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?