chapter 15 to accompany helping children learn math cdn ed, reys et al. ©2010 john wiley & sons...

Chapter 15Chapter 15

To accompany Helping Children Learn Math Cdn Ed, Reys et al.©2010 John Wiley & Sons Canada Ltd.

Guiding Questions

1. Why should elementary mathematics programs include geometry?

2. What should early childhood and elementary students learn about geometric shapes?

3. How do the van Hiele levels guide the development of geometric experiences for elementary children?

4. What types of explorations with geometry help build elementary children’s spatial reasoning and visualization skills?

5. What do location and movement have to do with geometry?

Why Geometry?

• It is a mathematics topic that engages children differently both in performance and persistence.

• It is natural to include other skills in the study of geometry such as following directions and for reasoning about shapes and their properties.

• Children can make and verify conjectures about geometric figures.

• Geometry is a topic that will help you teach many other mathematical topics.

Why Geometry?

• Understanding the properties or attributes of objects and their relationships among different geometric objects is an important part of elementary mathematics.

Levels of Geometric Thought

• Very young children often attend to only visual cues that are salient to them (Level 0, Prerecognition).

• Later, children focus on visual cues—it is a rectangle because it looks like a rectangle (Level 1, Visual).

Levels of Geometric Thought (cont.)

• They then begin to recognize and more carefully describe properties of all rectangles and move to Level 2 (Descriptive/Analytic) .

• More abstract thought is at Level 3 (Abstract/Relational). At this level, students establish relationships among properties and among figures.

Three-Dimensional Shapes

• Describing and Sorting • Constructing to Explore and Discover

Describing and Sorting: Beginning Activities

• The following questions ask students to describe and justify their answers when sorting shapes:– Who am I?– Who stacks?– How are we alike or different?– Who doesn’t belong?– How many faces do I have?

Describing and Sorting: Intermediate Activities

• Edges, vertices, and faces: Who am I?• Classifying Solids• Searching for Solids.

Describing and Sorting: Advanced Activities

• Parallel Faces• Perpendicular Edges• Right Prisms

Constructing to Explore and Discover

• It is essential to have models of the solids when studying three-dimensional geometry. While the models may be expensive to purchase they can easily be created using readily available materials.

Two Dimensional Shapes

• General things to keep in mind:– Children first recognize shapes in a holistic

manner-- that is a triangle is a triangle because it looks like a shape someone has called a triangle.

– Children should begin to recognize shapes through examples and non-examples, not through definitions.

Two Dimensional Shapes (cont.)

• General things to keep in mind:– Children need to be able to recognize geometric

shapes as models for real objects. – Children should know the names of common

shapes and words associated with shapes

Properties of Two Dimensional Shapes

• Number of sides and corners• Symmetry• Lengths of sides• Sizes of angles• Parallel and perpendicular sides• Convexity and concavity• Altitude• Classification schemes

Locations and Associated Representations

• Map of Locations• Coordinate Graphing

Transformations

• Translation (slide)

• Reflection (flip)

• Rotation (turn)

Congruence

• Two shapes are said to be congruent if they have the same size and shape.

Many middle school students would respond that the parallelogram shown here is congruent to the rectangle because they have the same area.

Similarity

• Two shapes are similar if the corresponding angles are equal and the corresponding sides are in the same ratio.

Visualization and Spatial Reasoning

• Geometric Physical and Pictorial Materials • Research shows that the use of physical

materials can be helpful in developing geometric representations but they must be used wisely.

• Children also need practice putting together and taking apart shapes to understand their component parts.

Visualization and Spatial Reasoning (cont.)

Using Mental Images• Children need help to develop visual images of

geometric shapes.• This is a developmental process with students

able to manipulate images in more sophisticated ways as they gain experience.

Visualization and Spatial Reasoning (cont.)

Using Mental ImagesHere is a great introductory activity:

Geometry Across Cultures

• Cultures have their own designs for their arts and crafts. • Many of these designs are geometric.• Homes and other structures also have distinctive geometric properties.• In today’s diverse classrooms, you will find students with

many different backgrounds. The visual nature of geometry helps you accommodate these differences in many geometric activities.

A Geometry Activity

Geoboard Quadrilaterals • Mark off a 3 by 3 section on your geoboard. All your

exploration will be done on these 9 pegs.• How many non-congruent quadrilaterals can you form?• Use your geoboard to explore and then record your

discoveries on geoboard recording paper.• Compare your set of quadrilaterals with your peers to

find those you may have missed.• Can you categorize the quadrilaterals in any way? How?

Copyright

Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.