7/29/2019 Producing Conjectures Using Maintaining Dragging: Instrumented Abduction

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Producing Conjectures Using Maintaining Dragging:

Instrumented Abduction

IntroductionBackground: 

Research carried out by Arzarello et al. (2002) and Olivero (2002) led to the description of ahierarchy 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 dynamic geometry system(DGS). Akey 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. Thestudy we report on in this poster (Baccaglini-Frank & Mariotti, 2009) was designed to shed

light onto this delicate moment; we proceeded by elaborating (from Arzarello et al. ’s classification) four dragging modalities  – in particular maintaining dragging  (MD), developedfrom dummy locus dragging  – and developing and testing a cognitive model that describes aprocess of conjecture-generation. The study makes use of two more notions present in theliterature: abduction, and instrument.

Abduction: Peirce was the first to introduce the notion of abduction as the “inference whichallows the construction of a claim starting from some data and a rule” (Peirce, 1960). Recently,there has been renewed interest in the concept of abduction. In particular, Magnani describesabduction as: “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 andevaluated” (Magnani, 2001, pp. 17-18).

The general experimental design was articulated in two phases, a pilot study followed by arefinement and revision phase that preceded the final study. For both the pilot study and the full-blown study subjects were high school students in Italian “licei scientifici”, 9 (3 single students and 3pairs) students for the pilot study and 22 (11 pairs) for t he final study. Since according to the literature(Olivero, 2002), spontaneous use of dummy locus dragging does not seem to occur frequently, first,we introduced our dragging schemes to the subjects during two lessons. We then interviewedstudents while working on open-problem activities. Data collected included: audio and video tapesand transcriptions of the introductory lessons; Cabri-files worked on by the instructor and the studentsduring the classroom activities; audio and video tapes, screenshots of the students’ explorations,transcriptions of the task-based interviews, and the students’ work on paper that was produced duringthe interviews.

Conclusions

Anna Baccaglini-Frank1,2 and Maria Alessandra Mariotti2

University of New Hampshire (USA)1, and Università degli Studi di Siena (Italy)2

Acknowledgments

Experimental Design

The MDS as a Psychological Tool

This research study was partly funded 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.

• The model of the MDS seems appropriate for describing the processes of conjecture-generationwhen MD is used, providing evidence to a correlation between the introduction of the dragging

schemes, and MD in particular, and a specific new (with respect to those in literature) cognitiveprocess described by the model. We have referred to such process as a form of instrumentedabduction, a new notion that we hope can be generalized to other contexts in which abduction issupported by another instrument.

• We seem to have captured the key idea which may lead to complete appropriation of the MDS,and described how it resides at a meta-level with respect to each specific exploration in whichMD is exploited as an instrument for conjecture-generation.

• We described how for expert solvers the MDS might be transformed into a way of thinking. Inthis sense it may lead to the construction of fruitful “mathematical habits of mind” (Cuoco, 2008)that may be exploited in various mathematical explorations leading to the generation of conjectures.

When such way of thinking is developed, the abductive reasoning has the advantage of involvinggeometrical concepts, like in the case of Francesco and Gianni. The geometrical concepts that emerge inthis case can become “bridging  elements” with respect to the proving phase, since they can be re-elaborated into the deductive steps of a proof. On the other hand, expert use of the MDS supported byMD seems to lead to conjectures in which no geometrical elements arise to “bridge the gap” between thepremise and the conclusion. In other words, although expert use of MD seems to offer the possibility of generating “powerful” conjectures that solvers might have trouble reaching without support of thedragging instrument (since the IOD which becomes the premise may be cognitively “quite distant” fromthe conclusion), generating conjectures “mechanically” through the MDS associated to the dragginginstrument, may hinder the proving phase in which these “bridging elements” are essential.

Finally, discussing whether “giving” students the tool of MD (with the associated utilization scheme, theMDS) is appropriate or not in the classroom setting is beyond our present scope, however it seems to bean important issue to consider in the near future.

The Notion of Instrumented AbductionConsider the following open-problem activity:

“Construct three points A, B, and C on the screen, the line through Aand B, and the line through AandC. Then construct the parallel line l to AB through C, and the perpendicular line to l through B. Call thepoint of intersection of these last two lines D. Consider the quadrilateral ABCD. Make conjectures onthe kinds of quadrilaterals can it become, trying to describeall the ways it can become a particular kindof quadrilateral.” 

Unlike Ila and Val, expert solvers seem to withhold the key for  “making sense” of their findings, whichseems to be conceiving the IOD as a “cause” of the III within the phenomenology of the DGS, and theninterpreting such cause as a geometrical “condition” for the III to be verified. In other words, the solversestablish a causal relationship between the two invariants generating  – as Magnani says  – anexplanatory hypothesis for the observed phenomenon. Moreover, as soon as they decide to use MD toexplore the construction, experts seem to “search for a cause” of the III in terms of a regular movement

of the dragged-base-point. This idea is key; it seems to lie at a meta-level with respect to each specificinvestigation the solvers engage in, and possessing it seems to allow complete exploitation of the MDS,culminating in the formulation of the conjecture. Moreover, as mentioned above, the process of conjecture-generation through MDS seems to become “automatic” for expert solvers.However automatic use of the MDS seems to condense and hide the abductive process that occursduring the process of conjecture-generation in a specific exploration: the solver proceeds through stepsthat lead smoothly to t he discovery of invariants and to the generation of a conjecture, with no apparentabductive ruptures in the process. Thus our research seems to show that, for the expert, the abductivereasoning that previous research described as occurring within the dynamic exploration occurs at ameta-level and is concealed within the MD-instrument. We introduce the new notion of instrumentedabduction to refer to the inference the solver makes through the MDS, leading to the formulation of aconjecture. Below is a representation of the two levels that constitute the MDS described by our model.

Development of a Conjecture Using the

MDS

Intentionally Induced Invariant (III): Property (or configuration) that the solver chooses to try tomaintain during dragging.

Path: Set of points with the following property: when the selected-base-point is dragged along thisset, the intentionally induced invariant (III) is (visually) verified.

Geometric Description of the Path (GDP): Characterization of the figure-specific path as ageometrical object.

Invariant Observed During Dragging (IOD): Property (or configuration) that seems to be maintainedby the Cabri-figure while an intentionally induced invariant (III) is being induced through maintainingdragging.

Critical Link (CrL): (implicit) link of causality between the IOD and the III perceived within thephenomenology of the dynamic geometry system.

Conditional Link (CL): (implicit) logical connection between the IOD and the III, interpreted asgeometrical properties.

Conjecture : (explicit) statement with a premise and a conclusion that expresses the conditional link(CL).

Key Words of the MDS Model

During this episode there seems to be the intention of looking for something , which we interpret asan attempt at “making the path explicit”. This can lead to perceiving a second invariant, that we callthe invariant observed during dragging (IOD), perceived as the movement of the dragged-base-point along a GDP. Both invariants, the IOD and the III, are perceived within the phenomenological domainof the DGS, where a relationship of  “causality” may also be perceived between them. Our model

refers to this first relationship of causality as a Critical Link (CrL). Of course such relationship can beformulated within the domain of Euclidean Geometry as a Conditional Link (CL) between geometricalproperties corresponding to the invariants, provided that the solver gives an appropriate geometricalinterpretation.Below is a representation of how the elements of the model might come into play during anexploration leading to the generation of a conjecture.

James seemed to be looking for  something , which hedescribed as a “pretty precise curve”. This intention seemedto indicate that James had conceived an object along whichdragging the base point Awould guarantee that the III wouldbe visually verified. This is what we call a  path. Moreover hewas trying to “understand” what such path might be. In other words he was searching for a geometric description of the

 path (GDP). To do this he suggested activating the trace tool.

The solvers activated trace on A and Simon performed MDagain. James seemed to be searching for a GDP byinterpreting the trace mark left by Aon the screen.

Two solvers, James and Simon, followed the steps that led to theconstruction of ABCD and soon noticed that it could become arectangle. Simon was holding the mouse, and followed James’ suggestion to use MD to “see what happens” when trying to maintainthe property “ABCD rectangle” while dragging the base point A. In suchsituation the selected property “ABCD  rectangle”, according to our model is called intentionally induced invariant (III). As Simon wasfocused on performing MD, James’ attention seemed to shift to themovement of the dragged-base-point.

Val: Move Aon the circle.Ila: You look to check that it stays...Val: There, it remains, it remains a parallelogram.Val: Yes, I mean a parallelo...it remains a rectangle.Ila: a rectangle.Val: Yes, more or less.Ila: Yes, ok. But...Val: Ok....why?Ila: Because...Val: Why?Val: So...I know that, uh, soIla: But B has to always be in that point there.Val: Where?Val: So I think...this remains a rectangle...when AB is perpendicular to DC, ok but in this caseit would also be BA is equ, perpendicular to CA.

Making Sense of an Exploration Above we illustrated an example of  “expert  behavior” in using the MDS. However reaching suchbehavior is not trivial, as shown by the fact that many solvers we interviewed did not seem able to

make sense of their discoveries even when they appeared to be using MD in a way that seemed tolead to the perception of an III and an IOD. In particular, even when invariants are properly perceived(Baccaglini-Frank et al., 2009) it seems that their simultaneous perception does guarantee theinterpretation of such phenomenon in causal terms. Moreover, putting the geometrical propertieswhich correspond to the III and the IOD in relationship with each other within the world of Euclideangeometry is not always straightforward. The excerpt below shows a case in which two solvers haveused MD maintaining the property “ABCD rectangle” as their III, dragging A, they have provided aGDP and perceived the invariant “A on the circle” as an IOD. However they do not seem to makesense of what they have discovered.

The solvers constructed the circle and dragged Aalong it , and then they wrote the conjecture:“ABCD is a rectangle when A is on the circle with diameter BC.” 

We now take our reflections on the MDS one step further and consider expert use of the MD. We havefound evidence that experts may use the MDS as a “way of  thinking” freeing it from the physicaldragging-support. In the following excerpts we will show how the MDS guided the process of conjecture-generation of Francesco (F) and Gianni (G) even though they were not able to reach an IOD throughMD. The solvers were working on the following open-problem activity:“Draw a point P and a line r through P. Construct the perpendicular line l to r through P, construct a pointC on it, and construct the circle with center in C and radius CP. Construct the symmetric point of C withrespect to P and call it A. Draw a point D on the semi-plane defined by r that contains A, and construct

the line through D and P. Let B be the second intersection with the circle and the line through P and D.Consider the quadrilateral ABCD. Make conjectures on the kinds of quadrilaterals can it become, tryingto describe all the ways it can become a particular kind of quadrilateral.” 

G: and now what are we doing? Oh yes, for the parallelogram?F: yes [as he drags D with the traceactivated] yes, we are trying to see when itremains a parallelogram.G: yes, okay the usual circle comes out.F: wait, because here…oh dear! where is itgoing?… F: So, maybe it’s not necessarily the case thatD is on a circle so that ABCD is theparallelogram.… F: Because you see, if we then do a kind of circle starting from here, like this, it’s good it’sgood it’s good it’s good, and then here… see,if I go more or less along a circle that seemedgood, instead it’s no good…so when is it anygood?

The solvers had chosen “ABCD parallelogram” as their III.

G: eh, since this is a chord, it’s a chord right? We have to, it means that this has to be an equal chord of another circle, in my opinion with center in A. Because I think if you do, like, a circle with center F: A, you say…G: symmetric with respect to this one, you have to make it with center A.F: uh huhG: Do it!F: with center Aand radius AP?G: with center A and radius AP. I, I think… F: let’s move D. more or less… 

G: it looks right doesn’t it?  F: yes.G: Maybe we found it!

 Although the “search for a cause” through use of MD with t he trace activated failed, the solvers were ableto overcome the technical difficulties and reach a conjecture by conceiving a new GDP without help fromthe instrument. In other words the solvers seemed to have interiorized the instrument of MD to the extentthat it has become a psychological tool which no longer needs “external” support. Moreover the abductiveprocess supported by MD in the case of an instrumented abduction now occurs “internally” and issupported by the theory of Euclidean geometry (BP and PD are conceived as chords of symmetriccircles). Taking a Vygotskian perspective (Vygotsky, 1978, p. 52 ff.), we can say that the MD has beeninternalized and the actual use of the MD-artifact has been transformed into a  psychological tool ,becoming internally oriented.

Instrument : In the current study weconsider  “dragging” in a DGS after theinstrumentation approach (Rabardel &Samurçay, 2001). A particular way of dragging, in our case maintaining dragging (MD), may be considered anartifact that can be used to solve aparticular task (in our case that of formulating a conjecture). When theuser has developed particular utilizationschemes for the artifact, we say that ithas become an instrument for the user.We will call the utilization schemesdeveloped by the user in relation toparticular ways of dragging, “dragging schemes.” In particular we studied the

maintaining dragging scheme (MDS).

The solvers’ search for a condition as thebelonging of D to a curve, defined throughother base points of the construction, wasnow complete. They constructed the circlewith center in A and radius AP andproceeded to link D to it in order to checkthe CL. The solvers seemed quitesatisfied and formulated their conjectureimmediately after the dragging test,proceeding coherently with the MDSmodel.

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