Sør-Trøndelag University College, March 2013

Learning mathematics in rich environments: solving problems by building efficient systems of instrumentsMathematical 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)


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.


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…



In this case: two sides of the same medal…



The digital metamorphosis: a set of tools in a single envelope, portable, tactile…


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…


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 artefactsA 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 didacticsA 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 didacticsA 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 didacticsArtefacts 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, oriented 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

Task to perform,

context of work

Task to perform,

context of work


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


Instrumentation

Task to perform,

context of work

Task to perform,

context of work




Instrumentalisation

Task to perform,

context of work

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



Instrumentalisation

Task to perform,

context of work

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


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 ?


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



First exercise



First exercise



First exercise




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.


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

BC

AB = AC = 5What 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

BC

AB = AC = 5What 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…

ReferencesEngeström, 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-paris8.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.

Documents

Sør-Trøndelag University College, March 2013