Name: Rolize van Tonder
Student number: 63238233
MAE102K Assignment 02
Unique number: 712516
Due date: 8 May 2020
, Question 1
1.1) According to Van Hiele’s theory, learners develop through levels of geometrical
thought. It begins with the most basic level where children display a Gestalt-like
visual approach and then progresses to increasingly advanced levels of description,
analysis, abstraction, and proof. It states that learning is a discontinuous process
and for learners to function sufficiently, they must have acquired most of the previous
level. Another characteristic of this theory is that each level has its own language.
This means that two people that function on different levels cannot understand each
other, even if they are uttering the same sentences.
1.2) Level 3: Formal deduction
The target of this level is the relationship between properties of geometrical objects.
Learners can establish theorems within an axiomatic structure. They can restructure
the differences among undefined terms, definitions, axioms, and theorems. This level
enables learners to construct original proofs. The result of their reasoning is the
establishment of second-order relationships.
Level 4: Rigour/mathematical
This is the highest level of Van Hiele’s theory. The target of this level is deductive
axiomatic systems for geometry. Learners reason about mathematical systems.
They are now capable of studying geometry in the absence of reference models, and
they can reason by manipulating geometric statements. The result of their reasoning
is the establishment, elaboration, and comparison of geometrical axiomatic systems.
1.3) 1.3.1) At level 0, learners will categorise shapes according to what they look
like. They will say something is a parallelogram because it looks like a parallelogram.
They will distinguish between different shapes, not by their properties, but by the way
they look.
1.3.2) At level 1, learners can recognise shapes according to their properties, for
example a parallelogram’s opposite sides are congruent. They will see the properties
of a shape in isolation and without any relationship with each other. For example,
they will say that all rectangles are parallelograms, but without reason.
1.3.3) At level 2, learners can see the relationships between shapes. They know the
requirements for a shape to be identified. Thus, they know that all rectangles are
parallelograms because they understand that all the same properties are included in
that definition. For example, that the diagonals of both a parallelogram and a
rectangle bisect each other.
Student number: 63238233
MAE102K Assignment 02
Unique number: 712516
Due date: 8 May 2020
, Question 1
1.1) According to Van Hiele’s theory, learners develop through levels of geometrical
thought. It begins with the most basic level where children display a Gestalt-like
visual approach and then progresses to increasingly advanced levels of description,
analysis, abstraction, and proof. It states that learning is a discontinuous process
and for learners to function sufficiently, they must have acquired most of the previous
level. Another characteristic of this theory is that each level has its own language.
This means that two people that function on different levels cannot understand each
other, even if they are uttering the same sentences.
1.2) Level 3: Formal deduction
The target of this level is the relationship between properties of geometrical objects.
Learners can establish theorems within an axiomatic structure. They can restructure
the differences among undefined terms, definitions, axioms, and theorems. This level
enables learners to construct original proofs. The result of their reasoning is the
establishment of second-order relationships.
Level 4: Rigour/mathematical
This is the highest level of Van Hiele’s theory. The target of this level is deductive
axiomatic systems for geometry. Learners reason about mathematical systems.
They are now capable of studying geometry in the absence of reference models, and
they can reason by manipulating geometric statements. The result of their reasoning
is the establishment, elaboration, and comparison of geometrical axiomatic systems.
1.3) 1.3.1) At level 0, learners will categorise shapes according to what they look
like. They will say something is a parallelogram because it looks like a parallelogram.
They will distinguish between different shapes, not by their properties, but by the way
they look.
1.3.2) At level 1, learners can recognise shapes according to their properties, for
example a parallelogram’s opposite sides are congruent. They will see the properties
of a shape in isolation and without any relationship with each other. For example,
they will say that all rectangles are parallelograms, but without reason.
1.3.3) At level 2, learners can see the relationships between shapes. They know the
requirements for a shape to be identified. Thus, they know that all rectangles are
parallelograms because they understand that all the same properties are included in
that definition. For example, that the diagonals of both a parallelogram and a
rectangle bisect each other.
