Digital representations of mathematical objects in the teaching-learning process:
a cross European research project
Jean-baptiste LagrangeLaboratoire de Didactique André Revuz
Université Paris-Diderothttp://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
20mn25mn10mn
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 responsibility1987: 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 “Technology-enhanced 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 Objectives1. To bridge the gap between technology and pedagogy2. 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.
MethodologyA cyclical process of
a) desiging and developing six state-of-the-art 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
UNISI Didirem
CruisletCasyopée
ETLalienalien familiar familiar alien
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 activitiesEnactive-Iconic Algebraic
Local Global Generational Transforma. GlobalMeta
levels
Covariation and dependency in a physical system__________Covariation and dependency between magnitudes or measures__________ Mathemati-cal Functions of one real variable
Small moves.Observing effect on elements________Small moves.Observing effect on values________Tracing graphs Browsing Tables
Moving elements Observing transfor-mations________ Graphs of measure against time or another magnitude________Perceiving properties of graphs and tables
________ Building pre-algebraic “geometrical” formula.Choosing an independent variable.___________Expressing algebraically a domain and a formula
___________Computing, recognising equivalent expressions.
Choosing an appropriate form
Considering ‘generic’ objects and measures.
Interpreting
___________Working on ‘families’ of functions
Parameters (animated or formal).Proving
PME 33
Casyopée
a, b, c, 3 parameters >0A(-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]
An optimisation problemCasyopée
Enactive-Iconic Algebraic Generational
Magnitudes Moving elements Observing effects on values
Building pre-algebraic “geometrical” formula.
CONNECTIONS BETWEEN ACTIVITIES
Casyopée
Algebraic Generational
Magnitudes Choosing an independentvariable
Mathematical Function
Expressing algebraically a domain and a formula
CONNECTIONS BETWEEN ACTIVITIES
Casyopée
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
CONNECTIONS BETWEEN ACTIVITIES
Casyopée
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
Theory of Didactical Situations
Anthropo. Theory of Didiactics
Background on
Functions
Semiotic Register
Theory of Semiotic
MediationsInstrumental Approach
DIDIREM
DIDIREM and UNISI Theoretical frameworks
ActivityTheory
UNISI
Casyopée cross-case study
DIDIREM
Anthropological theory of didactics (ATD) Ecological
perspective Sensibility to
institutional constraints and norms
Attention to (instrumented) techniques
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
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.
The DIDIREM scenarioThe UNISI scenario
Casyopée cross-case study
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?
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
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
Two scenarios giving high importance to the students’ autonomous work. A characteristic transcending the theoretical
diversity. But a different balance between
individual/group work and collective work, and a very different management of collective phases.
The evident influence of differences in theoretical frameworks.
Casyopée cross-case study
Similarities and differences
Summarizing the cross case study
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
• 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
Casyopée cross-case study
Context
DIDIREM difficulties with Cruislet
The DIDIREM culture
Controlled design
Anticipating the potential and limit of adidactic adaptations
Cruislet Characteristics
Epistemological concern
AAnticipating possible cognitive
outcomes
Implementation of mathematical
objects
Instrumental sensitivity
Technological distance
Curricular distance
Cruislet cross-case analysis
Context
Conclusion: Beyond ReMathDifferent 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 studyunderstanding the consistency of ‘alien’ choicesawareness of the crucial role of context in didactical
research, and the need for better conceptualizationAn inspiration for (young) researchers?