Pages: 36

No chapter abstract available.

Pages: 7

You have decided to join the teaching profession at a time when there is a great need for highly qualified, well-prepared mathematics teachers. Expectations of teachers are very different from what they were as little as a decade ago. The purpose of this chapter is to present a vision of high-quality teaching and outline the key considerations for achieving that level of instruction. A description of instructional practice in mathematics, complexities of teaching, challenges facilitating student-centered, assessment-based teaching, reflective practice, and professional development set the stage for the remaining chapters in this book.

Pages: 9

In this chapter, we demonstrate how to dissect a lesson, looking at its pieces and how they fit together. This approach orients your examination of others' teaching and helps you focus on important aspects of your own lessons. The first decomposition of a lesson is temporal and straightforward; we describe the parts of a lesson as it unfolds. The second decomposition is more complex; we discuss lesson dimensions and dimension indicators—the material of the lesson and how to think about it.

Pages: 7

In this chapter, we suggest ways to examine the aspects of teaching that are the driving force behind every lesson: teacher cognitions. This approach allows you to understand better your own and others' teaching, and it provides the mechanism with which you can engage in self-reflection and improvement. As we have argued, what counts is the teacher's cognition and how it comes into play at different times. Table 3.1 provides an overview of the nature of teacher cognition and when it is applied. The overarching aspects of teacher cognition are teachers' goals, knowledge, and beliefs. Other cognitions come into play at different times and in different ways: planning prior to the lesson, monitoring and regulating during the lesson, and evaluating and revising after the lesson.

Pages: 11

This is, for the most part, a practical book. Starting in the next chapter, we will suggest how to observe lessons (including your own), reflect on them, and improve them. But you have a right to ask whether this is a valid model for examining teaching, and we have an obligation to show that it is. This chapter provides the evidence. It describes research results in which we documented the specific relationships between teachers' cognitions and their instructional practices. We also present evidence of the usefulness of the model.

Pages: 88

No chapter abstract available.

Pages: 40

This chapter describes ten structured observations with examples from reports of actual lessons observed in elementary, middle, and secondary schools. Each of the first nine observations focuses on a critical aspect of instructional practice in mathematics: the nature of the mathematical content, questioning, motivation and teaching strategies, homework, use of class time, verbal behavior of students, task orientation of students, assessment, and teacher expectations and stereotyping. The observations have been arranged so that they first focus primarily on the content, then on the teacher, and then on the individual students. However, because all aspects of instruction are deeply interrelated, these observations can certainly be sequenced differently.

Pages: 34

The fact that teachers' cognitions are the driving force behind their behaviors in the classroom makes it essential that whenever you analyze your own teaching, you incorporate assessments of your own thoughts and decision-making as well as your instructional practices. In this book, the model we present provides the conceptual basis for approaches you, as a preservice teacher, can use to reflect on and assess your teaching in a structured way. As suggested in previous chapters, the lesson plan you write and your instructional practice are only part of your work. The cognitions that you have before you write your lesson plan, the thoughts you have while teaching your lesson, and the thoughts you have after you have completed your lesson are also critical components of teaching. To facilitate analyses of cognitions and instructional practices before, during, and after lessons, we recommend participating in written and oral pre- and postlesson activities. Although you cannot be expected to write pre- and postlesson thoughts on a daily basis, writing your plans and thoughts when your supervisor or a peer observes you or for a videotaped lesson can be an excellent tool for professional growth. Moreover, if you get accustomed to thinking about your lesson in such a structured manner, it is likely that it will become a natural part of your reflective practice (even though it may not be in written form). It is this application of the model, explained and exemplified in the sections “Reflection and Self-Assessment With the Assistance of a Supervisor or a Cooperating Teacher”, “Reflection and Self-Assessment With Peers”, and “Reflection and Self-Assessment Using Video”, that is the focus of the remainder of this chapter.

Pages: 12

A portfolio is a useful tool for both professional and personal development. You can use it for professional purposes: it is not unusual to be asked for a portfolio when you apply for a teaching position, and it is required for National Board certification. Some state certification examinations also require a portfolio. A portfolio is often required for tenure applicants. You may even wish to create a portfolio for personal use, as a means of documenting your professional growth and collecting your best pieces of work for future reference. Whatever your goal, you can use the guidelines given in this chapter for creating a portfolio. A description for each of the elements for the portfolio (i.e., lesson plans, assessment/data analysis, homework assignments, project, and unit plan) and suggested grading rubrics can be found in Appendix E, beginning with the overview of the portfolio assignment in Appendix E.1.

Pages: 17

No chapter abstract available.

Pages: 15

The fundamental premise of this book is that teachers' classroom practices as well as their underlying cognitions must be predominantly student-centered for students to learn mathematics with understanding. In the preceding chapters, we described the development and use of our model as a way to understand and appreciate the complexity of student-centered teaching. This chapter consists of five case studies that show various issues and concerns that arise when student teachers engage in the process of reflection and self-assessment about their cognitions and instructional practices. For each student teacher, Maria, Iris, Mindy, Elizabeth, and Kyungso, we describe their experiences and their professional growth by organizing their comments under the following headings: “Identifying the Problem”, “Owning the Problem and Searching for Resolution”, and “Growth in Self-Understanding”. Furthermore, we include a commentary section to share our own analysis of their experiences and to suggest probable trajectories for their progress in becoming student-centered professionals. As you read the comments of these student teachers, watch for the unspoken contradictions in what they say and what they do; feel their emotional struggles as they try to make their practice consistent with their cognitions; and share in their breakthrough moments when they confront and seek resolution of acknowledged problems. In shadowing their experiences, you get a glimpse of the complexity of the dynamic interplay among goals, knowledge, and beliefs and classroom practice and an appreciation of what it takes to grow in self-understanding of one's own teaching.