Digital representations of
mathematical objects in the teachinglearning process:
a cross European research
project
Jean-baptiste Lagrange
Laboratoire de Didactique André Revuz
Université Paris-Diderot
http://www.lar.univ-paris-diderot.fr
The ReMath project
 Representing Mathematics with digital
medias
 Communication, cooperation and
collaboration …for
 connecting ideas about representations
 networking theoretical frameworks
Plan
 ReMath: questions and working
plan
 The ReMath’s approach
 Cross case studies as a
methodology
 The Casyopée cross-case
 Presentation
 Small group work
 Report and Discussion
 Conclusion on ReMath
35mn
20mn
25mn
10mn
Representing Mathematics
with Digital Media
 STREP Number IST4-26751 (FP 6)
 42 months (Dec. 2005 - May 2009)
 Six teams






Instituto Technologie Didattiche, ITD Genova
Università degli Studi, UNISI Siena
National Kapodistrian University, ETL Athens
Institute of Education, IOE London
Université Joseph Fourier, Mehta Grenoble
Université Paris Diderot, Didirem Paris
FP6
 European research activities are
structured around consecutive
programmes, or so-called Framework
Programmes.
1984: First Framework Programme (1984-1987)
1987:“European Single Act” -science becomes a Community
responsibility
1987: SecondFramework Programme (1987-1991)
1990: Third Framework Programme (1990-1994)
1993:Treaty on European Union; role of RTD in the EU enlarged
1994: Fourth Framework Programme (1994-1998)
1998: FifthFramework Programme (1998-2002)
2002: SixthFramework Programme (2002-2006)
2007: Seventh Framework Programme (2007-2013)
The Sixth Framework Programme
(FP6) 2002-2006.
 the Priority – Information Society
Technologies
 IST 2005-06 Work Program.
 strategic objective 2.4.10 “Technologyenhanced learning (TEL)”
 To explore interactions between the learning of
the individual and that of the organisation …
 To contribute to new understandings of the
learning processes by exploring links between
human learning, cognition and technologies.
Key Objectives
1. To bridge the gap between technology and
pedagogy
2. A representations-based approach to cognition in
learning mathematics


we can only access and operate on Mathematical objects
by means of representations.
the potential impact of ICT tools on mathematical learning
seen through the filter of representations.
3. Support to teachers and learners
offering tools that address
 not only individual cognition
 but also the entire learning situation
4. Integration of efforts in the European context

how different theoretical frameworks deal with the
question of representations.
Methodology
A cyclical process of
a) desiging and developing six state-of-theart DYNAMIC DIGITAL ARTEFACTS for
representing mathematics,
b) developing scenarios for the use of these
artefacts for educational added value
c) carrying out empirical research involving
cross-experimentation in realistic
educational contexts
The project’s structure





WP1 : Theoretical integration
WP2 : Software developpement
WP3 : Scenarios (pedagogical plans)
WP4 : (cross) experimentations
WP5 : Multilingual repository and
communication platform (Math.Di.L.S.)
http://remath.cti.gr
D1 Integrated theoretical framework Version A
m6
D4 First Version of the Dynamic Digital Artefacts
m12
D7 Scenario Design, First Version
m15
D8 Release Version of Dynamic Digital Artefacts
m18
D9 Integrated theoretical framework Version B
m18
D10 Scenario Design, Refined Version
m18
D11 Research design
m20
D13 Design-Based Research: process and results
m30
D16 Refined Version of Dynamic Digital Artefacts
m33
D17 Scenario Design, Final Version
m36
D18 Integrated theoretical framework Version C
m36
DDAs: Digital Didactical
Artefacts
Diversity in ReMath: DDAs
 Domains and objects:
 algebra, functions, 3D geometry,
cinematic, geography…
 Representations
 connections between them,
 means of action
 possibilities of evolution
 Distance with
 usual systems of representation,
 usual software used in education,
 with the curriculum
Initial Diversity : Frameworks
ETL
Theoretical Integration
 Progressive elaboration of a shared
theoretical basis about representations
 Extension of the connections between
frameworks
 Specific common research tools
 Distinction between metaphoric and functional
use of theories
 The language of “concerns”
 The idea of “didactical functionalities”
(1)Tool features, (2) Educational goals, (3) Modalities
of employment
A special methodology:
the cross case studies
ETL
Didirem
UNISI
alien
familiar
Casyopée
alien
familiar
Cruislet
alien
Casyopée
Epistemological Profile
 Objects represented
mathematical function of one variable(families of)
 dependencies in a physical system (2D geometry)
 algebraic functions
 Curriculum compatible but innovative
 Connections and activities
 Crossing two entries
 Three different levels where functions can be
represented
 Two types of representations, with specific activities
Casyopée
Three different levels where covariation and
dependency can be experienced and/or
represented.
1. Physical systems (dynamic geometry)
2. Magnitudes and measures
3. Mathematical Functions
Casyopée
Representations and Types of activities
 Enactive-iconic Representations (Tall)
 Experience of movements inside physical systems
 Work on graphical or tabular representatives issued of
physical systems
 ‘Explorations’ on graphs and tables of mathematical
functions
 Algebraic Representations
 Semiotic Registers of representations (Duval 1999)
Treatments - Conversions
 Three categories of activities (Kieran 2004)
 generational
 transformational
 global / meta-level
Casyopée
Representations and Types of activities
Enactive-Iconic
levels
Covariation
and
dependency
in a physical
system
__________
Covariation
and
dependency
between
magnitudes
or measures
__________
Mathematical
Functions of
one real
variable
PME 33
Local
Global
Small
moves.
Observin
g effect
on
elements
________
Small
moves.
Observin
g effect
on values
________
Tracing
graphs
Browsing
Tables
Moving
elements
Observing
transformations
________
Graphs of
measure
against
time or
another
magnitude
________
Perceiving
properties
of graphs
and tables
Algebraic
Generational
________
Building prealgebraic
“geometrical
” formula.
Choosing an
independent
variable.
___________
Expressing
algebraically
a domain
and a
formula
Transforma.
GlobalMeta
Considering
‘generic’
objects and
measures.
Interpreting
___________
Computing,
recognising
equivalent
expressions.
Choosing an
appropriate
form
___________
Working on
‘families’ of
functions
Parameters
(animated or
formal).
Proving
Casyopée
An optimisation problem
a, b, c, 3 parameters >0
A(-a,0); B(0,b); C(c,0)
Find a rectangle MNPQ of maximal area
with M on [OA] ; Q on [OC] ; N on [AB] and P on [BC]
Casyopée
CONNECTIONS
BETWEEN
ACTIVITIES
Magnitudes
Enactive-Iconic
Algebraic
Generational
Moving elements
Observing effects on values
Building pre-algebraic
“geometrical” formula.
Casyopée
CONNECTIONS BETWEEN
ACTIVITIES
Algebraic
Generational
Magnitudes
Choosing an
independent
variable
Mathematical Expressing
Function algebraically a
domain and a
formula
Casyopée
CONNECTIONS BETWEEN
ACTIVITIES
Enactive-Iconic
Physical
system
Small moves
Observing effect on
objects
Magnitudes or
measures
Small moves
Observing effect on
values
Mathematical
Functions
Tracing graphs
Browsing Tables
Casyopée cross-case study
 Context in the two experiments
 A DDA innovative but highly compatible with the
curriculum.
 Close epistemological and didactic references
 Previous collaboration teachers/researchers
 Grade levels: Grade 11 (France), Grade 12 (Italy)
 Institutional pressure:
High (France)/Moderate (Italy).
 Teachers Familiar with DDA:
Yes (France)/No (Italy).
Casyopée cross-case study
DIDIREM and UNISI Theoretical frameworks
Theory of
Semiotic
Mediations
Activity
Theory
Instrumenta
l Approach
Semiotic
Register
Background
on
Functions
UNISI
Theory of
Didactical
Situations
DIDIREM
Anthropo.
Theory of
Didiactics
DIDIREM
 Theory of didactic
situations (TDS)
 Attention to
students’ a-didactic
interaction with the
milieu of the
situation.
 Careful choice of
tasks and control of
the didactic
variables
 Anthropological theory
of didactics (ATD)
 Ecological
perspective
 Sensibility to
institutional
constraints and
norms
 Attention to
(instrumented)
techniques
UNISI
 Theory of semiotic mediations (TSM)
 Gives much attention to the collective
progression of mathematics knowledge
 Through the progressive evolution of systems of
signs:
 students’ personal signs first linked to their
activity with the artifact
 shared during collective activities purposefully
designed,
 develop with the help of the teachers into
semiotic chains towards mathematical signs.
Casyopée cross-case study
The DIDIREM scenario
The UNISI scenario
Casyopée cross-case study
Questions:
1. What important similarities and differences
between the two scenarios? Hypotheses
about factors explaining these.
2. What research outcomes can be expected
from a cross-study with regard to:
(a) representations
(b) theoretical integration
(c) role of the context?
Casyopée cross-case study
Similarities and differences
 Two scenarios with slightly different educational
goals but favouring the same type of tasks:
 functions approached in terms of co-variation;
 functions approached as modeling tools for
problems arising in geometrical context.
 Two scenarios giving high importance to the
interaction between the different semiotic registers
offered by Casyopée.
The intertwined influence of differences in grade levels
and close epistemological views.
Casyopée cross-case study
Similarities and differences
 Two scenarios paying evident attention to the
process of instrumental genesis but managing it in
different ways
 in the UNISI scenario, an organization of the
instrumentalization process mainly concentrated in the
first session;
 in the DIDIREM scenario, a progressive organization of
the instrumentalization process along the whole
scenario.
The influence of a shared instrumental concern
combined with the differences induced by its
inscription into two different theoretical frames.
Casyopée cross-case study
Similarities and differences
Casyopée cross-case study
Summarizing the cross case study
• The Didirem team : several theoretical
frames.
• Attention to students’ instrumental
genesis
• Compatibility with institutional demand
• Process of learning designed through
a careful choice of mathematical
tasks, with an adidactical potential
• But the teacher's actions and role
escapes the PP’s design
Maracci M., Cazes C., Vandebrouck F., Mariotti M-A. (2009)
Casyopée in the classroom: two different theory-driven
pedagogical approaches, Proceedings of CERME 6
• The Unisi team has mainly
structured its pedagogical plan
according to the Theory of Semiotic
Mediation
• The teacher plays a crucial role
throughout the whole pedagogical
plan, especially for
• fostering the evolution of
students’ personal meanings
towards the targeted
mathematical meanings
• facilitating the students’
consciousness-raising of those
mathematical meanings
Context
Cruislet cross-case analysis
DIDIREM difficulties with Cruislet
Instrumental
sensitivity
Controlled design
The DIDIREM
culture
Epistemological
concern
Cruislet
Characteristics
Technological
distance
AAnticipating
possible cognitive
outcomes
Anticipating the
potential and limit of
adidactic adaptations
Curricular distance
Implementation
of mathematical
objects
Context
Conclusion: Beyond ReMath
Different conception of the theoretical work
connections based on concrete common practice on
“boundary objects” understanding
 the necessity of theoretical constructs
 their influence on tool and scenario design
Research practices as objects for study
 understanding the consistency of ‘alien’ choices
 awareness of the crucial role of context in didactical
research, and the need for better conceptualization
An inspiration for (young) researchers?