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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). Generalization of patterns.

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Synchronizing gestures words and actions in pattern generalizations
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)


Generalization of patterns
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?


Perception
Perception

What does it mean to perceive something?

An historical example:

The Platypus

How to interpret “this something”?


Perception1
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


Our focus

How is the process of perceptual semiosis accomplished by the students engaged in pattern generalizations?

language

gestures

Phenomenology of learning

actions

...


Methodology

DATA

Methodology

  • 6-year longitudinal study

  • classoroom activities (regular teaching lessons)

  • small groups work

  • classroom discussions (teacher)

  • written material (activity sheets, tests)

  • video-tapes

  • transcripts


The activity

grade 9

  • Observe the following pattern:

  • Draw Figures 4 and 5;

  • How many circles will Figure 10 have?

  • And Figure 100?


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



Jay

Rita

1. RITA: You have five here…

Mimi


Jay

Rita

1. RITA: You have five here…

Mimi

DEICTIC GESTURES + LANGUAGE:

qualitative and quantitative

way to apprehend the figure


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


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


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.


5. MIMI: But you must go six on the bottom…and five on the top

M's words

synchrony

J's action

V


…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.



7. JAY: Ok. Figure 4 has five on top, right?

8. MIMI: Yeah…

9. JAY: … and it has six on the bottom


7. JAY: Ok. Figure 4 has five on top, right?

8. MIMI: Yeah…

9. JAY: … and it has six on the bottom


10. MIMI: Oh yeah. Figure 10 would have …

synchrony

12. MIMI: There would be elevenand there would be tenright?


GESTURES

visual

geometrical

analogical

10. MIMI: Oh yeah. Figure 10 would have …

Two aspects

of the problem

synchrony

12. MIMI: There would be elevenand there would be tenright?

LANGUAGE

numerical

discrete

linear


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 elevenand there would be tenright?

LANGUAGE

TYPOLOGICAL

MEANING

numerical

discrete

linear


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 elevenand there would be tenright?

LANGUAGE

knowledge objectification

TYPOLOGICAL MEANING

numerical

discrete

linear


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 elevenand there would be tenright?

LANGUAGE

knowledge objectification

TYPOLOGICAL MEANING

numerical

discrete

linear


Signs synchronization
Signs synchronization

12. MIMI: There would be elevenand there would be ten, right?


Signs synchronization1
Signs synchronization

12. MIMI: There would be elevenand there would be ten, right?

synchrony

inter-personal

intra-personal


Signs evolution
Signs evolution

Referring to fig 4

Referring to fig 10

Referring to fig 100


Signs evolution1
Signs evolution

Referring to fig 4

Referring to fig 10

Referring to fig 100

Gestures:

Existential signification

Imaginative signification


Signs evolution

Referring to fig 4

Referring to fig 10

Referring to fig 100

Gestures that mime or

“iconize” the referent,

pinpointing and depicting

in an iconic way the

essential features of the

new referent

objectifying iconics


Signs evolution

Referring to fig 4

Referring to fig 10

Referring to fig 100

Simplification:

- Loss of movement

- Shortening of duration

objectifying iconics


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) andtwelve(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


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) andtwelve(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


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

TOPOLOGICAL/

ANALOGICAL

MEANING

TYPOLOGICAL

MEANING


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)


Implications for future research
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


Relevance
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


Research question
Research Question

Perception is continuously refined through signs

How is the process of perceptual semiosis accomplished by the students engaged in pattern generalizations?


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