Dr. Lee Wai Heng & Dr. Ng Kok Fu

1. SPATIAL SENSE • What and why Spatial Sense? • van Hiele Model Geometric Thinking Dr. Lee Wai Heng & Dr. Ng Kok Fu

2. Spatial sense is an intuitive feel for shape and space. It involves the concepts of traditional geometry, including an ability to recognize, visualize, represent, and transform geometric shapes. It also involves other, less formal ways of looking at 2- and 3-dimensional space, such as paper-folding, transformations, tessellations, and projections. WHAT IS SPATIAL SENSE?

3. Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement, and describes and classifies the physical world in which we live. Young children can learn about angles, shapes, and solids by looking at the physical world. NCTM: GEOMETRY & SPATIAL SENSE

4. NCTM: GEOMETRY & SPATIAL SENSE • Spatial sense gives children an awareness of themselves in relation to the people and objects around them

5. Spatial understandings are necessary for interpreting, understanding, and appreciating our inherently geometric world. Children who develop a strong sense of spatial relationships and who master the concepts and language of geometry are better prepared to learn number and measurement ideas, as well as other advanced mathematical topics. (NCTM, p. 48) WHY CHILDREN SHOULD LEARN GEOMETRY

6. The world is built of shape and space, and geometry is its mathematics. Experience with more concrete materials and activities prepare students for abstract ideas in mathematics Students solve problems more easily when they represent the problems geometrically. People think well visually. Geometry can be a doorway to success in mathematics WHY CHILDREN SHOULD LEARN GEOMETRY

7. Spatial relationships is connected to the mathematics curriculum and to real life situations. Geometric figures give a sense of what is aesthetically pleasing. Applications architectural use of the golden ratio tessellations to produce some of the world’s most recognizable works of art. IMPORTANCE IN DAILY LIFE

8. Well-constructed diagrams allow us to apply knowledge of geometry, geometric reasoning, and intuition to arithmetic and algebra problems. Example: Difference of 2 squaresa2 - b2 = (a-b) (a+b) Whether one is designing an electronic circuit board, a building, a dress, an airport, a bookshelf, or a newspaper page, an understanding of geometric principles is required. IMPORTANCE IN DAILY LIFE

9. van Hiele Model of Geometric Thinking

10. Background of van Hiele Model • Husband-and-wife team of Dutch educators (1950s): Pierre van Hiele and Dina van Hiele-Geldof noticed students had difficulties in learning geometry • These led them to develop a theory involving levels of thinking in geometry that students pass through as they progress from merely recognizing a figure to being able to write a formal geometric proof.

11. Levels of Thinking in Geometry • Level 1. Visual • Level 2. Analysis • Level 3. Abstract • Level 4. Deduction • Level 5. Rigor The development of geometric ideas progresses through a hierarchy of levels. The research of Pierre van Hiele and his wife, Dina van Hiele-Geldof, clearly shows that students first learn to recognize whole shapes then to analyze the properties of a shape. Later they see relationships between the shapes and make simple deductions. Only after these levels have been attained can they create deductive proofs.

12. Levels of Thinking in Geometry • The levels progress sequentially. • The levels are not age-dependent. • The progress from one level to the next is more dependent on quality experiences and effective teaching. • A learner’s level may vary from concept to concept

13. 1 - Visual Level Characteristics • The student • identifies, compares and sorts shapes on the basis of their appearance as a whole. • solves problems using general properties and techniques (e.g., overlaying, measuring). • uses informal language. • does NOT analyze in terms of components.

14. Visual Level Example It is a flip! It is a mirror image!

15. 2- Analysis Level Characteristics The student • recognizes and describes a shape (e.g., parallelogram) in terms of its properties. • discovers properties experimentally by observing, measuring, drawing and modeling. • uses formal language and symbols. • does NOT use sufficient definitions. Lists many properties.

16. Analysis Level It is a reflection!

17. 3 - Abstract Level Characteristics The student can • define a figure using minimum (sufficient) sets of properties. • give informal arguments, and discover new properties by deduction. • follow and can supply parts of a deductive argument. http://www.mathopenref.com/kite.html

18. Abstract Level If I know how to find the area of the rectangle, I can find the area of the triangle! Area of triangle =

19. 4 - Deductive Level Characteristics The student • recognizes and flexibly uses the components of an axiomatic system (undefined terms, definitions, postulates, theorems). • creates, compares, contrasts different proofs.

20. Deductive Level Example In ∆ABC, is a median. I can prove that Area of ∆ABM = Area of ∆MBC. ∆ABM ∆MBC.

21. 5 - Rigor The student • compares axiomatic systems (e.g., Euclidean and non-Euclidean geometries). • rigorously establishes theorems in different axiomatic systems in the absence of reference models.

22. References • Learning to Teach Shape and Space by Frobisher, L., Frobisher, A., Orton, A., Orton, J. • Geometry Module http://math.rice.edu/~rusmp/geometrymodule/index.htm • Mind map of van Hiele model http://agutie.homestead.com/FiLes/mindmap/van_hiele_geometry_level.html • van Hiele model at Wikipediahttp://en.wikipedia.org/wiki/Van_Hiele_levels