1

Bal Chandra Luitel

SMEC Student, Postgraduate Diploma in Education in Mathematics Education

Email: bcluitel@yahoo.combcluitel@yahoo.com

URL: http://www.ses.curtin.edu.au/~luitelb/http://www.ses.curtin.edu.au/~luitelb/

Date: 20 May, 2002

Time: 4:00-5:00 PM

Venue: SMEC Seminar Hall

2

Representation: revisited

The discussion will be about:The meaning of Representation Different perspectives of representation-

constructivism & constructionism, Piaget & Vygotsky, Ethnomathematics

Types of representation The role of representation in mathematics

learning

3

The meaning of representation-1

Internal abstraction of mathematical ideas

Mental reproduction of former mental state

Structurally equivalent presentation through pictures, symbols and signs.

Something in place of something

4

The meaning of representation-2

Hypothesized mental constructs-Goldin

Fusion(not confusion)-internalizing the external representation

“Mimetic”-the representational capacity of early man.

5

The meaning of representation-3

What happens if we say representation as “presentation” and “re-presentation”?

They represent the meaning of representation to some extent but not at all

Presentation- is an act of displaying

Re-presentation- is an act of re-displaying(repeating the presentation)

6

Constructivism and Constructionism

Representation-internalizing the external world-constructivism

Notions of viability in representation of knowledge –constructivism-the knowledge related to experiential world

Representation-shared, communal-constructionism

7

Vygotsky and Piaget Material reality to internalization of representation-Vygotsky as influenced by Marxism.Shared and sociocultural perspective of representation-signs. Schemata-interaction of individual and physical reality is a basis of representation –Piaget.Internalization-social or individual?

8

Ethnomathematics Ethnomathematics-culture and mathematics-representation through cultural perspective.

The myths, cultural artifacts,language and literature are the sources of representation in ethnomathematics

Looking at the different mathematical concepts and activities represented

9

Types of representationInternal representation-Psychological, affective representation,-verbal/syntactic, imagistic, formal notational and affective.

External Representation- concrete objects, formulas, equations, diagrams, signs, Computer generated representation of mathematical systems, ethno-source of representation, (shared)

Shared-representation constructed through classroom discourse, social interaction

Behaviorism-Deals with external representation-Does not exist internalization

10

The role of representation in mathematics learning-1

Two types of representation affects mathematics learningUnit circle-equation-shared representationUnit circle-writing journal about the classroom activities,problem solving-constructing representation.

11

The role of representation….2

What would be the role of the following model of representational system in learning?

Problem/problematicSituation

Developed subjective representation by a mathematician

Discussion,amendment and acceptance of representation

Shared representation

Historical perspective of development of mathematical representation

12

Seven roles of representation

source of communication

indicator of students' attitude towards mathematics

means of probing understanding

means of establishing links between the concepts

as developmental process

as a means of overcoming cognitive obstacles

as a part of process or means of constructing mathematical ideas

13

Seven roles of representation….1

Representation as a source of communication-

As a part of communicationEffective communication relies upon the

effective representational system Story and explanatory (verbal and written )Mathematics as a special form of language Problem representation and communication

14

Seven roles of representation….2

Representation as an indicator of students' attitude towards mathematics

Internal representation –psychological representation

Journal writing represents feeling of studentsMethods of solving problems

15

Seven roles of representation….3

Representation as a means of probing understanding

Diagrammatic, concept mapping, story(written and verbal), problem representation is a basis of probing understanding (Multiple representation)

Not only formal manipulation but to look at the students changing attitude towards mathematics and mathematical representation.

16

Seven roles of representation…4

Representation as a means of establishing links between the concepts

The following chart represents the types of triangle

Triangle

Angle Sides

Acute

Right

Obtuse Equilateral

Isosceles

Scalene

17

Seven roles of representation…5

Representation as developmental process that exists in the procedural-conceptual continuum

Phase

Represent Relates to Goal of

Procedural Performance Algorithm, mnemonic, facts , formula

Success orientation

Meta Procedural

Internal knowledge

Network of facts formula and previous experiences

Organization oriented behavior

Conceptual

Regulated knowledge by internal mental network

Relational and conceptual knowledge

Establishing control on external-internal continuum

18

Seven roles of representation…6

Representational system as a means of overcoming cognitive obstacles Cognitive obstacles-for instance-

multiplication as a repeated addition may be obstacle in multiplication of fraction.

xx 2xx 3

19

Seven roles of representation…7

Representation is not a method but as a part of process or means of constructing mathematical ideas

Way of putting into a system It is not a method-can be embedded in all

methods and techniquesCategories and subcategories of the ideas Means of putting the mathematical ideas

together

20

Some Research Aspects of representation People involved in research in the area of representation:

-Diezmann, Lowrie, Swafford and Langrall, Moritz, Lewis and Meyer, Versaffel, Goldin, Kammii, Kaput

Research area-problem solving, CL(consistent language)&IL(inconsistent language), levels of representation, representation of function, representation through retelling

Possible areas of research- multiple representation of mathematical problems.- Is the use of concrete representation less helpful to

develop internal representation?- The level of representation-lower to higher- How representations are made-Internal to external or vice

versa?