Transcript Devlopmental Approaches to Teaching Mathematics

Serow, UNE, 2008
DEVELOPMENTAL APPROACHES
TO TEACHING MATHEMATICS
Pep Serow
THE CONSTRUCTIVIST PERSPECTIVE
Ernest, P.(Ed) (1989) Mathematics Teaching: The State of The Art
(p.151)
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“The view that children construct their own
knowledge of mathematics over a period of time
in their own, unique ways, building on their preexisting knowledge”.
THE VAN HIELE THEORY
 Developed
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in the 1950’s
 The focus is on:
- the importance of insight in learning
Geometry
- Levels of thinking in Geometry identifying the thinking of the student
- Five phase approach to instruction.
INSIGHT
van Hiele (1986, p.161)
• Insight is acting in a new situation adequately
and with intention.
• The student must have a sense of ownership of
their mathematical ideas.
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“Insight is, as it were, the foundation for later
thought; success for a great part depends upon
it”.
THE VAN HIELE LEVELS
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
Level 1: Figures are judged by their appearance.
Level 2: Figures are identified by their
properties. These properties are independent of
one another.
Level 3: The properties of figures are no longer
seen to be independent.
Level 4: The place of deduction is understood.
Level 5: Comparison of deductive systems can
be undertaken.
EXAMPLES OF THINKING
 Level
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1 - A rectangle looks like a door.
 Level 2 - A square has four equal sides,
four right angles, and four axes of
symmetry.
 Level 3 - A minimum definition of a
square is that it as four equal sides and 1
right angle (and the student can explain
why this is the case).
- A square is a rhombus with
equal diagonals.
JUST A FEW FEATURES…
Hierarchical nature
 Different level - different language
 Crisis of thinking
 Level Reduction

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FACILITATING THE CRISIS - VAN HIELE
TEACHING PHASES
PHASES
AIM
1. Information
For students to become familiar with the
working domain
2. Directed Orientation
For students to identify the focus of the
topic through a series of teacher-guided
tasks.
3. Explicitation
For students to become conscious of new
ideas and new language.
4. Free Orientation
Tasks where students find their own way.
5. Integration
Overview of the material investigated.
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TEACHING EXAMPLE
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Construct 12 different triangles using the
Geoboards and record your triangles on dot
paper. (Directed orientation)
Cut your triangles out. Explore and record the
characteristics of your triangles (sides, angles,
symmetry) (Explicitation)
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
Brainstorm everything the class knows about
triangles. (Information)
SEQUENCE CONT …
 In
 Summary
of class findings - in students’
own language.
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pairs, classify your triangles. Record
your classification in a flow chart, tree
diagram, or concept map to share with the
larger group. (Free Orientation)
THE SOLO MODEL
Evaluates the quality of students responses.
 Involves:
- Five modes of functioning
- Series of five levels

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MODES OF FUNCTIONING
 Sensori-motor:
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involves a reaction to the
physical environment
 Ikonic: Internalisation of images and
linking to language
 Concrete Symbolic: application and use
of a system of symbols
 Formal: Consideration of abstract
concepts
 Post-Formal: challenging or questioning
abstract concepts.
SOLO LEVELS
Prestructural: below the target mode
“A square is like a box”
 Unistructural: focus on a single aspect
“a square has all sides equal”
 Multistructural: focus on more than one
independent aspect
“A square has all sides equal, four axes of symmetry
…”
 Relational: Focus on the integration of the
components. “A square has four equal sides and a
right angle”.
 Extended Abstract: beyond the domain of the
task.

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HOW DOES THE SOLO MODEL ASSIST THE
TEACHER
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• Basically a coat-hanger.
• Allows you to make informed judgments
about where students are on their
developmental journey
• Provides a window for understanding
conceptual development will all
curriculum areas.
• Assists in the selection and sequencing of
teaching strategies (Unit and lesson
plans).
• Informs your questioning in the
classroom.
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CHALLENGE …