A New Way of Teaching

Abstract

A new view of teaching is emerging from the work of the constructivists and mathematics education reform leaders. In particular, we examine here four aspects of teaching that must change: task selection, guidance of classroom discourse, setting the learning environment, and the analysis of teaching and learning. Several national curriculum projects are working to effect these changes and examples of their work are provided. This work has motivated individual teachers to similarly design investigations that engage students in the study of significant mathematics, and two examples are included. Assessment must also change and students must learn to become less dependent on “authority” for the correctness of answers. Finally, our present understanding of constructivism and its implications for teaching/learning must not be static; though that view now may be at the center, we must listen to those who are on the edges and expect to be changed again and again in the years ahead.

The Development of an Instrument to Measure Preservice Teachers' Attitudes about Discourse in the Mathematics Classroom

“…“Worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work” (NCTM, 2000, p. 19). These tasks should invite dialogue and student interaction concerning reasoning of mathematical ideas (NCTM, 1991; 2000; Ball, 1991; Prevost, 1996). Tasks that encourage more than one solution or alternative interpretations (Ellerton, 1991; NCTM, 1989, 1991, 2000) help foster this form of interaction where students are actively engaged in their learning (NCTM, 2000; Prevost, 1996).…”

“…These tasks should invite dialogue and student interaction concerning reasoning of mathematical ideas (NCTM, 1991; 2000; Ball, 1991; Prevost, 1996). Tasks that encourage more than one solution or alternative interpretations (Ellerton, 1991; NCTM, 1989, 1991, 2000) help foster this form of interaction where students are actively engaged in their learning (NCTM, 2000; Prevost, 1996). Additionally, when students are given a fewer number of rich tasks to occupy their minds, there is a tendency for more in‐depth concepts to be exchanged (Emenaker, 1995; Hiebert & Wearne, 1993).…”

“…Finally, teachers of mathematics should “sustain an atmosphere of ‘intellectual ferment’ in the classroom” (Chazan & Ball, 1995, p. 7) in order to promote discourse. At the core of a learning environment that is both rigorous and nurturing of students, teachers and students should share authority where sound mathematical reasoning dictates the validity of a solution (Ball, 1991; Brown & Renshaw, 1999; Chazan & Ball, 1995; Lappan, 1997; Prevost, 1996). Valuing student opinions, explanations, and justifications is one way to foster a more balanced proportion of mathematical authority and encourage students to rely on mathematical evidence (Ball, 1991; Borko et al, 2000; Lappan, 1997).…”