1. D EVELOPMENTAL A PPROACHES TO T EACHING M ATHEMATICS Pep Serow Serow, UNE, 2008

2. T HE C ONSTRUCTIVIST P ERSPECTIVE “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. T HE VAN H IELE T HEORY 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. I NSIGHT “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. T HE VAN H IELE L EVELS 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. E XAMPLES OF T HINKING 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. J UST A FEW FEATURES … Hierarchical nature Different level - different language Crisis of thinking Level Reduction Serow, UNE, 2008

8. F ACILITATING THE C RISIS - VAN H IELE T EACHING P HASES PHASESAIM 1. InformationFor students to become familiar with the working domain 2. Directed OrientationFor students to identify the focus of the topic through a series of teacher-guided tasks. 3. ExplicitationFor students to become conscious of new ideas and new language. 4. Free OrientationTasks where students find their own way. 5. IntegrationOverview of the material investigated. Serow, UNE, 2008

9. T EACHING E XAMPLE 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. S EQUENCE C ONT … 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. T HE SOLO M ODEL Evaluates the quality of students responses. Involves: - Five modes of functioning - Series of five levels Serow, UNE, 2008

12. M ODES OF F UNCTIONING 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-Forma l: challenging or questioning abstract concepts. Serow, UNE, 2008

13. SOLO L EVELS Prestructural : below the target mode “A square is like a box” Unistructural : focus on a single aspect “a square has all sides equal” Multistructura l: 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. H OW DOES THE SOLO M ODEL 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. Y OUR C HALLENGE … Serow, UNE, 2008