Dale Winter

Addressing difficulties with student-centered instruction

ABSTRACT We identify and analyze some recurrent problems with the implementation of cooperative and active learning strategies. Specifically, we address the use of questions, management of instructor-centered activities, management of in-class group activities, and giving up forms of control in the classroom. We provide suggestions for dealing with each of these difficulties.

Student Learning Objectives and Mathematics Teaching.

ABSTRACT The current work is the first article in a two-paper series exploring the role of explicit learning objectives in undergraduate mathematics instruction. A definition of student-learning objective (SLO) is introduced. We give examples of SLOs for topics from introductory college and university mathematics courses. We list potential advantages of a general program of grounding mathematics instruction in a set of such explicit SLOs. In the second paper in this series, an explicit, step-by-step algorithm for creating sets of student learning objectives will be described and its use illustrated.

Learning to Teach and Teaching to Learn Mathematics: The Professional Development Program

Overview We believe that it is important for a mathematics department to be actively involved in the professional development of its faculty and graduate students. By professional development we mean the training, assessment, and improvement of teaching skills and practices. Professional development that comes from within the mathematics department, rather than through non-disciplinary channels, is more likely to be meaningful and credible to the instructors who are involved. For example, sensibilities about teaching mathematics are sometimes quite different from those in other disciplines. Compare what is meant by “read the textbook” in a mathematics class to what is meant by the same statement in a literature or history class. In addition, the types and standards of argumentation and precision are different in the mathematics department than they are elsewhere. Moreover, there are certain obstacles that are more prevalent in mathematics than elsewhere. For instance, innumeracy and mathematical illiteracy are far more socially acceptable, or even favored, than is an inability to read or write. For these and other reasons, we feel that the best place for training and supporting mathematics instructors is within the mathematics department. We believe that an effective way for a mathematics department to train its inexperienced instructors, and to support and develop its other faculty members and graduate students is through an integrated professional development program that includes four components.

MENTAL MAPS AND LEARNING OBJECTIVES: THE FAST-SLO ALGORITHM FOR CREATING STUDENT LEARNING OBJECTIVES

ABSTRACT We review the processes of goal identification employed in teacher planning. A new type of graphical organizer (Fact-Action-Schema-Transcript or FAST) is introduced and its applications to teacher planning discussed. We describe a pair of algorithms for (a) creating FASTs for undergraduate mathematics and (b) using FASTs to create sets of student learning objectives (SLOs). Application of these algorithms to lesson planning are illustrated with several examples from introductory college mathematics courses.

Learning to Teach and Teaching to Learn Mathematics: Assessing and Evaluating Students' Work

The mathematics course that we envision utilizes a variety of assessment techniques, including team home- work, projects, student presentations and writing assignments. A common element of these assessment practices is the clear and precise communication of mathematical ideas. In many cases, students are expected to work cooperatively – to produce a single piece of work representing the collective efforts of three or four students. In order to provide sufficient challenge for a group of students and sufficient incentive for the students to work cooperatively (instead of simply dividing the work among themselves, and later compiling their individual contributions for submission) the problems assigned are more complicated, and are sometimes more “open-ended” than exercises typically assigned in traditional mathematics classes. The problems assigned typically require students to make appropriate assumptions, to try alternative avenues of inquiry, to try to understand the mathematics more thoroughly by recognizing its application to otherwise unfamiliar situations, etc. The work that students produce on these more complicated assignments is not simply pages of algebraic manipulations with boxed answers at the end. Instead, students are encouraged (and helped) to exhibit their understanding of the mathematics in multiple ways (such as graphs, written accounts of their assumptions and reasoning processes), instead of simply recording the algebraic steps that they performed. As may be expected, assessing and grading student work of this kind can be radically different from assessing and grading pages of algebraic manipulations with a conveniently highlighted answer at the end.