1 Bal Chandra Luitel SMEC Student, Postgraduate Diploma in Education in Mathematics Education URL: 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 representation Internal 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 learning Unit circle-equation-shared representation Unit 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/problematic Situation 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 communication Effective 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 students Methods 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 RepresentRelates toGoal of ProceduralPerformanceAlgorithm, 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.

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 techniques Categories 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?