How fragile is consolidated knowledge?

Journal of Mathematical Behavior 24 (2005) 15–38

How fragile is consolidated knowledge? Ben’s comparisons of infinite sets Pessia Tsamir∗ , Tommy Dreyfus School of Education, Tel Aviv University, P.O. Box 39040, Ramat Aviv, Tel Aviv 69978, Israel Available online 11 January 2005

Abstract This article builds on two previous ones in which we presented the processes of construction and consolidation of one student’s knowledge structures about comparisons of infinite sets, according to a recently proposed theory of abstraction. In the present article, we show that under slight variations of context, knowledge structures that have apparently been well-consolidated may become inactive and subordinate to more primitive ones. © 2004 Published by Elsevier Inc. Keywords: Abstraction; Consolidation; Fragility of knowledge structures; Comparison-of-infinite-sets

1. Introduction The processes by which abstract knowledge emerges are complex. These processes may be surmised to consist of an initial phase in which a knowledge structure is constructed in a specific context, followed by phases of consolidation which render the knowledge structure more conscious, more flexible, more available and more easily usable. The issue that stands at the center of the present article is the stability of consolidated knowledge structures. Our analysis uses a recently proposed theoretical model for processes of abstraction in context, which we summarize in Section 2. The model is apt to provide insight into these processes by means of a fine-grained description of the construction and consolidation of knowledge structures and has been used for this purpose by several research groups with learners of widely varying ages, learning a variety of mathematical topics in different learning environments and different social situations. The relevant references may be found in a recent paper by the authors (Dreyfus & Tsamir, 2004). ∗

Corresponding author. Tel.: +972 3 640 6513. E-mail address: [email protected] (P. Tsamir).

0732-3123/$ – see front matter © 2004 Published by Elsevier Inc. doi:10.1016/j.jmathb.2004.12.001

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The situation considered in this article is that of a single student, Ben, learning about comparisons of infinite sets in an interview situation. This is the third of a sequence of articles, in which we provide a detailed analysis of Ben’s construction and consolidation of his relevant knowledge structures. The two prior articles in the sequence are each based on one interview with Ben. They illustrate (Tsamir & Dreyfus, 2002) and considerably expand (Dreyfus & Tsamir, 2004) the proposed model by adding a theory of consolidation. These articles will be summarized, as far as they are relevant, in Section 3. In the present paper, we present a cognitive and didactic analysis of a third interview with Ben. We specifically address the question how stable or fragile consolidated knowledge may be. The design principles (Section 4) and the unfolding (Section 5) of the interview lead to this two-pronged analysis that is presented in Sections 6 and 7. Our analysis illustrates that even a didactically very carefully designed teaching episode is likely to contain minor inadvertencies that may lead to a situation in which a seemingly very well-consolidated knowledge structure turns out to be less stable than expected. More generally, even apparently wellconsolidated knowledge structures may become inactive and subordinate to more primitive ones under slight variations of context.

2. Theoretical framework Hershkowitz, Schwarz, and Dreyfus (2001) have proposed a theoretical framework for describing abstraction in context. Abstraction in context focuses primarily on process aspects of abstraction rather than on outcomes. Abstraction is defined as a process in which students vertically reorganize previously constructed mathematics into a new mathematical knowledge structure. Processes of abstraction are observed via epistemic actions, i.e., actions relating to the acquisition of knowledge (Pontecorvo & Girardet, 1993). The three epistemic actions related to processes of abstraction are recognizing, buildingwith and constructing. Recognizing, building-with and constructing are the elements of a model for processes of abstraction, called the dynamically nested RBC model of abstraction. According to the model, constructing is central to the process of abstraction while the other two actions are dynamically nested within it. A summary of the model and its epistemic actions is given in Dreyfus and Tsamir (2004). The model describes processes of abstraction of concepts, methods, strategies, relationships or, more generally, structures. Structures in this sense are mental outcomes that result from mathematical activity. Constructing refers only to the first time a learner reorganizes knowledge into a novel structure. Newly constructed structures may later become an integral part of the learner’s available knowledge by means of processes of consolidation that follow construction. A structure that has not been consolidated is likely to be fragile. It may be at the learner’s disposal only in a specific context, under certain circumstances, in certain representations and when dealing with a certain type of problem. According to Dreyfus and Tsamir (2004), consolidation is a process in which a previously constructed mathematical knowledge structure becomes progressively more familiar to the learner. Problem solving and reflective activities are likely mediators for consolidation. Consolidation occurs in complex sequences of recognizing the structure, building-with it, reflecting on it and potentially also constructing further structures with it. This happens in a way that promotes the following main characteristics of consolidation: Immediacy refers to the speed and directness with which a structure is recognized or made use of (built-with) in order to achieve a goal; self-evidence refers to the obviousness that the use of a structure has for the student;

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obviousness implies that the student feels no need to justify or explain the use of the structure, though (s)he is able to justify and explain it. Self-evidence is directly related to the confidence or certainty with which a structure is used. Frequent use of a structure is likely to support the establishment of connections, and thus contribute to the flexibility of its use. A student may be quite proficient in using a structure, even using it flexibly, but without being consciously aware that (s)he is doing so. The awareness of a structure enables the student to reflect on related mathematical and instructional issues, add to the depth of her or his theoretical knowledge and power and ease when using the structure. All in all, the structure is used with ‘increasing ease’ (Hershkowitz et al., 2001).

3. Ben’s initial structures In this section, we offer a summary description of Ben’s previous interviews, referring to the representations available to Ben as well as to our interpretation of his evolving knowledge structures. Ben dealt with comparison-of-infinite-sets tasks on three previous occasions: about a year and a half before the present interview, during a first comparison-of-infinite-sets interview (Tsamir & Dreyfus, 2002), next when volunteering to read a related research paper (Tsamir, 1999), and a year later, in a second interview (Dreyfus & Tsamir, 2004). The present interview took place about a month after the second one. We will concisely describe the development of Ben’s knowledge structures as we observed it during the previous two interviews. When he entered the first interview, Ben was familiar with some elementary set theory he had studied at school, and knew how to compare the sizes of finite sets by methods like counting, inclusion, one-to-one correspondence and by means of the consideration of intervals of numbers that were or were not included in the sets. We will collectively call this collection of methods for comparing finite sets Ben’s Finite Structure. From among this collection, the Inclusion Structure the One–One Structure and the Intervals Structure will play an important role in this paper. Ben’s first interview consisted of two stages that involved different representations of the same task; the task was to compare the numbers of elements in the set of natural numbers and the set of perfect squares. We used representations of these sets on three cards (see Fig. 1). We first highlighted the inclusion relationship by asking Ben to choose, mark and copy the perfect squares in the set of natural numbers

Fig. 1. The cards that were presented horizontally, interview 1, stage 1.

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(Card M). We presented the sets in a horizontal representation, putting cards A and B one beside the other. Indeed, Ben claimed that the number of elements in set A was larger than the number of elements in set B, and his explanation “set B is actually part and I mean really part of set A”, was rooted in the Inclusion Structure, and was further substantiated by the application of the Intervals Structure, indicating that “it is easy to notice that the further I go [in set B] the larger the intervals . . .. More elements are cut from the set than remain in it”. At this point in time, Ben seemed to be unaware that he was now using with infinite sets the Inclusion and Intervals Structures, which he had constructed and consolidated with reference to finite ones. In the second stage of the first interview, Ben was presented with a geometric representation (squares) of the same task, relating to a sequence of segments (length: 1, 2, 3, 4 cm, etc.) and the areas of corresponding squares (1, 4, 9, 16 cm2 , etc.) (see Fig. 2). He easily identified two One–One Substructures: The Bijective Function Substructure, based on the formal notion of a one-to-one function between sets A and B, and the Pairing Substructure, relating to the creation of pairs of matching elements, one from each set with no ‘leftovers’. Ben’s answers in each of the two stages expressed his confidence in the correctness of the respective solutions (using terms like “of course” and “no doubt”). Suddenly, however, he realized the mutual incompatibility of these solutions, and perhaps his confidence intensified the impact of this sudden realization. His reaction was intense and unambiguous: He was adamant that such a situation must not occur. We interpreted this as Ben having a well-developed and consolidated Consistency Structure. This

Fig. 2. The cards that were presented vertically, interview 1, stage 2.

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Fig. 3. The cards that were represented horizontally, interview 2, stage 1.

led him to construct the Infinity Structure—ruling out interchangeable use of one-to-one correspondence and inclusion when comparing infinite sets. However, he made no explicit reference to the possibility of interchangeable use of intervals and one-to-one correspondence. Between the first and second interviews, Ben read about Cantorian set theory and learned about the way to avoid contradiction by using only criteria based on one-to-one correspondence for comparisons of infinite sets. The second interview with Ben included four tasks. In each of the first three, he was asked to compare the set of the natural numbers and the set of the natural numbers larger than 2, given in a horizontal, a geometric and an explicit formula representation, in this order (see Figs. 3–5). The fourth task asked to compare the set of natural numbers and the set of positive even numbers, presented horizontally

Fig. 4. The cards that were presented geometrically, interview 2, stage 2.

Fig. 5. The cards that were presented, interview 2, stage 3.

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Fig. 6. The cards that were presented horizontally, interview 2, stage 4.

(see Fig. 6). Ben’s reaction to the horizontal representation in the first task reflected the coercive influence of his Inclusion Structure. However, he soon shifted to the use of one-to-one correspondence, labeled it “an equivalence”, and stated that the correspondence “n to n + 2 is the thing that matters”. In task 2, when presented with the geometric representation, Ben recognized that the given sets were the same as those in task 1. While he identified the inclusion relationship, he emphasized the need to rely on one-to-one correspondence, progressed from “there is a certain possibility to match elements”, and “this possibility is what counts”, to the actual matching rule defining a one–one and onto mapping, which he regarded as conclusive evidence that the number of elements is the same. In task 3, he immediately pointed to the fact that the tasks are identical, and to the equivalence correspondence. Task 4 presented a horizontal representation of a different problem, and Ben immediately said that the number of elements was “. . . equal. The matching rule is multiplicative instead of being additive . . . for each x in A there is 2x in E or for every x in E there is x/2 in A. It is actually a function. There is an equivalence correspondence.” He exhibited awareness of the need to use one-to-one correspondence, to avoid inclusion, as well as awareness of the pitfalls associated with the latter “[inclusion] is like an intuitive obstacle . . . I remind myself that even though there is inclusion this is not the way to determine whether the sets have the same number of elements, just equivalence correspondence.” We interpreted this awareness as an expression of Ben’s stable Cantorian Structure. During the interview, Ben repeatedly analyzed differences between finite and infinite cases, addressing the danger of contradiction in the infinite case: “. . . it cannot be that we’ll use both, as in the finite cases . . .. We have to decide either one–one correspondence or inclusion . . .. The existence of inclusion . . . it is irrelevant . . . it is actually dangerous.” We note that Ben made no reference to the Intervals Structure. Ben also repeatedly analyzed the impact of the different representations on his solution processes. He expressed that the horizontal representation is confusing because it triggers inclusion considerations, and explained how the two other representations support one-to-one correspondence. The explicit formula representation clarified the correspondence for Ben. The trapezoids in the geometric representation helped Ben pair the numbers in the two infinite sets by providing visual support: “The trapezoids grant a good picture of the role that each number plays in the equivalence correspondence . . .. Each trapezoid is actually the ordinal place that physically combined bases . . .. It provided a visual way to see the place, and see that the pairing of exactly one number from one set to exactly one number in the other can be natural”. In task 4, it turned out that the trapezoids were instrumental in Ben’s thinking, even though they intervened in such a smooth and flexible manner that the interviewer could not notice it without Ben pointing it out. His efficient use of the geometric representation was based on his ability to visualize it, thus making the structural relations self-evident. The self-evidence provided by this visualization contributed to Ben’s confidence.

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In summary, by the end of the second interview Ben successfully solved the comparison of sets where √ the matching rule was additive (n → n ± k), multiplicative (n → n ×/÷ k), and quadratic (n → n2 / n), and managed to deal with different representations of the same task, and with counter-intuitive horizontal representations of different tasks. He was sensitive to the mathematical objects in the tasks, to their representations, to the methods he used in his solutions and to the need to preserve consistency. His reflective discussion of the Infinity Structure was interwoven with repeated cautioning to use one-to-one correspondence and reject inclusion, with an analysis of the differences and similarities between finite versus infinite cases, and with reflections on the impact of the different representations on his reasoning. In a previous paper (Dreyfus & Tsamir, 2004), we showed that Ben’s Infinity Structure became selfevident during the interview, and he exhibited growing confidence when using it. Ben used this structure frequently and flexibly, being aware of his performance, and reflective on related mathematical and instructional issues. Ben became aware that he must continuously stand on guard, that is to say, recognize inclusion considerations as well as the horizontal representation as obstacles. He made his self-regulation an object of explicit attention, constantly reminding himself that although these ideas keep interfering, they are not valid for building solutions to comparison-of-infinite-sets tasks. Consequently, we concluded that he had consolidated his Infinity Structure. In the same study, we similarly arrived at the conclusion that Ben had a very well-consolidated Consistency Structure, which he presumably acquired independently of and before the sequence of interviews. We also made the point that Ben’s Cantorian Structure, while clearly available to him, was not equally firmly consolidated, among others because it was to some extent based on authority rather than on his own construction.

4. Design of the interview About a month after the second interview on comparison-of-infinite-sets, Ben was interviewed again, with two aims: We wanted to ascertain that Ben had indeed constructed and consolidated the structures we had identified as such; we also intended to give Ben an opportunity to expand and further consolidate these structures, mainly the One–One Structure whose many and varied substructures had not yet been the object of explicit attention. As in the two earlier interviews, the tasks presented to Ben in this interview dealt with the comparison of two countable infinite sets. The sets to be compared were presented in a horizontal representation, since this is the most common representation of such tasks, and also the least intuitive one. The tasks were designed so as to vary some of the conditions according to which the sets had been chosen, as well as to use tasks that combined some properties that had not been combined in the earlier tasks. Specifically, all the comparison-of-infinite-sets tasks that Ben had been presented with until this point had dealt exclusively with positive numbers. In every task, the set of natural numbers had been one of the two sets to be compared, and it had always included the other set. In the first task of the present interview, both these conditions were discarded, and Ben was asked to compare the set of natural numbers A with a set of P integers that strictly included A, and thus contained some non-positive numbers (see Fig. 7). This task is similar to tasks Ben had successfully solved previously insofar as it deals only with integers and insofar as one of the sets can be obtained from the other by adding a finite number of elements. Moreover, a proof that the two sets were equivalent could be based on an additive matching rule of the type Ben had identified in the previous interview.

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Fig. 7. The cards that were presented horizontally, interview 3, stage 1.

In the second task, a different combination of features of the sets was used in comparing the set of natural numbers to the set V (see Fig. 8). The set V has increasing intervals between consecutive elements, a feature that Ben had successfully dealt with in a previous interview (see Fig. 1), but it included, in addition to elements that were elements of A, the element 0, which was not an element of A. Thus, in this task, neither of the two sets was a subset of the other, a feature that was new for Ben. Moreover, even without the 0, it is not necessarily obvious how to find a matching rule. Nevertheless, based on our interpretation of Ben’s structures at the end of the previous interview, we had high expectations for Ben’s performance on these tasks. Ben is a talented youth with general interests in the arts and the sciences. He has obtained prizes for short stories he wrote, for his contribution to the life of his school, and for academic excellence. At the time of the interview, he studied in a regular high school, majoring in Mathematics and Physics. Ben is far more reflective than most students. He is at ease during the interviews. His responses tend to be driven by his eagerness to construct meaning. He clearly expresses his thoughts, relates to his own ideas and explained how he reached his conclusions. The two researchers independently predicted how we expected Ben to deal with the tasks. We both expected him to look for a matching rule and easily come up with the rule x → x − 4 (and its inverse) to justify equivalence of the sets in the first task, based on the Cantorian and One–One Structures as expressed in the explicit formula representation. Although the second task was expected to be more difficult, we both expected Ben to succeed. One of us expected him to use the geometric trapezoid representation (with the first trapezoid degenerating into a triangle), and to argue the equivalence of the two sets from the mapping between the trapezoid bases. The other researcher expected Ben to notice first that the sequence of differences of the elements of V was identical to the sequence of elements of A, and then use an argument based on the fact that each element of V has an ordinal place given by the

Fig. 8. The cards that were presented horizontally, interview 3, stage 2.

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corresponding element of A. As mentioned above, Ben had briefly alluded to such an argument in the first interview. Thus, task 2 was designed in the hope that it would not only help Ben consolidate the Infinity and the Cantorian Structures but to reconstruct and consolidate the Ordinal Substructure of the One–One Structure. We meant these two tasks to be starters for more complex ones, such as the comparison of the set of natural numbers to the set of whole numbers. However, we were surprised by the difficulties Ben encountered: Unexpectedly Ben’s work on these two tasks lasted about an hour, after which the interview ended naturally. The interview was audio-taped and transcribed.

5. The interview In this section we attempt to give a descriptive interpretation of those excerpts of Ben’s interview, which are most relevant for the analysis that we plan to undertake in the next two sections, namely those excerpts that yield insight into Ben’s use of his comparison-of-infinite-sets structures. Ben’s reaction to the first task was not immediate. After some hesitation, he said: B6.

Yes. I believe that it is equal, but it does not look equal.

In his familiar open and reflective manner, he went on to explain what was troubling him. Unsurprisingly, he had a problem with the negative numbers and the zero. However, this problem was not rooted in the numbers per se, but rather in his preference for using the trapezoid-representation for comparing infinite sets. B12.

I13. B14. I15. B16. I17. B18.

Well I try to create the trapezoids. So, the lower bases are 1, 2, 3, etc., [pause] but what can I do with the negative numbers? I cannot use them for lengths of bases [pause], and zero [pause], this is really strange. What kind of trapezoids would those be? Aha . . .. I considered leaving the negative numbers, for a moment, and trying to start by building the trapezoids without them. I see, [pause] and the zero? Oh, the zero, [pause] what am I going to do with the zero? [Pause] Perhaps I’ll ignore also the zero for a moment. And? I cannot find a way to construct trapezoids that will represent all the given numbers. If I could find one, then I would have the correspondence.

Ben attempted to solve the task by manipulating the given sets into the geometric representation. This is how he had successfully solved the tasks in the previous interview. However, here the negative numbers and zero impeded this approach. Ben was trapped by linking the geometrical, irrelevant demands of this representation with the numbers in the set. While in the previous interview, he had stated that the mere possibility of pairing the elements in such a manner is what counts, here we witness him treating the trapezoids and the length of their bases as central elements for the process of pairing. Ben tried to extricate himself by first handling the friendly (positive) numbers only (B14, B16). However, he soon returned (B18) expressing that only a pairing of all the given numbers would produce the

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correspondence needed to confirm the hypothesis he had made in B6. Upon a brief question by the interviewer, Ben turned to the explicit formula representation and found the matching rule, linking it to the notion of function: I19. B20.

I21. B22. I23.

B24.

Is there another way to address this problem? [Thinking] To try and express the numbers in a different manner. [Pause] To look for a way to match elements in the two sets: 1 to (−3), 2 to (−2), 3 to (−1), 4 to 0, 5 to 1, ehh, I think . . . [Pause; then writes down 1—1 − 4, 2—2 − 4, 3—3 − 4, 4—4 − 4, 5—5 − 4; then goes back to stress the 4s that were subtracted: 1—1 − 4, 2—2 − 4, 3—3 − 4, 4—4 − 4, 5—5 − 4 . . .]. It is again a function x → x − 4. What can you say about the number of elements in the two sets? It is equal. You showed how each number here [N] has a corresponding number here [N − 4]. How can I be sure that there are no unmatched leftovers among these elements [N − 4]? I mean can we be certain that all the elements in set P have matching elements in set A? Yes. We can reverse the function [draws two parallel lines and marks x on one of the lines and y on the other; then, marks the natural numbers on one of the lines and the [N − 4] numbers on the other; then draws arrows from n to n − 4, and adds heads in the opposite direction—from t in set P to t + 4 in set A]. It is an equivalence correspondence.

The interviewer’s further probing (I23) led Ben to produce the parallel axes description of the function relationship, which he had used earlier, and complete his explanation with the term “equivalence correspondence”. At this point, the interviewer gave Ben an opportunity to clarify the confusion caused by the trapezoid-representation. Ben’s reaction to her question was confused, discussing squares versus trapezoids and leading to: B36. I37. B38. I39. B40. I41. B42.

I still have four unmatched numbers. The negatives and zero are still problematic. What would you suggest? I do not know how to use this method. Is the number of elements in the sets equal or not equal? Equal. I showed it. But you also speak about four unpaired elements. What does that mean? It could mean that there are more elements here [in set P] than here [in set A], but it does not. There is a correspondence, and therefore I am sure that the number of elements is equal. I remember, I read about it and I also did it with you a while ago. I am positive—when there is an equivalence correspondence the sets are equal. The number of elements is equal. I know.

Ben’s response here was immediate and confident. His confidence was expressed in his tone as well as in his wording. A further attempt by the interviewer to close the open ends left with regard to the geometric representation revealed that Ben’s past experience had led him to expect that each one-to-one correspondence could be represented in the trapezoid-representation. He was therefore surprised by the mismatch he had encountered in this task. However, he confirmed his conclusion:

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B48. I49. B50.

I51. B52.

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Equal. There is a One–One function. In both directions. It is equal. Are you sure? Absolutely. The matching rule is additive or subtractive. We once had an “addition rule” and once a “multiplication rule”. It does not matter. They determine an equivalence correspondence and they determine equality. What about the trapezoids? They did not help, but they did not prove the opposite. I just could not use this method.

Ben now differentiated between the role of representations as tools and the solution to the mathematical problem. He expressed that the mere existence of an equivalence correspondence proves that the sets have the same number of elements. The depth of his understanding of this crucial matter transpires in B52, where Ben clearly states that the fact that the trapezoids are not helpful in finding an equivalence correspondence does not grant them the power of refuting the conclusion that the sets had an equal number of elements. When presented with the second task, Ben reviewed his options: B56. I57. B58.

Ehh . . . I am looking for a function to connect the two sets [pause] a One–One function. In both directions. Aha. Maybe I can start with the trapezoids [pause]; if I find them I have the matching rule.

In spite of the problems caused by the trapezoids in the previous task, Ben still turned to them. When he noticed the zero element of set V, he reacted in a similar manner as in task 1 (B14/16). Once more, the geometrical demand for positive lengths led him to a dead end: B64. I65. B66. I67. B68. I69. B70.

I cannot have a zero basis [pause] maybe I should ignore the zero. I can build trapezoids with bases 1—1 oops, ehh 2—3, 3—6 . . . [thinking]. Why did you say ‘oops’ after the 1—1? It is not a trapezoid. It is a square. So? I ignore it, like I ignored the zero [pause] for a moment [pause] I mean until I see the picture. I understand. But I do not understand. I do not see the picture [thinking quite some time] maybe I could accept the square, but the zero [pause] the zero remains problematic [pause]. The zero ruins it all. On the one hand, it is only one extra element against an infinite number of elements. It seems negligible. But it bothers me, especially since I have not found a way to match the elements. [Pause] I have not found one yet. It could be that there is a way, only that I have not found it yet.

Ben is frustrated by the inextricable situation generated by his clear grasp that a match between the sets would do the job, his inability to find that match, and his awareness that for an existence proof such as the present one, not being able to find an example proves nothing. The interviewer’s probing questions concerning the precise meaning Ben associated with “a way to match the elements”, were less compelling to Ben than his intuition that the sets under consideration were equivalent:

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I71. B72. I73. B74. I75. B76. I77. B78. I79. B80.

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If this is the case, what would you respond if you find one way to match the elements? To match properly? Can there be an improper way to match the elements? Sure. If it is not a One–One function, like the ones I showed with the parallel axes. OK. If you find a single, proper way to match the elements, but you still do not know what to do with the zero basis, what would you say? That the numbers of elements are equal. To be honest, my intuition triggers me to say even now, I mean before even finding such a matching rule, that the number of elements is equal. Do you have any idea, why is this your intuition? The sets are infinite, and my dilemma relates only to one element. It seems too tiny to make a difference. Is it? [Pause] I mean, too tiny? [Pause] I do not think so. It could be said that one set is included in the other, and therefore it has less elements. But I remember that inclusion is not the way to go about it [pause] even though, again and again, it seems very logical. On the other hand, these are infinite sets. [Pause] I could suggest that infinity is soooo large that a finite number of elements, not to mention one single element, can make no difference. This also seems logical. [Pause] I know that I have to be careful. Very careful.

Two ideas crossed Ben’s mind with clarity. One of them is based on the finite case: If one set is strictly included in another, then the other set is larger. The second idea, on the other hand, is closely linked to the infinite case, and has already appeared earlier in this interview (B14/16 and B64): A single element, even a finite number of elements cannot make a difference in terms of the equality of infinite sets. Ben appears to be aware that the two ideas lead to mutually contradictory conclusions, hence his warning to be careful, which arouses the interviewer’s interest: I81. B82. I83. B84 I85 B86

What do you mean? For instance, the latter two logical ideas—each of them could be wonderful by itself, but together they cause contradiction. I do not get it. If I use inclusion, then one element can cause two sets to be different, but if I say that one element cannot have an impact on an infinite set, then it contradicts my previous claim. I see. What would you suggest? To reject both considerations. [Confidently] I have to find a one-to-one correspondence.

Notwithstanding the interviewer’s ‘poker face’ questions, Ben rejects both intuitions in favor of finding a one-to-one correspondence between the two sets. After this theoretical intermezzo, and without being prompted, Ben turned to the explicit representation and tried to construct an algebraic matching rule:

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B90

I91 B92

I93 B94

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I try to find the matching rule. [Writes down 1—1 − 1, 2—2 − 1, 3—3 − 0, 4—4 + 2, 5—5 + 5, stops] the rule is not by addition or subtraction. I’ll try multiplication. [Writes down 1—1 × 0, 2—2 × 1/2, 3—3 × 1, 4—4 × 1.5, 5—5 × 2, 6—; stops with no match for 6.] It is not by multiplication. What now? [Writes down the numbers 0, 1, 3, 6, 10, 15, 21, 28, 36, and between each consecutive pair leaves some distance. Then, marks an arc to show the intervals between the consecutive elements, and writes down the differences on these intervals. Finally, he copies the numbers of the differences. 1, 2, 3, 4, 5, 6, 7, 8,] Amazing. I do not follow. There is a one-to-one correspondence between set A and the sequence of differences. It is not just one-to-one correspondence it is exactly the same set! But again I have one extra element. The number of intervals is smaller by one than the number of elements. [Pause] I’ll try to find a matching rule in the opposite direction. It is sometimes easier.

Ben thus made two different attempts at finding a matching rule, exhibiting his flexibility at solving difficult problems. However, in spite of making substantial progress in two potentially fruitful directions, he was not able to lead any of his attempts to a successful conclusion. In B90, Ben searched for a correspondence of the form n → n + C or n → n − C. Here, the ‘constant’ C had to be chosen different for each n. Two ways out of this difficulty could be imagined: If Ben had noticed the pattern Cn = (n − 1)/2 in the multiplicative case, he might have realized the similarity to the n → n2 task that he had successfully solved in an earlier interview. But even without noticing the pattern, Ben might have noted that the sequence n·Cn is strictly increasing, and therefore the match is an equivalence correspondence. However, this is not what happened. In B92/94, Ben reached a second golden opportunity, when he realized that the intervals between consecutive numbers in set V were exactly the natural numbers. This could conceivably have led him to a recursive or explicit expression for the elements of V; such an expression could have been used as basis for the one-to-one correspondence and the resulting equivalency between the two sets as expected by one of the researchers—could have but did not. Alternatively, he could have noticed that in set V there are as many intervals as there are elements, and thus the set V is equivalent to the set of intervals in V, which in turn is the same set as A—again, could have but did not: Ben made the analogy to the finite case, where the number of intervals is smaller by one than the number of elements (B94). Ben continued to look for a matching rule, in a quiet and thoughtful manner. He reported running various unsuccessful matching rules in his head, until suddenly he exclaimed: B100 I101 B102

. . . Oh. I see. What? The differences between successive elements get larger and larger. Wow! really larger. I

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I103 B104

I105 B106 I107 B108

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see. Set V consists of fewer elements. Really fewer. It is not a matter of one element more. Wow! It is a tremendous difference in the number of elements. Now, I know why I was confused. It is the zero! I felt bad about the zero. This is too fast for me. The zero does make a difference? No. Yes. I mean, mathematically it does not. But psychologically in gave me the impression, [pause] the intuitive impression that set V begins with a smaller number than the first number in set A. When I tried the trapezoids, and it was mainly zero that I could not pair, because there is no zero basis. So, I believed that the numbers of set V could be paired with the numbers in set A but the zero is an extra number in set V. When I examined the intervals, again it seemed as if zero was the only element to remain solo. It gave me the impression that somehow set V consists of more elements. Not many more, actually one more, still, more elements. And this is the catch! I was under the impression that I follow, but I lost you. [Triumphantly] Set V has fewer elements than set A. How is that? Because of the intervals. If we go far enough, the intervals here [in set V] get huge and the elements rare, while here [in set A] we advance in a constant, rather small pace. Here, there is no matching rule between the sets.

This is a crucial point in this interview. Ben knew that attempts to look for a matching rule could go on forever, like looking for a needle in a haystack. How can you prove that the needle is there? You must find it. How can you prove that it is not there? By means of a logical argument, showing that it cannot be there. And this is what Ben thought he had found—a logical argument proving that there could not be a matching rule. Indeed, upon the interviewer’s query about what he would say if she did find a matching rule, he confirmed his certainty that there is no one-to-one correspondence in this case and added: I119 B120

Can you explain what makes you so certain? The intervals. I think that this is the way to prove that there is no correspondence. Imagine, if I had no way to know when there is no correspondence I would go on looking for matching rules, either for ever or until I found one. If I find a rule, it is OK and the sets are equal. But if I do not find a rule, I really have a problem. The search might go on forever. How do I know when to stop? I think that what I found today is not less important than deciding to go by one-to-one correspondence. That is how to be sure that there is no one-to-one correspondence.

Clearly, at this stage, Ben was not aware of the increasing intervals in the n → n2 example that he had easily solved in an earlier interview by means of the matching rule, based on the geometric as well as on the explicit formula representations. He adopted the growing-intervals justification (that had never before been explicitly ruled out) as proof that the sets were not equivalent. He did not clarify to himself that he had not actually provided any steps of such a proof; rather he expressed his confident conviction that such a proof exists. The interviewer made a number of attempts to shake Ben’s confidence. She mentioned that he, himself, had earlier pointed out the contradiction evolving from the interchangeable use of one-to-one correspondence and interval considerations; she drew his attention to the fact that he acted in contradiction to his principle of using exclusively correspondence considerations; she even provided an example from a task that had earlier been successfully solved by Ben, in which one of two equivalent sets had larger intervals than the other. Ben countered all these attacks with confidence and with clearly formulated arguments:

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I121 B122 I123 B124 I125 B126 I127 B128 I129 B130

I131 B132

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You once told me that using inclusion and correspondence leads to contradiction. And then you read that only equivalence correspondence should be used for comparing infinite sets. Right? Sure. Should we stick to correspondence? Yes. Always? Yes. But just now you suggested another criterion. Does this seem OK? Yes. I used it to show that there is no correspondence. There has to be a way to prove that there is no correspondence otherwise I would conclude that all infinite sets are equal. What would happen if I suggest checking the intervals when comparing the set of the natural numbers and that of the even numbers? Nothing. The matching rule shows the equality [pause]. When we have a matching rule of addition the rate of change remains the same, like a shift. When the matching rule is multiplication by a constant, the intervals in one set are larger but constant and we advance in both sets at a constant pace. True, in one set the step is bigger. Still, since the sets are infinite, there is a way to pair the elements and, so to determine equivalence correspondence, which determines equality. I think that here I found a case where the infinite sets are not equal. And I explained why. How do you feel about your response? Good.

And thus the interview ended with Ben’s problem solved but not the interviewer’s. As an aside, we note that a couple of weeks later, Ben contacted the interviewer in order to tell her that he had been wrong and that he now saw the equivalence between the two sets V and A.

6. Analysis of Ben’s mental structures The conclusions of Dreyfus and Tsamir (2004), as recapitulated in Section 3, were that Ben had constructed the Consistency, Infinity and Cantorian Structures, with the Consistency and Infinity Structures being well-consolidated and the Cantorian Structure being fairly well-consolidated. Moreover, Ben had assembled a progressively richer repertoire of one-to-one correspondence considerations, pointing to his One–One Structure being under construction. In this section, we will confirm these conclusions on the basis of Ben’s work on task 1 and his accompanying explanations. We will then analyze his performance on task 2 in terms of his structures. In particular, we will present an attempt to explain his failure to come to the correct conclusion on task 2 in spite of his well-developed comparison-of-infinite-sets structures. In both tasks, Ben immediately started to search for a one-to-one correspondence between the two given sets (B12, B56); in other words, when presented with comparison-of-infinite-sets tasks, he acted without hesitation according to the Cantorian Structure. He exhibited his command of this structure by the immediate conclusions he drew from finding a one-to-one correspondence (B22, B48, and B50). Moreover, in B40, B42 and B52, he showed that he was confident enough to insist on these conclusions in view of attempts by the interviewer to make him use inclusion considerations. The elaborate nature of Ben’s Cantorian Structure was expressed in several ways: (i) Ben was acutely aware that the trapezoids are only a representation, and that they offer one way to attack things, but that other ways, in particular the explicit function representation, can be used as alternatives. We note

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that this awareness was not hampered by Ben’s strong preference for using the geometric (trapezoid) representation for trying to establish one-to-one correspondences in both tasks (B12, B58). We further note that the strength of this preference found its expression in the fact Ben insisted on going as far as he possibly could with the geometric representation and this in spite of the obstacles presented by the non-positive numbers (B14, B16, B64, B68). (ii) Ben stressed repeatedly that establishing one-to-one correspondence by any one method would obliterate the need for any further considerations (B38, B52). (iii) Even though Ben and the interviewer both often used the simplified term ‘one-to-one correspondence’, Ben was very conscious of the fact that the required correspondence also had to be surjective (‘onto’) (B24, B74). Ben’s work on the second task confirmed that his Cantorian Structure was fairly well-consolidated: Although he did not actually find a one-to-one and onto correspondence for this task, he was adamant that looking for such a correspondence is the way to go about the task (B70, B86) and he was able to theoretically discuss what would be the implications if he were to find such a correspondence (B76). There is less direct evidence in the interview to support the claim that Ben’s Consistency and Infinity Structures were well-consolidated. We nevertheless strongly make this claim: These structures were already well established in the previous interview. They relate to the kind of issue that, once dealt with, is so obvious that mentioning them is superfluous. Indeed, the use of the Cantorian Structure may and should be regarded as indirect evidence for the underlying Consistency and Infinity Structures. In addition, whenever there was some reason to relate to aspects of the Infinity Structure, these were obvious to Ben (B82, B84, B122, B130). With the Cantorian Structure quite firmly established, Ben’s undertaking, from our point of view as researchers and teachers, was to elaborate his One–One Structure. A person’s One–One Structure is the collection of all ideas (substructures) that the person has at her disposal for establishing one-to-one correspondences between two (infinite) sets, as well as the relationships between these substructures. Ben showed evidence of having constructed several such substructures in earlier interviews, among them the Pairing Structure and the Bijective Function Structure. In the present interview, Ben consolidated these structures by applying them in a somewhat different context. Specifically, his use of the trapezoid representation provides evidence of his Pairing Structure. Similarly, his use of the explicit function representation, for example in B20, provides evidence of his Bijective Function Structure. He recalled from earlier interviews that such functions might follow different algebraic rules, including additive as in n → n + 4 or multiplicative as in n → 4n (e.g., B90). He was also clearly in command of the requirement for such functions to be invertible (e.g., B24). In the present interview, Ben tried to adapt and extend these structures to suit the complexities of the tasks. For example, he again attempted to use the trapezoid representation as a tool to implement the Pairing Structure (B12, B58); the trapezoid tool, however, limits Ben’s Pairing Structure to sets with positive elements. He does attempt to circumvent this limitation by omitting of the non-positive elements, at least temporarily (B14, B16, B64); this could conceivably lead to the following extension of the Pairing Structure: If one needs to compare two (infinite) sets A and B, and there exists a pairing between subsets that have been obtained by omitting the same finite number of elements from each set, then this pairing can be extended to the sets A and B by means of the a (finite) pairing of the omitted elements. Although he had at least two opportunities for such an extension of the Pairing Structure, Ben did not construct it. Apparently, Ben’s Pairing Structure was too closely linked to the artifacts he used to establish the pairing, namely the trapezoids.

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Ben’s Bijective Function Structure is limited as well. In B90–B94, Ben searched for additive and multiplicative rules but could not come up with the more complex, quadratic rule (n → n(n − 1)/2), which would have done the job. This limitation of the Bijective Function Structure is different from the limitation of the Pairing Structure in that it does not directly stem from an artifact associated with the representation of the structure. Nevertheless, this limitation is related to artifacts as well, and this in two ways: (i) Ben appears to be aware of only one method to establish that a one-to-one correspondence exists, and that way is to explicitly construct a One–One and onto function; such a function is an artifact establishing the One–One correspondence, and thus allowing one to apply the Bijective Function Structure. (ii) Given that this is the method to apply the Bijective Function Structure, Ben’s repertoire of flexibly available functional expressions appears to be limited to additive and multiplicative functions. Thus, here the limitation is not inherent in a specific artifact, but rather in the lack of flexibility due to Ben’s restricted repertoire of artifacts. Similarly to his Pairing Structure, the construction of Ben’s Bijective Function Structure is in progress. It could be elaborated in two ways: in view of (ii) by expanding the repertoire of functions, and more importantly in view of (i) by the Ordinal Structure, whose essence is that for any two infinite countable sets, equivalence can be established via ordinal numbers by numbering the elements in each set and associating each element of each set with its ordinal number. Interestingly, the Ordinal Structure was not completely unknown to Ben. Indeed, Ben had, without even being asked, given quite an elaborate description of the Ordinal Structure in an earlier interview: When relating to the horizontal representation of the set N of natural numbers and the set T of natural numbers larger than 2 (see Fig. 2), and reflecting on the impression that T is included in N, Ben volunteered the following: . . . the two extra, unmatched elements stand out and trigger the conclusion that here we have infinity and here infinity plus two, which seems larger. Instead of matching numbers at the same ordinal place [pause]. I mean, assuming that if for each place n there is one and only one element in each of the two sets, then they go on hand in hand, corresponding, and extra elements are just in our imagination. The infinite nature makes it possible that no matter which number you chose in one of the sets, at the same ordinal place there is a matching specific number placed in the other set. It cannot be that the numbers in the second set are finished and cannot provide a matching element, because the set is infinite, and this behavior of plus two goes on, like, forever. Still, it is not obvious, because we are used to deal with finite situations and the ‘tail’ of 1 and 2 is significant in finite situations. (Dreyfus & Tsamir, 2004, p. 287) One might thus have concluded that Ben possessed quite an elaborate Ordinal Structure; however, his Ordinal Structure did not appear in the context of the present interview. Ben may have constructed it as a theoretical construct—but it had not become a freely available tool for his use in the comparisonof-infinite-sets. Constructing a structure, and even clearly verbalizing it in a specific context does not necessarily ensure that the same structure will be recognized (and can be used for building-with it) in another context, even if the other context is not very different from the previous one. In summary, Ben’s Infinity, Cantorian and One–One Structures, based on his comparison of numeric sets, can be presented as follows:

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Infinity Structure: I1: Using different methods (inclusion and one-to-one correspondence) to compare infinite sets may lead to contradictions. Cantorian Structure: C1: In Cantorian set theory, the method to be used for comparing infinite sets is one-to-one correspondence. In order to avoid contradictions such as in I1, other methods, in particular inclusion must be excluded. Ben is well aware of C1 but has no deep mathematical grounds for it—he remembers it (B80). C2: Using one-to-one correspondence for comparing two infinite sets means to establish an explicit computable mapping between the two sets. C3: In principle, one does not need to establish the correspondence—it is enough to show its existence, but the only way to show existence is to actually establish a correspondence. C4: If existence has been established in any manner whatsoever, the inclusion relationship becomes irrelevant; that is, even if one of the two sets is included in (is a subset of) the other, but there exists a one-to-one correspondence between the set and its subset, the sets are equivalent. One–One Structure: OP: Pairing: If the elements of one set can be paired with the elements of the other set with no leftovers in either set, a one-to-one correspondence has been established from one set to the other and vice versa. The way to do this is to build, or at least think about trapezoids whose base lengths are the elements of the sets, so that for each upper base there is a single lower base and for each lower base there is a single upper base. OB: Bijective Function: Another way to establish one-to-one correspondence is to construct a one-toone function from one set onto the other. The way to do this is to find an explicit formula such that substituting the nth elements of the first set yields the nth element of the second set and then checking that each element of the second set appears exactly once as image of the function. OO: Ordinal: If the elements in a set can be written in a sequence, then there is a one-to one and onto correspondence between the elements of this set and the set of natural numbers. If this can be done for two sets, than both of them are equivalent to the set of natural numbers, and therefore are equivalent to each other.

Task 2 put Ben into a more complex situation than the earlier tasks. While the task is similar to the earlier ones in that it asks for the comparison of the sizes of two explicitly listed sets, the task has several features, which the previous tasks did not have: (i) Each set has elements that do not appear in the other set. Up to this point in time, in every task Ben was given, one set had been a proper subset of the other. Ben’s challenge was thus not, not to fall into the inclusion trap (namely to follow the intuition that the subset is smaller than the full set) but rather to use C4, for example in combination with OP or OB to show that the two sets are equal. The new feature of task 2 may lead to a situation where the intuitive inclusion considerations work both ways thus creating confusion. We note that in this task as earlier, Ben was aware of his intuitive inclusion considerations and of the fact that he should avoid them (C1). (ii) The explicit algebraic rule is difficult to find: It is neither additive nor multiplicative. Up to now all rules had been either additive or multiplicative, with one exception, the easily recognized n → n2 rule. Now, a more complex quadratic rule was used, which is not immediately transparent. (iii) The ‘intervals of omitted elements’ in set V are increasing. It may be obvious to the reader that this is a consequence of the quadratic nature of the rule, but it was not obvious for Ben. While he had

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seen this feature in the n → n2 example, he had presumably ‘forgotten’ it. This new example did not remind him of the earlier one. Such recognition of the structural analogy might have completely changed the remainder of the interview. As expounded above, Ben tried to use pairing by means of trapezoids but soon mobilized his Bijective Function Structure, looking for an explicit formula, presumably in view of his successful experience with this approach in task 1. His search for an explicit formula turned out to be difficult and fruitless. Ben was thus in a bind: He had exhausted the immediate means at his disposal for establishing a correspondence, even after considerable effort. There were no further recognizable structures, which he could have used for building-with them, a solution to task 2. He faced a need for either expanding an existing structure or finding a new one. There are at least four ways in which he could have attempted such a construction: Adapt the Pairing Structure to the case at hand; make the Bijective Function Structure more powerful by enlarging the set of available explicit formulas; develop the Bijective Function Structure into the Ordinal Structure; or attempt to restructure the task data so as to see an unexpected pattern emerge. From Ben’s own point of view, he had already exhausted the first two options. Given that the Ordinal Structure was not within his reach at this time, and given that he had already seen many examples in which he was eventually able to show equivalence, he could reasonably be expected to think that the two given sets are possibly not equivalent, a reaction, which would focus his attention on the fourth option. And this is indeed what happened: In B92, Ben had written the numbers 1, 2, . . ., 8 and referred to them as differences. We may thus assume that he had computed them as the sequence of differences of the sequence of elements of V. In B100, he then made the observation that the sequence of differences represented the sizes of the omitted intervals. He inferred that these intervals grow and this provided him the fresh point of view, allowing an unexpected pattern to emerge, the pattern of the increasing intervals. While Ben had based one of his solutions to the n → n2 example in interview 1 on a very similar pattern of increasing intervals, it had then helped him to reach the conclusion that the two solutions he had arrived at were incompatible, and thus that either one-to-one correspondence or intervals (but not both) could be used for comparing infinite sets. Ben’s conclusions at that point of interview 1 were so impressive that the interviewer did not take the opportunity to highlight the problem inherent in using one-to-one correspondence and intervals. During the second interview, Ben regularly related to inclusion and oneto-one correspondence but not to intervals, and thus the increasing intervals, which he rediscovered in the present interview, appeared to be a novel way to solve the problem, i.e., to provide the argument that two given sets are not equivalent. We note that this switch of point of view amounts to a considerable change of context: The main task has changed from finding a correspondence to showing that there is no correspondence. Ben’s argument, based on his new pattern, is as follows: We know a way to prove that two sets are equivalent. There must also be a way to prove that two sets are not equivalent (B120, B128). If there were no way to prove that two sets are of different size, one would conclude that all sets are equal (B128). I, Ben, found such a way today (B120). It is the way to show that there is no correspondence (B120). I have even used the intervals criterion to show that there is no correspondence (B128). And upon the interviewer’s prodding: “I know that one should always use correspondence to establish equivalence. Nevertheless, I have just used a different way, namely intervals. But note that I have not used intervals for establishing equivalence; rather I have used them for establishing that there is no correspondence.” (B128). Ben is so strongly convinced by his argument that the increasing intervals are a method to prove the non-equivalence of two sets that he failed to notice that what he provided was not actually a proof and

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that the Intervals Substructure may be inconsistent with the Cantorian Structure. What, then, happened to Ben’s consolidated mental structures during this last part of the interview? Since the One–One Structure is used to prove equivalence of sets, it is irrelevant here. We need to relate to Ben’s Cantorian Structure and to his Infinity Structure. Were these two structures so weak that they collapsed? Was our interpretation of them as well-consolidated structures mistaken? We do not think so. Let us look at the Cantorian Structure. Ben’s argument can be interpreted as lying within the Cantorian Structure as C1–C4 are explicit only about the case where sets are equivalent. Ben had never yet been asked to compare two non-equivalent infinite sets. Possibly he was even expecting such a task (B120): “If all sets were always equivalent anyway, why would one need such an elaborate criterion to show equivalence?” What Ben did do in B100–B130 is to adapt the criteria in C1–C4 to the negative case: In order to show that two sets are not equivalent, his reasoning went, I have to show that no one-to-one correspondence can exist. It was clear to Ben that this is what had to be done, and this corresponds precisely to his Cantorian Structure (B120, B128). It was less clear to him how it could be done; in fact, this is not an easy task by any means. For example, the proof that the set of real numbers and the set of natural numbers have different powers is non-trivial and could not possibly have been expected from Ben. His purpose was to find a method within his reach that would allow him to show non-existence of a one-to-one correspondence. He used the increasing intervals consideration for this purpose (or at least expressed his conviction that this could be done). This is at odds with an ideal student’s hypothetical Infinity Structure, but not necessarily with Ben’s present Infinity Structure. Such an ideal Infinity Structure expresses that using any two different substructures of the finite One–One Structure in comparison-of-infinite-sets may lead to contradictions. However, all examples Ben had dealt with up to this point were such that one set was included in the other, and in every case the inclusion property had been singled out as the one not to be used. And this was what Ben had strongly consolidated: Inclusion and one-to-one correspondence must not be used simultaneously. In addition, a delicate logical implication of I1 seems to have escaped Ben (B128). According to I1, inclusion or interval considerations may lead to conclusions opposed to those from One–One correspondence considerations. Thus conceivably, inclusion or interval considerations could lead to the conclusion that two sets are not equivalent in spite of the fact that a One–One correspondence between them exists. Therefore, I1 implies that inclusion or interval considerations can never lead to the conclusion that there is no one-to-one correspondence. How could this logical implication have escaped Ben’s sharp mind? We attribute this to the complexity of the logical implication that was compounded by the complexity of the situation in which Ben was acting, as expressed in the following summary: (a) The entire argument was confounded by the two-sided containment. First Ben felt he had to show that V is not bigger than A, in spite of having one element that was not an element of A. Then he realized that things might be the other way: A might be bigger than V. (b) Next it struck him that according to intuitive inclusion considerations, A is not only bigger than V in the ‘usual’ sense (the sense he was used to from the examples he was familiar with) of a shift (such as in task 1) or constant intervals (such as in a multiplicative case like n → 2n). He realized that in the present case the intervals increase. (c) Hence, he was clearly aware that his task was to prove non-existence of a one-to-one correspondence—exactly following the Cantorian Structure. (d) In order to propose a proof for this non-existence, he did not work with inclusion but rather with increasing intervals, with which he had no explicit experience. He was thus not acting in danger of

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opposition to the explicit part of his Infinity Structure but only the much vaguer I1. (e) Ben’s argument was not even in opposition to I1 but only to a rather intricate logical consequence of I1. We strongly suspect that the combined complexity of (a–e) was too high for Ben to successfully handle all aspects of it in the context of this task. This complexity of the logical argument may also have been compounded by a psychological factor: Up to this point, Ben had only been presented with positive examples. Ben, however, may have felt the need for a non-example, a case of two infinite sets that are not equivalent. What can we learn from this analysis about processes of abstraction? (1) Mental structures arising from processes of abstraction may be very complex; this is true, for example, for the ones related to the comparison-of-infinite-sets. (2) Structures are often built on prototypes; even if a person who easily verbalizes and generalizes like Ben formulates things generally as in I1, his actions may well contradict I1 without contradicting any of the prototypical cases he has seen and worked with. (3) Ben’s Infinity Structure was less elaborate than we had thought when writing the previous paper.

7. Didactical review and conclusion In this section, we propose some thoughts regarding didactical aspects of the sequence of three interviews with Ben. The didactical considerations in the choice of interview tasks were based on our research aims, on the learning aims regarding the comparison-of-infinite-sets, on related research results about students’ reasoning (e.g., Tsamir, 1999, 2001, 2002, 2003; Tsamir & Tirosh, 1999), and on our growing familiarity with Ben’s personality, abilities and performance in the topic. While we were aware of the reported impact of inclusion and intervals considerations on students’ tendency to regard certain equivalent sets as being non-equivalent, our early impression of Ben’s command, based on his impressive line of decisions was exaggerated. The thoughts, hesitations and conclusions he shared with us gave us the feeling that he had no or few problems in comparing countable infinite sets. When repeatedly declaring that one-to-one correspondence and inclusion should never be used interchangeably for the comparison-of-infinite-sets, Ben exhibited clearly his need to reject the incompatibility of equal and not-equal solutions to the same pair of sets, as well as his knowledge of ways for preserving consistency. When repeatedly stating that one-to-one correspondence is the only method to use for the comparison-of-infinite-sets, Ben further conveyed a message of understanding what to do and what to accept when given a comparison-of-infinite-sets task. These two sides of the comparison-of-infinite-sets coin led us to assume that, if in a particular case Ben would fail to find a pairing of elements and also fail to refute the existence of such a pairing, he would conclude that he cannot decide, thus showing sensible and sensitive awareness of the boundaries of his current knowledge. This was the way he performed in the first interview. In the present interview, our high expectations led us to set the two tasks discussed in this paper as appetizers for more challenging comparisons: First, the natural versus whole number comparison, for which we naively expected Ben to be able to draw upon his n versus 2n comparison and come up with the correct ‘equal’ conclusion. Then, we intended to ask about the natural versus positive rational comparison, where we assumed Ben would not be able to provide a correct solution without additional clues.

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However, the present interview refuted our expectations in three respects: (1) In spite of his obvious talent, and the observed consolidation of his Infinity Structure, Ben did not realize the inconsistencies that arose between intervals and one-to-one correspondence considerations. (2) Ben’s Ordinal Substructure was not freely available to him in the present situation. Ben had earlier constructed a weak version of the Ordinal Substructure but he had not yet consolidated it. His observation from the second interview that the trapezoids of the geometric representation are just placeholders for sequencing the elements in the two sets, did not lead Ben to a definite conclusion. The Ordinal Structure has not become a usable tool for him. (3) At an unidentified moment, Ben became fixed on the trapezoids of the geometric representation, and was manipulated by extraneous aspects of the model rather than managing to manipulate the model for solving his problem. Several questions naturally arise: What if we would not have taken the didactical steps we took but others? What could we have done differently? In addition? Instead? In what stages could/should we make some changes and why? And more generally: How can we help students consolidate their knowledge? And how can we assess to what extent knowledge has been consolidated? How can we be sure that a slight change of context in the next task would not cause unanticipated problems? We will try to answer these questions while referring to the three points made above. (1) How to extend and deepen students’ Infinity Structure? The task n versus n2 presented in our first interview still seems as good a choice for constructing the Infinity Structure as we have assumed at first, since it entails the capacity to elicit the Inclusion Structure, the Intervals Structure (in the horizontal representation), and the One–One Structure (in the geometric representation). Still, a major change is required in the teacher’s guidance to promote students’ awareness of the problematic inherent in interchangeably using either one-to-one correspondence and inclusion or one-to-one correspondence and intervals. There appears to be a need to explicitly and separately draw the students’ attention to both of these incompatibilities. Similar remarks are valid with respect to two other criteria that students were found to use: ‘all infinite sets are equal’ and ‘infinite sets are incomparable’ (Tsamir, 2002). Perhaps, additional tasks like n versus n3 and n versus 4n would be useful to strengthen the Infinity Structure. That is to say, extra emphasis should be placed on the Intervals Structure and other misleading knowledge structures and lead to explicit discussion with the students what points may be misleading. In this spirit, our suggested tasks were ‘good’ springboards for raising and discussing conflicting ideas that may lead to mathematical contradiction. (2) How to draw students’ attention to, and promote their sensitivity for the Ordinal Structure? One might think of introducing the Ordinal Structure early on. But there is a lot of wisdom in not introducing the Ordinal Structure and the accompanying representation too early: Otherwise, students may adopt a superficial way of using it. Generally, students may initially hang on to external features of any structure. The external features of the Ordinal Structure might be too simple to allow students to problematize the issues, and thus enrich the structure, if they are introduced too early. It may therefore be wise to lead students to solve some comparison-of-infinite-sets tasks in several ways, including the Ordinal Structure, to illustrate the rich offering of One–One Substructures, which are interwoven to create the One–One Structure. (3) How to design task sequences so that artifacts like the trapezoids can be used where they are helpful, and at the same time so that they do not become all-encompassing; in other words so that the

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students can easily get rid of them and go beyond them when the need arises. In its purely geometric appearance, the trapezoids may introduce problems by being applicable only to positive numbers. In light of our findings, it seems right to use the ‘What if Not’ teaching approach (Brown & Walter, 1993), posing very early on questions like, what if it is not {1, 2, 3, 4, 5, . . .} versus {3, 4, 5, 6, 7, . . .}, but {0, 1, 2, 3, 4, . . .} versus {3, 4, 5, 6, 7 . . .} or {−3, −2, −1, 0, 1, . . .} versus {3, 4, 5, 6, 7. . .}, and using ‘the spirit of the trapezoids’ to illustrate their role of ‘placeholders’ in establishing the correspondence. This offers a good opportunity to ask for additional ways to prove the existence of a correspondence, such as, to temporarily put aside a few pairs or construct a suitable formula that may add to the richness of the One–One Structure. It should be noted that the discussion should explicitly refer to the contribution and the shortcoming of each approach. In particular, the intuitive scaffolding provided by various geometric models (e.g., square-areas, perimeters and mid segments in triangles; see Tsamir & Tirosh, 1999), and their intuitive pitfalls should be part of the discussion with the students. While having Ben in mind when making these specific suggestions, we contend that these didactical inferences are generalizable to other comparison-of-infinite-sets teaching situations since they are based on a profound mathematical analysis of the topic. Attempts to help students consolidate their knowledge should be adapted to the fine detail of knowledge structures in two senses: The potential richness and the actual richness of the students’ structure. The teacher should be a sensitive listener, hearing not only what the students say, but also what they fail to mention (e.g., in Ben’s case, that one-to-one correspondence is the way either to prove or refute the equivalency of given infinite sets). But even with these precautions we can never be sure that knowledge has been consolidated and that the next task would not cause the learner problems. Therefore, cycles of designing instruction and investigating its impact on students’ knowledge structures are necessary steps in promoting students’ knowledge. There exists broad agreement among mathematics educators that instructional design and teaching should be based on students’ preliminary knowledge structures. In this study, we used a design that took into account research about relevant student knowledge structures of the comparison-of-infinite-sets (e.g., Tsamir, 1999) as well as Ben’s related knowledge, structured as found on previous occasions (Dreyfus & Tsamir, 2004; Tsamir & Dreyfus, 2002). We took into account not only the expert goal structures but also alternative ones, and we adapted the design to the learner’s (Ben’s) history and to his exceptionally reflective personality. We therefore assumed that we could predict Ben’s successful solutions to the present tasks. But, in spite of all this, Ben’s sequence of processes of abstraction took an unexpected direction. Both, as researchers and as teachers, we found that this surprising turn of events yielded insight into the intricacies of the mathematical topic under consideration—the comparison-of-infinite-sets. At the same time, the analysis presented in this paper shows the complexity of the theoretical notion of consolidation proposed in Dreyfus and Tsamir (2004). On the basis of a substantial number of research studies, using the dynamically nested RBC model of abstraction (and cited in Dreyfus & Tsamir, 2004), we can now quite confidently identify structures that a student constructed. While this identification cannot necessarily be made during the constructing process, it can unquestionably be made later by means of the student’s recognizing and building-with actions using these structures. The situation with respect to consolidation, however, remains less definite: Both, in the previous and the first part of the present interview, there were clear signs that Ben had consolidated his Infinity Structure and, possibly to a lesser extent, his Cantorian Structure. Nevertheless, he then used the Intervals Structure, missing the so generated violation of the Infinity Structure and the related inconsistency with the Cantorian Structure. Although we have provided a

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detailed analysis of how and why this may have happened, further research is needed in order to elaborate and refine the notion of consolidation and to hone our ability to identify whether knowledge can be considered as consolidated.

Acknowledgement We would like to thank Angelika Bikner-Ahsbahs for her insightful comments based on careful reading of an earlier version of this paper.

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