Numerous investigations on errors in mathematical procedures have been made in recent years. In particular, insights from cognitive science and the production system approach have been useful for examining the typical errors that students make in mathematics. This detailed cognitive analysis of the systematic errors (bugs) in students' calculation procedures gives us the possibility to develop more adequate instructional strategies intended to help students develop procedural rules (Maurer, 1987; VanLehn, 1982, 1983). This approach, which concentrates on the errors in algorithmic procedures, involves some limitations, however, if we consider mathematics education. The serious problems that can be involved in students' automated and faultless mathematical procedures have been described by researchers over several decades (see Dreyfus, 1991; Schoenfeld, 1985; Selden, Mason, & Selden, 1989; Wertheimer, 1959). Besides the systematic process errors (bugs), many other, more general problems of mathe-

matics learning, such as lack of understanding (Greeno, 1987), inadequate beliefs and metacognitions (Schoenfeld, 1985, 1987), and insufficient or erroneous strategies (Wenger, 1987), have been highlighted in recent research.