1Dr. Lee Wai Heng & Dr. Ng Kok Fu

SPATIAL SENSE

• What and why Spatial Sense?

• van Hiele Model Geometric Thinking

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WHAT IS SPATIAL SENSE?

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.

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NCTM: GEOMETRY & SPATIAL SENSE

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.

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NCTM: GEOMETRY & SPATIAL SENSE

Spatial sense gives children an awareness of themselves in relation to the people and objects around them

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WHY CHILDREN SHOULD LEARN GEOMETRY

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)

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WHY CHILDREN SHOULD LEARN GEOMETRY

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

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IMPORTANCE IN DAILY LIFE

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.

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IMPORTANCE IN DAILY LIFE

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.

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van Hiele Model of Geometric Thinking

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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.

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Levels of Thinking in Geometry Level 1. Visual Level 2. Analysis Level 3. Abstract Level 4. Deduction Level 5. RigorThe 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.

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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

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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.

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Visual Level Example

It is a flip!

It is a mirror image!

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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.

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Analysis Level

It is a reflection!

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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

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Abstract LevelIf I know how to find the area of the rectangle, I can find the area of the triangle!Area of triangle =

1

2bh

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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.

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Deductive Level Example

In ∆ABC, is a median.

I can prove that

Area of ∆ABM = Area of ∆MBC.

M

CB

A

BM

∆ABM

∆MBC.

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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.

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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 Wikipedia http://en.wikipedia.org/wiki/Van_Hiele_levels