Synchronizing gestures, words and actions in pattern generalizations

1
Synchronizing gestures, words and actions in
pattern generalizations
Cristina Sabena, Luis Radford, Caroline Bardini
Laurentian University (Canada)
Research founded by the Social Sciences and
Humanities Research Council of Canada (SSHRC/CRSH)
2
Generalization of patterns
predicates something that holds for all the
elements of a class based on the study of a few
of them
What is it which enables the generalization to be
accomplished? What is that process that allows
the students to see the general through/in the
particular?
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Perception
What does it mean to perceive something?
An historical example
The Platypus
How to interpret this something?
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Perception
What does it mean to perceive something?
Perception as an active ongoing process of
adjustments and refinements
Perception as significantly dependent on the use
of signs
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Perception
What does it mean to perceive something?
To perceive something means to endow it with
meaning, to subsume it in a general frame that
makes the object of perception recognizable
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Our focus
How is the process of perceptual semiosis
accomplished by the students engaged in pattern
generalizations?
language
gestures
Phenomenology of learning
actions
...
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Methodology

• 6-year longitudinal study
• classoroom activities (regular teaching
  lessons)
• small groups work
• classroom discussions (teacher)

• written material (activity sheets, tests)
• video-tapes
• transcripts

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The activity
grade 9

• Observe the following pattern
• Draw Figures 4 and 5
• How many circles will Figure 10 have?
• And Figure 100?

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The data microanalysis
Jay
Mimi
They begin counting the number of circles in
the figures, and realize that it increases by
two each time. Now, Jay is about to draw figure
4
Rita
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The data microanalysis-1 Video
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Jay
Rita
1. RITA You have five here
Mimi
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Jay
Rita
1. RITA You have five here
Mimi
DEICTIC GESTURES LANGUAGE qualitative and
quantitative way to apprehend the figure
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Jay
Rita
1. RITA You have five here
Mimi
2. MIMI So, yeah, you have five on top and six
on the...
3. JAY Why are you putting...? Oh yeah, yeah,
there will be eleven, I think (He starts drawing
figure 4)
4. RITA Yep
5. MIMI But you must go six on the bottom and
five on the top
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Jay
Rita
1. RITA You have five here
Mimi
2. MIMI So, yeah, you have five on top and six
on the...
scheme of counting
3. JAY Why are you putting...? Oh yeah, yeah,
there will be eleven, I think (He starts drawing
figure 4)
4. RITA Yep
5. MIMI But you must go six on the bottom and
five on the top
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Jay
Rita
1. RITA You have five here
Mimi
2. MIMI So, yeah, you have five on top and six
on the...
DEICTIC GESTURE 1) participating in
the drawing process, to offer guidance 2)
depicting the spatial position of the rows in an
iconic way 3) clarifying the reference
of the uttered words.
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5. MIMI But you must go six on the bottomand
five on the top
M's words
synchrony
J's action
V
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the group work is interrupted While Mimi and
Rita pay attention to the announcement, Jay keeps
on working, writing 23 and 203 as the
answers for the number of circles in figures 10
and 100...
Jay
Rita
Mimi
6. Mimi (to Jay) I just want to know how you
figured it out.
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The data microanalysis-2 Video
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7. JAY Ok. Figure 4 has five on top, right?
8. MIMI Yeah
9. JAY and it has six on the bottom
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7. JAY Ok. Figure 4 has five on top, right?
8. MIMI Yeah
9. JAY and it has six on the bottom
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10. MIMI Oh yeah. Figure 10 would have
synchrony
12. MIMI There would be eleven and there would
be ten right?
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GESTURES
visual geometrical analogical
10. MIMI Oh yeah. Figure 10 would have
Two aspects of the problem
synchrony
12. MIMI There would be eleven and there would
be ten right?
LANGUAGE
numerical discrete linear
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GESTURES
TOPOLOGICAL/ ANALOGICAL MEANING
visual geometrical analogical
10. MIMI Oh yeah. Figure 10 would have
Two aspects of the problem
Two types of meaning-making
synchrony
Lemke (2003)
12. MIMI There would be eleven and there would
be ten right?
LANGUAGE
TYPOLOGICAL MEANING
numerical discrete linear
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through signs of different sorts (words,
gestures, rhythm, drawings, ), the students are
making apparent key traits of figure 100 a
figure that is not directly perceivable
10. MIMI Oh yeah. Figure 10 would have
Two aspects of the problem
Two types of meaning-making
12a
12b
Lemke (2003)
synchrony
12. MIMI There would be eleven and there would
be ten right?
TYPOLOGICAL MEANING
LANGUAGE
knowledge objectification
numerical discrete linear
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through signs of different sorts (words,
gestures, rhythm, drawings, ), the students are
making apparent key traits of figure 100 a
figure that is not directly perceivable
10. MIMI Oh yeah. Figure 10 would have
Two aspects of the problem
Two types of meaning-making
12a
12b
Lemke (2003)
synchrony
semiotic node
12. MIMI There would be eleven and there would
be ten right?
TYPOLOGICAL MEANING
LANGUAGE
knowledge objectification
numerical discrete linear
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Signs synchronization
12. MIMI There would be eleven and there would
be ten, right?
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Signs synchronization
12. MIMI There would be eleven and there would
be ten, right?
synchrony
inter-personal
intra-personal
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Signs evolution
Referring to fig 4
Referring to fig 100
Referring to fig 10
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Signs evolution
Referring to fig 4
Referring to fig 100
Referring to fig 10
Gestures
Existential signification
Imaginative signification
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Signs evolution
Referring to fig 4
Referring to fig 100
Referring to fig 10
Gestures that mime or iconize the
referent, pinpointing and depicting in an iconic
way the essential features of the new referent
objectifying iconics
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Signs evolution
Referring to fig 4
Referring to fig 100
Referring to fig 10
Simplification - Loss of movement
- Shortening of duration
objectifying iconics
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Signs evolution
12. MIMI There would be eleven (quick gesture
that points to the air) and there would be ten
(same quick gesture but higher up) right?
13. JAY Eleven (similar gesture but more
evident, with the whole hand) and twelve (same
gesture but lower).
14. MIMI Eleven and twelve. So it would make
twenty-three, yeah.
15. JAY 100 would have one-hundred and one and
one-hundred and two (same gestures as the
previous ones, but in the space in front of his
face).
Simplification - Loss of movement
- Shortening of duration
objectifying iconics
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Signs evolution
12. MIMI There would be eleven (quick gesture
that points to the air) and there would be ten
(same quick gesture but higher up) right?
deictic terms disappear
13. JAY Eleven (similar gesture but more
evident, with the whole hand) and twelve (same
gesture but lower).
14. MIMI Eleven and twelve. So it would make
twenty-three, yeah.
15. JAY 100 would have one-hundred and one and
one-hundred and two (same gestures as the
previous ones, but in the space in front of his
face).
Simplification - Loss of movement
- Shortening of duration
objectifying iconics
V
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Some conclusions
the process of perceptual semiosis was underlined
by two kinds of meanings
TYPOLOGICAL MEANING
quantitas LANGUAGE
TOPOLOGICAL/ ANALOGICAL MEANING
qualitas GESTURES
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Conclusions
phenomenological import of the diverse semiotic
means of objectification to which
the students made recourse in transcending the
particular
signs synchrony
inter-personal
intra-personal
objectifying iconics
TYPOLOGICAL MEANING
TOPOLOGICAL/ ANALOGICAL MEANING
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Synchronizing gestures, words and actions in
pattern generalizations
Thank you!
Cristina Sabena, Luis Radford, Caroline Bardini
Laurentian University (Canada)
Research founded by the Social Sciences and
Humanities Research Council of Canada (SSHRC/CRSH)
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Implications for future research

• dynamic of the semiotic node
• - in generalizations
• - in other domains of mathematics

• scope and role of objectifying iconics

• role of the synchronizations in the case the
  teacher is interacting with the students

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Relevance
From an educational viewpoint, what can be gained
by formulating and studying the problem of
generalization in this way?

• Mathematical thinking is much more rich than just
  writing
• the students mathematical thinking cannot be
  fully captured by paying attention only to what
  the students write (e.g. their formulas)
• in order to think mathematically, the students
  use, in fundamental ways, other semiotic systems
  that show the embodied component of mathematical
  thinking

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Research Question
Perception is continuously refined through signs
How is the process of perceptual semiosis
accomplished by the students engaged in pattern
generalizations?