Dr. Lee Wai Heng & Dr. Ng Kok Fu SPATIAL SENSE What and why Spatial Sense? van Hiele Model Geometric Thinking We will focus on 2 issues in this lecture: What and why of Spatial Sense? A learning model for geometry. Dr. Lee Wai Heng & Dr. Ng Kok Fu

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. Geometry is all around us in art, nature, and the things we make. Students of geometry can apply their spatial sense and knowledge of the properties of shapes and space to the real world.

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.

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

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)

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

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.

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 squares a2 - 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. a2-b2 = (a-b)(a+b)

van Hiele Model of Geometric Thinking The van Hiele model of geometric thought outlines the hierarchy of levels through which students progress as they develop of geometric ideas. The model clarifies many of the shortcomings in traditional instruction and offers ways to improve it. Pierre van Hiele and his wife, Dina van Hiele-Geldof, focused on getting students to the appropriate level to be successful in high school Geometry.

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. Their theory explains why many students encounter difficulties in their geometry course, especially with formal proofs. The van Hieles believed that writing proofs requires thinking at a comparatively high level, and that many students need to have more experiences in thinking at lower levels before learning formal geometric concepts.

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. 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.

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 The hierarchy for learning geometry described by the van Hieles parallels Piaget’s stages of cognitive development. One should note that the van Hiele model is based on instruction, whereas Piaget’s model is not. Piaget’s theory of cognitive development: Sensori-motor (0-2 yrs) Pre-operations (2-7 yrs) Concrete operations (7-12+ yrs) Formal operations (12+ yrs)

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. Language at the Visual Level serves to make possible communication for the whole group about the structures that students observe. The vocabulary representing the figures helps in describing the figures. Any misconceptions identified may be clarified by the use of appropriate language. The language of the next level, e.g., congruence, will not be understood by students who are at the Visual Level. ** Students recognize figures by appearance alone, often by comparing them to a known prototype. The properties of a figure are not perceived. At this level, students make decisions based on perception, not reasoning. Level 1. Visualization: children identify prototypes of basic geometrical figures (triangle, circle, square). They view figures holistically without analyzing their properties. At this stage children might balk at calling a thin, wedge-shaped triangle (with sides 1, 20, 20 or sides 20, 20, 39) a "triangle", because it's so different in shape from an equilateral triangle. Squares are called "diamonds" and not recognized as squares if their sides are oriented at 45° to the horizontal. Children at this level often believe something is true based on a single example.

Visual Level Example It is a flip! It is a mirror image! The term “reflection” is introduced. Students may note that a mirror might produce a reflection.

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. The language of the Descriptive Level includes words relating to properties within a given figure, e.g., parallel, perpendicular, and congruent. Various properties can be used to describe or define a figure, but this is usually a very uneconomical list of properties. A concise definition, using a sufficient number of properties rather than an exhaustive list, is not possible at this level. Students functioning at the Visual Level are not able to understand the properties of the Descriptive Level even with the help of pictures. ** Students see figures as collections of properties. They can recognize and name properties of geometric figures, but they do not see relationships between these properties. When describing an object, a student operating at this level might list all the properties the student knows, but not discern which properties are necessary and which are sufficient to describe the object. Level 2. Analysis: children can discuss the properties of the basic figures and recognize them by these properties, but might still insist that "a square is not a rectangle." Children do not see the relationships between the properties. They might reason inductively from several examples, but not deductively.

Analysis Level It is a reflection! The line of reflection bisects the parallel segments which connect corresponding points on the pre-image and image. This property describes and defines a reflection. The congruency symbols, the perpendicular symbol, and the corresponding symbols used in written descriptions, become part of the language of the Descriptive Level.

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. The language of the Relational Level is based on ordering arguments which may have their origins at the Descriptive Level. For example, a figure may be described by an exhaustive list of properties at the Descriptive Level. At the Relational Level it is possible to select one or two properties of the figure to determine whether these are sufficient to define the figure. The language is more abstract with its causal, logical and other relations of the structure. A student at the Relational Level is able to determine relationships among figures, and to arrange arguments in an order in which each statement except the first one is the outcome of previous statements. *** Students perceive relationships between properties and between figures. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Logical implications and class inclusions, such as squares being a type of rectangle, are understood. The role and significance of formal deduction, however, is not understood. Level 3. Abstraction: students begin to reason deductively. They understand the relationships between properties and can reason with simple arguments about geometric figures. Learners recognize relationships between types of shapes. They recognize that all squares are rectangles, but not all rectangles are squares, and they understand why. They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus. http://www.mathopenref.com/kite.html

Abstract Level If I know how to find the area of the rectangle, I can find the area of the triangle! Area of triangle = In this example, the triangle is divided into parts which transform to form the rectangle. Properties of rotation, congruence and conservation of area are utilized.

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. The Deductive Level is sometimes called the Axiomatic Level. The language of this level uses the symbols and sequence of formal logic. *** Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. At this level, students should be able to construct proofs such as those typically found in a high school geometry class. Level 4. Deduction: learners can construct geometric proofs at a high school level. They understand the place of undefined terms, definitions, axioms and theorems.

Deductive Level Example In ∆ABC, is a median. I can prove that Area of ∆ABM = Area of ∆MBC. At this level students are able to construct the steps of a proof using appropriate symbolic language. ∆ABM ∆MBC.

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. The student is able to establish proofs and reach conclusions using the symbolic language of the system without the aid of visual cues. *** Students at this level understand the formal aspects of deduction, such as establishing and comparing mathematical systems. Students at this level can understand the use of indirect proof and proof by contrapositive, and can understand non-Euclidean systems. Level 5. Rigor: learners understand axiomatic systems and can study non-Euclidean geometries.

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