Theoretical frames : development
and evolution - the case of the
French didactics
Michèle Artigue
Université Paris 7 Denis Diderot
Summary
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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:
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the evolution of concepts: medium and didactic contract
ostensive and no-ostensive, mathematical and didactical
praxeologies
Connections between theoretical frames
Introduction
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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
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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…
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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
The predominant
influence of
Piagetian
constructive
epistemology
Priority given to
the cognitive
dimension
MK
T
S
The first developments of educational
research in mathematics
The predominant
influence of
Piagetian
constructive
epistemology
Priority given to
the cognitive
dimension
MK
T
S
The first steps of the French
didactics
An original position integrating:
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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
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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:
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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
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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
M
8m
3m
What is the intensity of the force that
the bar creates on the mass M?
Some crucial points
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Analysing the characteristics of the medium and how
it shapes students’ relationships with mathematical
knowledge:
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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
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The institutionnalisation process
The devolution process
The didactic contract and its paradoxes
The TDS seen as a hierarchy of
models
The a-didactic situation
Insitutionnalisation
Didactic contract
The didactic situation
Epistemic
subject
Devolution
Institutional
subject
How to reach an adequate balance between a-didactic and
didactic processes of adaptation?
Some issues progressively
open to discussion
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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
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Rejecting a vision of taught knowledge
as a mere simplication of scholarly
knowledge
Trying to understand the specific
economy of taught knowledge
Scholar
knowledge
Knowledge
to be taught
Taught
knowledge
Some issues progressively
open to discussion
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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
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?
Niche,
Habitat
Trophic
chain
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
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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 failure
of adaptation courses
Some « easy »
explanations
A radical change in the problematics
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
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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
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The construction of a multidimensional grid of
analisis of algebraic competence aiming at the
determination of both curricular and cognitive
coherences with:
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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
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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
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The TDS:
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Refining the concept of medium through its
vertical organisation (Margolinas, Bloch)
Refining the concept of didactic contract
The TAD:
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The dialectics between ostensive and noostensive (Bosch & Chevallard)
The notion of praxeology
The concept of medium
Level
Medium
+3
Student position
Teacher position
Situation
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
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Mathematical objects are not ostensive
objects, but they develop and are
worked through ostensive objects
The development of ostensive and noostensive objects is a dialectic
development
Ostensive objects have both a semiotic
and instrumental valence
The notion of praxeology
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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
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Connecting notions and frames:
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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:
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The
The
The
The
The
The
level
level
level
level
level
level
of
of
of
of
of
of
the persons (students, teachers)
classroom (from short term to long term studies)
a particular school
a particular level of education
the educational system
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
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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:
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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:
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Sensitiveness to the risk of abusive inferences when developing educational
strategies from the analysis of the use of metaphors by professional
mathematicians
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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
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Analysing the potential offered by embodied
activities and metaphors carried out around «
local linearity » in order to make the « globallocal 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
x 3  7x  2
The mathematisation process
The perceptive invariant
Zooming out:
the spatial proximity
a
b
c
Looking for equations
The numerical proximity
Unifying through algebraic symbolic language
and new algebraic practices
The main results
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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 globallocal game, whose development is supported by a specific discourse
The problematic ecology of this approach in the present Italian
educational culture
References
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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)