Kenneth RuthvenUniversity of Cambridge

Faculty of Education

New directions for instrumentation

Guin, D., Ruthven, K., & Trouche, L. [Eds.] (forthcoming, 2004). The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument (Kluwer, Dordrecht) ~500 pp.

Instrumentation theory within cognitive ergonomics

Rabardel, P. (2002) People And Technology: a cognitive approach to contemporary instruments http://ergoserv.psy.univ-paris8.fr/

Broadening the empirical base– DGS/geometry in contrast to CAS/algebra– horizontal mathematisation in contrast to vertical – primary, technical-vocational & university-

scholarly study in contrast to secondary

Elucidating the theoretical frame– articulation with elements of didactical theory– clarification & systematisation of the theory– elaboration of micro- & macro-social aspects

New directions for instrumentation

Provocative analogy– ‘a geometric calculator’, effecting [re]drawing

(through [re]computing ‘geometrical functions’)– ‘a geometric laboratory’, aiding experimentation

(through dragging and redefinition) – ‘a modeling tool for geometrical structures’

(identified through invariance under dragging)(Laborde in response to Cuoco and Goldenberg, 2003)

Fundamental distinction– in (surface) semiotics and (target) mathematics,

despite similar (computational) formalisation

DGS/geometry v. CAS/algebra

Illustrative instrumentation studies– equivalence of algebraic expressions (CAS)– variation of functions (CAS)– solution of parameterised equations (CAS)– use of drag mode and trace tools in proof activity

(DGS)

Emphasis on– diffusing canonical mathematics over supporting

progressive mathematisation– internal theorisation over external application

Horizontal v. vertical mathematisation

Primary school mathematics– more connection to everyday & wider experience– emphasis on building broad flexible competence

Technical-vocational mathematics– emphasis on pragmatic over epistemic aspects– more strongly contextualised in specific practice

University-scholarly mathematics– emphasis on formal structural/symbolic modes

of argument over informal/analogic/graphic– locally framed curriculum and assessment

Other settings v. secondary school

Instrumentation & didactique: individual cognitionBoth instrumentation & didactique treat schemes as the basis for (already functioning & developing) conceptions & competencies [Piaget;Vergnaud]Didactique emphasises the grounding of concepts in interplay between reference situations, operational invariants & semiotic systems [Vergnaud]

Instrumentation & didactique: cultural practiceWithin didactique, mathematical activity is theorised in ‘praxeological’ terms[Chevallard]

– task & technique– technical & theoretical discourses

Within instrumentation, theorisation of cultural practice is less central, although passing reference [Rabardel] is made to:

– anthropological approach [Chevallard]– cultural-historical approach [Engeström]

Instrumentation & didactique: mathematical representation

Computational transposition [Balacheff] (adapted from didactical transposition [Chevallard]) establishes operational constraints, requires perceptual adaptation [Trouche], & creates phenomena of pseudo-transparency & double reference [Artigue]

Instrumentation & didactique: managed learning

An instrumental orchestration [Guin & Trouche] is part of a didactical exploitation scenario [Chevallard] designed to exploit a mathematical [(a-)didactical] situation [Brousseau]

An environmentA locus, a teacher, students, artifacts, instruments systems

A mathematical situation

A didactical exploitation scenario(depending on the environment and the situation)

Management of different situation stages

An instrumental orchestration(for each situation stage)

A new environment(Instruments and instruments systems move from one state to another one)

Instrumental mediation

(Rabardel)

Instrumentalization of ‘instrument’

Clarifying instrument“An instrument is defined as a system incorporating an artifact… and one or more utilization schemes:

–usage schemes are orientated towards secondary tasks corresponding to the specific actions and activities directly related to the artifact;–instrument-mediated action schemes (incorporating usage schemes) are directed towards those primary tasks constituting the subject’s main goal.”

Why identify ‘instrument’ with an artifact+scheme system, rather than (as in common usage) with an artifact by virtue of its place in (many) such systems?

Clarifying instrumenta[liza]tion“We will use the term instrumentation… to designate aspects of the instrumental genesis process oriented toward the subject him/herself. We reserve that of instrumentalization for processes directed toward the artifact…These two types of processes are born of the subject. Instrumentalization by attributing a function to the artifact results from his/her activity, as does the adaptation of his/her schemes.”If both types of process are born of the subject, why not take instrumentation (in its more normal sense) as designating a process oriented towards the interaction between subject and object?

Interactional limitations of instrumental mediationThe situationally inflexible response of machines, their lack of interpretive capacity, limits the scope of human interaction with them (Suchmann, 1987) Experts often underestimate complexity of instrumented tasks, and overestimate interpretability of machine ‘feedback’ (Laborde, 2001)Automation –like routinisation– curtails the kinds of accounting for mathematical actions which underpin classroom communication, including the inter-subjective processes of appropriation central to effective teaching and learning (Ruthven, 2002)

Collective instrumented activity

“Instrumental usage is often located in the context of a collective activity… A third level of schemes must, therefore, be considered: that of instrument-mediated collective activity schemes… These concern the specification of the types of action or activity, of the types of acceptable results… when the group shares a same instrument or works with a same class of instruments. They also concern the coordination of individual actions and integration of their results as a contribution to the achievement of common goals.”

Socialising instrumented activity“ Utilization schemes have both private and social dimensions. The private dimension is specific to each individual. The social dimension comes from the fact that schemes develop in the course of a process in which the subject is not isolated. Other users, as well as artifact designers, contribute to this emergence of schemes. Schemes are the object of… transmissions and transfers… The social nature of schemes can by no means be confused with the fact that some of them are relative to instrument-mediated collective activities.” Logically but not functionally distinct? Joint activity highly pedagogically significant?

Instrumented joint activityJoint instrumented activity

(Rabardel & Bourmaud)

Instrumental orchestration

(Guin & Trouche)

Disrupted didactical activity system transparent

Toolsalient

stable Norms

emergent

devalorised Distribution

of Labourrequired

expertSubjectpioneer

established Community

extended

known Object

reframed

Competing didactical activity systems epistemic

Toolpragmatic

adidactical Norms

didactical

level-raising Distribution

of Laboureffort-avoiding

Teacher Subject Student

professional Community

lay/peer

preconceived Object

emergent

kr18@cam.ac.uk