Conjecture-generation through Dragging and Abduction in Dynamic

Education in a technological world: communicating current and emerging research and technological efforts _______________________________________________________________________________________ A. Méndez-Vilas (Ed.)

Conjecture-generation through Dragging and Abduction in Dynamic Geometry A. Baccaglini-Frank1 & M.A. Mariotti2 1

Department of Education and Human Sciences, University of Modena and Reggio Emilia, Italy. Via Allegri 9, 42121, Reggio Emilia, Italy 2 Department of Mathematics and Computer Science, University of Siena, Italy. Piano dei Mantellini 44, 53100, Siena, Italy Dragging points on the screen in dynamic geometry systems can be considered a tool for reaching conjectures in openended activities in Geometry. This chapter introduces a model describing a particular dragging scheme, referred to as the maintaining dragging scheme, elaborated during a recent study [1, 2]. Such model can be used to produce cognitive analyses of the students’ problem-solving sessions and, in particular, to illustrate the role of abduction in processes of conjecture-generation in which the maintaining dragging scheme is used. Keywords abduction; conjecture; dragging; dragging schemes; dynamic geometry; instrument; instrumented abduction; maintaining dragging

1. Introduction and theoretical framework Mathematics education supervisors and leaders have been encouraging the use of technology, such as Dynamic Geometry Systems (DGSs), in the classroom (for example, [3, 4, 5, 6, 7]). Several studies in the teaching and learning of Geometry have shown that a DGS can foster learners’ constructions and ways of thinking, and they have shown how, thanks to the dragging tool, a DGS can be powerful for explorations in open problem situations [8, 9, 10]. In this chapter we focus on reasoning and conjecture-generation in Geometry when a DGS, and in particular, the dragging tool, is used. A feature offered by a DGS is the dragging tool, that can be exploited in various ways by the solver, and that can support conjecture-generation. Research carried out by Arzarello, Olivero, Paola, and Robutti [9, 11] led to the description of a set of dragging modalities, classified through an a posteriori analysis of solvers’ work, that can be observed while a solver is producing a conjecture in a DGS. A key moment of the process of conjecture-generation is described in Arzarello et al.’s model as an abduction which seems to be related to the use of dummy locus dragging (later referred to here as maintaining dragging (MD) [1, 2]). However, various aspects of the relationship between dragging and abduction needed further clarification. Moreover, since there seemed to be potential in this way of dragging, with respect to the task of conjecture-generation, it seemed important to investigate whether through minimum intervention it was possible to foster students’ use of it. A recent study carried out by the authors [1, 2] was designed to shed light onto these aspects. For this scope four dragging modalities were elaborated (from Arzarello et al.’s classification) and introduced to 31 Italian students between the ages of 15 and 17, and a model, the MDconjecturing model, that describes a process of conjecture-generation when maintaining dragging (MD) is used, was constructed and successively re-elaborated. Such model is characterized by the presence of abduction. Before introducing the model, we briefly describe the notion of abduction, highlight aspects of dragging in a DGS as parts of a theoretical framework, and then taking an instrumental approach, we analyze the use of dragging in processes of conjecture-generation. 1.1 Dragging in a DGS The dragging mode is an important feature of DGSs that makes these environments different from the traditional paper-and-pencil environments. The dragging mode allows the transformation of images on the screen by producing a sequence of new images. Each image is reconstructed after the user’s choice of a new position for a specific point she is dragging, by clicking on it and moving the mouse. The high number of images in this sequence and the speed at which they are produced on the screen give a visual effect of continuity, analogous to what is seen in a movie. The changes in the image on the screen are perceived in contrast to what simultaneously remains invariant, and this constitutes the base of the perception of “movement of the image” [12]. We will refer to figures constructed in a DGS as dynamic-figures. The points through which the dynamic-figure is constructed and upon which the other elements depend, are referred to as base-points. In general, and this is the case in a DGS like Cabri, the invariants are determined both by the geometrical relations defined by the commands used to accomplish the construction, and by the relationship of dependence between the original geometrical relations of the construction and those that are derived as a consequence within the theory of Euclidean Geometry. All these invariants

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appear simultaneously as the dynamic-figure is acted upon, and therefore “moves”. However there is an a-symmetry between the types of invariants, which is fundamental for conceiving logical dependency within the DGS. This logical dependency between the perceived invariants corresponds to a functional relationship of dependency between the geometric properties represented by the invariants. It can be useful to distinguish between invariants that arise from “direct” movement, that is those defined directly in the construction, and invariants that arise from “indirect” movement, that is properties that are a logical consequence of that construction. Research has shown that perceiving and interpreting invariants is a complex task for a non-expert geometry student (for example [13, 14, 15]). Therefore, when using a DGS, it is important to be aware of the difficulties that may arise before reaching appropriate perceptions and interpretations of the different types of induced movement. The induced movement may be:  direct: the point (or, more in general, object) moves as a direct effect of the dragging mode that acts on it through direct action of the mouse (in most DGSs this is obtained by clicking on the point and moving the mouse);  indirect: the point (or, more in general, object) moves as a consequence of the direct movement of another object. For example, consider the following construction: ABC is a triangle, and E and F are constructed as the midpoints of AC and CB, respectively (Fig. 1).

Fig.1 A, B, and C are base-points; E is constructed as the midpoint of AC, and F as the midpoint of BC. Dragging A, B, or C, EF remains parallel to AB.

The constructed midpoints E and F remain midpoints during dragging, that is the properties “AE congruent to EC” and “CF congruent to FB” are conserved. However also the property “EF parallel to AB” is conserved, since it is a logical consequence within the theory of Euclidean Geometry (according to which this DGS was programmed). The leap in complexity is achieved through awareness of the hierarchy induced on the properties of the construction and of the fact that such hierarchy corresponds to logical relationships between the properties of the “geometric figure”. This awareness can, for example, lead to the following conjecture1: “if a segment is constructed with endpoints at the midpoints of two sides of a triangle, then it is parallel to the third side of the triangle.” The interpretation of dragging in terms of conservation of the relationship between invariants corresponds to a logical/theoretical type of control over the generality of the relationship between properties of a given figure, and such control is expressed within a specific theoretical system. Moreover, invariants may appear to be “robust” or “soft” [16]. For example, the construction above gives rise to three invariants, “AE congruent to EC”, “CF congruent to FB” and “EF parallel to AB”, that are “robust”, that is present for random dragging of the base-points. On the other hand, an invariant like “EF perpendicular to AC” can be induced by particular ways of dragging the base-points, but, for a given base-point, it is not always an invariant of the dynamicfigure. This is an example of “soft” invariant. Soft invariants are particularly useful in activities that involve conjecturegeneration through maintaining dragging, as we will describe later in the chapter. 1.2 Dragging as an instrument The instrumental approach [17, 18] puts an artifact in relation with a task and a solver. The solver develops a utilization scheme in order to accomplish the task using the artifact. The combination of the artifact and the developed utilization scheme is the instrument. In the current study we consider dragging in a DGS from an instrumental perspective, as has been done fruitfully by other researchers (for example, [19, 20, 21]). Under such lens, particular ways of dragging can be seen as artifacts supporting the task of conjecture-generation. Together with a utilization scheme developed by the solver during a process of instrumental genesis, a particular way of dragging may become an instrument. We will call the utilization schemes developed by the user “dragging schemes”. Some dragging schemes have been identified in the literature (for example, [9, 19, 20]). From this perspective, the MD-conjecturing model we

1

Once the conjecture is proved in the Theory of Euclidean Geometry, it becomes a theorem within such theory.

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developed can be considered a description of a utilization scheme for the instrument maintaining dragging in a task of conjecture-generation.

2. The MD-conjecturing model This section illustrates our model through a paradigmatic example, describing how a conjecture can be formulated when MD is used. Consider the following open-problem activity (in [1]): Draw three points A, M, K, then construct point B as the symmetric image of A with respect to M, and point C as the symmetric image of A with respect to K. Construct point D as the symmetric image of B with respect to K. Drag M and make conjectures about ABCD, trying to describe all the ways in which it can become a particular type of quadrilateral. A first conjecture might consider relationships of dependency between robust invariants, and it could be: “Given this construction, ABCD is a parallelogram” or “ABCD is always a parallelogram”. The type of exploration that leads to a conjecture like this one is based on use of wandering dragging (dragging freely on screen, as described in [9]) and on the solver’s ability to perceive robust invariants contained in the steps of the construction as different from robust invariants that are derived from the first ones, and that may also be unexpected. Here we want to consider a second type of conjecture, which corresponds to identifying under which conditions a given configuration takes on a certain property. In terms of invariants it means establishing the invariance of a particular property with respect to a particular movement, that is inducing a soft invariant. The special movement corresponds to the figure’s assuming a specific condition. The difficulty in determining such trajectory of movement and with it the property to take as a condition can be overcome using a particular way of dragging, that is moving the base-point so that the interesting property is maintained. This is what we call maintaining dragging (MD). The model can be described through three (implicit) tasks with possible subtasks the solver addresses. Task 1: Determine a configuration to be explored by inducing it as a (soft) invariant. The solver may choose to drag base-points to look for interesting configurations and conceive them as potential invariants to be intentionally induced.

a)

b)

Fig.2 The quadrilateral ABCD as a result of the construction (a). The base-points of ABCD can be dragged until the quadrilateral “looks like” a rectangle (b).

Dragging in this manner students may notice that ABCD can become different types of parallelograms. In particular they might notice that in some cases it seems possible for ABCD become a rectangle. With the intention of maintaining this property as an invariant (III: ABCD rectangle), students might mark some configurations of M for which this seems to be true, and through the trace tool, try to drag maintaining the property, as shown in Fig. 3. Task 2: Look for a condition that makes the intentionally induced invariant (III) be visually verified through maintaining dragging. This can occur through  a geometric interpretation of the movement of the dragged base-point  or a geometric interpretation of the trace mark. The “condition” may be considered the movement of the dragged-base-point along a path which can also acquire a geometrical description. The belonging of the dragged-base-point to a path determines the invariant observed during dragging (IOD). When the two invariants are observed simultaneously, the solver will have direct control over the invariant observed during dragging (IOD) and indirect control over the intentionally induced invariant (III). This may

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guide the conception of a conditional link between the two invariants. Solvers may engage in the sub-task of finding a geometric description of the path (for example, “a circle of diameter AK”), which can be useful for Task 3.

a)

b)

Fig.3 Dragging with the trace tool can help a student reach a geometric description of the path (a). Once a geometric description of the path is reached it can be constructed robustly (b) and the solver can perform a dragging test.

Task 3: Verify the conditional link through the dragging test. This requires the accomplishment of at least some of the following subtasks:  representing the invariant observed during dragging (IOD) through a construction of the proposed geometric description of the path;  performing a soft dragging test by dragging the base-point along the constructed geometric description of the path;  performing a robust dragging test by providing (and constructing) a geometric description of the path that is not dependent upon the dragged-base-point and redefining the base-point on it in order to have a robust invariant, to then perform the dragging test. When a geometric description of the path (or more generally an IOD) has been reached, the solvers might be inclined to constructing it geometrically. Fig. 3 shows a construction of the geometric description of the path proposed above. In order to accept the geometric description of the path and confirm a conditional link the solvers may drag along the constructed geometric description of the path (soft dragging test), or reconstruct the figure imposing on the dragged base-point to become linked to the constructed geometric description of the path (robust dragging test). At this point, if the solver is convinced that her geometric description of the path and conditional link are appropriate, she may state a conjecture like the following: “If M is on the circle of diameter AK, then ABCD is a rectangle,” or: “If AKM is a right triangle, ABCD is a rectangle.”

3. Abduction in the MD-conjecturing model Now that the MD-conjecturing model has been introduced, we can focus on its relationship with abduction. Peirce was the first to introduce the notion of abduction. According to Peirce [22], of the three logic operations, namely deduction, induction, abduction (or hypothesis), the last is the only one “which introduces any new idea; induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis. Deduction proves that something must be; induction shows that something actually is operative; abduction merely suggests that something may be.” Peirce describes the general form of an abduction as follows: a fact A is observed if C were true, then A would certainly be true. So, it is reasonable to assume C is true. Peirce also proposed an example in which he compares abduction to induction and deduction: “Suppose I know that a certain bag is full of white beans. Consider the following sentences: A) these beans are white; B) the beans in that bag are white; C) these beans are from that bag. A deduction is a concatenation of the form: B and C, hence A; an induction would be: A and C, hence B; an abduction is: A and B, hence C.” [12]. Peirce refers to A as fact, to B as rule, and to C as hypothesis of the abduction. The product of an abduction, according to the definition and example above, is the hypothesis, or statement C in the example. In the context of conjecturegeneration we have been investigating, such description appeared to be problematic. Let us clarify. Suppose someone observes the occurrence of A (or that A is true) whenever she observes the occurrence of C, and she generates the conjecture “if C then A”. We can schematically represent the situation as follows:

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fact rule

Peirce’s Example A is observed if C was true then A would be true

hypothesis

C is true

Conjecture-generation Example A is observed when C is observed/true A is observed/true or if C then A (conjecture) C is observed/true

Table 1 The table compares abduction according to Peirce and abduction in particular tasks of conjecture-generation in dynamic geometry.

So in tasks of conjecture-generation such as those we have focused on, the hypothesis, in Peirce’s terms, is not what we consider to be the product of the abduction. Instead, the product in our case is the rule. Moreover while in Peirce’s paradigmatic example the rule “if C were true then A would be true” (that corresponds to statement B: “the beans in that bag are white”) is a simple piece of information from the solver’s bag of acquired knowledge, in our case such rule is more complex. In the process of production of the rule we can identify two main components: the first is the observation of two simultaneous occurrences, C and A; the second component consists in the choice of using maintaining dragging to search for a cause for the invariance of an interesting property A. The first component lies at the level of perception during the phenomenological experience, while the second lies at a meta-level with respect to the first, and can give the solver awareness of the type of control, direct or indirect, exercised on each invariant, strengthening in this way the conjectured conditional link between C and A. Given these considerations, the above interpretation of Peirce’s framework did not seem enlightening, although the maintaining dragging scheme did seem to have an abductive nature. The elaboration of the definition of abduction provided by Magnani offers a way of interpreting the solution process that is particularly insightful. In his definition Magnani describes abduction as [23]: “the process of inferring certain facts and/or laws and hypotheses that render some sentences plausible, that explain or discover some (eventually new) phenomenon or observation; it is the process of reasoning in which explanatory hypotheses are formed and evaluated.” According to Magnani’s definition, we may consider the product to be the conditional link between the hypothesis and the observation (if these beans are from that bag, then they are white, what Peirce called “rule”). The conditional link is by all means an “explanatory hypothesis” in Magnani’s words, developed to explain the observed phenomenon as a whole. In the context of dynamic geometry, within the process we studied, this rule arises from capturing the logical dependence of two (or more) invariants. When solvers explore an open problem situation in dynamic geometry and are asked to formulate conjectures on a certain geometrical object, they notice invariants, that is, properties of the figure that remain constant during the dragging of a point and try to logically link two (or more) of such geometrical invariants in a conditional statement. Such a statement constitutes an “explanatory hypothesis” for the observed phenomenon. In fact when the invariants are obtained through MD there is an order in their discovery: a first invariant is that which is to be maintained, and the second is that which seems to be invariant when the first invariant is maintained. Students try to express the link between the two invariants using phrases like: “Because/since/every time this property is true [the property is the second invariant], this property is true [the property is the first invariant]”; or like: “In order that/so that this property is true [the property is the first invariant], this property is/has to be true [the property is the second invariant].” If we describe the process as a whole, from the initial random dragging of points to the formulation of a conjecture, and therefore consider the final conjecture to be the end product of the process, Magnani’s description seems to be appropriate. Moreover, this interpretation is not specific to use of the maintaining dragging scheme and can be generalized to different types of explorations in which invariants are perceived and conditionally linked. Although we gained in generality through such interpretation, we were searching for a finer lens through which to analyze abduction in the context of MD in processes of conjecture-generation through use of MD. We eventually realized that the key lies in a meta-rule that emerges during the process of instrumental genesis for MD. Such meta-rule allows the solver to generate what in our first interpretation of Peirce’s framework was the rule “when C is observed/true A is observed/true or if C then A”, which is also the explanatory hypothesis in Magnani’s terms, or what we refer to as conditional link in the MD-conjecturing model. So we can introduce a meta-level at which such meta-rule can be used to derive the conditional link. fact A is observed + maintaining C, A is visually verified meta-rule

??

explanatory hypothesis conditional link)

(or

C “causes” A/ if C then A

Table 2 The table shows the meta-level at which the meta-rule is used to derive the explanatory hypothesis or conditional link.

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In Table 2 we show how through an abduction using a meta-rule developed during instrumental genesis, an explanatory hypothesis (or conditional link) is obtained. We recall that the MD scheme allows the solver to “feel” (through haptic perception) the conditional link between two soft invariants that appear simultaneously. In particular the two invariants will be perceived as asymmetric because of the type of control the solver can exercise on each of them: on one invariant the solver will be exercising direct control by dragging a point, while on the other she will exercise indirect control, that is she will be able to induce it without directly acting upon it, but instead by inducing the former invariant. As part of the maintaining dragging scheme, the solver may therefore elaborate the following meta-rule: simultaneity of soft invariants A and C

?? =

if C then A (I experience)

+ direct control over C + indirect control over A

With respect to the example above, dragging M to induce the property ABCD rectangle, the two invariants can be described as: ABCD rectangle (III), and M on the circle with diameter AK (IOD). The abduction performed by the solver can be illustrated as fact meta-rule explanatory hypothesis (or conditional link)

III (ABCD rectangle) and IOD (M on circle with diameter AK) are observed simultaneously + direct control over IOD + indirect control over III simultaneity of soft invariants III and IOD if IOD then III + direct control over IOD (I experience) + indirect control over III if IOD (M on circle with diameter AK) then III (ABCD rectangle)

Table 3: The table uses Peirce’s framework, to describe abduction at a meta-level leading to the generation of an explanatory hypothesis in the context of conjecture-generation that involves maintaining dragging.

Our data suggested that once the MD scheme has been appropriated, the process of conjecture-generation as described by the MD-conjecturing model seems to become “automatic”, and the solver proceeds through steps that lead smoothly to the discovery of invariants and to the generation of a conjecture, with no apparent abductive ruptures in the process. The abduction seems to be concealed within the MD-instrument. This can now be explained through the metarule developed during the process of instrumental genesis of the tool (in this case MD) that allows the solver to interpret the phenomenon in her experience in dynamic geometry in terms of logical dependency between invariants, and therefore to produce a conditional statement of the type: “if property A (the second invariant perceived) then property B (the first invariant induced)”. Such conditional statement is the product of the solver’s exploration, an explanatory hypothesis, but that becomes “automatic” once the process of instrumental genesis is complete (i.e. the MD scheme has been appropriated). In this sense the abduction seems to be of a different type: it occurs at a meta-level though a metarule developed during a process of instrumental genesis, and is encapsulated in the use of the MD-artifact in the task of conjecture-generation. This new type of abduction seemed worthy of a name, so we introduced the notion of instrumented abduction to refer to it. The final interpretation we have proposed, introducing a meta-rule developed during the process of instrumental genesis for MD, seems to be particularly significant if we want to generalize our notion to describe other instances of instrumented abduction. Research investigating utilization schemes of different instruments may help to further develop this notion.

4. The hypothesis on introducing dragging schemes The model provides insight into a utilization scheme that students seemed to develop for using MD in tasks of conjecture-generation in a DGS. During the introductory lessons we purposefully did not introduce maintaining dragging as more than “a way of dragging during which you can try to maintain an interesting property”. This choice was made so that we could test the appropriateness of the MD-conjecturing model as a description of a scheme spontaneously developed by students. The introductory lessons were carried out over two class periods. The first preliminary lesson aimed at helping students overcome some difficulties, well-documented in the literature (for example, [13, 15]). Some difficulties are related to the control of the different status of objects of Cabri-figures, control that is fundamental in processes of conjecture-generation in a DGS. This lesson was developed around recognition of base-points and dependent points of a Cabri-figure that originated from a step-by-step construction that the students were asked to make. The second lesson was focused on the introduction of the four dragging modalities we developed

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appropriately. As students explored Cabri-figures they were asked to drag points in particular ways and describe their observations and perceptions (for example, how they moved their hand while dragging) with respect to a particular configuration. Students were asked to share their perceptions with the whole class, in a collective discussion during which the instructor gave names to specific “ways of dragging” while the students explained how they used them. The four modalities we elaborated and introduced during the lessons are: (1) wandering/random dragging (Italian: “trascinamento libero”): randomly dragging a base-point on the screen, looking for interesting configurations or regularities of the Cabri-figure; (2) maintaining dragging (Italian: “trascinamento di mantenimento”): dragging a basepoint so that the Cabri-figure maintains a certain property; (3) dragging with trace activated (Italian: “trascinamento con traccia”): dragging a base-point with the trace activated; (4) dragging test (Italian: “test di trascinamento”): dragging base-points to see whether the constructed figure maintains the desired properties. In this mode it can be useful to make a new construction or redefine a point on an object to test a formulated conjecture. For about 30% of the students interviewed, the introductory lessons were sufficient for them to develop, within 2 exploratory activities (during the interviews they completed 3 or 4 activities), a utilization scheme for MD that could be described through our model. However, the other students seemed to struggle with the instrumental genesis of MD and not be able to reach conjectures through the process described by the model. These students had not yet appropriated the utilization scheme, so even if they appeared to be “maintaining a property during dragging” they would not actually be looking for an IOD as a “cause” for their III. These are encouraging data suggesting that it is possible through not-toointense intervention sessions to foster the development of the MD scheme. However we are convinced that since appropriation of the MD scheme involves the development of a meta-rule a specific intervention of the teacher is necessary. Such intervention should focus on the various components of the process and comments on them in terms of logical dependency between properties.

5. Conclusion and next steps Analyses of our data have led to confirmation and refinement of the model, which seems to be a useful tool for capturing and describing the process of conjecture-generation when MD is used. The model seems to help gain deeper insight into specific cognitive processes involved in conjecture-generation in a DGS and in particular into a relationship between abduction and MD used in tasks of conjecture-generation. The model also allows the analysis of the extent to which students have appropriated the MD scheme, and it provides terminology that can be introduced to students to address their misconceptions and deepen their understanding in the field of conjecture-generation. In fact our research has outlined a further potentiality of the dragging tool, that is the possibility it offers teachers to support students’ construction of specific mathematical meanings associated to the concepts of “conditional statement,” “premise” and “conclusion.” We can refer to such “possibility” as a didactic potential of MD. Such didactic potential is accessible to teachers thanks to the nature of the experience within a DGS (and of dragging in particular) that allows students to investigate through a physical experience building upon real bodily perceptions. In this chapter have described how the MD-conjecturing model together with the notion of instrumented abduction show how the distinction, based on perception during MD, of two simultaneously-observed invariant properties leads to interpreting one as “premise” and the other as “conclusion” of a statement. More precisely, at the same time, a “condition” (or premise) is perceived as the invariant upon which the user has direct control (linked directly to a basepoint of the construction), while a “consequence” (or conclusion) is perceived as an invariant property that the user can only control indirectly, that is maintained “whenever” the previous property is maintained. Therefore, a direction of research worth pursuing is how implementing the teaching of the dragging schemes in school curricula can support students’ construction of specific mathematical meanings associated to the concepts of “conditional statement,” “premise” and “conclusion.” In other words we propose to pursue research in how the didactic potential of MD that we have described can be successfully exploited by the teacher. We hypothesize that successive interventions will be necessary, the first of which should focus on the invariant properties induced and perceived during students’ explorations, and on commenting on them in terms of logical dependency between properties. Once the MD scheme has been appropriated, the teacher can foster awareness of the meta-rule developed during instrumental genesis and focus on the conditional link, again in terms of logical dependence. Eventually we expect the teacher to be able to mediate the mathematical meanings associated to the concepts of “conditional statement,” “premise” and “conclusion.” Acknowledgements The support by the University if New Hampshire Graduate School, and by PRIN 2007B2M4EK (Instruments and Representations in the Teaching and Learning of Mathematics: Theory and Practice) - Università di Siena & Università di Modena e Reggio Emilia are gratefully acknowledged.

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