STUDENTS’ GAZES:
NEW INSIGHTS
INTO STUDENTS’
INTERACTIONS
PETER LILJEDAHL - SIMON FRASER UNIVERSITY (CANADA)
CHIARA ANDRÀ - UNIVERSITY OF TORINO (ITALY)
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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!
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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.
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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”
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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
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SOCIO-MATHEMATICAL
INTERACTION
WE EXPLORE THE ROLE OF EMOTIONS,
MOTIVATION, AS WELL AS KNOWLEDGE AND
LEARNING VISIBLE, AND INVISIBLE, WITHIN THE
GROUP INTERACTION
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THE TRANSCRIPT
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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?
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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
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INTERACTIVE FLOWCHART
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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?
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FLOWCHART + GAZES
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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 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 DAVIDE IS
ASKING A QUESTION
• 00:36
WHILE MARCO IS LOOKING AT THE PAPER
WHY IS MARCO SO INTENT ON LUCA?
WHY IS LUCA IGNORING MARCO?
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FLOWCHART + INTENSITY GAZES
casual glances
intense and longer gazes (stares)
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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
MARCO STARES INTENTLY AT LUCA
• 00:42
MARCO 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?
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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
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FICTIONAL WRITING
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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
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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
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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
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• 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'.
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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-PONTY 2002
THANK YOU
liljedahl@sfu.ca
chiara.andra@gmail.com
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