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PME 2014 Research Report by Anna Baccaglini-Frank, Samuele Antonini, Elisabetta Robotti and George Santi, talk delivered July 16, 2014, in Vancouver, Canada.
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Juggling Reference Frames in the Microworld Mak-‐Trace:
The Case of a Student with MLD Anna Baccaglini-Frank
Dip. di Educazione e Scienze Umane, Università di Modena e Reggio Emilia, Italy
Elisabetta Robotti
Dip. di Scienze Umane e Sociali, Università della Valle d’Aosta-Université de la Vallée d’Aoste, Italy
1
Samuele Antonini Università di Pavia, Italy
George Santi
I.I.S. “E. Majorana”, Bologna, Italy
Aim: to study aspects of the educaConal potenCal of Mak-‐Trace (free app for iPad and iPhone) for visuo-‐spaCal empowerment in cases of MLD. In parCcular: • To analyze processes in managing different systems of reference in the
student-‐arCfact interacCons. • To study aspects of the semioCc potenCal of Mak-‐Trace, in parCcular
with respect to developing awareness and becoming able to manage different systems of reference.
2
The Study
The study is part of the line of research on the use of arCfacts in processes of teaching and learning MathemaCcs in cases of
Learning DisabiliCes
Aim: to study aspects of the educaConal potenCal of Mak-‐Trace (free app for iPad and iPhone) for visuo-‐spaCal empowerment in cases of MLD. In parCcular: • To analyze processes in managing different systems of reference in the
student-‐arCfact interacCons. • To study aspects of the semioCc potenCal of Mak-‐Trace, in parCcular
to develop an awareness and ability to manage systems of reference different from one’s own.
3
The Study
The study is part of the line of research on the use of arCfacts in processes of teaching and learning MathemaCcs in cases of
Learning DisabiliCes
“...on the one hand, personal meanings are related to the use of the arCfact, in parCcular in relaCon to the aim of
accomplishing the task; on the other hand, mathemaCcal meanings may be related to the arCfact and its use. This double semioCc relaConship will be named the semio)c
poten)al of an ar)fact.” (Bartolini Bussi & MarioV, 2008, p.754)
Methodology: exploraCve case-‐study in mathemaCcs educaCon. CollaboraCon between teachers and university researchers: the teacher and researchers met weekly to adjust (and possibly design new) acCviCes. The acCviCes were conducted with a student diagnosed with severe LD in his regular school seVng, and they were video-‐recorded and analyzed weekly.
The Study
Case analysis Context : Filippo, 15 years old, is in the second year of high school (technical insCtute), he has been diagnosed with severe specific LDs.
The Study
According to recent studies, students with MLD may present different
mathemaCcal profiles, characterized by the presence of mulCple deficits including those of a visuo-‐spaCal
nature (Andersson & Östergren, 2012; Karagiannakis et al., 2014)
Case analysis Context : Filippo, 15 years old, was in the second year of high school (technical insCtute), he has been diagnosed with severe specific LDs. Filippo was helped 4 hours a week by a special educaCon teacher, whom he would work with individually, outside the classroom. The teacher had noCced that Filippo was not able to give or interpret direcCons (giving instrucCons to go from home to school).
The Study
his diagnosis: a comorbidity of severe dyslexia
and severe dyscalculia
Case analysis Context : Filippo, 15 years old, was in the second year of high school (technical insCtute), he has been diagnosed with severe specific LDs. Filippo was helped 4 hours a week by a special educaCon teacher, whom he would work with individually, outside the classroom. The teacher had noCced that Filippo was not able to give or interpret direcCons (giving instrucCons to go from home to school).
The Study
A working definiCon of visuo-‐spaCal abiliCes can be to perceive and operate in
physical space or on mental representaCons, based on
spaCal coordinates (Grossi & Trojano, 2002)
Case analysis Context : Filippo, 15 years old, was in the second year of high school (technical insCtute), he has been diagnosed with severe specific LDs. Filippo was helped 4 hours a week by a special educaCon teacher, whom he would work with individually, outside the classroom. The teacher had noCced that Filippo was not able to give or interpret direcCons (giving instrucCons to go from home to school).
The Study
Filippo seemed to have a Visual-‐SpaCal Disability.
This implies the neuro-‐cogniCve funcCons: • orientaCon of spaCal acenCon
• integraCon of visual-‐construcCve and visual-‐motor abiliCes
• visual-‐spaCal working memory; although a definiCon has not yet been agreed upon
(Rourke, 2002; Forrest, 2004)
Case analysis Context : Filippo, 15 years old, was in the second year of high school (technical insCtute), he has been diagnosed with severe specific LDs. Filippo was helped 4 hours a week by a special educaCon teacher, whom he would work with individually, outside the classroom. The teacher had noCced that Filippo was not able to give or interpret direcCons (giving instrucCons to go from home to school).
The Study
From the “Indicazioni Nazionali per il I ciclo” (MIUR, 2012, p. 50) Learning Goals for the end of grade III of primary school
Space and figures -‐ Communicate the posiCon of objects in physical space, both with respect to the subject, and with respect to other people or objects […]
-‐ Execute a simple path starCng from its verbal descripCon or from a drawing, describe a path that you are execuCng and give direcCons to some one for them to execute the same path.
10
Mak-‐Trace
!
immediate feedback of a programmed sequence is not
given
A first analysis: Filippo’s difficulCes became apparent in: -‐ considering systems of reference for spaCal relaCons independent of the self (allocentric)
-‐ transiConing between representaCon with reference to the self (egocentric) to reference external of the body (allocentric)
But …
11
We needed a more refined framework to elaborate a fine grained analysis of the difficulCes observed
Looking for a tool of analysis
12
Tool of analysis
Grush (2000) (cogniCve psychology) divided “allocentric” into: (A) egocentric space with a non-‐ego object reference point (decentred egocentric); (B) object-‐centred reference frames; (C) virtual points of view (i.e., maps); (D) “nemocentric” maps.
13
Points of view
Lucia Giovanni
14
Points of view
EGOCENTRIC: Lucia is in front
of me
15
Points of view
EGOCENTRIC DECENTERED (type A):
Lucia is to the right of Giovanni
16
Points of view
ALLOCENTRIC (type B): Lucia is to
Giovanni’s len
Three of the tasks
17
1) program the snail to draw a given path;
2) program the snail to draw a
square; 3) complete the mazes.
ExploraCon of the meaning of the command-‐icons
Task 1 Program the snail to draw a given path Filippo: it is a bit hard. It’s never what it… […] I am not
understanding anything […] wait…I didn’t tell him to go right and he went right. […] These two [R and L] are inverted […] I am not understanding anything […] if this arrow [L] makes it turn right, this one [R] makes it turn len.
Filippo: Every three squares, every three..., every three squares, every three down arrows, the fourth makes it go up...I don’t know. […] if we look only at these arrows here, you can’t really understand much.
Teacher: why does an arrow poinCng to the len make it go to the right?
Filippo: Ask the person who designed the game!
19
(B) object-‐centred reference frames It is necessary to consider a snail-‐fixed frame of reference BOTH at the origin AND in terms of orientaCon
(A) decentred egocentric frame
Teacher: ... This is the key-‐word [...] so what do the arrows refer to?
Filippo: it went backwards, not
upwards […]
Filippo: ... it depends on how the snail is oriented.
21
(B) object-‐centred reference frames
(A) decentred egocentric frame
Filippo constantly bounces
back and forth
22
Filippo correctly writes a sequence of commands to
make the snail follow a predefined track. We observe that the track is made up of horizontal
lines and verCcal lines “upward”
(the snail is never oppositely oriented)
Three of the tasks
23
1) program the snail to draw a given path;
2) program the snail to draw a
square; 3) complete the mazes.
(the snail is pointing upwards) Filippo constructs the sequence of command icons FFFFLFFFFL [short pause, he says “yes”]
Task 2 Program the snail to draw a square:
Acempt 1
L R F
B
He seems to be able to pass to an ALLOCENTRIC
perspecCve (type B)
(the snail is pointing upwards) Filippo constructs the sequence of command icons FFFFLFFFFL [short pause, he says “yes”] B [short pause] BBBR [long pause] FFFF.
Task 2 Program the snail to draw a square:
Acempt 1
L R F
B
He seems to be able to pass to an ALLOCENTRIC
perspecCve (type B)
He gets lost and goes back to the
egocentric decentered perspecCve
Filippo: he inserts the commands FFFFLFFFFL
[short pause, he inserts S, he deletes it, says “eh, alright, it’s enough to make it turn like this”, with his right index he draws a len turn in the air] FFFF [he says “I have to always keep” he gestures an anCclockwise rotaCon with his right hand] RFFFF.
Task 2: Acempt 2 Filippo corrects the third side but he makes a mistake on the turn when the snail-‐fixed system is oppositely oriented with respect to his own:
29
Task 2 Program the snail to draw a square:
Acempt 3
30
Filippo starts programming again from the beginning FFFFLFFFFL [he gestures an anCclockwise rotaCon with his right hand corresponding to the len turn that he inserts] LFFFF [he rotates the iPad to see the snail’s perspecCve] “There, I found it”… “I got…I get lost” “When it’s turned around…it goes backwards [clockwise rotaCon gesture with his right hand] so…if I want it to go here [gesture from len to right with len hand]” “uh, I don’t know, let’s try like this” RFFFF (he looks at the sequence) “No wait because otherwise it’s like before” he changes the R with a L [second correcCon without tesCng]
31
Filippo starts programming again from the beginning FFFFLFFFFL [he gestures an anCclockwise rotaCon with his right hand corresponding to the len turn that he inserts] LFFFF [he rotates the iPad to see the snail’s perspecCve] “There, I found it”… “I got…I get lost” “When it’s turned around…it goes backwards [clockwise rotaCon gesture with his right hand] so…if I want it to go here [gesture from len to right with len hand]” “uh, I don’t know, let’s try like this” RFFFF (he looks at the sequence) “No wait because otherwise it’s like before” he changes the R with a L [second correcCon without tesCng]
Awareness of the necessity to embrace the snail-‐fixed system (ALLOCENTRIC, type B)
compensatory strategy
compensatory strategy
• Program the snail to re-‐trace going backwards the sequence it traced going forwards.
• Dictate sequences to the teacher with a more rapid notaCon (for ex. 3f r 2f r 3f l ...) and use of this notaCon to compare sequences;
• Formulate a general rule such that, given any path, it allows to find the inverse path.
• AcCviCes with mazes and con macros. 32
Later acCviCes
Three of the tasks
33
1) program the snail to draw a given path;
2) program the snail to draw a
square; 3) complete the mazes.
34
Task 3 compensatory
strategy
Conclusion
36
(B) object-‐centred reference frames
to recognizing this perspecCve and trying to embrace it
from not being able to perceive the snail’s perspecCve, and trying to find a way of making sense of the command icons
The student’s confusion seemed to depend on his simultaneous use of the two incompaCble frames.
(A) decentred egocentric frame
37
(B) object-‐centred reference frames
to recognizing this perspecCve and trying to embrace it
from not being able to perceive the snail’s perspecCve, and trying to find a way of making sense of the command icons
The student’s confusion seemed to depend on his simultaneous use of the two incompaCble frames.
(A) decentred egocentric frame
In the end the student is aware that he can change frames of reference by mentally trying to “be the snail”
38
This occurred thanks to: • the specific tasks proposed, • the intervenCons of the teacher • the funcConaliCes of Mak-‐Trace, which required conCnuous juggling between two reference frames.
right for whom? ...the snail turns
refer to...
!
the commands are icons that can be spontaneously
interpreted in the egocentric frame
but that refer to the snail-‐fixed frame
immediate feedback of a programmed sequence is not
given
39
• resorCng to gestures that bridge one reference frame to the other.
compensatory strategies
• trial and error (since there are two
choices for the turns)
• trying to define a rule without embracing the snail’s perspecCve (when the snail is upside down everything is opposite)
• rotaCng the iPad or swivelling a pencil pretending it was the snail (changing his own perspecCve )
Mak-‐Trace appeared to help the student to “remain absorbed in a task for a period of Cme; … tolerate a period of confusion (with appropriate support);… use errors as a source of informaCon about what to try next”
(Russell, 1986, p. 103)