# Elementary Education (K-8)

## NCTM Standards

Expectations for the children enrolled in grades Pre K-2 :

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:

• recognize, name, build, draw, compare, and sort two- and three-dimensional shapes;
• describe attributes and parts of two- and three-dimensional shapes;
• investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

• describe, name, and interpret relative positions in space and apply ideas about relative position;
• describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance;
• find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.

Apply transformations and use symmetry to analyze mathematical situations

• recognize and apply slides, flips, and turns;
• recognize and create shapes that have symmetry.

Use visualization, spatial reasoning, and geometric modeling to solve problems

• create mental images of geometric shapes using spatial memory and spatial visualization;
• recognize and represent shapes from different perspectives;
• relate ideas in geometry to ideas in number and measurement;
• recognize geometric shapes and structures in the environment and specify their location.

In order to teach these topics successfully one needs a solid background in geometry. Many of these topics are related to those covered in the sections about Symmetry and Tessellations.

Expectations for the children enrolled in grades 3-5 

• Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes;
• classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids;
• investigate, describe, and reason about the results of subdividing, combining, and transforming shapes;
• explore congruence and similarity;
• make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.
• describe location and movement using common language and geometric vocabulary;
• make and use coordinate systems to specify locations and to describe paths;
• find the distance between points along horizontal and vertical lines of a coordinate system.
• predict and describe the results of sliding, flipping, and turning two-dimensional shapes;
• describe a motion or a series of motions that will show that two shapes are congruent;
• identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
• build and draw geometric objects;
• create and describe mental images of objects, patterns, and paths;
• identify and build a three-dimensional object from two-dimensional representations of that object;
• identify and draw a two-dimensional representation of a three-dimensional object;
• use geometric models to solve problems in other areas of mathematics, such as number and measurement;
• recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

Many of these topics are directly addressed in this course. The work in Symmetry and Tessellations will give one the background needed to teach young children about slides, flips and turns for instance. The general structure of the course shows in a natural way how these topics in geometry relate to other disciplines and the world around us.

Expectations for the children enrolled in grades 6-8 

• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
• use coordinate geometry to represent and examine the properties of geometric shapes;
• use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling;
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
• draw geometric objects with specified properties, such as side lengths or angle measures;
• use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume;
• use visual tools such as networks to represent and solve problems;
• use geometric models to represent and explain numerical and algebraic relationships;
• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

## The van Hiele Model of Geometric Thought

Two Dutch educators, Dina and Pierre van Hiele, suggested that children may learn geometry along the lines of a structure for reasoning that they developed in the 1950s. educators in the former Soviet Union learned of the van Hiele research and changed their geometry curriculum in the 1960s. During the 1980s there was interest in the United States in the van Hieles' contributions; the Standards of the National Council of Teachers of Mathematics (1989) brought the van Hiele model of learning closer to implementation by stressing the importance of sequential learning and an activity approach.

### Levels of Understanding

The van Hiele model asserts that the learner moves sequentially through five levels of understanding. Different numbering systems are found in the literature but the van Hieles spoke of levels 0 through 4.

Level 0 (Basic Level): Visualization

Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).

Level 1: Analysis

Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.

Level 2: Informal Deduction Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.

Level 3: Deduction At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof is seen. The possibility of developing a proof in more than one way is seen.

Level 4: Rigor Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

### Phases of Learning

The majority of high school geometry courses is taught at Level 3. The van Hieles also identified some characteristics of their model, including the fact that a person must proceed through the levels in order, that the advancement from level to level depends more on content and mode of instruction than on age, and that each level has its own vocabulary and its own system of relations.The van Hieles proposed sequential phases of learning to help students move from one level to another.

Phase 1: Inquiry/Information

At this initial stage the teacher and the students engage in conversation and activity about the objects of study for this level. Observations are made, questions are raised, and level-specific vocabulary is introduced.

Phase 2: Directed Orientation

The students explore the topic through materials that the teacher has carefully sequenced. These activities should gradually reveal to the students the structures characteristic at this level.

Phase 3: Explication

Building on their previous experiences students express and exchange their emerging views about the structures that have been observed. Other than to assist the students in using accurate and appropriate vocabulary, the teacher's role is minimal. It is during this phase that the level's system of relations begins to become apparent.

Phase 4: Free Orientation

Students encounter more complex tasks - tasks with many steps, tasks that can be completed in more than one way, and open-ended tasks. They gain experience in resolving problems on their own and make explicit many relations among the objects of the structures being studied.

Phase 5: Integration

Students are able to internalize and unify relations into a new body of thought. The teacher can assist in the synthesis by giving global surveys of what students already have learned.

Reference: Teppo, Anne , "Van Hiele Levels of Geometric Thought Revisited." , Mathematics Teacher , March 1991, pg 210-221.

## Teaching materials

These are just a handful of examples of lesson plans as they are used in elementary and middle school. There are many more examples available. The goal of a university geometry course is of course not to teach you how to use these specific lesson plans. The goal of a geometry course is to give you a sufficiently strong background in geometry that you will be able to easily use and adapt any such plan to your current teaching situation.

• [Escher in the Classroom] Escher-based activities used with middle school children (grades 5 to 8) to promote mathematics as the study of patterns. Excellent website by Jill Britton.
• [Tantalizing Tessellations] This unit integrates math with language arts and arts education. ... This site outlines 10 lesson plans aimed at grade 5. By Vivian Archambault, Danielle Desjardins and Terry Wood.