Synchronizing gestures words and actions in pattern generalizations
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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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

The activity

grade 9

  • Observe the following pattern:

  • Draw Figures 4 and 5;

  • How many circles will Figure 10 have?

  • And Figure 100?


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

The data microanalysis-1° Video


Synchronizing gestures words and actions in pattern generalizations

Jay

Rita

1. RITA: You have five here…

Mimi


Synchronizing gestures words and actions in pattern generalizations

Jay

Rita

1. RITA: You have five here…

Mimi

DEICTIC GESTURES + LANGUAGE:

qualitative and quantitative

way to apprehend the figure


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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.


Synchronizing gestures words and actions in pattern generalizations

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

M's words

synchrony

J's action

V


Synchronizing gestures words and actions in pattern generalizations

…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.


Synchronizing gestures words and actions in pattern generalizations

The data microanalysis-2° Video


Synchronizing gestures words and actions in pattern generalizations

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

8. MIMI: Yeah…

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


Synchronizing gestures words and actions in pattern generalizations

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

8. MIMI: Yeah…

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


Synchronizing gestures words and actions in pattern generalizations

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

synchrony

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

Signs evolution

Referring to fig 4

Referring to fig 10

Referring to fig 100

Simplification:

- Loss of movement

- Shortening of duration

objectifying iconics


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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


Synchronizing gestures words and actions in pattern generalizations

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

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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