NEW FRAMEWORKS FOR LOOKING AT STUDENT INTERACTIONS PETER LILJEDAHL - SIMON FRASER UNIVERSITY (CANADA) CHIARA ANDRÀ - UNIVERSITY OF TORINO (ITALY)

AN INTRODUCTION Luca Fabio Davide Marco

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.

THE TRANSCRIPT HOW DO WE EXPLAIN THIS?

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 SOCIO- MATHEMATICAL INTERACTION

SOCIO-MATHEMATICAL INTERACTION 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)

SOCIO-MATHEMATICAL INTERACTION 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 GROUP INTERACTIONS ARE COMPLEX SOCIALLY AND AFFECTIVELY CHARGED ENVIRONMENTS SO, HOW DO WE CODIFY THIS?

THE TRANSCRIPT

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?

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 SOLID ARROWS ARE ON-TOPIC AND DASHED ARROWS ARE OFF-TOPIC

INTERACTIVE FLOWCHART

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?

FLOWCHART + GAZES where the speaker is looking where non-speaker is looking P paper (new interlocutor)

ANALYSIS 3.0 LUCA IS NOT BEING IGNORED BY MARCO 00:25 DAVIDE IS ASKING A QUESTION WHILE GAZING AT THE PAPER. BUT MARCO IS NOT LOOKING AT DAVIDE – HE ISLOOKING AT LUCA 00:27MARCO RESPONDS TO DAVIDE’S QUESTION WHILE HE CONTINUES TO LOOK AT LUCA 00:34MARCO RESPONDS TO DAVIDE’S QUESTION WHILE HE IS LOOKING AT LUCA LUCA IS IGNORING (AVOIDING) MARCO 00:15LUCA LOOKS AT MARCO WHILE REACTING TO HIM 00:25 LUCA LOOKS AT MARCO WHILE DAVIDE IS ASKING A QUESTION 00:36WHILE MARCO IS LOOKING AT THE PAPER WHY IS MARCO SO INTENT ON LUCA? WHY IS LUCA IGNORING MARCO?

FLOWCHART + INTENSITY GAZES casual glances intense and longer gazes (stares)

ANALYSIS 3.1 SOMETHING INTERESTING HAPPENING AT 00:25 – 00:45 00:25DAVIDE A QUESTION; LUCA LOOKS AT MARCO; MARCO STARES INTENTLY AT LUCA → LUCA LOOKS AWAY 00:34MARCO STARES INTENTLY AT LUCA 00:36LUCA GLANCES AT MARCO WHILE MARCO IS LOOKING AT THE PAPER 00:37MARCO STARES 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.

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

FICTIONAL WRITING

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

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

DISCUSSIONS DIFFERENT CODIFICATIONS OF THE DATA ALLOW US TO SEE DIFFERENT THINGS IN THE DATA: 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

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'.

THANK YOU