Educators generally agree that familiarity with students' conceptions and ways of thinking about mathematical topics are essential for teachers (e.g., Australian Education Council, 1991; Even & Tirosh, 2002; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; NCTM, 1991; 2000). In much the same vein, it is important for teacher educators to be familiar with teachers' conceptions and ways of thinking about mathematical topics and about pedagogical issues. In this paper we describe our initial explorations of elementary school teachers' perspectives on one central, pedagogical issue: The role of students' mathematical errors in mathematics instruction.
Students' errors were traditionally perceived either as signals of the inefficiency of a particular sequence of instruction or as a powerful tool to diagnose learning difficulties and to direct the related remediation (see, for instance, Ashlock, 1990; Fischbein, 1987; Greeno, Collins & Resnick, 1996). Several researchers, including Avital (1980) and Borasi (1987, 1992, 1994) suggested an alternative approach to the role of errors in the learning endeavor. Borasi (1987) argued that errors could be used to foster a more complete understanding of mathematical content, as a means to encourage critical thinking about mathematical concepts, as springboards for problem solving, and as a way to motivate reflection and inquiry about the nature of mathematics. Borasi pointed to the need to reconstruct the role of errors in mathematics instruction in order to make full use of the educational potential of errors. Similar opinions were presented by Avital (1980). He recommended to present error-triggering tasks (i.e., tasks that are known to elicit incorrect responses) in mathematical classes. Moreover, he argued that the best way to address common mistakes is to intentionally introduce them and to encourage a mathematical exploration of the related definitions, theorems and concepts.
Avital (1980) and Borasi's (1987) challenging approaches to the role of errors in mathematics instruction are the ones that we aimed to encourage in our three years, in-service elementary school mathematics specializing project. We were aware that knowledge about the participating teachers' perspectives on using errors in mathematical instruction could significantly contribute to our attempts to promote this viewpoint. In this paper we describe and discuss our initial attempts in this direction.
In 2002 a national, public committee examined the situation of mathematics education in Israel and recommended that mathematics should be taught only by mathematics specialists from Grade 1 on. In light of this recommendation a massive, three-year national program for specialized mathematics teachers for elementary schools started in 30 institutes for higher education in 2002. The principals of about one third of the elementary schools in Israel were asked to recommend three to five teachers that would participate in the course and become the mathematics specialists in their schools. This program is now in its second year.
Tel-Aviv University is one of the institutes that participate in this endeavor. The sessions (30 weekly meetings of four hours each year) were mainly devoted to enhancing the participants' own understanding of mathematical concepts and structures. Alongside the weekly sessions, we conducted individual interviews and small group meetings in which we discussed specific difficulties that individual participants faced and their views of various mathematical and pedagogical issues.
We developed, for this course, materials that were aimed to encourage the participants to come up with different solutions to the mathematical tasks and to examine them in light of the related concepts, rules and definitions. The tasks were designed to stimulate participants to pose questions such as: "Why is this so?", "How can we justify this? …