STUDENTS’ GAZES: New Insights into Students’ Interactions

1. STUDENTS’ GAZES: New Insights into Students’Interactions Peter Liljedahl - Simon Fraser University (Canada) Chiara Andrà- University of Torino (Italy) P

2. the setting • Setting: • series of four lessons on probability: • grade 9 (14-15 year olds) • low socio-economic neighborhood in Bologna, Italy • Data gathered: • Probability task in the second lesson • Four students (Luca, Fabio, Davide, and Marco) videotaped by a grade 12 student • session lasted 50 minutes – first 5 minutes were transcribed • data selected for analysis: • first 45 seconds were selected • This subset of the data was selected because it exemplified individual intuitions within social interaction … but that is not what we did with it! C

3. the problem A robot walks along a corridor, it turns right with probability 1/3 and it turns left with probability 2/3. The map shows the labyrinth where the robot has to move. Compute the probability for the robot to be in each of the rooms. C

4. what we did with it! • some very interesting and turbulent undercurrents of group interactions • individual and social engagement with the mathematics and a social interaction around the mathematics • we wanted to codify this • and to analyze this • to understand better the undercurrents of “socio-mathematical interaction” P

5. socio-mathematical interaction • dialogue as scaffolded metacognition (Sfard, 2001) • but group interactions are complex socially and affectively charged environments • we need to overcome the dualistic approach between the individual’s interior space and his social interaction, and focus more on sociocultural conditions (Roth & Radford, 2011) • Learning occurs in and through relations with others driven by collectively motivated activity • activity is a process with inner contradictions, differentiations, transformations, as well as emotions—necessary for the activity and responsible of its development C

6. socio-mathematical interaction we explore the role of emotions, motivation, as well as knowledge and learning visible, and invisible, within the group interaction P

7. the transcript P

8. analysis 1.0 • students are making sense of the task. • Marco is dealing with fractions, he is interested in the procedure • Luca seems more interested in understanding the overall sense of the activity (“Why don’t we first compute how many probabilities are there in all?” 00.36) • Davide is still grasping the sense of the task (“What do we have to compute?” 00:28), and he is struggling to follow Marco’s reasoning (“Why?” 00:37) • Both Luca (00:11) and Marco (00:42) come to notice that the highest probability is related to the first room: • Luca – intuition • Marco – computation How is it that Marco does not see Luca’s contribution? C

9. interactive flowcharts • TWO TYPES OF SPEAKER’S META-DISCURSIVE INTENTIONS: THE WISH TO REACT TO A PREVIOUS CONTRIBUTION OF A PARTNER OR THE WISH TO EVOKE A RESPONSE IN ANOTHER INTERLOCUTOR • SFARD AND KIERAN, 2001 • A VERTICALLY OR DIAGONALLY UPWARD ARROW IS CALLED A REACTIVE ARROW AND POINTS TOWARDS A PREVIOUS UTTERANCE • A VERTICALLY OR DIAGONALLY DOWNWARD ARROW IS CALLED A PROACTIVE ARROW AND IT POINTS TOWARDS THE PERSON – OR PEOPLE (RYVE, 2006) – FROM WHOM A REACTION IS EXPECTED P

10. interactive flowchart P

11. analysis 2.0 • proactive statements: • Marco (n=7) • Luca (n=3) • Davide(n=0) • reactive statements: • Marco (n=5) • Davide (n=5) • Luca (n=1 not counting the self-talk as a reaction) • statements made that are reacted to: • Marco (n=6) • Davide(n=3) • Luca (n=1, not counting the self-talk) Luca and his solution are being ignored! … or is he? C

12. Flowchart + Gazes where the speaker is looking where non-speaker is looking P paper (new interlocutor) P

13. analysis 3.0 • Luca is NOT being ignored by marco • 00:25 Davideis asking a question while gazing at the paper. But Marco is not looking at Davide – he is looking at Luca • 00:27 Marco responds to Davide’s question while he continues to look at Luca • 00:34 Marco responds to Davide’s question while he is looking at Luca • Luca is ignoring (avoiding) Marco • 00:15 luca looks at marco while reacting to him • 00:25 luca looks at marco while Davideis asking a question • 00:36 while Marco is looking at the paper Why is Marco so intent on Luca? why is Luca ignoring Marco? C

14. flowchart + intensity gazes casual glances intense and longer gazes (stares) P

15. analysis 3.1 • something interesting happening at 00:25 – 00:45 • 00:25 Davide a question; Luca looks at marco;marco stares intently at luca→ luca looks away • 00:34 marco stares intently at luca • 00:36 luca glances at marco while marco is looking at the paper • 00:37 marcostares intently at luca • 00:42marco stares intently at luca there is an affective aspect to the interaction between Luca and Marco. There are emotions, efficacy, will, and motivation in how Luca and Marco are interacting with each other. but, what are the emotions, efficacy, will, and motivation? C

16. fictional writing Fictional writing is a technique that can help the researcher to go beyond the external and visible into the students’ inner subjective experience - Hannula, 2003 • envisioning the inner monologue of the student • creating likely impressions, and connections that do not exist in the original data • subjective in nature – but not wholly so • can help shed light on the students’ emotional disposition, attitudes and beliefs about mathematics good data + extensive analysis → inner monologue consistent with the empirical data P

17. fictional writing P

18. analysis 4.0 luca • feeling a sense of avoidance about fractions • tries to think on another level—a level that provides him with an overarching view of the task • Any time Marco uses fractions, Luca escapes • avoiding Marco’s gazes marco • has a procedural view of mathematics • is concerned mostly with computations with fractions—the whole sense of the task is to do computations • computation provides him with a sense of likely success—a sense of self-confidence and pleasure • pretends to have understood everything and spread his knowledge to his classmates davide • is aware that he is not a good student in math • has a willingness to understand C

19. analysis 4.0 Marco vs. Davide • Davide gives in to Marco luca vs. Marco • Luca is not prone to concede to Marco • differing views of mathematics • students’ gaze to each other but do not listen to each other • impede the interaction between Marco and Luca • Marco trying to catch luca’s attention and willing him to agree • luca trying to avoid marco and not willing to bend to him • they cannot really interact C

20. discussions this presentation has a rather methodological soul: we are exploring a set of methodologies to be able to see what is invisible at a first glance Transcript • observing the students’ utterances from a cognitive perspective • inferred their emotional dispositions • how knowledge emerges and is shared amongst the group is seen by means of the words the students say interactive flowchart (+ gazes) • looked at the students’ interactions through a behavioral lens and from an embodied mind paradigm • students’ gestures, postures and glances are seen as constitutive components of the meaning making process • the ideas that emerge from the activity are in their gestures and glances––to the point that if we discard these elements as we did at the beginning of the paper we miss many relevant facts fictional writing • provides a lens that helps us go deeper inside the students’ thoughts and will • in order to open a window on the students’ inner world it is necessary to repeatedly, patiently, and carefully look at their interactions, their words, and their postures P

21. conclusions • Knowledge and emotions are distributed throughout the students’ hands, eyes, mind, and body, and are inseparable from the development of the activity • Gazes give us insights into this inner world and allow us to write a version of the inner monologues of each participant • Other monologues can be constructed from the data just like other conclusions can be extracted from different analyses • Regardless of what monologues result, however, one thing is clear—the interactions between these four students have a turbulent undercurrent of emotions and intentions • The use of interactive flowcharts documenting the verbal interactions and the gazes gives a window into these emotions and intentions • Consciousness is in the first place not a matter of 'I think that' but of 'I can'. C

22. conclusions Consciousness is being-towards-the-thing through the intermediary of the body. To move the body is to allow oneself to respond to the call. - Merlau-Ponty2002

23. thank you liljedahl@sfu.ca chiara.andra@gmail.com P