Greg Henry’s Final Project and Presentation

Background

For my Masters project, I am developing tasks that engage Geometry students in problem solving in order to help them progress from a van Hiele level 1 to a van Hiele level 2. Student thinking is classified as level 1 when “the student reasons about geometric concepts by means of an informal analysis of component parts and attributes” and is classified as level 2 when “the student logically orders the properties of concepts” (Burger & Shaughnessy, 1986, p. 31). For example, a student at level 1 would be able to identify the properties of a rectangle: opposite sides are parallel and congruent, all angles are right angles, and diagonals are congruent and bisect each other. A student at level 2 would recognize that a rectangle is a parallelogram, but with the added property that the angles must be right angles. The progression from level 1 to level 2 is not automatic, nor does it simply come with age and maturity. Rather, it comes from instruction that guides the student through five phases, as described below in Table 1:

Table 1—Five phases to progress from one van Hiele level to the next (Fuys, Geddes, & Tischner, 1988, p. 7)

Phases / Description
Information / The student gets acquainted with the working domain (e.g., examines examples and non-examples).
Guided
Orientation / The student does tasks involving different relations of the network that is to be formed (e.g., folding, measuring, looking for symmetry).
Explicitation / The student becomes conscious of the relations, tries to express them in words, and learns technical language which accompanies the subject matter (e.g., expresses ideas about properties of figures).
Free
Orientation / The student learns, by doing more complex tasks, to find his/her own way in the network of relations (e.g., knowing properties of one kind of shape, investigates these properties for a new shape, such as kites).
Integration / The student summarizes all that he/she had learned about the subject, then reflects on his/her actions and obtains an overview of the newly formed network of relations now available (e.g., properties of a figure are summarized).

For my final project, I will use Geometer’s Sketchpad as my tool and quadrilaterals as my content to guide students through the five phases in order to help them progress from a van Hiele level 1 to a van Hiele level 2.

Phase 1 (Information) Objective: Identify examples and non-examples of quadrilaterals.

To meet this objective, first I will define quadrilateral as a four-sided polygon for the students. Then I will put up an overhead of several examples and non-examples of quadrilaterals and ask the students in groups to decide which ones are quadrilaterals. If a figure is not a quadrilateral, they will explain why. This activity will take 5-10 minutes.

Phase 2 (Guided Orientation) Objective: Measure the sides, angles and diagonals of six quadrilaterals and make conjectures about the properties of the shapes.

To meet this objective, the students will have the opportunity to explore different quadrilaterals using Geometer’s Sketchpad. They will open QuadrilateralExploration.gsp and first encounter a quadrilateral (page 1). The first shape has no specific properties, so we will discuss as a class what property we can add to the shape to get a different kind of quadrilateral:

If they ask for… / We will explore…
One pair of parallel sides / A trapezoid (page 2)
One pair of opposite congruent sides / No specific quadrilateral (page 3)
One pair of adjacent congruent sides / No specific quadrilateral (page 4)
One right angle / No specific quadrilateral (page 5)
Two right angles / Another trapezoid (page 6)
Three or Four right angles / A rectangle (page 7)
Two pairs of parallel sides / A parallelogram (page 8)
Two pairs of adjacent congruent sides / A kite (page 9)
Two pairs of opposite congruent sides / A parallelogram (page 8)
Four congruent sides / A rhombus (page 10)
Four congruent sides and four right angles / A square (page 11)

For each shape we explore, there will be a contest to find as many properties as possible by measuring sides, angles, diagonals, and anything else they want to explore. They will record their conjectures in an Excel file to keep themselves organized.

Phase 3 (Explicitation) Objective: Describe the properties of each quadrilateral using correct terminology (opposite sides are congruent, right angles, etc.)

As a class, we will move back and forth between phase 2 and 3. After the students explore each quadrilateral, we will discuss as a class the conjectures they made of possible properties. As we discuss the conjectures and write them down, we will focus on correct terminology.

This lesson will not be completed in one week. The first three phases will most likely take two class periods. The last two phases will be taught several weeks later after they have lots of opportunities to experience van Hiele level 1 activities.

Phase 4 (Free Orientation) Objective: Build a hierarchy for quadrilaterals, explaining why certain shapes are special cases of other shapes.

For this phase, the students will be given time to explore the WhatAmI.gsp sketch in order to build a hierarchy for quadrilaterals:

Page 1: Nine square shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like squares. Is a square a special case of each of those shapes?
  3. Are we interested in the properties of all of those shapes? Which ones do we want to appear on our hierarchy?

Page 2: Five rectangle shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like rectangles. Is a rectangle a special case of each of those shapes?
  3. Are we interested in the properties of all of those shapes? Which ones do we want to appear on our hierarchy?

Page 3: Three rhombus shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like a rhombus. Is a rhombus a special case of each of those shapes?
  3. Are we interested in the properties of all of those shapes? Which ones do we want to appear on our hierarchy?

Page 4: Three parallel-ogram shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like parallelograms. Is a parallelogram a special case of each of those shapes?
  3. Are we interested in the properties of all of those shapes? Which ones do we want to appear on our hierarchy?

Page 5: Two kite shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like kites. Is a kite a special case of the other shape?
  3. Are we interested in the property of the other shape? Do we want it to appear on our hierarchy?

Page 6: Two trapezoid shapes /
  1. Identify the properties of each shape and then identify the shape by name.
  2. Explain why each of these shapes could be made to look like a trapezoid. Is a trapezoid a special case of the other shape?
  3. Are we interested in the property of the other shape? Do we want it to appear on our hierarchy?

After answering these questions, each student will build a hierarchy of how they think all of the shapes relate to each other. Once this has been completed individually and then in groups, we will discuss the hierarchies as a class and determine which one we think is best.

Phase 5 (Integration) Objective: Write a 2-page paper discussing the properties of quadrilaterals and the relationships identified in the hierarchy.

The last phase involves synthesizing the new learning that has taken place. Writing is an excellent way to organize thoughts and explain what they now understand about quadrilaterals, their properties, and the quadrilateral hierarchy. They can include figures and tables of information in their paper. The most important thing is evidence that they have internalized this knowledge.

Other Technologies

In this class, we have used technologies in addition to Geometer’s Sketchpad. These include web applets, graphing calculators and Excel. I am fairly certain that sketches similar to mine are available on the web in an applet form. It would be good for me to locate these if they are indeed on the web because this would give students the ability to work on identifying the properties at home. On some graphing calculators, Geometry software is included, giving students the ability to construct the same figures and analyze them in a similar fashion. This would also allow students the ability to work on identifying properties at home. Although Excel may not be very helpful in analyzing the quadrilaterals, it is helpful in organizing information and could also be used to build the quadrilateral hierarchy. Excel could be matched up with any of the other technologies to facilitate the organization of the exploration.