From "easy" to "difficult" or vice Versa: the case of infinite sets.

It is commonly agreed that students' ways of thinking should be taken into account when planning instruction and that teachers should choose or construct sequences of instruction for use and discussion in class. The NCTM document, "Principles and Standards for School Mathematics" (2000, p. 17), for instance, specifies that teachers "need to understand the different representations of an idea, the relative strengths and weaknesses of each, and how they are related to one another ... They need to know the ideas with which students often have difficulty and ways to help bridge common misunderstandings". However, the shift from such theoretical knowledge to the design of teaching sequences and to practice is not a trivial one in the least.

For example, one important facet of mathematical knowledge is the ability to move flexibly among different representations of a given mathematical notion, process or problem. Still, it has been widely documented that different representations of the same mathematical problem often trigger different and sometimes even conflicting solutions (e.g., Even, 1998; Janvier, Girardon, & Morand, 1993). Research findings clearly indicate how students tend to respond to the different representations of the same mathematical task and their justifications for these responses (Duval, 1983; Hart, Johnson, Brown, Dickson, & Clarkson, 1989; Tirosh & Tsamir, 1996). A question that arises is how the accumulated data on students' reactions to different representations of given mathematical tasks can be applied to instruction.

1. Using Different Representations in Instruction

Two methods that use the accumulated findings indicating that different representations lead to different results, are teaching by analogy and teaching by cognitive conflict. In both methods, students are presented with different representations of the same task: one is the target task, known to commonly trigger an incorrect response (a difficult task), and the other is the anchoring task--a task known to intuitively trigger a correct response (an easy task). Teachers then attempt to promote students' awareness of the fact that they are being presented with the same task, albeit in different representations.

When going "From Difficult to Easy" in the cognitive conflict teaching approach, students are first asked to solve the target task and then the anchoring 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 that the two tasks are essentially the same, and that their responses are incompatible with each other, then to lead them to resolve the conflict in accordance with the formal, mathematical framework (see, for instance, Swan, 1983; Tirosh & Graeber, 1990). It should be noted that, in the literature the terms "anchoring task" and "target task" were defined and used with reference to the From Easy to Difficult teaching (by analogy) approach.

When going "From Easy to Difficult" in the teaching by analogy approach, the students are first asked to solve the anchoring task and then the target task. The teacher preferably chooses or constructs a sequence of bridging tasks that gradually lead from the anchoring task to the target task. Such a sequence of steps may assist the students in reaching the same correct solution to the same task presented in different ways. Indeed, it has been reported that by following such a "From Easy to Difficult" sequence of instruction, students frequently reach correct solutions to all tasks presented in the sequence, including the target task. The challenge is to find suitable anchoring and bridging tasks, and to verify that at each stage students make the mathematical connections that allow them to grasp the "sameness" of the tasks (see, for instance, Clement, 1993; Stavy, 1991).

The skill of sensibly using students' reactions to different representations for instruction seems very demanding. Some prerequisites are: (1) familiarity with different representations of given mathematical tasks, (2) awareness of students' common reactions to each representation, and (3) acquaintance with possible ways to include this information in instruction. The following illustrations demonstrate how different representations were used to exemplify such instruction in a preservice training course for secondary-school mathematics teachers.

The topic chosen for this purpose, "Comparing Infinite Sets within the Cantorian Set Theory", is part of an advanced mathematics course. It should be noted that the expression "comparing infinite sets" denotes "comparing the number of elements" (i.e., the power). This topic was chosen because of the central role that infinity plays in mathematics, its central role in the mathematical curriculum of prospective secondary-school teachers in Israel, and the reported difficulties that students encounter when dealing with the infinite (e.g., Duval, 1983; Martin & Wheeler, 1987; Tsamir, in press).

This paper describes a study that investigated prospective teachers' reactions to two sequences of different representations of comparison-of-infinite-set tasks. To put things in context, the presentation starts with a brief survey of students' common reactions to different representations of specific comparison-of-infinite-set tasks, as reported in the literature. Then, the construction of two teaching sequences is presented and followed by the description of a study that investigated students' reactions to these two different sequences. In conclusion, some comments regarding possible educational implications and some areas for further pertinent research are presented.

2. Responses to Different Representations of Comparison-of-Infinite-Sets Tasks

As mentioned before, studies on students' ideas of infinity report a tendency to use various methods for comparing the number of elements in infinite sets. The most prevalent ones are: "A proper subset of a given set contains fewer elements than the set itself; "There is only one infinity"; and "Infinite quantities are incomparable" (e.g., Borasi, 1985; Martin & Wheeler, 1987; Tsamir, 1999). Research has also indicated that students often use several of the above-mentioned methods when relating to different representations of the same comparison task, and that they reach contradictory conclusions without being aware of it (e.g., Duval, 1983; Tsamir & Tirosh, 1999).

Tirosh and Tsamir (e.g., 1996) presented 16-18-year-old students and prospective teachers with different representations of problems that required comparing the number of elements in pairs of infinite sets. For example, here are four representations of the task asking to compare the number of natural numbers with that of the natural numbers larger than 2, namely, {1,2,3,4,5,6,7,8,9,10,...} and {3,4,5,6,7,8,9,10,...}.

* Horizontal representation -- the sets are presented beside each other.

A = {1,2,3,4,5,...} B = {3,4,5,6,7,...}

* Vertical representation -- the sets are presented one below the other.

A = {1,2,3,4,5,...}

B = {3,4,5,6,7,...}

* Explicit representation -- the sets are presented one below the other so that it is visually easy to match each element of one set with one element of the other set. Here the pairs are 1[left and right arrow]1+2, 2[left and right arrow]2+2, 3[left and right arrow]3+2, 4[left and right arrow]4+2, 5[left and right arrow]5+2 etc.

A = {1, 2, 3, 4, 5,...}

B = {1+2, 2+2, 3+2, 4+2, 5+2,...}

* Geometrical representation -- the elements of the given sets are related to geometrical figures that trigger the matching of each element of one set with one element of the other set. For example, consider the two sets of numbers, and the set of trapezoids:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...}

B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...}

[ILLUSTRATION OMITTED]

Set M is an infinite set of trapezoids. The upper base of each trapezoid is shorter by 2cm than the bottom base. The first upper base is 1 cm long and each following segment is 1 cm longer than the previous one. The first bottom base is 3 cm long and each following bottom base segment is 1 cm longer than the previous one. Set A presents the numbers that express the lengths (in cm) of the upper bases; and set B presents the numbers that express the lengths (in cm) of the bottom bases in the trapezoids of set M.

For each representation, students were asked to judge whether the numbers of elements in sets A and B were equal, and to explain their answer.

It was found that the horizontal representation encouraged "more natural numbers" responses, usually justified by part-whole arguments, explaining, for instance, that "set A consists of all the elements that are included in set B in addition to two extra elements. Thus, there are two more elements in set A than in set B." The other representations evoked "the same number of elements" responses, accompanied by various types of justifications. The vertical representation triggered "single infinity" justifications, claiming that "all infinite sets have the same number of elements, so the presented sets are also equal." The explicit and geometrical representations triggered one-to-one correspondence considerations, pointing to a way to pair matching elements. Hence, students reached contradictory responses to the different representations of the same task, for instance, "equal number of elements" to the geometrical and explicit representations, and "unequal number of elements" to the horizontal representation.

In conclusion, it is notable that the topic of comparison-of-infinite-sets satisfied the three above-mentioned prerequisites for using research findings about students' reactions to different representations of the same mathematical task in instruction. That is to say, different representations of comparison-of-infinite-sets tasks were available, and students' common responses to each representation were reported in the literature. The latter information, together with the previously cited teaching methods ("From Easy to Difficult" and "From Difficult to Easy") was used in developing instructional sequences.

3. Using Common Reactions to Different Representations for Developing Instructional Sequences

An analysis of related research findings (e.g., Tirosh & Tsamir, 1996) served to determine that the explicit and the geometrical representations of the tasks could be considered "easy tasks" (i.e., tasks to which high percentages of correct responses based on one-to-one correspondence considerations were reported). These same findings also assisted in determining that the horizontal representation of the comparison-of-infinite-set tasks could be considered "difficult tasks" (i.e., tasks to which high percentages of incorrect responses were reported). This information was applied in designing two comparison-of-infinite-sets teaching sequences--"From Easy to Difficult" (i.e., teaching by analogy) and "From Difficult to Easy" (i.e., teaching by cognitive conflict) (see Figure 1). The included instructions are presented in the Appendix.

The present study investigated the impact of the two teaching sequences on prospective teachers' responses to comparison-of-infinite-sets tasks. The questions raised here were how participation in the "From Easy to Difficult" sequence or in the "From Difficult to Easy" sequence would influence (1) prospective teachers' tendency to use one-to-one correspondence, and (2) their awareness of the need to use only one method for such comparisons.

3.1. The Study

The "From Easy to Difficult" and "From Difficult to Easy" sequences were tried with two groups of prospective secondary-school mathematics teachers--15 prospective teachers in the "From Easy to Difficult" group and 16 in the "From Difficult to Easy" group. These prospective teachers had not yet studied Cantorian set theory, and they had not dealt with the comparison-of-infinite-sets.

The "From Easy to Difficult" and "From Difficult to Easy" sequences, were presented via three-stage infinite-set card activities relating to the comparison of natural numbers to perfect squares (see Figure 2). The tasks were presented to each prospective teacher individually during an oral interview that lasted about 20 minutes. The researcher conducted all interviews, and she gained the participants' serious attitude by promising that their solutions will serve as a basis for discussing psycho-didactical issues. The participants were presented with the cards of a stage only after responding to the tasks of the previous stage. During the entire interview, all the cards remained on the table, so that at Stage III students could easily re-examine the responses they had given during the first two stages.

About a week after the completion of the interviews, all prospective teachers were interviewed individually again. Each of them was first asked to respond to the difficult, horizontal representation of the following "N vs. 3N" comparison-of-infinite-sets task:

  Given two sets:
  N = {1, 2, 3, 4, 5, 6, 7,...}  T = {3, 6, 9, 12, 15, 18, 21,...}
  Is the number of elements in set N larger than/equal to/smaller than
  the number of elements in set T? (circle your choice).
  Explain your answer.

In reaction to his or her responses, each participant was challenged to react to a kind of "What-if-Not" task, i.e., a suggested solution that was based on a method different from the one given by the interviewee, and triggering a different conclusion. The "What-if-Not" task was presented as if another prospective teacher had used it to solve a similar ("N vs. 2N") comparison-of-infinite-sets task. The aim of the What-if-Not task was to examine whether the participants generally regarded as valid the application of different methods for the comparison of infinite sets, rather than to trigger a conflict between incompatible solutions to the same task. Therefore, when asked to reflect on the different solution, the participants were not presented with exactly the same task. For example, a participant who claimed that the number of elements in set N is larger than the number of elements in set T because set T is included in set N was presented with the following question:

[FIGURE 2 OMITTED]

[FIGURE 2 OMITTED]

  Another student was asked to compare the number of elements in the
  sets:
  N = {1, 2, 3, 4, 5, 6, 7,...}  E = {2, 4, 6, 8, 10, 12, 14,...}
  The other student responded that the number of elements in set E is
  equal to the number of elements in sets N. She explained that it is
  possible to pair an element from set N with a matching element in set
  E: 1 with 2, 2 with 4, 3 with 6, 4 with 8, 5 with 10, etc. For each
  element n is set N there is exactly one matching element 2n in E. The
  same is true for the opposite direction: for each element x in set E
  there is exactly one matching element x/2 in set N.
  In your opinion,
  - Is her solution correct?
  - Is it acceptable to use both your method of "inclusion" and hers of
  "matching elements" when comparing infinite sets?

It should be noted that participants who solved the N vs. 3N task based on "equal" and "infinite sets are always equal" were presented with an "unequal" and "inclusion" solution. Those who claimed they could not solve the N vs. 3N task were just asked whether, in their opinion, it was acceptable to use different methods, such as one-to-one correspondence and inclusion, for the comparison of infinite sets.

The interviews in which the "N vs. 3N" and "What-if-Not" questions were presented also lasted about 20 minutes. All interviews (of both stages) were audiotaped and transcribed.

This research structure facilitated the investigation of prospective teachers' responses to comparison-of-infinite-sets tasks during the interventions of the "From Easy to Difficult" and "From Difficult to Easy" sequences. It also permitted the impact of each of the two sequences on the participants' solutions, a week after the intervention, to be examined.

3.2 Results

This section presents the participants' reactions to each of the teaching sequences. First, the response of the "From Easy to Difficult" group are presented, and then, those of the "From Difficult to Easy" group. In each subsection, I start by reporting the participants' solutions to the three stages of the sequence (first interview), and conclude by reporting on their reactions to the "N vs. 3N" and "What-if-Not" tasks that were given a week later in the second interview.

3.2.1 Solutions of the "From Easy to Difficult" Group

First Interview: Solutions to the "From Easy to Difficult" sequence

In Stage I. when presented with the geometrical and the explicit representations, all the participants claimed that these tasks consisted of sets that were equal, and used one-to-one correspondence considerations to justify their responses (Table 1). Typical explanations of Task 1 were: "The number of segments is equal to the number of squares. Each segment has a specific length that contributes one element to set A, and each square has a specific area that contributes one element to set B; therefore, the number of elements in sets A and B is equal". The explanations of Task 2 were, for instance: "It is possible to go on pairing elements like 1 and [1.sup.2], 2 and [2.sup.2], 3 and [3.sup.2], one element from each set matching another, so that no element is left unmatched."

In Stage II, when presented with the vertical representation, these same participants argued again that the sets had an equal number of elements. In response to Task 1 in this stage, all participants identified set B and set P as including exactly the same elements (numbers). They usually said, "These are two ways of writing the perfect squares"; or "It is actually the same set, therefore the number of elements in B and in P must be equal"; or "1 is exactly the same as [1.sup.2], 4 is exactly the same as [2.sup.2], 9 is exactly the same as [3.sup.2], and so on. Each number in either of the sets is of the type [n.sup.2], therefore all numbers appear in both sets." Four participants added comments about the triviality of this problem.

All the solutions given in Task 2 in this section were again "equal" and the explanations were based on one-to-one correspondence considerations. Six participants commented that this task was the same as Task 2 in Stage I.

In Stage III, when responding to the horizontal representation, the prospective teachers repeated their "equal" argument and their explanations. All of them mentioned in one way or another that they noticed that they were being asked to respond to essentially the same task. Their comments included: "Actually, I already solved this problem here [in Stage II]"; or "It's the same problem over and over again"; or "I see that you found many ways to ask the same questions."

In conclusion, all "From Easy to Difficult" participants responded "equal" in all three stages, based their answers on one-to-one correspondence considerations, and spontaneously identified the "sameness" of the given tasks.

Second Interview:

Solutions to the "N vs. 3N" Task: Most (9) of the prospective teachers who had experienced the "From Easy to Difficult" sequence used part-whole considerations to justify their claim that set N consisted of more elements than set T. Typical explanations were: "Set T is part of set N, therefore it has fewer elements." The other six prospective teachers claimed that the sets had an equal number of elements, and all of them referred to their conclusions in the previous interview (in the sequence), saying, for instance: "This is similar to the task we did in our previous talk [first interview]". Still, while three of them identified one-to-one correspondence between matching elements, the explanations of the other three participants expressed the idea that "infinite sets probably have the same number of elements". The latter explained, for instance, that "I saw that all infinite sets are equal last time"; or "We showed previously that all infinite sets have the same number of elements."

Solutions to the "What-if-Not:" Task: All of the "From Easy to Difficult" participants regarded the What-if-Not solution presented as if suggested by another student as correct. They further concluded that it was acceptable to use different methods for the comparisons of infinite sets (Figure 3). A typical explanation was: "Depending on the problem, it is sometimes easier to use one method and sometimes another. So different problems are naturally solved in different ways."

3.2.2. Solutions of the "From Difficult to Easy" Group

First Interview: Solutions to the "From Difficult to Easy" sequence

In Stage I. when presented with the horizontal representation, all of the participating prospective teachers argued that the two sets did not have the same number of elements. They used part-whole considerations to explain their responses (Table 2). A typical explanation was: "Card A includes all the elements that are in card B as well as all the elements that were erased in the formulation of card B; therefore there are more elements in card A than in card B". Four participants elaborated and said, "I myself chose only part of the elements of set A to create B, so the number of elements in set B must be smaller"; or "I chose part of the elements, and the part always includes less than the whole."

In Stage II, all these participants argued that sets N and S had the same number of elements, basing their claims on one-to-one correspondence considerations. Typical explanations at Stage II related either to the geometrical representation--"Segments and their lengths, and squares and their areas correspond in a way that each length can be paired with a single area. Therefore the number of elements in sets N and S is equal"; or to the numeric-explicit representation--"Each number in set N has a matching number in set S, which is the square of this very number, like n and [n.sup.2]. Therefore the number of elements is equal"; or "I can match each number x in N with [x.sup.2] in S. All numbers will be paired in this way".

In Stage III, five participants spontaneously volunteered remarks such as: "Something is wrong here. I just gave two different solutions to the same question", expressing awareness of the "sameness" of the tasks presented in the two initial stages. The other 11 participants were led to this realization by the interviewer--for instance, by asking them to reexamine their solutions in previous stages. Consequently, all the participants noticed that they gave different solutions to essentially the same task mentioning that this was problematic.

Each of them was then asked to suggest a way of resolving the problem. Six participants were unable to provide such suggestions. They gave responses such as: "The different solutions I gave are definitely contradictory, and therefore it is impossible that both are correct ... Still, I have no idea what to do about it ..." The other ten participants suggested one of the methods are preferable for the comparison of infinite sets. While eight of them regarded "inclusion" as the natural way for solving comparison-of-infinite-sets tasks, the other two were more impressed by one-to-one correspondence and suggested that only this solution be used. Among the justifications for the choice of inclusion were: "Clearly, when one set is included in the other, it has fewer elements"; or "There is no doubt that a relationship of inclusion [of the sets] determines the relationship of the smaller within the larger". On the other hand, justifications for the choice of one-to-one correspondence included, for instance, "I can actually see pairs of elements that show that the number of elements is equal"; or "Here we have built a function that proves that the number of elements is equal. This seems a convincing way for determining the equality of the sets."

In conclusion, while all "From Difficulties to Easy" participants provided "unequal" and "equal" responses in Stages I and II respectively, they were all aware of the "sameness" of the different representations and of the contradiction they arrived by the end of Stage II. They made three types of suggestions to resolve the problem. Most of them suggested using only one method for such comparisons, and the others expressed a need for further study in order to be able to make any suggestions.

Second Interview:

Responses to the "N vs. 3N" Task: Among the prospective teachers who participated in the "From Difficult to Easy" sequence, seven said that they "could not answer" because, as they had concluded in the first interview, infinite sets were "different", "strange", and "different methods led to contradictory solutions". These participants expressed a need to study more about infinite sets in order to be able to answer. The nine other prospective teachers used "inclusion" to compare the number of elements in the given sets. However, five of them added expressions of uncertainty such as, "I am not sure about my solution. It can be problematic because the sets are infinite and I have already seen that they are strange."

Responses to the "What-if-Not" Task: All but two of the "From Difficult to Easy" participants rejected the What-if-Not methods, remained with their own solutions, and generalized that using different methods for the comparisons of infinite sets was problematic (Figure 3). A typical explanation was: "We saw [in the first interview] that the application of different methods for the comparison of infinite sets may lead to contradictions".

The two prospective teachers who accepted the different one-to-one correspondence solution offered in the What-if-Not task had originally solved the N vs. 3N task by "inclusion". Both of them commented that generally, "it may be OK to use inclusion in cases where I cannot find a way to match the elements"; or "When a certain method is inefficient for the solution, a different method should be considered."

4. Concluding Comments

The main question posed in the introduction was what the impact of the different instructional sequences, "From Easy to Difficult" and "From Difficult to Easy", on prospective teachers' solutions to comparisons-of-infinite-sets tasks would be. More specifically, how would participation in either of these activities influence (1) in participants' tendency to use one-to-one correspondence and (2) their awareness of the need to use only one method for the comparison of infinite sets? The discussion of the findings relates to these two questions.

The findings show that during all stages of the "From Easy to Difficult" sequence, all of the participants used one-to-one correspondence considerations, and reached solutions of "equal" to the different representations. This was not the case with the "From Difficult to Easy" participants. Here, all the participants gave "unequal" as solutions based on inclusion considerations to the horizontal representation, and "equal" solutions based on one-to-one correspondence considerations to the geometric and explicit representations. At the end of this sequence, only two of them suggested using one-to-one correspondence for the comparison of infinite sets. Moreover, while among the "From Easy to Difficult" participants, three used the one-to-one correspondence consideration to solve the N vs. 3N task in the second interview, none of the "From Difficult to Easy" participants offered such as solution.

Had we stopped here, it would have appeared that participation in the "From Easy to Difficult" sequence was significantly more effective for the promotion of one-to-one correspondence ideas. However, a deeper examination of the participants' solutions to the N vs. 3N task revealed a different situation. Quite surprisingly, most (9) participants in both interventions reached "unequal" as solutions based on inclusion considerations. Also, three of the "From Easy to Difficult" participants who reached "equal" as solutions based their explanations on an overgeneralization of their "equal" as solutions in the previous activity, rather than on a generalization of the one-to-one correspondence method they had experienced. They concluded, therefore, that "all infinite sets are equal". Moreover, those who did provide "equal" and "one-to-one correspondence" as solutions easily accepted a peer's inclusion solution to the similar N vs. 2N task. Therefore, it seems that participation in the "From Easy to Difficult" sequence was not significantly influential in promoting prospective teachers' application of one-to-one correspondence.

In conclusion, while the "From Easy to Difficult" sequence did trigger short-term one-to-one correspondence ideas, these ideas did not persist. A week later, when given a similar task, which was presented in the horizontal (difficult) representation, and not embedded in the guiding setting of the "From Easy to Difficult" sequence, most participants did not give any one-to-one correspondence solutions. The non-intuitive, horizontal representation triggered incompatible inclusion ideas.

The second question related to the participants' tendency to use only one method when solving comparison-of-infinite-sets tasks. The findings here indicated that the "From Difficult to Easy" participants identified the use of both inclusion and one-to-one correspondence as being the reason for their incompatible "unequal" and "equal" solutions in Stages I and II of the sequence. They subsequently referred to the application of the two methods for the comparison of infinite sets as being problematic. The contradiction they had experienced led most of them to suggest the application of only one (specific) method for the comparison of infinite sets. The other six participants, while also being aware of the problematic situation encountered by using both inclusion and one-to-one correspondence considerations, admitted that they had no idea of how to resolve their contradictory results.

In their reactions to the N vs. 3N task, all the "From Difficult to Easy" participants again referred to the contradiction they had encountered during this sequence. Then all but one of those who had suggested resolving the problem by using only inclusion did in fact do so in this comparison. The others (7) explained that although they were aware of the contradiction caused by the use of different methods, they had no idea how to solve such tasks. All of the "From Difficult to Easy" participants mentioned the impression made by the contradiction they had reached, convincing them that "the infinite behave differently from the finite", and that when comparing infinite sets, "different ways lead to contradictory solutions".

In the last task, when asked to reflect on a "different approach" used by a peer and on the validity of using different methods when comparing infinite sets, almost all (14) of these participants explicitly prohibited the use of different methods for the comparison of infinite sets. Even more significant was the "danger of contradiction" they mentioned as being the reason for their claim. One may wonder whether it was the "From Difficult to Easy" sequence that led these prospective teachers to their awareness of contradictions rooted in the use of different methods for the comparison of infinite sets. Perhaps any involvement with comparison-of-infinite-sets tasks would lead to such conclusions.

The findings here clearly indicate that participation in the "From Difficult to Easy" was special in the sense that participation in the "From Easy to Difficult" sequence granted no such awareness. As mentioned before, in their own solutions to the three stages of the "From Easy to Difficult" sequence, all participants reached the same "equal" solution based on one-to-one correspondence. Thus, their completion of the sequence provides us with no clue regarding existent awareness of the need to beware of using any additional method. In their solutions to the N vs. 3N task, most of these participants shifted to the use of inclusion, or to the idea that all infinite sets are equal, with no clear reference to the impact that such a change in the methods might have on the consistency of the solutions. Then, upon the interviewer's probing, most of these participants explicitly expressed acceptance of a different method, leading to a conflicting solution that was suggested by a peer in the last task. Not only did they regard the suggested conflicting solution as being valid, they also generalized that different methods are acceptable for the comparison of infinite sets, and stated that the choice of a method is made of the basis of local considerations of convenience and availability.

As already stated by a number of mathematics educators (e.g., Fischbein, 1987), awareness plays a crucial role in mathematics learning. Similarly, of the two approaches for teaching the comparison of infinite sets discussed in this paper, i.e., "From Easy to Difficult" and "From Difficulty to Easy", the results clearly indicate that the "From Difficult to Easy" had a greater impact on the participants' understanding. However, this does not mean that the "From Easy to Difficult" sequence should be set aside. It may, for instance, be beneficial as a starting point for a "positive" experience leading to the more frustrating "From Difficult to Easy" approach. How this should be implemented and what impact various implementations may have are topics for further research.

Moreover, the skill of sensibly using different representations for instruction seems very demanding. In order to make an effort and effectively integrate such non-standard, demanding teaching methods, which the teachers themselves did not necessarily experience as students, teachers must be convinced that such an effort is worthwhile. One way to go about this is to take advantage of opportunities where teachers themselves are studying mathematics. It therefore seems reasonable to promote teachers' attitudes toward and competence in using different representations in instruction in the initial stages of teacher education. The mathematics courses for prospective teachers usually include topics that they have never studied before. This setting therefore offers a natural environment for the prospective teachers to experience, as students, the type of teaching methods they will be expected to use in their future classes.

In addition, it is suggested that teacher educators share with prospective teachers various didactical possibilities and dilemmas that they themselves encountered while designing their instructional tools, and highlight the criteria for choosing the instructional sequences they eventually adopted for teaching specific mathematical topics. However, the impact of such activities on prospective teachers' mathematical performance, pedagogical views, and practice should be investigated further.

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Tsamir, P. (2002). When "the same" is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48, 289-307.

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Pessia Tsamir

Tel-Aviv University

"From Difficult to Easy"        From Easy to Difficult"

Stage I: The Target Task        Stage I: The Anchoring Task Encouraging
Encouraging "Part-Whole" Ideas  1-1 Correspondence
  Task 1-4: The horizontal        Task 1: The geometric representation
    representation                Task 2: The explicit representation
Stage II: The Anchoring Task    Stage II: The Bridging Tasks
Encouraging 1-1 Correspondence  Tasks 1-2: The vertical representation
  Tasks 1-3: The geometric      Stage III: The Target Task
    representation                Task 1: The horizontal representation.
  Task 4: The explicit
    representation
Stage III: Confronting
Inconsistencies

Figure 1. The "From Easy to Difficult" and "From Difficult to Easy"
Teaching Sequences

Table 1 Frequencies of Responses of the "From Easy to Difficult" group
(N=15)

                                                     Second Interview
                "From Easy to Difficult" Sequence   N    What-if-Not
                I: Geometric  II:       III:        vs.  Accept peer's
                & Explicit    Vertical  Horizontal  3N   solution

Method of Solution

One-to-one      15            15        15          3     9
correspondence
Inclusion       --            --        --          9     6
Infinities are  --            --        --          3    --
equal

       A single method  Many methods

FDTE   87.5              12.5
FETD:   0               100

FDTE: "From Difficult to Easy"
FETD: "From Easy to Difficult

Figure 3. Frequencies of the "From Difficult to Easy" and "From Easy to
Difficult" groups' call for a single method (in %).

Note: Table made from bar graph.

Table 2 Frequencies of Responses of the "From Difficult tp Easu" Group
(N=16)

                                                       Second Interview
                "From Difficult to Easy" Sequence           What-if-Not
                            II:         III:           N    Accept
                I:          Geometric   Resolve        vs.  peer's
                Horizontal  & Explicit  Contradiction  3N   solution

Method of Solution

One-to-one      --          16          2              --    2
correspondence
Inclusion       16          --          8               9    7
Cannot answer   --          --          6               7   --