Shaping a multidimensional analysis of signs

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Shaping a multi-dimensional analysis of signs
F. Arzarello, F. Ferrara, D. Paola,
O. Robutti, C. Sabena  
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FOCUS ON

• The relationships among
• gestures
• speech
• artefacts
• in communicating/thinking mathematics

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Two clips will show some episodes from the
activities carried out in the 11th grade of a
scientifically oriented high school in Italy.
These students are (early) introduced to the
fundamental concepts of Calculus since the
beginning of high school (9th grade).
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The students are used to work in small groups and
to participate to collective discussions
orchestrated by the teacher. They are also
accustomed to use technological devices, e.g.
sensors to investigate motion experiments,
symbolic calculators, and so on.
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THE TASK
Slope function
f(x) 0,5 x3 - 5x2 3
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GRAPHIC CALCULUS (by D. Tall)
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Specific question (asked by the teacher while
discussing with the students) Imagine that you
have not the red curve, but you see the tangent
while moving. Can you have information on the
concavity of the slope function?
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Simone
Teacher
Ciro
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• What relationships between gesture and speech?
• 
  
• Does the computer influence gesturing? How?

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CLIP 1
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CLIP 2
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2. Questions Discussion
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• What relationships between gesture and speech?
• 
  
• 2. Does the use of the computer influence
  gesturing?

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3. Comments
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• What relationships between gesture and speech?
• 
  
• 2. Does the use of the computer influence
  gesturing?

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1
13
3
CLIP 1
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CLIP 2
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Towards a theoretical framework

• Parallel and serial processes
• A semiotic analysis

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Parallel processes
!
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Serial processes
C we can say that the slope is going towards
zero degrees
!
SLOPE 0 JOIN NEAR POINTS

MATCH (redundant g.)
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Gestures and words achieve a coordination of
time, space, and movement leading to the social
objectification of abstract mathematical
spatial-temporal relationships. (L. Radford)
C we can say that the slope is going towards
zero degrees
!
SLOPE 0 JOIN NEAR POINTS

MATCH (redundant g.)
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The semiotic nature of gestures
Parallel processes illustrate aspects of
gestures, which are coherent with the analysis
made by psychologists for everyday gesturing
(e.g. the classification of McNeill,
Goldin-Meadow, e.g. matching mismatching). Seri
al processes frame a genetic aspect of gestures,
which depends on the essential and specific use
of signs and artefacts in mathematics.
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The semiotic nature of gestures
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IN SHORT
The mathematical meaning emerges gradually out of
repeated focusing on particular events
(movements, shapes,). It is generated through
students actions, gestures, words, interactions
(with the teacher, with the computer, with
peers), imaging in a dynamics of repeated
parallel and serial processes. 
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Thank you !
lim Df/Dx Dx?0