Basing Instruction on Theory and Research: What Is the Impact of an Extreme Case?
Tsamir, Pessia, Focus on Learning Problems in Mathematics
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). …