Research about technology in mathematics education: an evolution Jean-baptiste Lagrange Equipe de didactique des mathématiques, université Paris 7

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

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)

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 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. 36 30 24 18 12 6 6 30 25 20 15 10 5 5 24 20 16 12 8 4 4 18 15 12 9 6 3 3 12 10 8 6 4 2 2 6 5 4 3 2 1 1 6 5 4 3 2 1

Multiplication is not an operation

It does not commute

T1: the role of artefacts in conceptualisation Commutativity Operation Calculator Table

Frameworks Technologies Instrumental and Anthropological approaches 3. Spreadsheets and Computer Symbolic Systems New challenges and a need for new approaches 4. Quickly progessing uses of the Internet, Uses by ‘ordinary’ teachers Constructionism Situated cognition 2. Microworlds Reification theories 1. Programming and visualizing

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

« 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).

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"

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"

APOS Operation in the computer language Cognition Steps Collection of objects "once constructed, objects and processes can be interconnected in various ways .. Schemas New objects added to the programming language "when an individual … becomes aware of the process as a totality, realizes that transformations can act on them" Objects Programmation of procedures "when an individual reflects on an action scheme and interiorizes it" Processes Command mode execution "the individual requires complete instructions " Actions

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.

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.

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

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

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.

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".

“ 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

Weblabs Train 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.

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.

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

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.

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’.

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’.

An example: framing the graphic window. C onsider the function Use th e graphic calculator to obtain an accurate representation, M ake conjectures on its properties, T est and prove these conjectures

Instrumentation Distinction between tool, artefact, instrument Instrumental genesis (Rabardel) Interwoven mathematical and instrumental genesis An artifact Its constraints Its potentialities A human being Her/his knowledge Her/his work method An instrument Part of the artifact + schemes Instrumentation Instrumentalization

Distinction between tool, artefact, instrument

Instrumental genesis (Rabardel)

Interwoven mathematical and instrumental genesis

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

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).

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.

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.

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 Problems Tasks Techniques Theories Institution pragmatic epistemic

a manner of solving a type of task in an institution

a complex assembly of reasoning and routine.

a pragmatic value

an epistemic value

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

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

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

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

Context

Upper secondary level

Non scientific students

Reformed curriculum (sequences)

Spreadsheet compulsory