Possible Sequence of Geometrical Development

The purpose of teaching geometry in the elementary and middle school is to help children acquire abilities to be used in describing, comparing, representing, and relating objects in the environment as well as learn mathematically to justify and explain discoveries they make. The development of the studentís abilities relies on the kinds of experiences children have with real objects and on the ways in which they respond to those experiences. Their experiences should begin with real things not pictures of real things or names of objects. Many geometry experiences should involve unstructured play activities in which children are encouraged to experiment, to find out, and to tell why and what. These activities should focus on the following abilities. Notice that each of these abilities fits into some level of the van Hiele model.

Preschool (age 4) - Grade 2 (age 7)

Before age four, children begin to acquire some topological concepts (open/closed, inside/outside/boundary, separation/connectedness, order, and proximity).  Activity 16-5

Between ages, four and seven children begin to develop some Euclidean concepts.  Euclidean geometry studies size, shape, direction, and angle.  Activity 16-6

Children first become aware of direction and angularity to enable them to make some broad distinctions among simple closed curves (distinguish circles from triangles and rectangles).  Activity 16-5

Children from age four to seven become aware that there is structure to space.  They learn to express this awareness by sometimes using ideas of left, right, and sometimes using horizontal references.

Conserve length and area.  Students begin to realize that the length and area of a figure donít change when the figure is relocated in spaces.

Geometric activities for the four to seven age groups should be of three kinds:
  • Activities designed to help children refine topological ideas they have begun to develop;
  • Activities designed to help children extend their geometric knowledge to include simple Euclidean ideas as well as topological ideas.
  • Activities designed to help children discover properties of space figures and their relationships to plane figures.

Grade 2 (age 7) - Grade 4 (age 9)

These are the most dramatic ages of development of geometric ideas.

Horizontal and vertical references are not enough for interpreting the world.

Children are concerned with filling space and discover parallel planes.

Projections for geometric figures can be predicted; graphic representations can be discriminated and represented.

Maturing mental capabilities enable them to become more conscious of details and relationships.

Ability to find class relationships improves.

Able to consider a number of related concepts at one time.

Understands relative position, proximity, and distance.

Skills in creating rectangular coordinate systems increases.

Scale drawings and maps become more accurate.

Use and rely on logical argument to explain or demonstrate ideas and answers.

Geometric activities for the four to seven age groups should be such that students

  • Learn through exploration and experience
  • Learn to use appropriate names
  • Create various shapes, angles
  • Use logical mathematical abilities.
  • Students make models to observe important attributes.
  • List attributes of both plane and space figures.
  • Notice when faces and edges are parallel, perpendicular, or neither.
  • Name space figures appropriately and use names to identify figures.
  • Analyze properties.
  • Learn about space and objects in it.
  • Construct generalized notions about objects in space.
  • Manage the space in which we live and create structures in space.
  • Visualize objects in space from many vantage points.

Grades 4 (age 9) - 8 (age 13)

Geometric Ideas:

Attributes of plane figures should be more carefully studied.

Used to classify relationships among figures.

Study of space figures should become more structured.

Parallelism and perpendicularity used to classify space figures and make graphic representations of them.

Develop coordinate systems to reproduce arrangements of objects; create Scale Drawings

Activities involving measurement of length and area should be more precise.

Relationship between perimeter and area should be studied.

Study volume and angular measure in greater depth.

Classifying Shapes: very heart of geometric definitions

Learn to build criteria that accurately define sets of shapes.

Able to appreciate subtleties

Able to understand and use precise mathematical language.

Classify many different types of plane and space figures beginning with the polygons.

Learn to recognize figures that have turning symmetry.

Learn to demonstrate this symmetry.

Indicate the angles of symmetry and how many coincidences will occur within a complete turn.

Debate and draw conclusions concerning how line symmetry and turning symmetry are related.

Cross Sections

Ruler and compass constructions

Know names of figures, develop definitions, state generalizations, and practice skills.

Recognize Euclidean shapes that are plane figures in cross sections, turned, rotated, or manipulated. 

identify size, placement, and perspective

See how objects relate to one another

Identify, select, draw and arrange figures to express representations.

Dr. Robert Sweetland's Notes ©