Learning mathematics in rich
environments: solving problems by
building efficient systems of
instruments
Mathematical modelling as an approach to teaching and learning
mathematics in (lower secondary) school
Sør

Trøndelag University College, March 2013
Luc Trouche
French Institute of Education, ENS de Lyon
Early instruments for navigation.
Plate XX from N. Bion
‘
s
The
Construction and Principal Uses
of Mathematical Instruments.
Translated from the French.
To Which Are Added The
Construction and Uses of Such
Instruments as Are Omitted by
M. Bion; Particularly of Those
Invented or Improved by the
English. By Edmund Stone. . .
(London, 1723).
http://libweb5.princeton.edu/visual_materials/maps/websites/pacific/introduction/introduction.html
Plan
Mathematics and tools, a very ancient common story
Elements of an instrumental approach of didactics
A first example
Orchestration, a teaching challenge
Examples to work with
Discussion and perspectives towards new collaborative tools
Mathematics and tools, a
very ancient common story
Two sides of an old Babylonian tablet (2000 BC), 10cm x
10cm, highly structured (5 levels), bearing about one hundred
of mathematical problems…
Mathematics and tools…
The Mohr
–
Mascheroni theorem states that any geometric construction
that can be performed by a compass and straightedge can be
performed by a compass alone (Mohr, 1672, Maschieroni1797)
Mathematics and tools…
The four colors theorem states that, given any separation of a plane
into contiguous regions, producing a figure called a
map
, no more
than four colors are required to color the regions of the map so that
no two adjacent regions have the same color.
It was proven in 1976 by K. Appel and W. Haken. It was the first
major theorem to be proved using a computer.
Mathematics and tools…
A permanent coexistence of
several artefacts influencing
the way of
doing
and
thinking
mathematics…
In this case: a new way of
computing (
“
indian
computation
”
) and an old way
(abacus)…
A transition during, from the
south to the north of France,
several centuries…
Mathematics and tools…
A permanent coexistence of several artefacts influencing the way
of
doing
and
thinking
mathematics…
In this case: two sides of the same medal…
Mathematics and tools…
A permanent coexistence of several artefacts influencing the way
of
doing
and
thinking
mathematics…
The digital metamorphosis: a set of tools in a single envelope,
portable, tactile…
Mathematics and tools…
Finally, what artefacts are involved in the practice of mathematics?
Material
or
symbolic
: language, semiotic registers (integer numbers,
plane geometrical figures…); algorithms; compass and rulers;
calculators; various software…
At three levels:
primary
artefacts, mode of use, internal
representation of the artefact itself
Mathematical artefacts, or artefacts use for mathematical purpose…
Used by an individual as an isolated artefact, or in combination with
other artefacts
Implicit
or
explicit
use
Individual or collective artefact…
Mathematics and tools…
A set of artefacts always
changed by the adding of
new artefacts, leading to an
internal reorganization
For today: the toolkit will
include Geogebra and a
Pad, specific artefact
dedicated to collaborative
work (and reflective
practice)
A set of
artefacts
An instrumental approach of didactics
Nec manus nuda, nec
intellectus sibi permissus,
multum valet; instrumentis et
auxiliis res perficitur; quibus
opus est, non minus ad
intellectum, quam ad manum.
Neither the naked hand nor
the understanding left to itself
can effect much. It is by
instruments and helps that the
work is done, which are as
much wanted for the
understanding as for the hand.
Francis Bacon, London, 1561

1626
An instrumental approach of didactics
A tradition :
•
The idea of
technè
(Plato)
•
Working, tools and learning,
existence and conscience
(Descartes, Diderot, Marx)
Heir of this tradition, Vygotski
(quoting Bacon) situates each
piece of learning in a world of
culture where the
instruments
(material as well as
psychological) play an essential
role.
Same idea in the Activity Theory
(
Engeström 1999) [who refers to
the word
tätigkeit,
implying the
principle of
historicity
]
An instrumental approach of didactics
A tradition :
•
The idea of
technè
(Plato)
•
Working, tools and learning,
existence and conscience
(Descartes, Diderot, Marx)
Heir of this tradition, Vygotski
(quoting Bacon) situates each
piece of learning in a world of
culture where the
instruments
(material as well as
psychological) play an essential
role.
Same idea in the Activity Theory
(
Engeström 1999) [who refers to
the word
tätigkeit,
implying the
principle of
historicity
]
An instrumental approach of didactics
Artefacts are only
propositions
exploited or not by users
(Rabardel
1995/2002)
Two processes closely interrelated,
instrumentation and
instrumentalisation :
“
Students
’
activity is
shaped
by the tools, while
at the same time they shape the
tools to express their arguments
”
(Noss & Hoyles 1996)
An instrument as a result of an
individual and
social
construction
,
o
riented by tasks, then
context
dependent
, in a given community
A subject An artefact
Instrumentation
Instrumentalisation
An instrument
(to do something) =
an artefact (or a part of) and
an instrumented scheme
Instrumental
genesis
Task
to
perform
,
context
of work
An instrumental approach of didactics
Instrumentation
is a process
through which the constraints
and potentialities of an artifact
shape
the subject
’
s activity.
It develops through the
emergence and evolution of
schemes while performing tasks
A subject An artefact
Instrumentation
Instrumental
genesis
Task
to
perform
,
context
of work
An instrumental approach of didactics
An instrumental approach of didactics
A subject An artefact
Instrumentalisation
Instrumental
genesis
Task
to
perform
,
context
of work
A process of personalisation and
transformation of the artefact
Externalization, vs.
internalization.
“
Vygotsky (…) not
only examined the role of
artefacts as mediators of
cognition, but was also interested
in how children
created
artefacts
of their own to facilitate their
performance
”
(Engeström 1999)
Neither a diversion, nor a
poaching… But an essential
contribution of users to the
conception of artefacts
An instrumental approach of didactics
A subject An artefact
Instrumentalisation
Instrumental
genesis
Task
to
perform
,
context
of work
A process of personalisation and
transformation of the artefact
Externalization, vs.
internalization.
“
Vygotsky (…) not
only examined the role of
artefacts as mediators of
cognition, but was also interested
in how children
created
artefacts
of their own to facilitate their
performance
”
(Engeström 1999)
Neither a diversion, nor a
poaching… But an essential
contribution of users to the
conception of artefacts
An instrumental approach of didactics
A set of artefacts intervening in
each mathematical task
Being able to articulate them,
an essential objective of
mathematics learning
A challenge for
conceptualisation
(coordinating several semiotic
registers, a need to distinguish
a concepts and its
representations
–
see the case
of function)
A powerful way for solving
problems
A subject
Several artefacts
Instrumentation
Instrumentalisation
A set, or a system of
instruments ?
Instrumental
geneses
An instrumental approach of didactics
Coordinating several semiotic registers, a need to distinguish a
concepts and its representations
First exercise
Three circles have the
same radius, and pass
through the same point O.
What about the three
other intersection points I,
J and K?
First exercise
Three circles have the
same radius, and pass
through the same
point O.
What about the three
other intersection points I,
J and K?
First exercise
Three circles have the
same radius, and pass
through the same
point O.
What about the three
other intersection points I,
J and K?
First exercise
Three circles have the
same radius, and pass
through the same
point O.
What about the three
other intersection points I,
J and K?
First exercise
Three circles have the
same radius, and pass
through the same
point O.
What about the three
other intersection points I,
J and K?
Orchestration, a teaching
challenge
A great diversity of
environments, a very
rapid evolution
A necessity to think how
to monitor students
instrumental geneses,
according to the
mathematical situations
that student face, and to
the technological
environments where
these mathematical
situations take place.
Orchestration, a teaching
challenge
A crucial need to think the space and
time of the students’ mathematics
work.
A crucial need to organize the
artefacts (available, or to be
introduced), in relation with the
problem, the phases of its solving,
the didactical variables, the learning
objectives.
A
“
milieu
”
for
mathematics
learning
An
orchestration
(= a scenario)
A mathematical
situation
An environment
(= a set of artifacts)
A
B
C
AB = AC = 5
What is the
aera of the
triangle ABC?
Second exercise…
A
“
milieu
”
for
mathematics
learning
An
orchestration
A mathematical
situation
An environment
(= a set of artifacts)
A
B
C
AB = AC = 5
What is the
area of the
triangle ABC?
A problem to solve, in a reflective
way (what artefacts could be used,
what combination of artefacts…)
Then, some elements of a possible
orchestration to design, for
implementation of this situation in a
mathematics classroom (grade 10
students)
Different scenarios, according to
different pedagogical objectives…
Second exercise…
Objective : the concept of function
Environment: rulers and compass, and
network of calculators
Measures of the different data (BC,
height), computation of the
corresponding aera, and gathering by the
teacher of the couples (BC, area) on the
shared screen
Second exercise…
Looking for a formula, co

elaboration
of a solution modelling the given
problem
Is there a maximum, where and why?
Measure of BC
area
Second exercise…
Second environment
Objetivo: the concept of function
Environment including Geogebra
Students working by pairs
Second exercise…
Second exercise…
Second exercise…
Extension of the problem
AB = 5, AC = 4
Second exercise…
Discussion and perspectives
Orchestration in a double perspective:
Articulating the different instruments beeing developed
by all the students in a given classroom
Articulating the different instruments being developed
by a given student in his/her mind (instrument for
analysing the variation of a function, instrument for
analysing a geometrical figure, etc.)
Complex processes, needing to careful prepare un
teaching session…
Dynamic + collaborative artefacts: to be carefully
implemented…
References
Engeström, Y. & al. (
1999
).
Perspectives on Activity Theory
. Cambridge: Cambridge University Press
Gueudet, G., & Trouche, L. (
2011
). Mathematics teacher education advanced methods: an example in
dynamic geometry.
ZDM, The International Journal on Mathematics Education,
43
(
3
),
399

411
.
Guin, D., & Trouche, L. (
1999
). The Complex Process of Converting Tools into Mathematical
Instruments. The Case of Calculators.
The International Journal of Computers for Mathematical
Learning
,
3
(
3
),
195

227
.
Maschietto, M., & Trouche, L. (
2010
). Mathematics learning and tools from theoretical, historical and
practical points of view: the productive notion of mathematics laboratories.
ZDM, The International
Journal on Mathematics Education,
42
(
1
),
33

47
.
Noss, R., & Hoyles, C. (
1996
).
Windows on Mathematical Meanings:
Learning Cultures and Computers.
New York: Springer
Rabardel P. (
1995
,
2002
).
People and technology, a cognitive approach to contemporary instruments
(retreived from
http://ergoserv.psy.univ

paris
8
.fr/
)
Trouche, L., Drijvers, P., Gueudet, G., & Sacristan, A.
I. (
2013
). Technology

Driven Developments and
Policy Implications
for Mathematics Education. In A.J.
Bishop, M.A. Clements, C. Keitel, J.
Kilpatrick, &
F.K.S. Leung (Eds.),
Third International Handbook of Mathematics Education
(pp.
753

790
). New York:
Springer.
Trouche, L., & Drijvers, P. (
2010
). Handheld technology for mathematics education, flashback to the
future.
ZDM, The International Journal on Mathematics Education,
42
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7
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667

681
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Trouche, L. (
2004
). Managing the complexity of human/machine interactions in computerized learning
environments: guiding students
’
command process through instrumental orchestrations.
The
International Journal of Computers for Mathematical Learning,
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