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Developmental Approaches to Teaching Mathematics

Developmental Approaches to Teaching Mathematics. Pep Serow. 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”.

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Developmental Approaches to Teaching Mathematics

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  1. Developmental Approaches to Teaching Mathematics Serow, UNE, 2008 Pep Serow

  2. 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) Serow, UNE, 2008

  3. 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. Serow, UNE, 2008

  4. 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. Serow, UNE, 2008

  5. 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. Serow, UNE, 2008

  6. 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. Serow, UNE, 2008

  7. Just a few features… • Hierarchical nature • Different level - different language • Crisis of thinking • Level Reduction Serow, UNE, 2008

  8. Facilitating the Crisis - van Hiele Teaching Phases Serow, UNE, 2008

  9. 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) Serow, UNE, 2008

  10. 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. Serow, UNE, 2008

  11. The SOLO Model • Evaluates the quality of students responses. • Involves: • Five modes of functioning • Series of five levels Serow, UNE, 2008

  12. 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. Serow, UNE, 2008

  13. 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. Serow, UNE, 2008

  14. 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. Serow, UNE, 2008

  15. Your Challenge … Serow, UNE, 2008

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