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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.

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Synchronizing gestures, words and actions in pattern generalizations

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  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?

  3. Perception What does it mean to perceive something? An historical example: The Platypus How to interpret “this something”?

  4. 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

  5. Our focus How is the process of perceptual semiosis accomplished by the students engaged in pattern generalizations? language gestures Phenomenology of learning actions ...

  6. 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

  7. The activity grade 9 • Observe the following pattern: • Draw Figures 4 and 5; • How many circles will Figure 10 have? • And Figure 100?

  8. 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

  9. The data microanalysis-1° Video

  10. Jay Rita 1. RITA: You have five here… Mimi

  11. Jay Rita 1. RITA: You have five here… Mimi DEICTIC GESTURES + LANGUAGE: qualitative and quantitative way to apprehend the figure

  12. 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

  13. 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

  14. 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.

  15. 5. MIMI: But you must go six on the bottom…and five on the top M's words synchrony J's action V

  16. …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.

  17. The data microanalysis-2° Video

  18. 7. JAY: Ok. Figure 4 has five on top, right? 8. MIMI: Yeah… 9. JAY: … and it has six on the bottom

  19. 7. JAY: Ok. Figure 4 has five on top, right? 8. MIMI: Yeah… 9. JAY: … and it has six on the bottom

  20. 10. MIMI: Oh yeah. Figure 10 would have … synchrony 12. MIMI: There would be elevenand there would be tenright?

  21. 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

  22. 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

  23. 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

  24. 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

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

  26. Signs synchronization 12. MIMI: There would be elevenand there would be ten, right? synchrony inter-personal intra-personal

  27. Signs evolution Referring to fig 4 Referring to fig 10 Referring to fig 100

  28. Signs evolution Referring to fig 4 Referring to fig 10 Referring to fig 100 Gestures: Existential signification Imaginative signification

  29. 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

  30. Signs evolution Referring to fig 4 Referring to fig 10 Referring to fig 100 Simplification: - Loss of movement - Shortening of duration objectifying iconics

  31. 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

  32. 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

  33. 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

  34. 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)

  35. 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

  36. 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

  37. 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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