An Introduction to Chapter 9: Geometric Figures MTH 232 An Introduction to Chapter 9: Geometric Figures

Philosophy of this Course To study shapes in geometry in an informal manner while at the same time being mathematically careful with the material.

Historical Perspective Before 600 B.C., geometry was both informal and practical. From 600 B.C. to 300 B.C., geometry was transformed into a theoretical science. Euclid’s Elements introduced theorems (statements requiring proof) that were deduced from axioms (statements accepted without proof).

Intuition and Experience Undefined terms are identified and careful definitions of other concepts are made from those terms. Geometric facts are then discovered by explorations of pictorial representations and physical models.

Activity-based Learning: the Van Hieles The van Hieles were Danish mathematics teachers in the late 1950s. Their research concluded that the knowledge construct for themselves through hands-on activities is essential to learning geometry. Learning geometry progresses through five levels.

van Hiele Levels Level 0: Recognition of shape. Children recognize shapes holistically. Only the overall appearance of the shape is observed, with no attention given to the parts of the figure. Level 1: Analysis of single shapes. Children are aware of the parts of certain figures (e.g., a rectangle has four straight sides that meet at “square” corners).

Continued Level 2: Relationships among shapes. Children understand how common properties create abstract relationships among figures, and can make simple deductions about figures. Level 3: Deductive reasoning. The student at this level views geometry as a formal mathematical system and can write deductive proofs.

Finally Level 4. Geometry as an axiomatic system. This level is highly abstract and is reached only in high-level university courses. No dependence is placed on concrete or pictorial models.

Your Challenge As an elementary school teacher, you want your students to progress through the first three van Hiele levels. To do this, you must create interesting activities that support each child’s progression.

NCTM Principles and Standards Students first learn to recognize a shape by its appearance. In grades Pre-K through 2, geometry begins with describing and naming shapes. Formal terminology is gradually introduced in order to focus attention and clarify ideas. Teachers must provide materials (geoboards, tangrams, pattern blocks and geometric solids). The learning environment must encourage students to explore shapes and their attributes.

Continued Students need to see both examples and non-examples of shapes that correspond to a particular geometric concept. Interactive computer programs provide a rich environment for activities in which students compose (put together) and decompose (take apart) shapes.