Learning mathematics in a CAS environment : the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work : Learning mathematics in a CAS environment : the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work Michèle Artigue
Université Paris 7 & IREM
Summary : Summary Introduction
The progressive development of the theoretical frame
The unexpected complexity of instrumental genesis
The institutional status of « instrumented techniques »
Revisiting the relationships between technical and conceptual work: the epistemic value of instrumented work and techniques
Introduction : Introduction An evident contrast between:
the impregnation by technology of most social and professional practices
the limited penetration of these into the educational world
Behind this constrast, an opposition of values that CAS technology makes specially clear
Slide4 : From the professional and social sides:
A pragmatic relationship to technology,
Recognising that technology is not of an evident access,
Accepting the fact that technology strongly shapes our practices and perspectives
Slide5 : From the educational side:
A vision of technology which remains mainly a pedagogical one, technology being asked to serve values defined independently from it
An attitude that leads to see technology as something transparent with respect to knowledge and values
Some dangerous associations : Some dangerous associations Technology, by taking in charge the technical side, leaves the mind free for conceptual reasoning and provides new means for that Reality is much more complex… The use of technology favours conceptual reasoning
Computer technologies desqualify teaching pratices orientated towards skill acquisition,
and support constructivist approaches
In fact, our first observations of students working with DERIVE clearly showed: : In fact, our first observations of students working with DERIVE clearly showed: 1. The existence of two opposite tendencies :
One favouring reflexive and strategic work
One tending to save reflection or reduce the global coherence of action
2. The fact that technical work changes but does not disappear at all
3. The essential role played by the didactic situations at stake and by the teacher’s management of these
This was the starting point of a collective reflection on: : This was the starting point of a collective reflection on: the relationships between conceptual and technical work which made us more and more sensitive to instrumental issues and to the key role these play in learning and teaching processes in CAS environments, both at individual and institutional level ERES (Montpellier) DIDIREM (Paris) EQUIPE TICE (Rennes)
The development of a theoretical frame : The development of a theoretical frame Taking some distance from the dominant constructivist perspectives
And:
Engaging in an approach that would force us
to integrate the institutional dimension,
not to underestimate the role of techniques and instrumental mediations to mathematical knowledge
Some key points in the anthropological approach (Chevallard) : Some key points in the anthropological approach (Chevallard) Mathematical objects arise from institutional practices : « praxeologies » Praxeologies can be seen as complexes of tasks-techniques-technology-theory Techniques have both a pragmatic and epistemic value The advance of knowledge goes with the routinisation of tasks and techniques
The technological evolution breaks the traditional balance between conceptual and technical work: : The technological evolution breaks the traditional balance between conceptual and technical work: by reducing the cost of the technical work, and thus the routinisation needs,
by changing the pragmatic and epistemic values of techniques,
By introducing new conceptual needs through the computer transposition of mathematics knowledge
But the sensitivity to these changes depends on the research frames one adopts
Complementing the didactic anthropological approach by an ergonomic one (P. Rabardel) : Complementing the didactic anthropological approach by an ergonomic one (P. Rabardel) To the instrument Instrumental genesis From the artefact Instrumentalisation Instrumentation Constraints New potential
One paticular example : framing schemes : One paticular example : framing schemes f(x)=x(x+7)+9/x
The unexpected complexity of instrumental genesis (L. Trouche) : The unexpected complexity of instrumental genesis (L. Trouche) The diversity of students’ profiles Theorist Rationalist Scholastic Tinkerer Experimentalist Resources
Meta-knowledge
Validation
The unexpected complexity of instrumental genesis (L. Trouche) : The development of specific schemes and their evolution when passing from graphic calculators to symbolic calculators
Their interaction with global schemes such as the scheme of « approximative detour »:
anticipation and control – substitute – stratageme (guessing) The unexpected complexity of instrumental genesis (L. Trouche)
The instrumental genesis of variation (B. Defouad) : The instrumental genesis of variation (B. Defouad) First interview : understanding the variations of f(x)=x(x+7)+9/x
The second step: symbolic computations : The second step: symbolic computations CAS gives you everything you need…
Then, coming back to the graphic application : Then, coming back to the graphic application
Further verifications using tables and zooms : Further verifications using tables and zooms
The third interview : The third interview
Slide21 :
The instrumental genesis of variation : The instrumental genesis of variation A slow progression from the graphic calculator culture towards the CAS culture
The resulting change in the status of the different applications (Home, Graphic, Table)
An evident dependence of this progression on the evolution of mathematical knowledge
Specific phenomena : zapping, over-verification strategies, explosion-reduction phases
How to explain such results ?
The ordinary life of techniques in their relationship with conceptualisation : The ordinary life of techniques in their relationship with conceptualisation Solving new problems
Exploratory phase:
Craft work Selection, improvment,
institutionalisation
of some techniques Routinisation and
investment in more
complex situations Development of
a « theoretical »
discourse Personal techniques Offical techniques
What changes with instrumented techniques? : What changes with instrumented techniques? During the first experimentation:
no official selection,
legitimation but not institutionalisation,
a « theoretical discourse » reserved to paper and pencil techniques
Instrumented techniques remain private objects which are not officially worked out
Some specific difficulties… : Some specific difficulties… The diversity of commands and possible techniques
The mixture of computer and mathematics knowledge engaged in explanation and jusitications, including new math. knowledge
The problematic accessibility of technical knowledge
The distance with ordinary norms and values of mathematics teaching
Becoming aware of such constraints and difficulties: the second experimentation : Becoming aware of such constraints and difficulties: the second experimentation Some essential changes:
drastic selection
official work of institutionalisation and routinisation
management of the didactic contract taking into account its necessary evolution
With evident positive effects
Revisiting the dialectics technical/ conceptual: the epistemic value of instrumented work and techniques : Revisiting the dialectics technical/ conceptual: the epistemic value of instrumented work and techniques
Standard environment CAS environment Immediate results Step by step solving Multiplicity of accessible results Surprising results New mathematical needs
Slide28 : Understanding discretisation processes and their graphic effects : f(x)=sin(x)/x
Slide29 : Understanding CAS algebraic transformations and simplifications and learning to efficiently use these An opportunity for deepening knowledge about algebraic equivalence, relationships between « sense » and « denotation », and for addressing syntactic issues
Slide30 : Understanding exact – approximate computation modes
Slide31 : Two different kinds of situations Those arising from the use of the technology itself, and especially from the new mathematical needs induced by the computer transposition of mathematical knowledge
Those which take benefit from the pragmatic potential of CAS for introducing generalisation issues, modelling activities, and solving more complex problems
Balancing the pragmatic / epistemic valences of instrumental use for linking in a dialectic way technical and conceptual work
Slide32 :
What is expected from technology? : What is expected from technology? Must help to understand
mathematical concepts,
must increase students’ math power Must help to renew pedagogical practices,
must provide new teaching tools,
for visualising, communicating… Must help students to adapt
to the technological world Must save teaching
and learning time Must make teaching and learning easier and better Must allow students
to master current math
practices