Theoretical frames : development and evolution - the case of the French didactics

Michèle ArtigueUniversité Paris 7 Denis

Diderot

Summary Introduction Some characteristics of the French didactics

community that have influenced its relationship with theory

The birth of a systemic approach through the theories of didactic situations (TDS) and didactic transposition (TDT)

From the TDT to the anthropological approach (TAD): the integration of an institutional point of view

Some further developments of these theories: the evolution of concepts: medium and didactic contract ostensive and no-ostensive, mathematical and didactical

praxeologies Connections between theoretical frames

Introduction The essential role of theoretical frames in didactic

research A crucial issue: the diversity and heterogeneity of

the current theoretical landscape The interest of developing an « historical »

reflection, trying to understand the rationale for theoretical constructs, for their evolution, to look for possible connections between theoretical frames and to understanding also the limits of these connections

One essential aim of these two lectures: working on these issues through a particular case: the case of the French didactics

Some characteristics of the French didactics community A community attached from the begining to

the development of the didactic field as an autonomous scientific field

A community which, very early, attached strong importance to its institutional development and to its scientific coherence

A community attached to its links with the mathematical community

A community attracted by epistemological and theoretical reflections

Some dates…

1979: creation of the national seminar

1980: creation of the journal RDM and of the summer school

1984: creation of a RCP at the CNRS, becoming then a GDR

1991: creation of the ARDM

The first developments of educational research in mathematics

Priority given to the cognitive

dimension

The predominant influence of Piagetian

constructive epistemology

MK

T S

The first developments of educational research in mathematics

Priority given to the cognitive

dimension

The predominant influence of Piagetian

constructive epistemology

MK

T S

The first steps of the French didactics

An original position integrating:

A cognitive approach developed by G. Vergnaud which will lead to the theory of conceptual fields

But also a theory of didactic situations initiated by G. Brousseau whose central object is not the student but the situation where students interact with others and with mathematical knowledge

And also, very soon, a theory of didactic transposition initiated by Y. Chevallard that problematizes taught knowledge

And strong debates reflecting the existing tensions at that time between cognitive and systemic approaches

The theory of didactic situations A theory relying on the constructivist

epistemology, but not a cognitive theory The central object is the didactic situation,

and what is aimed at that time is: the understanding of the relationships that can

occur between teachers, students and knowledge in such situations, and their influence on learning processes

the development and control of « fundamental situations » for the development of mathematical knowledge in school context through a process that combines three different dialectics : dialectics of action, of formulation and of validation, associated to three functionalities of knowledge

One paradigmatic example: the race to 20

Two players: player 1 starts with 0 or 1 or 2, player 2 can add 1 or 2, player 1 can add 1 or 2 and so on. The first saying 20 wins.

1 3 5 6 8 10 11 13 15 16 17 18 20 Action: pupils play, progressively building

winning strategies Formulation: the elaboration of a specific

language and assertions « winner numbers »

Validation: what is now at stake is the validity of assertions

A fundamental situation for introducing rational numbers

Comparing the thickness of

sheets of paper

Couples of numbers which

can be compared leading to (Q+,<)

Then added, leading to (Q+,<,+)

A fundamental situation for the Riemann integral

8m 3m

What is the intensity of the force that the bar creates on the mass

M?

M

Some crucial points Analysing the characteristics of the medium

and how it shapes students’ relationships with mathematical knowledge:

Possible strategies and their respective cost and efficiency

Feedback provided by the medium Determining the didactic variables of the

situation Trying to optimize the choice of these variables

in order to make the mathematical knowledge aimed at, the knowledge underlying both an optimal and accessible strategy

Evolutions and reconstructions

The institutionnalisation process The devolution process The didactic contract and its

paradoxes

The TDS seen as a hierarchy of models

The a-didactic situation

The didactic situation

Devolution

Insitutionnalisation

Didactic contract

Epistemic subject

Institutional subject

How to reach an adequate balance between a-didactic and didactic processes of adaptation?

Some issues progressively open to discussion Does there always exist fundamental

situations? How to grasp through the TDS

situations where the relationships with the a-didactic medium are not enough for producing the expected knowledge?

What is the power of the TDS for analysing and understanding the functioning of ordinary school situations?

The theory of didactic transposition

Rejecting a vision of taught knowledge as a mere simplication of scholarly knowledge

Trying to understand the specific economy of taught knowledge

Scholar knowledg

e

Knowledge to be

taught

Taught knowledge

Some issues progressively open to discussion

Is scholar knowledge the unique source of legitimation of taught knowledge in mathematics?

Are the characteristics of taught knowledge presented by Chevallard all necessary characteristics?

What is the real field of validity of the laws governing the didactic transpositive process initially identified in reference to the new math reform movement?

The ecological vision

Niche, Habitat

Trophic chain

Where do mathematical objects live in the educational system and what functions do they play? With what other objects do they have to

compete? From what other objects do their existence depend?

A situation more complex that it can appear at a first sight as the ‘didactic time’ does not coincide

with the ‘learning time’

Towards an anthropological approach A radical change in the gravity center of

the theorisation: the central point becomes the institution

Mathematical knowledge emerges from institutional practices

The meaning attached to « knowing something » is institutionally dependent

Teaching and learning processes cannot be understood without taking into account this institutional dependence

One example: the thesis by B. Grugeon

The initial problem: the failureof adaptation courses

A radical change in the problematics

Some « easy » explanations

A problem of institutional transition

Vocational high school

General high school

Two institutions that have with algebra different institutional relationships

Most problematic differences are differences in institutional relationships as regard common objects, as they are source of

misunderstanding between teachers and students

What is the real source of the students’ failure?

The research work within this problematics Characterising the algebraic culture of

the two institutions Analysing the similarities and

differences between these Identifying possible sources for

transitional misunderstanding Finding ways for helping students and

teachers to build a bridge between the two cultures

The methodological tools The construction of a multidimensional grid of

analisis of algebraic competence aiming at the determination of both curricular and cognitive coherences with: 

a dimension focusing on the arithmetic-algebra transition,

a dimension focusing on the building and management of algebraic expressions

a dimension focusing on the functionalities of algebra and on algebraic rationality  

a dimension focusing on the connection between the settings and semiotic registers involved in algebraic work

The construction of a diagnostic set of tasks

The results and posterior developments Proving the existence of differences in institutional

relationships as regard common object, and their relative invisibility

Proving, thanks to the test diagnostic, the existence of coherences in the students’ algebraic functioning generating a more positive vision of these

Identifying didactic levers more appropriate for these students for progressing in algebra and overcoming their difficulties, relying more in their previous culture: enrichment of the work on formulas, connection with the functional world

Developing a didactic enginnering design that allowed the majority of these students overcome their failure state and develop a relationship with algebra compatible with the institutional values of algebra in general high school

Thanks to a collaborative work with researchers in AI, developing a computer version of the test and computer tools for instrumenting teaching practices

Some further developments of the TDS and the TAD

The TDS: Refining the concept of medium through

its vertical organisation (Margolinas, Bloch)

Refining the concept of didactic contract The TAD:

The dialectics between ostensive and no-ostensive (Bosch & Chevallard)

The notion of praxeology

The concept of mediumLevel Medium Student position Teacher position Situation

+3 Planning medium Noospherian teacher Noospherian situation

+2 Design medium Teacher planning Planning situation

+1 Didactic medium Reflexive student Teacher designing Design situation

0 Learning medium Student Teacher acting Didactic situation

-1 Reference medium Student learning Teacher supporting Learning situation

-2 Objective medium Student acting Teacher observing Reference situation

Ostensive and no-ostensive Mathematical objects are not

ostensive objects, but they develop and are worked through ostensive objects

The development of ostensive and no-ostensive objects is a dialectic development

Ostensive objects have both a semiotic and instrumental valence

The notion of praxeology Every human activity consists of doing some

task t of a certain type T, by the way of a technique , which is justified by a technology , which can be itself justified by a theory .

[T, , , ] is called a praxeology, and is formed of two blocks: the practical block and the theoretical one

This notion is used both for analysing mathematical and didactical organisations and their actual or potential life in educational institutions, and also infer the knowledge that can emerge from these

A crucial point of attention: the students’ topos in these praxeologies

Coming back to more general issues: internal connections

Connecting notions and frames: The same name but different underlying concepts:

conception, medium Different names but close objects:

conception - personal relationship to knowledge, ostensive – semiotic register of representation, students’ topos and type of didactic contract

Connecting different levels of analysis from the micro to the macro-level:

The level of the persons (students, teachers) The level of classroom (from short term to long term studies) The level of a particular school The level of a particular level of education The level of the educational system The level of the global society

External connections

Opposite tendencies: A global evolution of the field which

could favour the establishment of connections

But, at the same time, The multiplication of local theoretical

constructs, The diversity of educational cultures

and the necessary influence of this diversity on research cultures

Some interesting examples of mixity in theoretical frames

The thesis by Michela Maschietto relying both on embodied cognition and on the TDS

The thesis by Paul Drijvers relying both on RME, on the instrumental approach, and on the process-object duality

The thesis by M. Maschietto

The interaction between two didactic and educational cultures: the Italian culture and the French culture with:

Different curricular organisation and views

Different relationships with technology Different theoretical focus Different relationships with classroom

experimentation, between teachers and researchers

The negotiation of a research problematic from: Epistemological views on the field of analysis and the role played in it

by the dialectic interplay between local and global points of view Epistemological views on the transition between algebra and analysis

and the associated cognitive reconstructions Cognitive views on the role played by embodied activities and

metaphors in the development of mathematical knowledge Didactic views on the role played instudents’ knowledge development

by the didactic situations proposed to students and the potential they offer to a-didactic functioning

But also: Sensitiveness to the risk of abusive inferences when developing

educational strategies from the analysis of the use of metaphors by professional mathematicians

Sensitiveness to the danger of producing uncontrolled meta-cognitive slides if the management of metaphors in the classroom remains the usual one

The research project Analysing the potential offered by

embodied activities and metaphors carried out around « local linearity » in order to make the « global-local game » a fundamental dimension of high school analysis, from the begining

A symbolic calculator environment A methodology based on didactic

engineering

The crucial steps of the engineering design

The construction of a perceptive invariant and

the birth of a metaphor

The mathematisation of this perceptive invariant

The development of an associated algebraic langage and the enrichment of algebraic practices

The mathematisation process

273 xx 273 xx

a b c

The perceptive invariant

Zooming out: the spatial proximity

Looking for equations The numerical proximity

Unifying through algebraic symbolic languageand new algebraic practices

The main results The potential of the constructed fundamental situation for an

a-didactic construction of the perceptive invariant The fundamental role played by gestures and discourse

(zoomatalineare) The quality of the students’ engagement in the

mathematization game The crucial role played by the teacher in the successful

development of this game, the mathematical and didactical expertise it requires

The difficult control of the mathematical charge of the metaphor

Encouraging results as regard the symbolic dimension of the global-local game, whose development is supported by a specific discourse

The problematic ecology of this approach in the present Italian educational culture

References Perrin-Glorian (history of the TDS – connection

between frames) Margolinas (medium) Bloch (a-didactic/didactic) Brousseau (didactic contract) Chevallard (TAD) Bloch & Chevallard (semiotic dimension of the

TAD) Artaud (ecology of knowledge) Maschietto (thesis) Grugeon (thesis)