Yess4 Jean-baptiste Lagrange

http://jb.lagrange.free.fr/site

Research about technology in mathematics education:

an evolution

Jean-baptiste Lagrange

Equipe de didactique des mathématiques, université Paris 7


These themes are not independant. They differ by a specific entry into our general questionning, but they are convergent. …We adopt different theoretical orientations (TSD, TA, activity theory …). Our specificity is to cross these approaches and to analyse the implications of theoretical choices.


An example

Vandebrouk (1999)L’utilisation du tableau noir par des enseignants de mathématiques

Cazes & Vandebrouk (2006)An Emergent Inquiring Field: the Introduction in the Classroom of Online Exercises Set


A lecture about research in technologies.For what purpose ?

The role of artefacts in human Knowledge Cognition Social activity

An Artefact Is a product of art or industry, Expresses a fundamental property of a living

being: to have a project and to inscribe this project in a production. (Monod)


Artefacts influence conceptualisation Multiplication is not an operation It does not commute

Artefacts

1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

Task: The table and the calculator are two artefacts that can play specific roles in the conceptualisation of multiplication. - as an operation, as a commutative operation.

Please specify these roles.


Table Calculator

Operation

Commutativity

T1: the role of artefacts in conceptualisation


Technologies Frameworks

1. Programming and visualizing

Reification theories

2. Microworlds ConstructionismSituated cognition

3. Spreadsheets and Computer Symbolic Systems

Instrumental and Anthropological approaches

4. Quickly progessing uses of the Internet,

Uses by ‘ordinary’ teachers

New challenges and a need for new approaches


Reification

« Many theoretical and empirical arguments may be employed to show that in mathematics, operational conception precedes the structural. What is conceived as a process at one level becomes an object at a higher level. »

Sfard and Linchevski

The gains and the pitfalls of reification


Programming

Important feature of computer technology

Specific languages proposed as means to manipulate mathematical entities.

A central assumption :programming Helps learners to reflect on actions Favours conceptualisation (reification).


ISETL and APOS A programming language associated with

a specific theory “An individual's mathematical knowledge is her or

his tendency to respond to mathematical problem situations by

reflecting on them… constructing or reconstructing

mathematical actions processes and objects

organizing these in schemas to use in dealing with the situations"


APOSSteps Cognition Operation in the

computer language

Actions "the individual requires complete instructions "

Command mode execution

Processes "when an individual reflects on an action scheme and interiorizes it"

Programmation of procedures

Objects "when an individual … becomes aware of the process as a totality, realizes that transformations can act on them"

New objects added to the programming language

Schemas "once constructed, objects and processes can be interconnected

in various ways..

Collection of objects


Procepts (Tall)

An elementary procept is the amalgam of a process, a related concept produced by that process a symbol which represents both the process

and the concept. A procept consists of a collection of

elementary procepts which have the same object.


A more flexible approach The process involved must not first be given

and “encapsulated” before any understanding of the concept can be derived.

In introducing the notion of solving a differential equation, I have designed software to show a small line whose gradient is defined by the equation, encouraging the learner

to stick the pieces end to end to construct a visual solution through sensori-

motor activity.


This builds an embodied notion of the existence of a unique solution through every point,

It provides a skeletal cognitive schema for the solution process before it need be filled out with the specific methods of constructing solutions.

It uses the available power of the brain to construct the whole theory at a schema level rather than follow through a rigid sequence of strictly mathematical action-process-object.

http://www.Bibmath.Net/dico/index.Php3?Action=affiche&quoi=./C/champ.Html


Theories about visualization

To take advantage of the multiple representations of mathematical entities allowed by computer

To favor more flexible approaches to conceptualization


The idea of micro-world A more or less virtual space for learners

freely conceptualise by considering questions and constructing solutions.

Powerful enough as to evolve from the first vision linked to ‘turtle geometry’

(Papert 1980) to recent projects like Mathlab (Noss & Hoyles

2006), based on the idea of building new representations.


Papert “constructionism shares

constructivism's connotation of learning as "building knowledge structures" (and)

then adds the idea that this happens especially effectively when learners are engaged in construction for a “public” audience".


Weblabs


WeblabsTrain a robot to enumerate the

natural numbers.

Generate basic number sequences and their partial

sums.

Pose and solve number sequence challenges.

Group reflection on number sequence explorations.

Add-up challenge

Web Report

Add-a-number challenge

Guess my Robot activity


Guess my robot Nasko posted his response. He had built a robot that produced

Rita's five terms, So, he posed a two-part challenge back at Rita: Could she use his robot to generate a new sequence of five terms? Could she use her robot to generate the same sequence?

Rita was totally surprised: Nasko and Ivan had solved her challenge, but their robots seemed completely different from hers.

She worked out what inputs Nasko must have given his robot, and showed that her robot could in fact generate the same output as his.

She has made a new robot that subtracted one stream of outputs from the other and had watched the robots create a stream of zeros. She had generated thousands of zeros in this way and was convinced that this was a 'proof' of her conjecture that the sequences were the same.


Situated cognition

Because computer objects and representations generally differ from usual mathematics, Math Educators

became aware that conceptualisation always depends on situations

questioned the notion of abstraction (Noss & Hoyles 1996), introducing the idea of connection.


Computer symbolic systems Raised a lot of attention, Assumption: Quick and easy actions in

problem should dramatically reduce the part of

‘meaningless technical manipulation’ favor conceptualization


The spreadsheet

Specific notation to express relationship between entities

Dynamic execution Great potential for

Introducing younger students to algebra,

Preparing them to notions like variables, equations and functions.


Difficulties when implementing tools in the classroom.

To benefit of the tool’s potential, a learner needs knowledge intertwining mathematical understanding and awareness about the tools functioning.

Acquiring this knowledge is a non-obvious and time-consuming process, ‘instrumental genesis’.


Consider the function

Use the graphic calculator to obtain an accurate representation,

Make conjectures on its properties,

Test and prove these conjectures

An example: framing the graphic window.


Instrumentation

Distinction between tool, artefact, instrument

Instrumental genesis (Rabardel)

Interwoven mathematical and instrumental genesis

An artifact

Its constraintsIts potentialities

A human being

Her/his knowledgeHer/his work method

An instrument

Part of the artifact + schemes

Instrumentation

Instrumentalization


Concepts first, then skills ??

“If mathematics instruction were to concentrate on meaning and concepts first, that initial learning would be processed deeply and remembered well. A stable cognitive structure could be formed on which later skill development could build.” (Heid 1988, p. 4).

The anthropological approach


Techniques and concepts

Not so simple relationship Suppressing paper-pencil techniques

also suppresses the possibility of reflection on these, useful for conceptualisation,

brings difficulties related to teachers’ systems of values.


Ruthven (2002) In the experimental classes, constitution of a quite different

system of techniques

The shift to “reasoning in non algebraic modes of representation [which] characterized concept development in the experimental classes” (p. 10) created new types of task, encouraged systematic attention to corresponding

techniques The experimental course

exposed students to (…) wider techniques; helped them to develop proficiency in what had become

standard tasks, even if they were not officially recognized as such, and had

not been framed so algorithmically, taught so directly, or rehearsed so explicitly as those deferred to the final ‘skill’ phase.


Techniques a manner of solving a type

of task in an institution a complex assembly of

reasoning and routine.

a pragmatic value an epistemic value

Tasks

Techniques

Theories

pragmatic

epistemic


New challenges to mathematics education

Today fast developing web based technologies Internet based communication and social

interaction. Self learning Learning in different institutions

The position of the teacher using technology


Observations of Gaps Strong institutional demand/

few actual uses Potentialities/

actual uses by teachers Teacher expectations/

actual carrying-out of the lesson in the classroom


Hypotheses Discrepancy between

potentialities underlined by researchers from a didactical analysis

teachers’ expectations towards supposed effects of technology,

marked by aspirations regarding students’ activity

Episodes marked by improvisation and uncertainty Hidden constraints and obstacles


Task 4. Hidden obstacles

Context Upper secondary level

Non scientific students

Reformed curriculum (sequences)

Spreadsheet compulsory

Yess4 Jean-baptiste Lagrange

Education

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