Basing instruction on theory and research: what is the impact of an extreme case?

Introduction

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.

Participants

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 the lengths of the remaining sides. This theorem is usually presented in school only with reference to triangles, even though it can easily be proved for all polygons. Moreover, the correct solution is also in line with the intuitive rule more A-more B.

Task 2 dealt with the comparison of the length of two sides of a pentagon with the sum of the lengths of its remaining three sides. This is a mathematically undecidable situation, since there is no theory to support a single generalized conclusion. Such comparisons are case-dependent, i.e., each case must be examined to determine which of the sums is larger. However, this questionnaire dealt with a specific story accompanied by an illustration, which actually could serve to solve the problem. In the drawing for Task 2, the sum of three sides was actually shorter than that of the remaining two sides. Indeed, it was possible (and even necessary) to simply measure the sum of the lengths of each suggested route in order to determine which one was shorter. This task was deliberately presented in such a way that the intuitive rule led to incorrect responses. That is, the correct solution here clashed with the intuitive rule more A (number of sides) -more B (longer route), because the route with the fewer sides was longer.

Results-Study A

No significant differences were found between the solutions of the 9th and the 11th graders, and therefore the findings relate to all participants.

As expected, almost all students who answered the problem represented by the quadrilateral drawing argued that the one-sided route is shorter than the three-sided route. Many students offered no explanation to their answers. Yet several participants, who did justify their answer, drew a diagonal and correctly explained their claim by twice applying the argument "In any triangle, the sum of the lengths of two sides is always larger than the length of the third side." Figure 2 shows David's solution. (2)

[FIGURE 2 OMITTED]

Other participants ambiguously claimed that "this is a generalization of the theorem relating to triangles." Only a small number of participants based their answers on measurements.

The 'pentagon program' like the previous one, was illustrated by means of a relevant drawing, but only about 20% of the students gave the correct answer. This means that only these students correctly observed that the three-sided route was shorter than the two-sided route. As before, students usually did not justify their responses. Yet, those who did provide explanations, either said, "it seems shorter," or performed (accurate) measurements "I have measured both routes and found this one to be longer."

Another interesting response, given by about 10% of the participants, was that the information presented in the problems did not allow determining which sum was greater. These students analyzed the given specific case from an (over) generalized perspective, claiming that it is impossible to determine which sum was larger: "One cannot determine whether or not two sides are larger than the other three. It depends on the given pentagon."

Nevertheless, a substantial number of participants (about 40%) argued, in line with the intuitive rule more A-more B, that the three sided route was longer than the two sided one. Moreover, when providing justifications, they usually claimed, "the larger the number of sides that are summed up- the longer the route created." These explanations referred explicitly to the intuitive rule more A-more B. A number of participants provided an unclear explanation: "It seems shorter/larger."

Several participants presented an over-extension of the intuitive rule more of A - more of B: "... in a triangle two sides are longer than the third, so, here, three sides are longer than the other two." Sheila and Joan, for instance, came up with an interesting combination of formal knowledge and the intuitive rule to explain that the three-sided route would be longer than the two-sided one.

  Sheila: It [the pentagon] is actually an extension of the theorem
  regarding triangles, that: "In any triangle the sum of any two sides
  is larger than the third side"; Accordingly "in any pentagon the sum
  of three sides is larger than the sum of the other two sides". As in
  algebra, adding an equal increment (1) to both sides of a given
  inequality (2>1), preserves the inequality (3>2).

  Joan: The pentagon is a generalized case of the triangle. One should
  try to prove this mathematical characteristic by using "induction."
  The theorem is true for the triangle, which case should be understood
  as a substitution of n=1 (one side is smaller than the sum of the
  others). The pentagon is actually the case of n=2 (the sum of two
  sides is smaller than the sum of the others); the heptagon, n=3 etc.
  Correctness should be assumed for n=k (the sum of k sides is smaller
  than the sum of the others, where there are 2k+1 sides). Then, based
  on this assumption, correctness should be proved for n=k+1".

Diana fabricated another invalid proof (see Figure 3):

[FIGURE 3 OMITTED]

On the whole, it seems that the intuitive rule more A-more B played a major role in students' comparisons, leading them to conclude, "the three sided route is longer than the two sided one." These claims were commonly accompanied by explanations, such as, the larger the number of the vertices--the bigger the sum of the sides, which were categorically consistent with the intuitive rule. In other cases students justified these responses by a mixture of intuitive and formal yet invalid ideas.

What Can We Learn from Study A?

As in previous cases (Stavy & Tirosh, 1996a; 2000; 2002), the present findings indicate that the intuitive rule more A-more B is indeed a good tool for analyzing students' responses to geometric comparison tasks. A substantial number of students from both grade levels tended to view all routes consisting of a greater number of sides to be the longer ones. These responses were correct in one case but incorrect in the other case.

Among the most remarkable observations was the students' tendency to support their intuitive claims by nonexistent formal explanations based on mathematical theories they had learned, such as "recursive proof", "drawing diagonals", and imaginary rules of "algebraic inequalities". They provided strange, incomplete, invalid, seemingly formal justifications for the answers, whose sole actual basis was the intuitive rule more A-more B.

A substantial number of participants concluded that "it is impossible to determine which route is shorter", indicating their "seeing the general in the particular" (a la Mason & Pimm, 1984). That is to say, rather than meeting the researcher's expectation of focusing on the particular, these students probably interpreted the problem differently. It seems that the conventional context and the sociomathematical norms that usually guided them also played a major role in their performance (see also Yackel & Cobb, 1996). This questionnaire was administered in students' geometry lesson, and therefore it is likely that the "norms" of this class were applied. According to these norms students are guided NOT to trust the drawing, NOT to trust measurements, and NOT to base their solutions on specific diagrams and measurements. Being used to dealing with abstract, general mathematical problems, and relating to illustrations merely as representative examples of various possibilities and not as a basis for the solution--these students failed to consider the concrete task at hand seriously.

Moreover, being aware of the need to formally justify every mathematical claim, a number of those who provided more A-more B solutions constructed invalid "proofs", which looked like the normative geometry proofs, in support of their predetermined responses. It seems that these students were pretty sure of their answers and treated the proof as "something one needs to present" rather than as the reliable way to examine and check their ideas.

Consequently, the intuitive rule more A-more B seems remarkably influential in directing students' reasoning. It was even more powerful than the relevant drawing provided in the task, which graphically presented the correct answers, and it was also more powerful than students' formal knowledge of geometry.

A question that naturally arises is, How can this information be implemented in instruction?

Study B: Basing Instruction on Research Findings, Extreme Cases and Cognitive Conflict

The literature reports that in various cases, such as that of Study A, different representations of essentially identical mathematical tasks often trigger in students different and even conflicting solutions (even, 1998; Janvier, Girardan, & Morand, 1993; Tsamir, 2003). Which representation is likely to trigger a correct solution and which an incorrect solution has been determined in many cases. One type of task that triggers correct solutions is the "extreme case" task. Such tasks are designed in such a way as to describe an extreme condition for which a correct judgment is almost unavoidable.

For example, Dembo, Levin and Siegler (1997) presented students aged 12-18 with a series of tasks, addressing different geometric figures when using a given (constant) perimeter. Among other figures, a square was transformed in front of the students into a rhombus, and a circle into an ellipse. The students tended to claim, incorrectly, that the area of both figures was the same because the perimeter was the same. Dembo and his colleagues asked part of these students to imagine what would happen to the respective areas of the two shapes if the transformations were taken to their extreme, and found that those students significantly outperformed the control group. Here, as in our study, in order to trigger conflict and re-evaluation of their original incorrect solution, the extreme case was presented after the students had given an incorrect response.

The above mentioned cognitive conflict approach can be characterized as a "from difficult to easy" teaching approach, where students are first asked to solve a difficult, counter-intuitive task and then an easy, related intuitive task. The students frequently reach two different solutions. Thus, in a subsequent discussion, teachers try to lead them to identify the conflicting elements in their different solutions, and to resolve the conflict according to the relevant mathematical theory. The challenges are to promote students' awareness of the following facts: that the two tasks essentially relate to the same mathematical issue, that the given responses are incompatible, and that the conflict must be resolved in accordance to a formal, mathematical framework (see, for instance, Swan, 1983; Tirosh & Graeber, 1990; 1994).

Study B examined the impact of extreme cases on students' tendency to use more A-more B considerations in their solutions to the geometric tasks presented in study A, when experiencing the cognitive conflict approach. Study B included three main stages: a pre-test, an extreme case intervention, and a post-test.

Participants

Eighteen 10th graders, who had studied Euclidean geometry, participated in the first stage of Study B by answering the two tasks in Study A. Nine of the students, i.e., those who incorrectly answered the pentagon route task in line with the intuitive rule more A-more B, participated in the extreme-case intervention and in the following post-test.

Tools and Process

Stage 1: The pre-test. During a geometry lesson, participants were asked to solve, in writing, and as part of a 10-task assignment, the two geometry tasks presented in Study A. There was no time limit. Based on the intuitive rules theory and on my findings in Study A, the prediction of Study B was that students would tend to answer correctly in the case of Task 1, but incorrectlyin line with the intuitive rule more A-more B, in the case of Task 2.

Stage 2: The intervention. Two days after the pre-test, the nine students who incorrectly answered Task 2 in line with the intuitive rule more A-more B, were individually interviewed. They were first presented with their previous, erroneous solution to the pentagon task, and then with the related extreme case situation (Figure 4). Finally, they were presented again with the original pentagon-task. All interviews were semi-structured, audiotaped and transcribed.

[FIGURE 4 OMITTED]

It was planned that, at this stage, students would either volunteer, or be encouraged, to reexamine their responses to the counter intuitive representation.

Stage 3: The post-test. About a month after the interview, these nine students were asked to solve the following tasks in writing and to elaborate on their explanations (Figure 5).

[FIGURE 5 OMITTED]

The tasks was designed so that Task 1 here was similar, but not identical, to Task 1 in the pretest, Task 2 was identical to Task 2 in the pretest, and Task 3 was counter-intuitive like Task 2, but with a different type of illustration. In the drawings in Tasks 2 and 3 the sums of the larger numbers of sides (three sides in Task 2 and four sides in Task 3) were smaller than the sums of the remaining (two or three) sides.

Results--Study B

Stage 1: Results of the pre-test. All eighteen participants correctly answered that the one-sided route is shorter than the three-sided route in Task 1, where the correct solution was in line with the intuitive rule more A-more B. This was not the case in Task 2. Only six students correctly answered that the two-sided route was longer, and five of them explained that they measured the routes. The other twelve participants incorrectly answered either that the three-sided route is longer than the two sided one (nine students), or that it is impossible to determine which route is longer (three students.

Stage 2: The intervention. This is the intervention in which the nine students who responded in line with the intuitive rule more A-more B participated. When shown their original solutions and asked to comment, none of the nine participants changed their mind. While three of them just said that they believed they had answered correctly, the other six elaborated on their original explanations. Typical comments were, "The sum of a larger number of sides is always greater", or "When you add up more sides, then you get a larger sum".

When presented with the extreme case drawing, all participants expressed awareness of the sameness of the idea underlying the two pentagon tasks, the one presented in the pre-test and the extreme-case presentation. They related to the incompatibility of their two solutions, to this being problematic, and consequently they rejected their initial, erroneous solutions. Ronny reacted in a typical way:

   Wow ... I can't believe it. I was so sure that here I had answered
   correctly [pointing to the counter-intuitive pentagon task] ... So, I
   felt like BOOM, when I saw this task [pointing to the extreme case].
   It definitely showed the same situation in a new way. I should have
   thought about that option here [in the first counter-intuitive case]
   too. My first solution is incorrect. I mean that it is not as obvious
   as I imagined, and I can't be sure that my solution is correct for
   this particular drawing.

It should be noted that all students expressed amazement at their revelation and referred to the obviousness and confidence they attributed to their initial, erroneous solution.

Five participants immediately went on to present their revised solutions, while the other four, like Ronny, stopped after realizing that their initial solution was erroneous. The interviewer further probed the latter students, and consequently all participants revised the initial pentagon-solutions. The participants were asked whether they were sure about their corrections and all of them replied that they had no doubts about them.

The nine revised solutions included seven suggestions to measure the two routes, and when offered a ruler, participants correctly pointed to the two-sided route as being longer. Most students said, like Daffy, that:

  Via this problem [the extreme case] I realized that there are cases
  where the route with the fewer sides is longer. So ... I gathered that
  here [the counter-intuitive representation] I have to measure the
  routes in order to see which one is longer.

Phillip, who previously provided a "general proof", by incorrectly extending the case of the triangle to explain why the two-sided route in the pentagon was shorter, added the following clarification,

  ... The drawing of Danny's two available routes home is actually a
  map. Like a picture of the "real thing" ... It is not presenting a
  general situation ... like for any polygon ... It is a specific case
  ... Therefore, in order to compare the two ways, I need to measure
  each of the two routes [used a ruler, and released a quiet whistle of
  surprise]. Here, too, the route with the fewer sides is longer.

Not only did Phillip present his revised solution (measure and judge), he also realized that the generalization he had previously made from the case of the triangle was an overgeneralization, which is irrelevant and inapplicable here.

The revised solutions of the last two students, who did not measure the two routes, were that it is impossible to determine which route is longer. Joy said,

  What a surprise ... I was sure that the number of sides determines the
  length ... like in the triangle.... Well, it is clear now [after
  seeing the extreme case] that in a pentagon [giggles] ... I understand
  that Danny's story was created to examine the sides of a pentagon ...
  It is impossible to give a conclusive solution. It depends on the
  given pentagon. So, here, the answer is that it is impossible to
  determine which route is longer.

In conclusion, all the nine participants in the extreme-case intervention had provided more -more solutions to the initial, counter-intuitive representation, and, were subsequently surprised to see that the two-sided route was longer than the three-sided route in the extreme case task. They were all aware of the basic similarity of the different representations they had solved and of the contradictions that arose. They made two types of suggestions to resolve the problem when changing their initial, erroneous responses. Most of them now correctly related to Danny's story and its accompanying drawing, as a specific case that should be examined and solved by means of a ruler. The others ignored the specific nature of Danny's schematic routes home, and treated the task as representing a general pentagon. While the latter failed to grasp the specific nature of the story, they too managed to get over their intuitive tendency to use more A-more B considerations.

Stage 3: Results of the post-test. In reaction to Task 1, seven of the nine students who participated in the extreme-case intervention answered correctly that the two-sided route was longer than the one-sided route. However, both Ruth, who had measured the routes by the end of the extreme-case intervention, and Joy, who had reached the "impossible to determine" conclusion, here provided an uninvited generalization, i.e., an "impossible to determine" response. Interestingly, both of them mentioned the extreme-case intervention in their explanations. Ruth, for instance, said:

  We saw ... in the interview ... that we cannot ... it is impossible to
  answer such questions in a general way. It [the solution] depends on
  the specific case we are dealing with ... So, it is impossible to say
  which route is longer.

In reaction to Task 2, all nine participants volunteered remarks that suggested recognition that they had dealt with this task in the extreme-case intervention. Six answered correctly that they needed to measure in order to determine which route was longer, and went on to actually use their rulers. Before taking measurements, two of them remarked that if it was really the same task, then the two sided route had to be longer, but they added that after their previous experience (in the intervention) they were "not taking any risks".

The other three students, Ruth, Joy and Jane (Jane also reached the "impossible to determine" conclusion in the extreme case intervention), said that it is never possible to give a conclusive solution to such tasks.

Most interesting, however, were the responses given to Task 3. While the latter three students, Ruth, Joy and Jane, argued that it was not possible in the case of the heptagon too, to determine whether the three sided route was longer, equal or shorter than the four-sided one, the other six students claimed the four-sided route was longer. That is to say, all the participants incorrectly solved the heptagon-route task, and most of them regressed to their initial more A (sides) - more B (length) way of thinking. Sigal gave a typical response,

  This route is longer because it has four sides ... I mean more sides
  ... and turning points.

What Can We Learn from Study B?

Study B examined the impact of the extreme-case intervention, i.e., the impact of using extreme cases, on students' tendency to use more A-more B considerations in their solutions to geometric tasks, when faced with a cognitive conflict.

When examining students' reactions to the pentagon task, the data here show that, indeed, the different representations of the task (in the pretest vs. the extreme case in the intervention) elicited incompatible solutions. As predicted, the counter-intuitive representation usually triggered erroneous solutions, mostly in line with the intuitive rule more A-more B, while the extreme case always triggered correct solutions. During the extreme case intervention, the participants spontaneously identified the extreme case and the related counter-intuitive task as being essentially identical. This identification was in all cases accompanied by a remark relating to the incompatibility of the participants two solutions or to the contradiction reached. In conclusion, students suggested ways to resolve the contradiction, usually by pointing to the correct solution. That is to say, through the cognitive conflict approach, the students went through the process detailed in the introduction to Study B. The students were first asked to address a difficult, counter-intuitive representation of the task and then an easy, related intuitive representation of it. Consequently, they reached incompatible solutions, which usually led them to resolve the conflict according to the mathematical theory (see also, Swan, 1983; Tsamir, 2003).

All the participants' responses to the counter-intuitive pentagon task in the post-test, drew upon the extreme-case intervention and were freed from the intuitive impact of the more A-more B rule. Seven of the nine students responded it was necessary to measure the routes, and the remaining two incorrectly claimed it was impossible to determine which route was longer. It seemed that all the participants abandoned their intuitive, more A-more B ideas.

Had we stopped here, it would have appeared that the extreme-case intervention was extremely effective for controlling the problematic impact of the intuitive rule more A-more B on students' solutions. However, further examination of the participants' solutions to the heptagon and triangle tasks in the posttest revealed a different situation. Quite surprisingly, all the participants who reached the correct solution to the pentagon task, when solving the similar (for us) heptagon task, reverted to their initial intuitive more A-more B ideas. Even the proximity (both in time and place) to the pentagon task in the questionnaire did not promote the participants' awareness of the need to reexamine such intuitive solutions.

Only the three students who as a result of the extreme-case intervention ignored the specific cases presented, remained resistant to the intuitive impact of the more A-more B rule. Although they blundered into a new type of error by imparting generality to specific cases, they benefited from the extreme-case intervention by acquiring "immunity" to the coerciveness of intuitive impact. However, an examination of these students' solutions to the triangle task indicated that the impact of the extreme case intervention on their solutions was more pervasive than expected. These students became convinced that even in the self-evident case of the triangle, where the sum of the two sides is always larger than the third one, it is impossible to determine which route is longer. This means that participation in the extreme-case intervention led these students to develop new pseudo-generality, imparted to all such comparison tasks, by erroneously over-generalizing from a difficult, irrelevant situation to the easy, self-evident one.

Consequently, it seems that students were able to use their solution to the extreme case for revising their initial, erroneous solutions to the counter-intuitive representation of the same task. However, this beneficial impact was restricted to a very limited range of problems--i.e., only pentagons. On the one hand, the expected extrapolation to the heptagon was not made, and on the other hand, unexpected (erroneous) connections were made with the triangle. Obviously, the question is, How can we deal with the misconceptions induced by this intervention, in order to improve students' performance?

Summing Up and Looking Ahead

This paper illustrates the use of the intuitive rules theory for analyzing students' reactions to geometry tasks regarding polygons and the use of theory, research findings and the "extreme case" method for teaching by cognitive conflict.

In Study A, the Intuitive Rules Theory shed much light on the analysis of the data. It was found that the task, which was in line with the intuitive rule more A-more B, triggered many correct solutions, while the counterintuitive task triggered many incorrect, more (sides) -more (length), responses. In addition to the influence of the intuitive rules, the findings revealed another phenomenon, namely that of the tendency to provide erroneous, "impossible to determine" solutions.

In Study B, the findings of Study A and familiarity with the intuitive rules theory served in the design of the extreme-case intervention. It was found that the sequence designed, indeed, gave rise to the expected conflict in students' reasoning. The extreme case provided a visual anchor which could compete with, and even overcome, the intuitive more A - more B considerations. However, the students' revised solutions indicated limited improvement. Study B verifies Fischbein's (1987) claim that intuitive knowledge is robust and tends to survive even when contradicting formal instruction. It is reasonable to assume that here, in Study B, like in Zazkis's (1999) study, some students did not give up their more - more intuitions, but treated the case of the pentagon as an exception. Clearly, there is a need to address the shortcomings of this intervention and investigate the impact of any subsequent teaching.

This paper illustrates the usefulness of using theory to analyze data. However, while the vast majority of the findings of Study A could be explained by means of the intuitive rule more A - more B, some findings revealed a hitherto unobserved, "cannot determine" phenomenon. This phenomenon became more evident and problematic in the case of the triangle in Study B, even after the extreme-case intervention.

Similarly, this paper illustrates the usefulness of applying theory, research findings and familiar teaching methods in the design of instruction, but demands a careful assessment of the students' resulting knowledge. A critical examination of students' reactions to a wide range of related tasks, with reference to the instructional experience they have had, may assist in reflecting on past and future instructional sequences.

It seems that the cycle of research and instruction is a never-ending one: Students' primary knowledge, analyzed through a certain theoretical framework, serves as a basis for designing instruction. The knowledge students thus gain is again analyzed, and the data serves in assessing the impact of the intervention--its pros and cons--so that further instruction or alternative instruction may be considered. In this paper we might consider addressing the general vs. specific dilemma, which emerged as a result of the extreme-case intervention, as the next-step for instruction. In the future, it would be interesting to consider an alternative intervention based on the intuitive rules theory, on research findings, and on teaching methods such as the analogy approach (Clement, 1993; Stavy, 1991; Tasmir, 2003). Clearly, the contribution of any instructional approach should be analyzed and assessed by means of further research.

Notes

(1) For the sake of simplification, the phrase "the sum of lengths of a number of sides of a polygon" is occasionally shortened to "the sum of a number of sides of a polygon".

(2) All names used in this paper are pseudonyms.

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Pessia Tsamir

Tel-Aviv University