New directions for
instrumentation
Kenneth Ruthven
University of Cambridge
Faculty of Education
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/
New directions for instrumentation
Broadening the empirical base
– DGS/geometry in contrast to CAS/algebra
– horizontal mathematisation in contrast to vertical
– primary, technical-vocational & universityscholarly 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
DGS/geometry v. CAS/algebra
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
Horizontal v. vertical
mathematisation
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
Other settings v. secondary school
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
Instrumentation & didactique:
individual cognition
Both 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 practice
Within 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 e n vi ron m e n t
A locus, a t eacher,
students, art ifact s,
instrument s
syst ems
A m ath e mati cal
si tuati on
A di dacti cal e xpl oi tati on sce nario
(depending on t he environment and t he sit uation)
Man age me n t of
di ffe re nt
si tuati on stage s
An i n stru me ntal
orch e stration
(for each sit uation
stage)
A n e w e nviron me n t
(Inst rument s and inst ruments systems move from
one st at e t o 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 mediation
The 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 intersubjective 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 activity
Joint instrumented activity
(Rabardel & Bourmaud)
Instrumental orchestration
(Guin & Trouche)
Disrupted didactical activity
transparent
system
Tool
salient
expert
known
Subject
Object
pioneer
reframed
devalorised
stable
Norms
emergent
established
Community
extended
Distribution
of Labour
required
Competing didactical activity
systems
epistemic
Tool
pragmatic
Teacher
preconceived
Student
emergent
Subject
Object
level-raising
adidactical
Norms
didactical
professional
Community
lay/peer
Distribution
of Labour
effort-avoiding
kr18@cam.ac.uk