A growing number of postsecondary mathematics educators are exploring teaching strategies other than lecture (Holton, 2001). The motivations for such change include personal dissatisfaction with student learning, students' poor retention of knowledge, student dissatisfaction with their undergraduate experiences in science, mathematics, and engineering (National Science Foundation, 1996; Seymour & Hewitt, 1997), as well as efforts to rethink core courses such as calculus, linear algebra, and differential equations. As postsecondary educators make changes to their practice they often struggle with many of the same issues that K–12 mathematics teachers encounter as they attempt to change their practice. In this chapter we address one of these issues, namely the role of teacher lecture (or telling) and strategies that teachers can use to balance student discovery and teacher telling.

Navigating a new terrain of teaching practice is particularly tricky for any teacher, elementary or university, who may never have experienced as a learner an approach to teaching other than lecture and demonstration. For example, some teachers believe that changes in practice must be dramatic and involve a total abandonment of lecture (where the teacher has all the responsibility for developing the mathematics) to a form of practice that leaves students to discover all ideas and techniques for solving problems. These are two ends of a continuum from all student discovery to all teacher telling. How, why, and when a teacher positions him or herself along this continuum is a source of tension for teachers.

Students in many undergraduate mathematics courses tend not to readily and appropriately use theorems as tools for making arguments and solving problems (Schoenfeld, 1989; Hazzan & Leron, 1996). Students' reluctance to use theorems as tools is a problem that is not only cognitive in nature (that is, the difficulty is in how students conceptualize particular mathematical ideas), but also social in nature (that is, the nature of class discussion, the interpretation of tasks and ideas, etc.). In this chapter we highlight results from a classroom-based research program in differential equations that has resulted in some positive progress on the problem of students' reluctance to use theorems as tools for reasoning and solving problems.

The main result of the analysis of student learning and use of the uniqueness theorem for first order differential equations is the delineation of four interrelated cognitive and social factors that help account for why students actually made progress in using the uniqueness theorem as a tool for making arguments and solving problems (Rasmussen, 2004). The intention is that readers might, after understanding the details of this particular case, find the four factors useful more generally as an orienting framework for thinking about ways in which they can promote their students' use of theorems as tools for reasoning in other content areas. Thus, even if one does not regularly teach differential equations, this chapter intends to offer useful information for those who want their students to develop and use formal mathematics with understanding.

This volume is intended for mathematicians and mathematics instructors who want to enhance the learning and achievement of students in their undergraduate mathematics courses. The chapters in this volume are based on research that closely examines the cognitive and social complexities of how learners build mathematical ideas.

Academic mathematicians spend much of their time teaching, preparing to teach, or thinking about how to improve their teaching. Their conversations with colleagues and personal reflections often raise difficult questions about teaching and student learning. All the while the undergraduate mathematics education research community produces theories, models, curricula, and learning materials that speak to the questions mathematics instructors ask. To date, however, few vehicles have been available to assist instructors in using this research knowledge to better understand students' conceptual growth and to facilitate their reflection on teaching practice.

This volume is intended to bring some of the knowledge created by mathematics education researchers to the attention and service of mathematics instructors. The 23 chapters in the volume are divided into two sections. In Part 1, “Student Thinking,” the chapters describe perspectives and findings derived from investigations about how people learn central ideas in the undergraduate mathematics curriculum. Part 2, “Cross-Cutting Themes,” contains chapters that focus on the teaching of mathematics and various ways to frame issues that are inextricably related to the art of teaching. In the table of contents, for each article there is a brief annotation describing what issues the article addresses.