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What can we learn about the differences between experts and novices from a teaching simulation?
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The primary goal of this paper is to investigate whether a computer-based simulation can detect the difference between novice and expert teachers’ decision-making in mathematics instruction, which is complex in nature. The design of the simulation is grounded in a sociological perspective on practical rationality of mathematics teaching. The simulation consists of classroom scenarios, in the form of cartoon-based storyboards, with a series of decision moments to simulate the instructional situation of doing proofs in geometry. Empirical data helped verify and revise our design hypotheses/principles and showed that the simulation was able to detect some differences between novice-teacher and expert-teacher decision-making. Results of this study could inform the development of more advanced, computational models of mathematics teachers’ decision-making.
The analysis reported in this paper contributes to the effort to understand how mathematics teachers might proactively support their students' mathematical learning by documenting one first-grade teacher's role in guiding the development of sociomathematical norms in her classroom. We highlight the learning opportunities that arose for both the teacher and her students during this process. The analysis therefore serves to clarify what teachers might actually do to support the emergence of the type of mathematical disposition advocated in reform documents. In addition, the analysis describes the decision-making processes that were involved in setting the teacher's agenda. This aspect of the paper focuses on the teacher's interactions as part of a research team whose dual focus was to support both the students' mathematical learning and their development of mathematical autonomy. The analysis was informed by and builds on Yackelk and Cobb's (1996) discussion of sociomathematical norms.
Teachers who attempt to use inquiry-based, student-centered instructional tasks face challenges that go beyond identifying well-designed tasks and setting them up appropriately in the classroom. Because solution paths are usually not specified for these kinds of tasks, students tend to approach them in unique and sometimes unanticipated ways. Teachers must not only strive to understand how students are making sense of the task but also begin to align students' disparate ideas and approaches with canonical understandings about the nature of mathematics. Research suggests that this is difficult for most teachers (Ball, 1993, 2001; Leinhardt & Steele, 2005; Schoenfeld, 1998; Sherin, 2002). In this article, we present a pedagogical model that specifies five key practices teachers can learn to use student responses to such tasks more effectively in discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses. We first define each practice, showing how a typical discussion based on a cognitively challenging task could be improved through their use. We then explain how the five practices embody current theory about how to support students' productive disciplinary engagement. Finally, we close by discussing how these practices can make discussion-based pedagogy manageable for more teachers.
Conventional mathematics instruction has failed to help many students in urban schools to develop desirable forms of mathematical proficiency. A promising alternative is offered by instruction that is oriented toward helping students develop a meaningful understanding of mathematical ideas through engagement with challenging mathematical tasks. For 5 years, teachers involved in the QUASAR project have engaged in developing and implementing this alternative style of mathematics instruction in urban middle schools. In this article, some critical features of QUASAR instruction are described and exemplified, and some findings regarding the positive impact of this instruction on students are discussed. Finally, some of the challenges and obstacles encountered in attempting to build the capacity of teachers and schools to offer meaning-oriented mathematics instruction are identified.
In this article we examine the relationship between teaching as a knowledge endeavor and teaching as a moral enterprise, using episodes from our own elementary school teaching as sites for our analysis. One episode concerns the teaching of social studies, the second the teaching of mathematics. We first describe the episodes themselves, highlighting the ways in which they shed light on issues of pedagogical content knowledge and reasoning. We then revisit each episode with a different lens: that of teaching as moral work. Our framework consists of two essential components: concerns for subject matter and for students. This analysis is meant to be neither a complete delineation of teaching as a moral enterprise nor an exhaustive analysis of pedagogical content knowledge. It is meant to show that, in teaching, concerns for the intellectual and the moral are ultimately inseparable.
In recent years, it has been reported that an alarming number of teachers are leaving the profession in the first three years after graduation from a pre-service program. This phenomenon is common in North America and it is essential that educators identify the challenges surrounding new teachers and provide supports to assist them. The vast majority of literature surrounding new teacher induction and mentorship support is void of the Canadian context and the novice teacher voice. In this study, Ontario graduates from a two year pre-service program were surveyed and 5 teachers were selected for case studies. Participants found administrative leadership, refining the mentorship selection process, hiring practices, and district-sponsored supports as positive factors necessary for them to grow into the profession.
This paper analyzes the tension between the traditional foundation of efficacy in teaching mathematics and current reform efforts in mathematics education. Drawing substantially on their experiences in learning mathematics, many teachers are disposed to teach mathematics by "telling": by stating facts and demonstrating procedures to their students. Clear and accurate telling provides a foundation for teachers' sense of efficacy--the belief that they can affect student learning--because the direct demonstration of mathematics is taken to be necessary for student learning. A strong sense of efficacy supports teachers' efforts to face difficult challenges and persist in the face of adversity. But current reforms that de-emphasize telling and focus on enabling students' mathematical activity undermine this basis of efficacy. For the current reform to generate deep and lasting changes, teachers must find new foundations for building durable efficacy beliefs that are consistent with reform-based teaching practices. Although productive new "moorings" for efficacy exist, research has not examined how practicing teachers' sense of efficacy shifts as they attempt to align their practice with reform principles. Suggestions for research to chart the development of, and change in, mathematics teachers' sense of efficacy are presented.
The author is a scholar of teaching practice and also an elementary mathematics teacher. Her work, like that of her colleagues at the Institute for Research on Teaching, focuses on teaching practice from the point of view of the practitioner. Here, in two case studies, she views the teacher as dilemma manager, a broker of contradictory interests, who "builds a working identity that is constructively ambiguous." To emphasize her conviction that teaching work is deeply personal, the author makes herself the subject of one of these studies. She concludes with an examination of how her view contrasts with prevalent academic images of teachers' work.
Despite general agreement that students should consider more than one way to solve a complex mathematics problem, this practice is rarely observed in U.S. classrooms. In this paper, we examine data that trace the year-long professional development experience of 12 veteran teachers of middle grades mathematics — experienced users of a standards-based mathematics curriculum — to unearth some issues and concerns that appeared to serve as obstacles to their use of multiple solutions in the classroom. We also identify key transformations that appear to account for the success of some teachers in embedding this practice in their instructional repertoire.
Beyond being told not to tell. For the learning of mathematics
- D Chazan
- D Ball
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Methods matter: Improving causal inference in educational and social science research
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