Piaget and Van Hiele: Geometric Concept Development
Piaget’s Theory
Perception: Knowledge of objects resulting from direct contact with them.
Representation: Involves the evocation of objects in the absence of recognition. The order is as follows:
- Recognizes the ring (it has a hole, which is a topological property).
- Recognizes the circle (clearly differs from the straight sides, projective property).
- The square and finally the parallelogram (which has right angles).
The credit for reaching almost all children at age three and a half and the order to recognize is: circle, square, triangle, and rhombus. However, the ability to reproduce a drawing is acquired more slowly and only completed at about seven years (with the figures presented here).
The Van Hiele Model
The model has two aspects:
- Prescriptive: Proposes guidelines in the organization of education, which are specified in a sequence of steps to move from one level to the next.
- Descriptive: Attempts to explain the learning of geometrical concepts through the division of geometric knowledge of an individual in a series of levels that can categorize the potential of the individual in this field. These levels are considered as links in a chain in which, increasingly, higher levels of competition and abstract geometry are achieved.
Level 0: Visualization
The objects of thought at this level are figures that are designed for their appearance. It is a global perception. It often includes non-essential attributes. The perception of the figures is individual. They do not generalize to other figures of the same class.
The products of thought at level 1 are the properties of the figures. Through observation and experimentation, the individual begins to discern the features of the figures.
Level 1: Analysis
The reasoning at this level includes the discovery and generalization from a few cases. The definitions make no sense, in these conditions are missing and can include other irrelevant ones. The ratings are in many cases exclusive.
Level 2: Informal Deduction
The objects of thought at this level are the properties of figures. Using the definitions in a mathematical sense, you understand their rankings are not exclusive. For example, when a diamond is required to have a right angle, then it has four 90° angles, thereby recognizing the square as a kind of diamond.
Level 3: Formal Deduction
The objects of thought at this level are relations between properties of shapes. They are able to work with abstract statements about geometric properties and draw conclusions based more on logic than on intuition. For example, at this level one can recognize that the diagonals of a rectangle are cut at its midpoint, as well as a level 2 student, but at level 3 shows the need to prove this from deductive arguments. At this level, one is able to build a demonstration or proof of a property knowing certain preconditions.
Level 4: Rigor
The objects of thought at this level are axiomatic systems for geometry. You can see the differences between different axiomatic systems and can make comparisons between them. For example, you can set the axiom that two parallel lines are cut, which can produce results in projective geometry and compare them with those obtained in Euclidean geometry. Geometry is managed at the highest level of abstraction.
To get from one level to another, Van Hiele proposes five stages in the teaching of content to be covered at each moment, these are:
- Inquiry/Exam: The teacher asks students to know their background, we propose simple activities to explore the information they have on the contents to try and then introduces the specific vocabulary. For example, what is a rectangle? What is a square? What do they look like?
- Orientation Addressed: The proposed activities that the teacher has prepared and sequenced carefully so that the success of the student is guaranteed. At the time, such activities should gradually reveal the properties of the objects of study.
- Explication: From past experience, students must express and exchange their findings and observations, so that language is empowered. The teacher’s role at this stage is minimized.
- Free Guidance: Students will be offered increasingly complex tasks which can be accomplished in different ways in which they must decide what to do from time to time, establish conjectures, verify or refute them, etc. It is open or closed to propose activities that encourage their interest in the investigation of unknown issues.
- Integration: Students analyze and summarize what they have learned to take a new and broader perspective of the new network of objects and relationships that have been studied. The teacher’s work here is essential to aid in the synthesis of the most important points.
At the end of this phase, students must have reached the next level of understanding of geometric concepts.
Vinner and Concept Formation
In the theory of Vinner, a distinction is established between a concept and the image of that concept.
Concept: Is what emerges from the mathematical definition. Example: a square is a quadrilateral with four equal sides and four equal angles.
Image of a concept: The idea that a person has of what the concept is. The features (properties or attributes) in any example of a concept may be:
- Relevant Attributes: One of the characteristics that must necessarily be met by a case to be an example of this concept. Example: for the square, relevant attributes are being a polygon with four sides, with all sides equal and all angles equal.
- Irrelevant Attributes: They are the features that do not meet all instances of that concept. Example: for the parallelogram, irrelevant attributes are that the sides have the same length or have right angles.
Definition: There are only relevant attributes. It must be a minimum set of attributes that is sufficient to obtain all the others.
Example: You have to have all relevant attributes, although other attributes may be irrelevant.
Counterexample: Not all relevant attributes, but also may have other irrelevant attributes.