Representation in School Mathematics: A Unifying Research Perspective

1
Representation in School Mathematics A
Unifying Research Perspective

• Jill Zarestky
• EDCI 628
• Fall 2009

2
Representations

• Examples
• Numerals
• Graphs
• Equations or Functions
• Relationships are two-way

3
Systems of Representation

• Components
• Characters or symbols
• Configurations
• Structure
• Meaning is based on the relationships within the
  system

4
Internal and External Systems

• Where does the Mathematics live?
• Internal understanding
• External expression
• Translation between systems

5
Questions Behind the Research

• How can we build powerful internal representation
  systems?
• How can we infer or measure students internal
  representational capabilities?
• What is the best use of structured external
  representations to foster the development of
  internal representations?
• What are the obstacles?

6
Types of Representational Systems

• Internal
• Verbal/syntactic
• Imagistic
• Formal notational
• Executive control
• Affective
• External Media
• Dynamic vs Static
• Interactive vs Inert
• Recording vs Nonrecording

7
Stages of Development

• Inventive/Semiotic
• Assign meaning to internal constructs based on
  established representations
• Structural Development
• Relationships of newer constructs are based on
  prior knowledge
• Autonomous
• System detaches from prior knowledge to function
  with flexibility

8
Research Directions

• Specific Conceptual Domains
• Investigate in terms of a specific mathematical
  concept
• Representations as Problem-Solving Tools
• Affect as Representation
• Feelings about mathematics are used by students
  to direct effort

9
Research Directions

• Task-based Interviews Classroom Activities
• Thoughtful design
• Challenging but doable
• Rich and varied representational structures
• Plan for contingencies
• Encourage free problem solving
• Maximize interaction with the external
  environment
• Multiple, ongoing assessments
• Build a library of successful activities
• Watch for novel student strategies
• Balance as appropriate

10
Implications for Theory and Practice

• Representational systems unify cognitive theories
  and mathematical structure.
• Balances behavioral observations and mathematical
  formalism
• Supports knowledge constructed over time
• Incorporates sociocultural factors

11
Reference

• Goldin, G. A. (2003). Representation in school
  mathematics A unifying research perspective. In
  J. Kilpatrick, W. G. Martin, D. Schifter
  (Eds.), A research companion to principles and
  standards for school mathematics (pp. 275-285).
  Reston, VA National Council of Teachers of
  Mathematics.