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DEVELOPMENTAL APPROACHES TO TEACHING MATHEMATICS
Pep Serow
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THE CONSTRUCTIVIST PERSPECTIVE
“The view that children construct their own knowledge of mathematics over a period of time in their own, unique ways, building on their pre-existing knowledge”.
Ernest, P.(Ed) (1989) Mathematics Teaching: The State of The Art (p.151)
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THE VAN HIELE THEORY
Developed 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.
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INSIGHT
“Insight is, as it were, the foundation for later thought; success for a great part depends upon it”.
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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THE VAN HIELE LEVELS
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.
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EXAMPLES OF THINKING
Level 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.
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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.
Serow, UNE, 2008
TEACHING EXAMPLE Brainstorm everything the class knows about
triangles. (Information)
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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SEQUENCE CONT …
In pairs, classify your triangles. Record your classification in a flow chart, tree diagram, or concept map to share with the larger group. (Free Orientation)
Summary of class findings - in students’ own language.
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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: 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.
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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
• 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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YOUR CHALLENGE …
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