Instrumentation - Educmath

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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
(
7
),
667
-
681
.

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,
9
,
281
-
307
.