Dragging in dynamic geometry: from a utilization scheme to a psychological tool

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Dragging in dynamic geometry:from a utilization scheme to a

psychological tool

Anna Baccaglini-Frank

[email protected]



...so what will I be talking about?

• it will be related to dynamic geometry

• “dragging” will be looked at from different

theoretical perspectives

– the instrumental approach (Rabardel)

– a Vygotskian lens

• examples of choosing and developing

framework elements for “seeing” what one

wants• some findings from the study for my

dissertation



structure I chose for today

• presentation of the problem

• and of the framework I started from and

elaboration of theoretical constructs to build a

lens for “seeing” what I wanted

• presentation of a model for conjecture-generation in dynamic geometry

• proposal of a new theoretical construct (still

needs to be generalized)

• surprising episode and new hypothesis withadditional framework elements.



Conjecturingand Proving

in Geometry

Open Problems

Dynamic GeometrySystems

Focusing on “The Problem”




in Geometry

Open Problems


Conjecturing Phase:exploration and

argumentation

leading to astatement to be

validated

Focusing on “The Problem”



A

s c e n

d i n

g C

o n t r o l

D e s

c e n d i n g

C o n t r o

l

Wandering Dragging

Bound Dragging

Guided Dragging

Lieu Muet(or Dummy Locus) Dragging

Line Dragging

Linked Dragging

Dragging Test

Arzarello et al.’s Cognitive Analysis of Dragging



ABDUCTION

guiding the

transition

It is the different

control to change

the relationshipsamong the

geometrical objects,

both in the way they

are ‘drawn’ and in the way

they are

‘seen.’

“...[abduction] Themost delicate

cognitive point...”(Arzarello et al.,

1998)

The transition from the ascending

to the descending control is promoted by abduction, which putson the table all the ingredients of

the conditional statements

Arzarello et Al.’s Cognitive Analysis of Dragging



A

s c e n

d i n

g C

o n t r o l

D e s

c e n d i n g

C o n t r o

l

Wandering Dragging

Bound Dragging

Guided Dragging


Line Dragging

Linked Dragging

Dragging Test

abduction conjecture

??

?



A

s c e n

d i n

g C

o n t r o l

D e s

c e n d i n g

C o n t r o

l

Wandering Dragging

Bound Dragging

Guided Dragging


Line Dragging

Linked Dragging

Dragging Test

abduction conjecture

??

?

“The role of lieu muet in the dynamic of ascending/

descending control supports and helps students to

produce abductions and provokes the switching

between ascending and descending controlmodalities.” (p. 38)

(Arzarello, Micheletti, Olivero, Robutti, Paola, &

Gallino, 1998)



What is Dummy Locus Dragging ?

figure

dragging a base point to

maintain a property of the dynamic-figure

Maintaining Dragging



“...abduction looks at facts and looks for a theory to explain

them, but it can only say a "might be", because it has aprobabilistic nature. The general form of an abduction is:a fact A is observed;if C was true, then A would certainly be true;so, it is reasonable to assume C is true”(Peirce, 1960, p. 372).

Abduction

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. ...an abduction is: A and B, hence C

(Peirce called hypothesis the product of the abduction)



What is the product of the abduction in the case of

conjecture-generation?

Abduction in Conjecture-generation

A first problem

Peirce’s Example Conjecture-generation

Example

fact A is observed A is observed

rule if C was true then A would

be true

when C is observed/true

A is observed/true or if Cthen A (conjecture)

hypothesis C is true C is observed/true



What is the product of the abduction in the case of

conjecture-generation?

Abduction in Conjecture-generation

A first problem

Peirce’s Example Conjecture-generation

Example

fact A is observed A is observed

rule if C was true then A would

be true

when C is observed/true

A is observed/true or if Cthen A (conjecture)

hypothesis C is true C is observed/true

so the product can be a

conditional statement

containing the abductivehypothesis itself!!



“…an abductive inference may serve to

organize, reorganize and transform problem

solvers’ actions.” (Cifarelli, 1999) abduction...“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)




in Geometry

Open Problems



argumentation


validated




in Geometry

Open Problems



argumentation


validated

A r z a r e l l o

e t a l. ’ s

c o g n i t i v

e m o d e l




in Geometry

Open Problems



argumentation


validated

A r z a r e l l o

e t a l. ’ s

c o g n i t i v

e m o d e l



(Some) Research Questions

1) What relationship do the forms of reasoning (in

particular abduction) used by solvers during the

conjecturing stage of an open problem in a DGS, have

with the ways in which solvers use the dragging tool?

2) When a solver uses maintaining dragging in open

problems there seems to be a common process used

to generate conjectures. How can this be described?



I still needed to focus on solvers’ use of

maintaining dragging

so I did some homework, got good advice

and turned to...



The Instrumental Approach

it looks at an artifact with respect to...

• a task

• a solver

• the solver develops utilization schemes



The Artifact



Solver Task

Artifact



Solver Task

Artifact

UtilizationScheme



Solver Task

Artifact

UtilizationScheme




in Geometry

Open Problems



argumentation


validated

A r z a r e l l o

e t a l. ’ s

c o g n i t

i v e m o d e l



I used this framework

• to develop a model for conjecture-generation when maintaining dragging is

used by the solver

(that is, the description of a utilization

scheme for the instrument maintaining

dragging in a task of conjecture-

generation)

• to elaborate on in case something elseinteresting and related to my problem

“came up”.



Let A, M, K be three base-points, and construct B as the symmetric point of

A with respect to M, and construct C as the symmetric point of A with

respect to K. Construct the parallel line l to BC through A, and theperpendicular line r to l through C. Let D be the intersection of l and r .



Use of MD in the Exploration



Leading to These Conjectures

If K ∈ the perpendicular bisector of AB ⇒ ABCDrectangle

If M ∈

the circle with center Z (midpoint of AK)and radius AZ then ABCD rectangle

If A belongs to the ⊥ line to l through M ABCDrectangle



The MD-conjecturing Model

We can describe the process through 3 (implicit) tasks

and the different invariants the solver perceives.




Task 1: Determining an

Invariant to Induce Intentionally (III)

Determine a configuration to be explored by

inducing it as a (soft) invariant.

Through wandering dragging the solver can look for

interesting configurations and conceive them as

potential invariants to be intentionally induced (III).Intentionally Induced

Invariant (III)




Task 2: Searching for a Condition through MD

Look for a condition that makes the intentionally

induced invariant (III) be visually verified throughmaintaining dragging.

This can occur • through a geometric interpretation of the movement

of the dragged base point

• or through a geometric interpretation of the trace.







This can occur • through a geometric interpretation of the movement

of the dragged base point

• or through a geometric interpretation of the trace.

pathgeometric

interpretation

of the path

(GDP)







Invariant Observed during

Dragging (IOD)

Intentionally Induced

Invariant (III)

Conditional

Link

(CL)




Task 3: Checking a Conditional Link (CL)

between the Invariants

Verify the conditional link (CL) through the dragging test.

Possible subtasks:

• construct the GDP;

• perform soft dragging test ;

• perform robust dragging test constructing the IOD

robustly.

Express a Conjecture

(making the CL explicit)



III

task 1: determine aninteresting property to

intentionally induce

as an invarianttype of dragging:

wandering dragging

IODtask 2 (part 1): search for

a “condition” thatmakes the III visually

verified, as movementof a base point along

a pathtype of dragging:

maintaining dragging

and dragging withtrace activated

CLtask 2 (part 2): interpret

(within the world of geometry) the IOD as a

condition under which theIII is verified

through a

geometric

interpretation of

the trace or of

the movement

of the draggedbase point

CL checktask 3: verify the CL

through a dragging test

type of dragging: soft or

robust dragging test




So now we have a model that seems to well

describe what happens when a solver uses MD

for conjecture-generation...

The Model contains the description of a

possible utilization scheme for MD in tasks of conjecture-generation.

...what happened to the abduction?



The utilization

scheme allows thesolver to solve the

specific task

During the phase of

instrumental genesis thesolver seems to develop

“general” aspects of the

scheme transcendingspecific tasks

Proposing a new theoretical construct



Proposing a new theoretical construct

I nst r ume nt e d

abd uc t i o n



ActivityConstruct:

• a point P• a line r through P

• the perpendicular to r

through P

• a point C on the

perpendicular

• point A symmetric to C

wrt P

• a point D on the semiplane

opposite to C wrt r

• line through D and P

• circumf. center C, rad. CP

• B as 2nd intersection of the

circumference

with line through DP.

• quadrilateral ABCD

Interpretation of a surprising episode



MD as a Psychological ToolInterpretation of a surprising episode



Taking a Vygotskian perspective (Vygotsky, 1978, p. 52ff.), the process that was external, supported by the

MD-instrument in the case of instrumented abduction,now can be seen as “transformed” into an internal

process.

We can say that the MD-instrument has beeninternalized and it can now be used as a psychological

tool (Wertsch & Addison Stone, 1985; Kozulin, 1998)

to solve a conjecture-generation problem.

MD as a Psychological ToolInterpretation of a surprising episode




in Geometry

Open Problems



argumentation


validated

A r z a r e l l o

e t a l. ’ s

c o g n i t i v e

m o d e l



The Oral Proof A new hypothesis on conjecturing and proving



The Oral Proof • the circles are symmetric so AD is congruent to AP

which is congruent to PC and to therefore to BC;• the isosceles triangles APD and PBC are congruentbecause they have congruent angles,since the angle DPA is opposite at its

vertex to CPB;• therefore PD is congruent to PB,• so ABCD has diagonals thatintersect at their midpointsand therefore it is

a parallelogram.

A new hypothesis on conjecturing and proving



Hypothesis

Expert use of MD can become automatic andit seems to generate conjectures with a gap,while use of MD as a psychological tool

seems to generate conjectures with bridgingelements.

Therefore use of MD as a psychological toolmay foster the solver’s construction of aproof of the statement of his/herconjecture.

A new hypothesis on conjecturing and proving



Reasoning about the use of MD and fostering awareness of

the process of conjecture-generation achieved with its

support can be used by the teacher to trigger a process of semiotic mediation centered on the use of dragging with

respect to mathematical meanings like “premise”,

“conclusion”, “implication”, and “conjecture”.

With respect to the issue of teaching and learning proof ...

we advance a hypothesis:

• MD as a psychological tool geometrical concepts canbecome “bridging elements”

with respect to the proving

phase

• MD used “automatically” tends to lead to conjectures with

no geometrical elements to“bridge the gap” between the

premise and the conclusion.

There are various implications – here are a couple



Conjecturing

and Proving

in Geometry

Open Problems

Dynamic Geometry

Systems


argumentation


validated

A r z a r e l l o

e t a l. ’ s

c o g n i t i v e

m o d e l

Documents

Dragging in dynamic geometry: from a utilization scheme to a psychological tool