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?

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 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?

FLOWCHART + INTENSITY GAZES

casual glances intense and longer gazes

(stares)

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.

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 YOUliljedahl@sfu.ca