We are not the first to argue that there may be discordances and conflicts between many advanced mathematical concepts and ‘‘naı¨ve mathematics.’’ Fischbein (1987) was one of the first mathematical educators to notice that intuitive beliefs may be the cause of students’ systematic errors in mathematics, a fact also noted by Stavy and Tirosh (2000) in their intuitive rules theory (see also Tirosh & Tsamir, this issue), by researchers such as Greeno (1991), and Verschaffel and De Corte (1993) in the case of addition and subtraction, and pointed out by many other math educators such as Vergnaud (1989) and Sfard (1987). Other researchers have argued that incompatibility between prior knowledge and incoming information may be the source of students’ difficulties in understanding algebra (Kieran, 1992), fractions (Hartnett & Gelman, 1998), rational numbers (Merenluoto & Lehtinen, 2002), etc. The conceptual change approach has the potential to enrich a social constructivist perspective and provides the needed framework to systematize the above-mentioned widespread findings and utilize them for a theory of mathematics learning and instruction.
Some of the more obvious advantages of exploring the instructional implications of the conceptual change approach are the following: It can be used as a guide to identify concepts in mathematics that are going to cause students great difficulty, to predict and explain students’ systematic errors and misconceptions, to provide student-centered explanations of counter-intuitive math concepts, to alert students against the use of additive mechanisms in these cases, to find the appropriate bridging analogies, etc. In a more general fashion, and as it is being discussed by the individual contributions to this volume, it highlights the importance of developing students who are intentional learners and have developed the metacognitive skills required to overcome the barriers imposed by their prior knowledge (Schoenfeld, 1987; Vosniadou, 2003).
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