instructor wanted the teacher candidates to confront and question the inequities in the distribution and packaging of knowledge by socioeconomic class. But Janice, a young woman from a rural background
whose firsthand experience with migrant workers had helped to shape her views of children's differential academic needs, understood Anyon in prescriptive rather than critical terms. She interpreted the author as showing what pupils of different social classes needed:
It just made me think that, maybe, some things maybe aren't important. Certain things should be stressed in certain schools, depending on where they're located. (p. 247)
Feiman-Nemser and Buchmann (1986) argue that, in general, because the "pull of prior beliefs is [so] strong"
very little normatively correct learning can be trusted to come about without instruction that takes the preconceptions of future teachers into account . . . In learning to teach, neither firsthand experience nor university instruction can be left to work themselves out by themselves. Without guidance, conventional beliefs are likely to be maintained and new information or puzzling experiences absorbed into old frameworks. (p. 255)
The Knowledge Use in Learning to Teach study also suggests some of the ways in which prospective teachers' knowledge of subject matter interacts with their understandings and assumptions about teaching and learning and about pupils to shape what they learn from their courses as well as what they do as they begin teaching. For example, urged to avoid textbooks and to create their own instructional materials instead, the prospective elementary teachers we were studying designed their own substantively thin or even incorrect curriculum, often misrepresenting or even sidestepping the content. They, however, did not see that but felt proud that they were doing something of their "own" (Ball & Feiman-Nemser, in press). In addition to the fact that little research has been conducted on what teacher candidates learn from their professional education, most of it has also been generic — that is, not focused on learning to teach in particular subject areas. An exception is the Knowledge Growth in a Profession project, conducted by Lee Shulman and his colleagues at Stanford University between 1984 and 1987. Researchers in this project studied how beginning secondary school teachers began learning to transform their own understandings of academic subjects in order to teach them. Interesting subject matter differences emerge from this work, suggesting the need to examine the process of learning to teach in specific subject areas.
For example, secondary social studies teachers come from majoring in a range of social science disciplines, such as anthropology, sociology, history, and political science, and then are assigned to teach particular social studies courses, often outside of their area of specialization. Wilson and Wineburg (in press) show how beginning teachers' "disciplinary lenses" shaped their teaching of social studies. For example, a former political science major, Fred viewed history as dry knowledge of facts and dates, while Jane, a history major, saw facts as "part of history woven together by themes and questions, and . . . embedded in a context that lends meaning and perspective" (p. 5). Wilson and Wineburg write:
What is interesting in our findings is the way in which our teachers' undergraduate training influenced their teaching. The curriculum they were given and the courses they subsequently taught were shaped by what they did and didn't know. (p. 14)
The study's findings about beginning mathematics teachers were quite different. Conceptual understanding of the content was a bigger issue than it was with English or social studies majors. While undergraduate mathematics majors study similar mathematics topics, they do not necessarily seem to develop the same deep understandings of either the substance or the nature of their subject as do majors in other disciplines. Some of the beginning math teachers in the study struggled to teach high school content because their own understanding was weak. At times, they made mistakes or misrepresented the content, and some were unable to provide "why" explanations. Steinberg, Haymore, & Marks (1985) suggest that their results "contrast with the popular belief that any person who goes through calculus masters the high school content very well" (p. 22). They argue that difficulties in teaching high school mathematics may well stem from problems in teachers' own understanding of the subject matter.
If research on what teacher candidates learn from teacher education is scant, investigations of what prospective teachers bring to teacher education is non-existent. Virtually no research has been done to investigate what prospective elementary teachers know and believe relative to the teaching of mathematics before they enter formal programs of teacher education. Further study has to be done to represents a foray into that domain.
Prospective mathematics teachers bring lots of ideas with them to teacher education. They may believe in assigning plenty of homework or they may have a repertoire of bulletin board displays. Reading may be a passion and penmanship a weakness. Obviously not all of what teacher candidates bring with them is equally related to the knowledge and beliefs most critical to this approach to learning to teach mathematics from the perspective of mathematical pedagogy. A framework for looking has helped to focus my inquiry, exploring the knowledge and beliefs most critical to this approach to teaching mathematics. The appropriate framework for investigating what teacher candidates know and believe thus derives from the logical requirements of this approach to teaching. What does it take, and where are entering teacher candidates in relation to that standard?
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