The integration of computer technologies in mathematics education: what is offered by an instrumental approach?

Michèle Artigue

Université Paris 7 & IREM

The begining of the story : a command from the Ministry of Education asking me

To enter the CAS group created by the Ministry two years before for identifying the potential of CAS for mathematics teaching and learning, and for reflecting on the necessary curricular changes required by their integration into mathematics teaching

To help this group draw the lessons of the innovative and empirical work it had developed in this area, to articulate the kind of knowledge it had built, and to make it widely accessible.

The first phase: entering the group, listening and learning…

A group of 15 experts in the educational use of computer environments familiar with CAS,

Strongly convinced that CAS could make their teacher’s life as well as their students’ mathematical life better,

Having developed a lot of reflections on CAS and a lot of classroom situations using these,

But meeting evident difficulties at identifying what could be considered as the results of these.

Six months later, a first research project:addressing the issue of CAS potential by

Investigating what was identified by experts as potential of CAS, with what kind of evidence through questionnaires and survey literature.

Investigating how these potential actualized in experts’ teaching practices through the observation of selected classroom sessions.

What results ?

The homogeneity of experts’ discourse about the potential of CAS for maths teaching and learning, and the limited evidence underpinning it,

The evident gap between discourses and practices.

An homogeneous discourse about the potential of CAS

Supporting experimental approaches, the management of more complex problems, of more realistic problems.

Providing immediate feed back to students’ actions. Scaffolding for students meeting difficulty with

algebraic techniques. Allowing the devolution to students of mathematical

work on syntactic aspects of algebra. Supporting conceptual work and learning by freeing

students from the technical algebraic burden.

But, contrasting with this…

At the exception of two cases, answers to questionnaires evidencing not a real integration of CAS, but only an episodic use in the best cases.

Evident discrepancies between predictions, teachers’ visions and the reality of students’ functionning in classroom observations.

Observations of students in classroom sessions showed:

1. The existence of two opposite tendencies :• One favouring reflective and strategic work,• One tending to save reflection or reduce the

global coherence of action.

2. The cognitive cost of interpreting feed-back / the diversity of possible actions.

3. The fact that technical work changed but did not disappear at all.

What lessons?

The sign of an evident under-estimation of crucial phenomena such as the computer transposition of mathematical knowledge, of instrumental issues,

The sign of an inadequate vision of the relationships between conceptual and technical work in such environments

The second research project around the TI92: an opportunity for investigating

The relationships between conceptual and technical work,

Instrumental issues and the key role these play in learning and teaching processes, both at individual and institutional level.

ERES (Montpellier) DIDIREM (Paris)

EQUIPE TICE (Rennes)

The need for an adequate theoretical frame

Taking some distance from the dominant perspectives in CAS research at that time

And: Engaging in an approach that would force us:

not to underestimate the role of techniques and instrumental mediations to mathematical knowledge,

to integrate the institutional dimension of learning processes into the reflection.

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

some tasks and techniques

Some resulting conjectures: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 knowledgeWhat are exactly these changes in the case of CAS technology and how to efficiently cope with these ?

Complementing the didactic anthropological approach by an ergonomic one (Rabardel)

To the instrument

Instrumental genesis

From the artefact

Instrumentalisation Instrumentation

Constraints New potential

One particular example : framing schemes

f(x)=x(x+7)+9/x

Why choosing to rely on cognitive ergonomy?

A long tradition of scientific cooperation with the Rabardel’s team

The publication of a synthetic book about instrumental approaches at the time the research was begining

But nevertheless a clear awareness of essential differences as regards instrumented learning between professional and educational contexts: the crucial problem of legitimacy and values

New research questions

What about instrumental genesis in the specific case of CAS technology ?

Can such a genesis be efficiently managed in the current secondary mathematical culture ? Under what conditions ?

Focusing on one particular emblematic theme: the theme of functional variation at grade eleven in scientific sections (Defouad).

The methodology

A qualitative study based on (the selection and following up all the year long of 8 students, through regular questionnaires, classroom observations, interviews, and systematic collect of written productions (beyond the data collected for the global project).

The interviews

Informations about the use out of school, in the most recent assessment, about personal structuration…

Conjecturing the variations of a particular function and trying to prove the articulated conjectures – free use of the TI92 – a function out of the range of the students’ familiar objects

What results ? The unexpected complexity of instrumental genesis

First interview : understanding the variations of f(x)=x(x+7)+9/x

The second step: symbolic computations

CAS gives you everything you need but…

Then, coming back to the graphic application

Further verifications using tables and zooms

The third interview

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

Solving new problems Exploratory phase:

Craft work

Selection, improvment,institutionalisationof some techniques

Routinisation andinvestment in more complex situations

Development of a « theoretical »

discourse

Personal techniques

Offical 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 remained private objects which were not officially worked out

Why? Some specific difficulties…

The diversity of commands and possible techniques

The mixture of computer and mathematics knowledge engaged in explanation and jusitication, 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

Some essential changes: drastic selection of commands officially used Institutionalisation of a selected set of instrumented techniques

and development of an associated ‘technological’ discourse official work of routinisation of instrumented techniques management of the didactic contract as regard instrumented

techniques and their relationships with paper/pencil techniques taking into account its necessary evolution

All of this with evident positive effects

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

Understanding discretisation processes and their graphic effects : f(x)=sin(x)/x

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

Understanding exact – approximate computation modes

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 efficiently

linking technical and conceptual work

Summarizing

A progressive and dialectic development of research questions and theoretical frames

Identifying phenomena, finding reasonable explanations to these, integrating these phenomena in coherent systems

A better understanding of the cost and conditions for effective integration of CAS

New conceptual tools in order to address issues linked to learning and teaching processes in computer environments more generally