Post on 27-Dec-2015
transcript
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)