In the last decade, it has been widely recommended that mathematics teaching consider and address students' correct and incorrect ideas concerning the subject matter (NCTM, 1991; 2000). Teachers are expected, among other things, to design instruction based on data regarding students' relevant conceptions and misconceptions, and to be alert to them in the course of class discussion. It is clear that this requires familiarity with students' common errors, with what makes a problem easy or difficult for them, and with possible ways to address their difficulties. Helpful sources for such teaching may be found in theoretical frameworks that explain students' correct and incorrect ideas, as well as in general teaching approaches.
The Intuitive Rules Theory is one such theoretical framework, and the cognitive conflict approach is one such teaching approach. This paper illustrates the use of the intuitive rules theory for analyzing students' reactions to geometrical tasks regarding polygons, and the use of the cognitive conflict approach for subsequent teaching. The paper consists of three main sections, (a) Study A: using the intuitive rules theory to analyze students' solutions, (b) Study B: basing instruction on research findings, extreme cases and cognitive conflict, and (c) summing up and looking ahead.
Study A: Using the Intuitive Rules Theory to Analyze Students' Solutions
The intuitive rules theory that accounts for many of the incorrect responses students present to scientific and mathematical tasks, was formulated and investigated by Stavy and Tirosh (Stavy & Tirosh, 1994; 1996a; 1996b; 2000; Tirosh & Stavy, 1999). The main claim of the intuitive rules theory is that students tend to react in a similar, predictable manner to various, unrelated scientific, mathematical and daily tasks that share some external features. One intuitive rule, which has been extensively investigated, is more A-more B, and its strong explanatory and predictive power has been widely reported (Stavy & Tirosh, 2002). All tasks that elicit responses in line with the intuitive rule more A-more B are comparison tasks, describing two objects differing with regard to a certain salient quantity, A (A1>A2). Students are asked to compare these two objects with respect to another, given quantity B, where [B.sub.1] is not necessarily larger than [B.sub.2]. It was found that students tended to claim that [B.sub.1] > [B.sub.2] because [A.sub.1] > [A.sub.2].
For example, Fischbein (1993) presented students with two points: Point A--the intersection point of two lines--and Point B--the intersection point of four lines. Students tended to view Point B as larger and heavier than Point A. They explained the more lines that intersect--the larger the intersection point, and that the more lines that intersect--the heavier the intersection point. In another research Klartag and Tsamir (2000) found that high school students tended to claim that for any function f(x), if f([x.sub.1]) is larger than f([x.sub.2]) then f' ([x.sub.1]) is larger than f' ([x.sub.2]). These claims were also evident when the students were presented with specific functions, given in an algebraic representation, where it was easy to refute this claim by substituting a suitable value. In both cases, students tended to claim that [b.sub.1] > [b.sub.2] because [a.sub.1] > [a.sub.2], or more A (number of intersecting lines, value of the function f(x)) more B (size of intersection point, value of the derivative of the function f' (x)).
Stavy and Tirosh (2000) claimed that the rule more A-more B is intuitive in the sense that Fischbein (1987) used the word, i.e., reactions based on it are immediate and confident, and the correctness of the associated solutions seems self-evident. Indeed, studies in mathematics and science education indicate that more A-more B is often intuitively used by students in relation to various topics (Noss, 1987; Stavy & Tirosh, 1996a; 2000; Tsamir, 1997; Zazkis, 1999).
In the first study that is described in this paper, the intuitive rule more A-more B was used to analyze students' responses to comparison tasks regarding polygons. More specifically, this study dealt with the sum of lengths of a number of sides of a polygon as compared to the sum of lengths of its remaining sides (1). It seems quite obvious that when given a triangle the sum of lengths of two sides is larger than the length of the third side. However, what happens, for instance in the case of a quadrilateral, or a pentagon, or a heptagon? This study aimed to investigate whether secondary school students, who had studied Euclidean geometry, would tend, in all cases, to view the sum of the lengths of more sides as being larger. That is, would the intuitive rule more A-more B be applied only in cases where applicable, or also in other cases where its use is inappropriate.
One hundred and thirty six secondary school students, who had studied triangles, quadrilaterals, and circles in the framework of Euclidean geometry, participated in this study. Seventy of them were 9th graders and the other 66, 11th graders.
Tools and Process
The participants were asked to relate in writing to a questionnaire which they filled out during a geometry lesson (see Figure 1). No time limitation was determined. The questionnaire included a story about Danny, who in each task has two alternative routes by which to return home. In this way, the students were asked to examine two polygons--a quadrilateral and a pentagon--and to compare the lengths of one/several sides of the polygon to the sum of the lengths of its remaining sides.
[FIGURE 1 OMITTED]
Task 1 dealt with the comparison of the length of one side of a quadrilateral with the sum of the lengths of its remaining three sides. In this case, the "triangle inequality" theorem indicates that in a triangle the length of a single side is always shorter than the sum of …